Handbook Of Coastal And Ocean Engineering

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HANDBOOK OF COASTAL AND OCEAN ENGINEERING

edited by

Young C Kim California State University, Los Angeles, USA

World Scientific NEW JERSEY

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LONDON

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SINGAPORE

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BEIJING

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SHANGHAI

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HONG KONG

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TA I P E I

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CHENNAI

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Published by World Scientific Publishing Co. Pte. Ltd. 5 Toh Tuck Link, Singapore 596224 USA office: 27 Warren Street, Suite 401-402, Hackensack, NJ 07601 UK office: 57 Shelton Street, Covent Garden, London WC2H 9HE

British Library Cataloguing-in-Publication Data A catalogue record for this book is available from the British Library.

HANDBOOK OF COASTAL AND OCEAN ENGINEERING Copyright © 2010 by World Scientific Publishing Co. Pte. Ltd. All rights reserved. This book, or parts thereof, may not be reproduced in any form or by any means, electronic or mechanical, including photocopying, recording or any information storage and retrieval system now known or to be invented, without written permission from the Publisher.

For photocopying of material in this volume, please pay a copying fee through the Copyright Clearance Center, Inc., 222 Rosewood Drive, Danvers, MA 01923, USA. In this case permission to photocopy is not required from the publisher.

ISBN-13 978-981-281-929-1 ISBN-10 981-281-929-0

Typeset by Stallion Press Email: [email protected]

Printed in Singapore.

YHwa - Hdbk of Coastal & Ocean.pmd

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Preface Although coastal and ocean engineering is a very ancient field with the construction of Port A-ur near the mouth of the Nile in 3,000 BC, significant advances in this field have been made in the last several decades. The rise of interest in this field can be seen from the number of attendees by academics and practitioners in international conferences. The first International Conference on Coastal Engineering was held in Long Beach, California in 1950 with less than 100 people. When the same conference was held in San Diego, California, in 2006, over 1000 delegates attended. In the last several decades, the world has seen significant coastal and ocean engineering projects, one of which is the Delta Project in the Netherlands. This project was designed to shorten and strengthen the total length of coast and dykes washed by the sea by closing off the sea arms in the Delta region. Other noteworthy coastal engineering projects include the Kansai Airport Project in Japan and, in recent years, the construction of mobile barriers at inlets to regulate tides in the Venice Lagoon known as the Venice Project. Interest in coastal and ocean engineering has arisen in recent years as humankind experiences coastal disasters that derive from coastal storm, hurricane and coastal flooding and seismic activities such as tsunamis, and the impacts of climate change which result in sea-level rise. The tsunami activity in Sumatra in December 2004 affected countries throughout the Indian Ocean and resulted in the loss of thousands of lives. Hurricane Katrina in New Orleans also claimed many lives with property damage exceeding $63 billion. Global warming and sea-level rise will affect shoreline retreats, inundate low coastal areas, damage coastal structures, and accelerate beach erosion. The need for better understanding of our coastal and ocean environment has risen considerably in recent years. This handbook contains a comprehensive compilation of topics that are the forefronts of many technical advances in ocean waves, coastal and ocean engineering. It represents the most comprehensive reference available on coastal and ocean engineering to date, and it also provides the most up-to-date technical advances and latest research findings on coastal and ocean engineering. More than 70 internationally recognized authorities in the field of coastal and ocean engineering contributed papers on their areas of expertise to this handbook. These international luminaries are from highly respected universities and renowned research and consulting organizations from all over the world.

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Preface

This handbook provides a comprehensive overview of shallow-water waves, water-level fluctuations, coastal and offshore structures, ports and harbors, coastal sediment processes, environmental problems, sustainable coastal development, coastal hazards, physical modeling, and coastal engineering practice and education. This book is an essential source of reference for professionals and researchers in the areas of coastal engineering, ocean engineering, oceanography, meteorology, and civil engineering, and as a text for graduate students in these fields. This handbook will be of immediate, practical use to coastal, ocean, civil, geotechnical, and structural engineers, and coastal planners and managers as well as marine biologists and oceanographers. It will also be an excellent source book for educational and teaching purposes, and would be a good reference book for any technical library. I would like to express my indebtedness to those who guided me and supported me as a mentor and a colleague throughout my professional life. They are: Professor Robert L. Wiegel, University of California, Berkeley Professor Joe W. Johnson, University of California Berkeley Professor Robert G. Dean, University of Florida Professor Fredric Raichlen, California Institute of Technology Professor Raymond C. Binder, University of Southern California Professor Shoshichiro Nagai, Osaka City University Dr Basil Wilson, Science Engineering Associates Dr Lars Skjelbreia, Science Engineering Associates Dr Bernard LeMehaute, University of Miami Professor Richard Silvester, University of Western Australia Mr Orville T. Magoon, Coastal Zone Foundation Professor Billy L. Edge, Texas A&M University Professor Michael E. McCormick, US Naval Academy Professor Yoshimi Goda, Yokohama National University and ECOH Corporation Professor Philip L.F. Liu, Cornell University Professor Forrest M. Holly, The University of Iowa Dr Etienne Mansard, National Research Council, Canada Professor J. Richard Weggel, Drexel University Mr Ronald M. Noble, Noble Consultants, Inc. I also wish to express my indebtedness to those who nurtured me from my early teen years and changed my course of life. They are: Dr Helen Miller Bailey, East Los Angeles College Mr H. Karl Bouvier, Jet Propulsion Laboratory I extend my gratitude to my wife, Janet, for her constant support, encouragement, patience, and understanding while I was undertaking this task and to my daughter, Susan Calix, for proofreading some of the materials.

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Finally, I wish to express my deep appreciation to Ms Kimberley Chua of World Scientific Publishing Company who gave me invaluable support and encouragement from the inception of this handbook to its realization. Young C. Kim Los Angeles, California January 2008

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Contents

Preface. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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Contributors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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Section 1: Shallow-Water Waves 1. Wave Setup . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . R. G. Dean and T. L. Walton

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2. Wavemaker Theories . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . R. T. Hudspeth and R. B. Guenther

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3. Analyses by the Melnikov Method of Damped Parametrically Excited Cross Waves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . R. B. Guenther and R. T. Hudspeth

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4. Random Wave Breaking and Nonlinearity Evolution Across the Surf Zone . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Y. Goda

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5. Aeration and Bubbles in the Surf Zone . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . N. Mori, S. Kakuno and D. T. Cox

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6. Freak Wave . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . N. Mori

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7. Short-Term Wave Statistics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . A. Kimura

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Section 2: Water-Level Fluctuations 8. Generation and Prediction of Seiches in Rotterdam Harbor Basins . . . . . M. P. C. de Jong and J. A. Battjes

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9. Seiches and Harbor Oscillations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . A. B. Rabinovich

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10. Finite Difference Model for Practical Simulation of Distant Tsunamis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . S. B. Yoon

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Section 3: Coastal Structures 11. Tsunami-Induced Forces on Structures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . I. Nistor, D. Palermo, Y. Nouri, T. Murty and M. Saatcioglu

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12. Nonconventional Wave Damping Structures . . . . . . . . . . . . . . . . . . . . . . . . . . . H. Oumeraci

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13. Wave Interaction with Breakwaters Including Perforated Walls . . . . . . . . K.-D. Suh

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14. Prediction of Overtopping . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . J. van der Meer, T. Pullen, W. Allsop, T. Bruce, H. Sch¨ uttrumpf and A. Kortenhaus

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15. Wave Run-Up and Wave Overtopping at Armored Rubble Slopes and Mounds . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . H. Sch¨ uttrumpf, J. van der Meer, A. Kortenhaus, T. Bruce and L. Franco

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16. Wave Overtopping at Vertical and Steep Structures . . . . . . . . . . . . . . . . . . . T. Bruce, J. van der Meer, T. Pullen and W. Allsop

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17. Surf Parameters for the Design of Coastal Structures . . . . . . . . . . . . . . . . . . D. H. Yoo

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18. Development of Caisson Breakwater Design Based on Failure Experiences . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . S. Takahashi

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19. Design of Alternative Revetments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . K. Pilarczyk

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20. Remarks on Coastal Stabilization and Alternative Solutions . . . . . . . . . . . K. Pilarczyk

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21. Geotextile Sand Containers for Shore Protection. . . . . . . . . . . . . . . . . . . . . . . H. Oumeraci and J. Recio

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22. Low Crested Breakwaters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . A. Lamberti and B. Zanuttigh

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23. Hydrodynamic Behavior of Net Cages in the Open Sea . . . . . . . . . . . . . . . . Y.-C Li

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Section 4: Offshore Structures 24. State of Offshore Structure Development and Design Challenges . . . . . . . S. Chakrabarti

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Section 5: Ports and Harbors 25. Computer Modeling for Harbor Planning and Design . . . . . . . . . . . . . . . . . . J.-J. Lee and X. Xing

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26. Prediction of Squat for Underkeel Clearance . . . . . . . . . . . . . . . . . . . . . . . . . . . M. J. Briggs, M. Vantorre, K. Uliczka and P. Debaillon

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Section 6: Coastal Sediment Processes 27. Wave-Induced Resuspension of Fine Sediment . . . . . . . . . . . . . . . . . . . . . . . . . M. Jain and A. J. Metha

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28. Suspended Sand and Bedload Transport on Beaches . . . . . . . . . . . . . . . . . . . N. Kobayashi, A. Payo and B. D. Johnson

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29. Headland-Bay Beaches for Recreation and Shore Protection . . . . . . . . . . . J. R.-C. Hsu, M. M.-J. Yu, F.-C. Lee and R. Silvester

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30. Beach Nourishment . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . R. G. Dean and J. D. Rosati

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31. Engineering of Tidal Inlets and Morphologic Consequences . . . . . . . . . . . . N. C. Kraus

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Section 7: Environmental Problems 32. Water and Nutrients Flow in the Enclosed Bays . . . . . . . . . . . . . . . . . . . . . . . Y. Koibuchi and M. Isobe

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Section 8: Sustainable Coastal Development 33. Socioeconomic and Environmental Risk in Coastal and Ocean Engineering . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . M. A. Losada, A. Baquerizo, M. Ortega-S´ anchez, J. M. Santiago and E. S´ anchez-Badorrey 34. Utilization of the Coastal Area . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . H.-H. Hwung

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Section 9: Coastal Hazards 35. Ocean Wave Climates: Trends and Variations Due to Earth’s Changing Climate . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . P. D. Komar, J. C. Allan and P. Ruggiero

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36. Sea Level Rise: Major Implications to Coastal Engineering and Coastal Management . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . L. Ewing

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37. Sea Level Rise and Coastal Erosion. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . M. J. F. Stive, R. Ranasinghe and P. J. Cowell

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38. Coastal Flooding: Analysis and Assessment of Risk . . . . . . . . . . . . . . . . . . . . P. Prinos and P. Galiatsatou

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Section 10: Physical Modeling 39. Physical Modeling of Tsunami Waves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . M. J. Briggs, H. Yeh and D. T. Cox

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40. Laboratory Simulation of Waves. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . E. P. D. Mansard and M. D. Miles

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Section 11: Coastal Engineering Practice and Education 41. Perspective on Coastal Engineering Practice and Education . . . . . . . . . . . J. W. Kamphuis

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Contributors Jonathan C. Allan Coastal Field Office Oregon Department of Geology and Mineral Industries Newport, Oregon [email protected] William Allsop Technical Director HR Wallingford Wallingford, UK [email protected] Elena Sanchez Badorrey Associate Professor CEAMA — Universidad de Granada Granada, Spain [email protected] Asuncion Baquerizo Associate Professor CEAMA — Universidad de Granada Granada, Spain Jurjen A. Battjes Emeritus Professor Environmental Fluid Mechanics Section Delft University of Technology Delft, The Netherlands [email protected] Michael J. Briggs Research Hydraulic Engineer Coastal and Hydraulics Laboratory U.S. Army Engineer Research and Development Center Vicksburg, Mississippi [email protected]

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Tom Bruce School of Engineering and Electronics University of Edinburgh Edinburgh, UK [email protected] Subrata Chakrabarti Joint Professor, Civil and Mechanical Engineering University of Illinois at Chicago Chicago, Illinois [email protected] Peter J. Cowell Associate Professor School of Geosciences Institute of Marine Science University of Sydney Sydney, Australia Daniel T. Cox Professor School of Civil and Construction Engineering Oregon State University Corvallis, Oregon [email protected] Robert G. Dean Graduate Research Professor of Coastal Engineering, Emeritus Department of Civil and Coastal Engineering University of Florida Gainesville, Florida [email protected]fl.edu Pierre Debaillon Research Hydraulic Engineer Centre d’Etudes Techniques Maritimes Et Fluviales (CETMEF) Compiegne, France [email protected] Martijn P. C. de Jong Formerly at Environmental Fluid Mechanics Section Delft University of Technology Presently at Delft Hydraulics Delft, The Netherlands

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Lesley Ewing Senior Coastal Engineer California Coastal Commission San Francisco, California [email protected] Leopoldo Franco Professor of Coastal Engineering Department of Civil Engineering University of Rome 3 Rome, Italy [email protected] Panagiota Galiatsatou Research Associate Department of Civil Engineering Aristotle University of Thessaloniki Thessaloniki, Greece [email protected] Yoshimi Goda Professor Emeritus Yokohama National University Adviser to ECHO Corporation Tokyo, Japan [email protected] Ronald B. Guenther Professor Emeritus Department of Mathematics Oregon State University Corvallis, Oregon [email protected] John Rong-Chung Hsu Professor Department of Marine Environment and Engineering National Sun Yat-sen University Kaohsiung, Taiwan Honorary Research Fellow School of Civil and Resource Engineering University of Western Australia Nedland, Australia [email protected]

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Robert T. Hudspeth Professor and Director, Emeritus Coastal and Ocean Engineering Program Oregon State University Corvallis, Oregon [email protected] Hwung-Hweng Hwung Professor of Hydraulic and Ocean Engineering Director of Tainan Hydraulics Laboratory Department of Hydraulic and Ocean Engineering National Cheng Kung University Tainan, Taiwan [email protected] Masahiko Isobe Professor and Special Adviser to the President Department of Sociocultural Environmental Studies Graduate School of Frontier Sciences The University of Tokyo Chiba, Japan [email protected] Mamta Jain Coastal Engineer Halcrow Inc. Tampa, Florida [email protected] Bradley D. Johnson Coastal and Hydraulics Laboratory U.S. Army Engineer Research and Development Center Vicksburg, Mississippi Shohachi Kakuno Professor and Vice President Department of Civil Engineering Osaka City University Osaka, Japan [email protected] J. William Kamphuis Professor of Civil Engineering, Emeritus Department of Civil Engineering Queen’s University Kingston, Ontario, Canada [email protected]

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Akira Kimura Professor Department of Social Systems Engineering Tottori University Tottori, Japan [email protected] Nobuhisa Kobayashi Professor and Director Center for Applied Coastal Research University of Delaware Newark, Delaware [email protected] Yukio Koibuchi Assistant Professor Department of Sociocultural Environmental Studies Graduate School of Frontier Sciences The University of Tokyo Chiba, Japan [email protected] Paul D. Komar Professor of Oceanography College of Oceanic and Atmospheric Sciences Oregon State University Corvallis, Oregon [email protected] Andreas Kortenhaus Leichtweiss-Institute for Hydraulics Technical University of Braunschweig Braunschweig, Germany [email protected] Nicholas C. Kraus Senior Scientist Coastal and Hydraulics Laboratory U.S. Army Engineer Research and Development Center Vicksburg, Mississippi [email protected] Alberto Lamberti Professor Department of Civil Engineering University of Bologna Bologna, Italy [email protected]

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Fang-Chun Lee Department of Marine Environment and Engineering National Sun Yat-sen University Kaohsiung, Taiwan Jiin-Jen Lee Professor of Civil and Environmental Engineering Sonny Astani Department of Civil and Environmental Engineering University of Southern California Los Angeles, California [email protected] Yu-Cheng Li Professor School of Civil Engineering Dalian University of Technology Dalian, China [email protected] Miguel A. Losada Professor Research Group on Environmental Flux Dynamics CEAMA — Universidad de Granada Granada, Spain [email protected] Etienne P. D. Mansard Executive Director Canadian Hydraulics Centre National Research Council Canada Ottawa, Ontario, Canada [email protected] Ashish J. Mehta Professor of Coastal Engineering Department of Civil and Coastal Engineering University of Florida Gainesville, Florida [email protected]fl.edu Michael D. Miles Canadian Hydraulics Centre National Research Council Canada Ottawa, Ontario, Canada

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Nobuhito Mori Associate Professor Disaster Prevention Research Institute Kyoto University Kyoto, Japan [email protected] Tad S. Murty Adjunct Professor Department of Civil Engineering University of Ottawa Ottawa, Ontario, Canada [email protected] Ioan Nistor Assistant Professor Department of Civil Engineering University of Ottawa Ottawa, Ontario, Canada [email protected] Younes Nouri Department of Civil Engineering University of Ottawa Ottawa, Ontario, Canada Miquel Ortega Associate Professor CEAMA — Universidad de Granada Granada, Spain [email protected] Hocine Oumeraci University Professor Leichtweiss-Institute for Hydraulic Engineering and Water Resources Technical University of Braunschweig Braunschweig, Germany [email protected] Dan Palermo Assistant Professor Department of Civil Engineering University of Ottawa Ottawa, Ontario, Canada [email protected]

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Andres Payo Graduate School of Science and Technology University of Kumamoto Kumamoto, Japan Krystian W. Pilarczyk (Former) Manager, Research and Development Hydraulic Engineering Institute Rykswaterstaat Delft, The Netherlands HYDROpil Consultancy Zoetermeer, The Netherlands [email protected] Panayotis Prinos Professor of Hydraulic Engineering Department of Civil Engineering Aristotle University of Thessaloniki Thessaloniki, Greece [email protected] Tim Pullen Senior Engineer HR Wallingford Wallingford, UK [email protected] Alexander B. Rabinovich P.P. Shirshov Institute of Oceanology Russian Academy of Sciences Moscow, Russia Department of Fisheries and Oceans Institute of Ocean Sciences Sidney, B.C., Canada [email protected]. Roshanka Ranasinghe Associate Professor UNESCO-IHE/Delft University of Technology Delft, The Netherlands [email protected] Juan Recio Leichweiss-Institute for Hydraulic Engineering and Water Resources Technical University of Braunschweig Braunschweig, Germany

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Julie D. Rosati Research Hydraulic Engineer Coastal and Hydraulics Laboratory U.S. Army Corps of Engineers Mobile, Alabama [email protected] Peter Ruggiero Assistant Professor Department of Geosciences Oregon State University Corvallis, Oregon [email protected] Murat Saatcioglu Professor Department of Civil Engineering University of Ottawa Ottawa, Ontario, Canada [email protected] Juan M. Santiago Associate Professor CEAMA — Universidad de Granada Granada, Spain [email protected] Holger Schuttrumpf Professor and Director Institute of Hydraulic Engineering and Water Resources Management RWTH — Aachen University Aaachen, Germany [email protected] Richard Silvester Professor Emeritus School of Civil and Resource Engineering University of Western Australia Nedland, Australia [email protected]

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Marcel J. F. Stive Professor and Director Delft Water Research Centre Department of Hydraulic Engineering Delft University of Technology Delft, The Netherlands [email protected] Kyung-Duck Suh Professor Department of Civil and Environmental Engineering Seoul National University Seoul, Korea [email protected] Shigeo Takahashi Executive Researcher and Director Tsunami Research Center Port and Airport Research Institute Yokosuka, Japan takahashi [email protected] Klemens Uliczka Research Hydraulic Engineer Federal Waterways Engineering and Research Institute (BAW) Hamburg, Germany [email protected] Jentsje van der Meer Principal Van der Meer Consulting Heerenveen, The Netherlands [email protected] Marc Vantorre Professor Division of Maritime Technology Ghent University Ghent, Belgium [email protected]

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Todd L. Walton Director Beaches and Shore Resource Center Florida State University Tallahassee, Florida [email protected] Xiuying Xing Graduate Research Assistant Sonny Astani Department of Civil and Environmental Engineering University of Southern California Los Angeles, California Harry Yeh Professor School of Civil and Construction Engineering Oregon State University Corvallis, Oregon [email protected] Dong Hoon Yoo Professor Department of Civil Engineering Ajou University Suwon, Korea [email protected] Sung Bum Yoon Professor Department of Civil and Environmental Engineering Hanyang University Ansan, Korea [email protected] Melissa Meng-Jiuan Yu Department of Marine Environment and Engineering National Sun Yat-sen University Kaohsiung, Taiwan Barbara Zanuttigh Assistant Professor Department of Civil Engineering University of Bologna Bologna, Italy [email protected]

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The Editor Young C. Kim, PhD, is currently a Professor of Civil Engineering, Emeritus at California State University, Los Angeles. Other academic positions held by him include a Visiting Scholar of Coastal Engineering at the University of California, Berkeley (1971); a NATO Senior Fellow in Science at the Delft University of Technology in the Netherlands (1975); and a Visiting Scientist at the Osaka City University for the National Science Foundations’ US–Japan Cooperative Science Program (1976). For more than a decade, he served as Chair of the Department of Civil Engineering and recently he was Associate Dean of the College of Engineering. For his dedicated teaching and outstanding professional activities, he was awarded the university-wide Outstanding Professor Award in 1994. Dr Kim was a consultant to the US Naval Civil Engineering Laboratory in Port Hueneme and became a resident consultant to the Science Engineering Associates where he investigated wave forces on the Howard-Doris platform structure, now being placed in Ninian Field, North Sea. Dr Kim is the past Chair of the Executive Committee of the Waterway, Port, Coastal and Ocean Division of the American Society of Civil Engineering (ASCE). Recently, he served as Chair of the Nominating Committee of the International Association of Hydraulic Engineering and Research (IAHR). Since 1998, he served on the International Board of Directors of the Pacific Congress on Marine Science and Technology (PACON). He currently serves as the President of PACON. Dr Kim has been involved in organizing 10 national and international conferences, has authored three books, and has published 52 technical papers in various engineering journals.

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Chapter 1

Wave Setup Robert G. Dean Department of Civil and Coastal Engineering University of Florida, Gainesville, FL, USA [email protected] Todd L. Walton Beaches and Shores Resource Center Florida State University, Tallahassee, FL, USA [email protected] Wave setup is the increase of water level within the surf zone due to the transfer of wave-related momentum to the water column during wave-breaking. Wave setup has been investigated theoretically and under laboratory and field conditions, and it includes both static and dynamic components. Engineering applications include a significant flooding component due to severe storms and oscillating water levels that can increase hazards to recreational beach goers and can contribute to undesirable oscillations of both constructed and natural systems including harbors and moored ships. This chapter provides a review of the knowledge regarding wave setup and presents preliminary recommendations for design. It will be shown that wave setup is not adequately understood quantitatively for engineering design purposes.

1.1. Introduction Wave setup was brought to the attention of coastal engineers and scientists in the 1960s (i.e., see Ref. 1, p. 245) after the initial theoretic developments of LonguetHiggins2 and Longuet-Higgins and Stewart3,4 along with limited field observations and laboratory studies supported the existence of wave setup, the magnitude of which was observed to be in the order of 10–20% of the incident wave height. It was noted in early field observations that water levels on the beach were higher than those recorded by a tide gauge at the end of a pier suggesting a wave setup physically forced by wind waves and swell. 1

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Fig. 1.1. Definition sketch. Energy and momentum are transferred from winds to waves in the generating area. The waves convey energy and momentum to the surf zone where the waves break. Upon breaking, the energy is dissipated and the momentum is transferred to the water column resulting in longshore and onshore forces exerted on the water column.

Wave setup is the additional water level that is due to the transfer of wave-related momentum to the water column during the wave-breaking process. As waves approach the shoreline, they convey both energy and momentum in the wave direction. Upon breaking, the wave energy is dissipated, as is evident from the turbulence generated; however, momentum is never dissipated but rather is transferred to the water column resulting in a slope of the water surface to balance the onshore component of the flux of momentum (see Fig. 1.1). If waves are irregular, in addition to a steady wave setup, the setup includes a dynamic component that oscillates with the wave group period and there may be a weak resonance within the nearshore amplifying this oscillating component. These have been termed infragravity waves and are more dominant for narrow banded spectra both in frequency and in directional spreading. The oscillatory component is denoted “dynamic wave setup” in this chapter. This chapter discusses the significance of wave setup to coastal engineering design, provides a review of the classical linear wave theory of wave setup, reviews results from laboratory and field studies, summarizes results and recommends preliminary design approaches for the static component. To provide a “look ahead,” we will see that the phenomenon of wave setup is not yet adequately understood for satisfactory engineering calculations and that the effects of profile slope are very

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significant. The interested reader is also referred to an earlier review article on wave setup by Holman.5 1.2. Engineering Significance of Wave Setup Wave setup (both static and dynamic components) is relevant to a number of engineering applications. The contributions of wave setup under extreme storm events can be substantial, adding several feet to the elevated water levels. The interaction of wave setup with vegetation differs from wind surge and thus it is important to differentiate the two components, for example in ascertaining the benefits of wetlands in reducing wave setup. Finally, the oscillating component of wave setup is relevant to beach safety in some locations and to many natural and constructed coastal systems that have the capability to resonate including harbors and moored ships. 1.3. Terminology and Related Considerations Standard terminology defines the water level in the absence of wave effects as “still water level,” whereas wave setup will cause a departure from the still water level and this water level including the effects of the waves is the “mean water level.” As implied, the mean water level is determined as the average of the fluctuating water level over a suitable time frame usually taken as a number of multiples of the short wave period, say the spectral peak. In considering wave setup, often the location of interest is that of the maximum wave setup at the shoreline. This raises the question of whether wave setup is defined at elevations above the maximum rundown, say on the beach face where the water is present over only a portion of the wave period. Since wave setup is defined as the mean water level, over what period should the water surface be averaged on the beach face which is “wetted” over only a portion of the wave period? If the time average is over only the portion of the period that water is present, in the upper limit, the maximum setup will be the maximum runup. For purposes here, wave setup will usually be defined only for conditions where water is present over a full wave period. When calculating wave runup on a structure such as a levee or revetment, the question arises whether it is appropriate to first calculate wave setup and then add the wave runup which is usually empirically based on model results. In the more recent empirical results (e.g., the TAW method, see Ref. 6), the runup is expressed as a proportion of the significant wave height at the base of the steeper slope (e.g., at a revetment or levee). The wave runup determined in the model on which the method was based generally included some wave setup (or setdown) seaward of the toe of the slope and included wave setup landward of the toe of the slope. Thus, in the application of interest, the most appropriate approach is to calculate and include wave setup at the toe of the slope; however, recognizing that the measured landward runup includes setup, no additional setup should be added explicitly landward of the toe of the slope.

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1.4. A Brief Review of Wave Setup Mechanics 1.4.1. Static wave setup for monochromatic waves Longuet-Higgins2 and Longuet-Higgins and Stewart3,4 were the first to formalize the notion of wave momentum flux and its relationship to wave setup. The momentum flux, Sij , is a second-order tensor given by 
1 , Sxx = E n(cos2 θ + 1) − 2 
1 2 (1.1) Syy = E n(sin θ + 1) − , 2 E sin 2θ, 2 where E is the wave energy density, n is the ratio of wave group velocity to wave celerity and θ is the angle between the wave direction and the x-axis. The term Sxy reads “the flux per unit width, in the x-direction, of the y-component of momentum,” etc. The steady-state equations of motion obtained by time averaging over the short wave period are, including the effects of wind stress and bottom friction:   1 ∂Sxx ∂Sxy ∂ (η wind + ηwave ) =− + − τsx + τbx ∂x ρg(h + η) ∂x ∂y Sxy = Syx =

and ∂ (η wind + η wave ) 1 =− ∂y ρg(h + η)





(1.2)

∂Syy ∂Syx + − τsy + τby . ∂y ∂x

In the above, ηwind is the surge component due to the wind stress, ηwave is the wave setup, ρ is the mass density of water, g is the gravitational constant, h is the local water depth, τsx and τbx are the surface and bottom shear stresses, respectively, and similarly for the y-direction. The coordinate direction, x is oriented shoreward and a right-handed coordinate system is considered. The most simple solution is for waves propagating directly shoreward (Sxy = 0) in which the surface and bottom stresses are considered negligible, and all variables are considered uniform in the y-direction. The resulting equation is   1 ∂Sxx ∂η wave =− . (1.3) ∂x ρg(h + η) ∂x To proceed, we need to determine a boundary condition for η wave ,a at the seaward end of the surf zone. Longuet-Higgins7 has shown that in the absence of energy dissipation, the following general relationship for η applies η=C−

1 2 (u − w2 )η=0 , 2g

(1.4)

a For purposes of convenience, hereafter the subscript on η wave will be omitted such that the wave setup is simply denoted as η.

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where u2 and w2 represent the time averages of the square of the first-order horizontal and vertical wave velocities evaluated at the mean water surface, respectively. Equation (1.4) is a type of a Bernoulli equation for unsteady flows which, when evaluated at the break point and considering no wave setup in deep water to evaluate the constant, C = 0, the setup is negative (setdown) and given by ηb = −

Hb2 kb , 8 sinh 2kb hb

(1.5)

where Hb is the breaking wave height and kb is the wave number at breaking. For shallow water conditions and depth limited breaking (Hb = κ(hb + η b )), Eq. (1.5) yields ηb = −

κHb . 16

(1.6)

As an example, for a κ value of 0.78, the wave setdown is approximately 5% of the breaking wave height. With the seaward boundary condition now established, for the case of shallow water wave-breaking and the consideration of depth limited breaking across the surf zone, the wave setup is η=−

3κ2 /8 κHb + (hb − h) . 16 (1 + (3κ2 /8))

(1.7)

It is noted that in the above equation, the bottom shear stress has been taken as zero and that a shoreward directed bottom shear stress on the water column as would occur due to undertow would increase the wave setup. As examples, the ratio of wave setup to breaking height at the still water line (h = 0) and at the location of maximum wave setup (η = − h) for a κ value of 0.78 are
 5κ η(h = 0, κ = 0.78) = = 0.198 (1.8) F0 |κ=0.78 ≡ Hb 16(1 + (3κ2 /8)) κ=0.78 and Fmax |κ=0.78

η(h = − η, κ = 0.78) ≡ = F0 Hb

   3κ2  5κ  1+ = = 0.244 . 8 κ=0.78 16 κ=0.78 (1.9)

It is seen that the wave setup is strongly dependent on the value of the breaking ratio κ which will be shown to decrease with decreasing beach slope. Figure 1.2 presents the ratios, F0 and Fmax versus κ. It is useful to relate κ in an approximate manner to beach slope. Although there is not a one-to-one correspondence, Fig. 1.3 is based on the Dally et al.8 wave-breaking model and provides an approximate correspondence between uniform profile slope and the associated κ value. It is evident that the Dally et al. model provides reasonable κ values for smaller beach slopes (say less than about 0.06), but the κ values are too large for steeper slopes.

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0.4

Fo Fmax

Fo and Fmax

0.3

0.2

0.1

0.0 0.0

0.2

0.4

0.6

0.8

1.0

Kappa Fig. 1.2.

Values of F0 and Fmax versus wave-breaking index, κ (kappa).

2.0

Kappa

1.5

1.0

0.5

0.0 0.00

0.02

0.04

0.06

0.08

0.10

Profile Slope

Fig. 1.3. Relationship between profile slope and κ (kappa) value. Based on Dally et al.8 wavebreaking model.

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1.4.2. Effects of wave nonlinearity Wave nonlinearity depends on the following parameters: H/L0 and h/L0 . The nonlinearity is exhibited in the wave profile by peaked crests and flatter troughs and increases with wave height and shallow water. Somewhat surprisingly, the momentum fluxes in shallow water are less for nonlinear waves than for linear waves of the same height. This is primarily because the momentum fluxes are proportional to wave energy (Eq. (1.1)) and the wave energy is proportional to the root-mean square of the water surface displacement that is less for nonlinear waves with long troughs and peaked wave crests. Figure 1.4 presents the ratio of nonlinear to linear momentum fluxes as determined by Stream Function wave theory.9–11 The reason that the quantities for nonlinear waves are greater in deep water than for linear waves is that the nonlinear calculations extend up to the actual free surface whereas the linear quantities only extend up to the mean free surface. 1.4.3. Role of wave directionality

1.2

b =1

H/ H

b

/H H

b=

b =0

0.6

H/ H

.2 5

0. 50

=0 .7 5

0.8

.0

1.0

H/ H

Ratio of Nonlinear to Linear Momentum Flux

Equation (1.1) demonstrates that for a given wave height, the maximum shoreward flux of onshore momentum occurs for normally incident waves (θ = 0◦ ). Thus as expected for directional waves, the Sxx term is reduced. However, this reduction is relatively small as can be demonstrated by considering a breaking wave direction of 30◦ relative to a beach normal (this represents a reasonably large wave obliquity

0.4

0.2

0.0 10-3.000

2

3

4 5 6

10-2.000

2

3

4 5 6

10-1.000

2

3

4 5 5

100.000

2

3

4 5 6

101.000

h/Lo Fig. 1.4. Ratio of nonlinear to linear wave momentum flux, Sxx , for forty stream function wave combinations.12

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at breaking). The reduction in Sxx for a wave of given height and for shallow water conditions is 16.7%. 1.4.4. Effects of vegetation The effects of vegetation have been shown to result in a reduced setup and, in some cases, may cause a setdown.12 For linear waves, vegetation protruding through the water surface experiences a net drag force (quadratically related to velocity) on the vegetation in the direction of wave propagation and, of course, there must be an equal and opposing force exerted on the water column. This opposing force acting on the water column partially counteracts the force due to momentum transfer and thus reduces the wave setup (similar to an offshore directed wind stress). For linear waves and vegetation which is submerged during the entire wave passage, no net vegetation-related force exists on the water column and thus there is no effect on the wave setup. However, due to the character of nonlinear waves with higher and shorter shoreward velocities under the wave crests, even if the vegetation is fully submerged during the passage of the wave, a net drag force is induced on the vegetation in the wave propagation direction again resulting in a reduction in the wave setup and, for some cases, a wave setdown. 1.4.5. Dynamic wave setup It is noted that theoretical formulations of the dynamic wave setup must include the time dependent terms in the counterparts of Eq. (1.2). The dynamic wave setup or “surf beat” was first identified through field observations and measurements by Munk13 and Tucker.14 A number of theoretical treatments of dynamic wave setup based on various hypotheses have been developed with each focusing on a different mechanism. These include Symonds et al.15 (time-varying breakpoint), Symonds and Bowen16 (trapping of long waves by longshore bars), Schaffer and Svendsen17 (reinforcement of incoming and reflected long waves), etc. Kostense18 conducted laboratory experiments to investigate the dynamic setup component and found that the results were in qualitative agreement with the theory of Symonds et al.15 We can apply the results for monochromatic waves to investigate the approximate dynamic wave setup for a simple irregular wave case. Consider a bichromatic wave system with wave heights H1 and H2 (H1 > H2 ) and a small frequency difference between the two components. If the resulting wave group varies so slowly that static conditions occur within the surf zone, Eq. (1.7) applies and is written as η = F Hb ,

(1.10)

where Hb is the breaking wave height and F is a proportionality factor depending on whether the referenced setup is at the still water shoreline or the maximum wave setup (see Eqs. (1.8) and (1.9)). The maximum and minimum wave setup values are: η max =

F (H1 + H2 ) ,

ηmin =

F (H1 − H2 ) .

(1.11)

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Table 1.1. Static and dynamic wave setup characteristics for a biharmonic wave system. H2 /H1 0.2 0.4 0.6 0.8 1.0

η max /F H1

η avg /F H1

(η max − η avg )/η avg

1.2 1.4 1.6 1.8 2.0

1.01 1.04 1.09 1.16 1.27

0.19 0.35 0.47 0.55 0.57

It can be shown that the average wave setup depends on the ratio H2 /H1 as shown in Table 1.1. The fourth column presents the ratios of the maximum dynamic wave set amplitude to the average wave setup component. In the case with H2 = H1 , the dynamic wave setup displacement from the mean setup equals 57% of the average wave setup (Table 1.1, Column 4). In the above, we have examined the dynamic wave setup for the case of a simple bichromatic wave system in which the difference in frequencies of the two components was fairly small. For the case of a wave spectrum, the situation is much more complex with, for the case of a narrow spectrum, the group envelope varying according to the Rayleigh distribution. For the case of a wide spectrum, the dynamic component is reduced considerably.

1.5. Laboratory and Field Measurements of Wave Setup Having reviewed the theory of wave setup and its relationship to various factors, the two sources available for evaluation are laboratory and field data. 1.5.1. Laboratory experiments on wave setup Many laboratory investigations of static and dynamic wave setup have been conducted. The results of an early laboratory investigation with monochromatic waves by Bowen et al.19 are shown in Fig. 1.5. For this study, the ratio of maximum wave setdown and wave setup on the beach face to breaking wave height are − 0.035 and + 0.316, respectively, compared with − 0.049 and + 0.244 on the beach face for a κ value of 0.78. The effect of beach slope has been noted earlier and the relatively large beach slope of 0.082 in these experiments is undoubtedly a contributor to the large setup value. Later, laboratory investigations have included examination of irregular waves including measurements of water particle velocities and pressures which form the basis of the Sxx momentum flux component. Battjes20 conducted one of the earliest laboratory studies of wave setup due to irregular waves. Setup was measured through bottom mounted manometers and it was found that the wave setup was less than predicted. It was hypothesized that this difference was possibly due to air in the water column of the manometers. The entire setup was shifted landward relative to the theoretical and this delay was later attributed to a “roller” that is transported along with the wave crest region and

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Fig. 1.5.

Measured wave setup and setdown in the laboratory.19

conveys wave energy and momentum landward prior to transfer of the momentum to the water column and the associated wave setup.21 Later, Stive and Wind22 conducted a very detailed laboratory investigation in which they demonstrated the role of wave nonlinearity. In this study, the momentum flux components (velocities and pressures) were measured to the degree possible and it was found that the calculated wave setup based on nonlinear wave theories was in much better agreement with measured wave setup than calculations based on linear wave theories. In these comparisons, it was not necessary to introduce the roller concept. The two laboratory studies reviewed above have focused on static setup and it has been noted that irregular waves also produce dynamic wave setup. Hedges and Mase23 have presented an interesting reanalysis of earlier runup laboratory measurements by Mase24 in which irregular waves provided the forcing.b The planar slopes represented in the data were: 1:5, 1:10, 1:20, and 1:30. Figure 1.6 presents an example of the form in which the data were plotted where the horizontal axis is b Walton25

was the first to analyze the Mase data to extract the static wave setup.

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Fig. 1.6.

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Variation of nondimensional runup with Iribarren number.23

the Iribarren number, ξ0 defined as tan α ξ0 =  H/L

(1.12)

in which tan α is the profile slope. The interpretation of Fig. 1.6 is that for a zero slope (zero Iribarren number), there would be no short wave runup; therefore, the intercept represents the sum of the static and dynamic components of wave runup. Equations of the following form were fit to plots of the type of Fig. 1.6: Rchar Schar = + cchar ξ0 , H1/3 H1/3

(1.13)

where the subscript “char” refers to the percent associated with the variable; for example, the 2% runup is defined as R2% . It was found that both Schar and cchar were Rayleigh distributed with S1/3 and c1/3 equal to 0.27 and 1.04, respectively, where only the first term represents wave setup and is of interest here. The results for Schar can be interpreted in terms of the static and dynamic wave setup components. As an example, Smean = 0.17 and S2% = 0.37. Thus, the 2% value of the nondimensional dynamic setup defined here as ∆S2% is ηdyn,2% = (S2% − Smean ) = (0.37 − 0.17) = 0.20 . (1.14) ∆S2% = H1/3 Thus, the mean setup at the still waterline is 17% of the significant wave height measured at the toe of the slope and the 2% dynamic component at the still waterline is 20% of the significant wave height at the toe of the slope or slightly larger than the mean wave setup. These results are interesting and of reasonable magnitudes; however, there are two problems with recommending them for universal application. First, we know that the mean setup depends on the slope (through the κ dependency

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as discussed earlier), and the second is that the oscillating wave setup component should depend on the width of the input spectrum. Referring to Fig. 1.6, which is one of several similar plots presented in the Hedges and Mase paper, since each plot may include a mix of beach slopes, the slope dependency is not resolved in the Smean results which of course are derived from the y-intercept of these graphs. Secondly, these experiments were not designed to evaluate the effect of spectral width and the spectral characteristics included in the experiments are not known. However, it is of interest to identify the “representative” κ value and beach slope associated with a Smean = η avg /H1/3 of 0.17. Referring to Fig. 1.2, we see that the associated κ value is approximately 0.63 for F0 . Based on the Dally et al.8 breaking wave model, the associated beach slope from Fig. 1.3 is 1:29 compared to the beach slopes in the Mase experiments ranging from 1:30 to 1:5. 1.5.2. Field experiments on wave setup The paragraphs below describe several field experiments and observations of wave setup. An early study of wave setup comprised a pair of observations at an exposed coastal site (Narragansett Pier, RI) and a calmer water site (Newport, RI) where, at the latter, wave action was assumed not a factor and was found to show an approximate 3 foot water level difference during the peak of the 1938 hurricane storm surge.26 In a second early field experiment on wave setup at Fernandina Beach, Florida, Dorrestein27 placed transparent plastic tubes with lightweight floats to track the water surface on the beach in the zone of wave setup. To obtain the setup records, 16 mm movie film recorded the tracked surface of the floats. A float type tide gage on the end of a fishing pier provided offshore water level records. Through analysis of the tide gage records and the beach placed setup gages, Dorrestein27 evaluated the setup (with respect to the end of the pier) and compared observational results to existing setup theory. He found the measured setup in four of five experiments to be larger than the computed setup. One shortcoming of Dorrestein’s work is that the water level records were only 72 s in length and thus subject to considerable scatter and large standard deviation as later noted by Holman and Sallenger.28 Although rationale was provided by Dorrestein27 for possible differences between measured and computed setup in this early experiment, large discrepancies between measured and analytically or numerically computed setup still exist today. A North Sea field wave setup experiment was conducted on the Island of Sylt by Hansen.29 Utilizing a combination of ultrasonic wave gages and pressure sensor wave gages out to a distance of 1280 m from shore (10 m depth), Hansen29 found good correspondence of data to an empirical expression provided by: η = 0.3Hos = 0.42Horms .

(1.15)

Hansen also noted the maximum wave setup to be approximately 50% of the significant breaking wave height. It is not clear as to the methodology utilized to obtain η max in this field experiment.

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A wave setup field experiment was conducted as part of the Nearshore Sediment Transport Study at Torrey Pines Beach, San Diego, California by Guza and Thornton.30 The Torrey Pines Beach face was gently sloping (beach slope ≈ 0.02) and the beach material was a moderately sorted fine grain sediment (≈ 0.1 mm). A dual wire resistance runup meter was used for the recording and estimation of the wave setup. It should be noted that the measurements of the wave setup were considered to be the average runup determined from wires placed approximately 3 cm above the beach level rather than an actual water level at one location in these experiments. Offshore pressure sensors outside the surf zone at mean depths of 7 to 10.5 m were used for estimating wave height with recording lengths of 4096 s. Guza and Thornton30 note specific problems in the data set, which are typical of field measurements, i.e., the difficulty in obtaining a common datum for the offshore wave measurements and the beach wave setup measurements. Results of their measurement program suggest an empirical relationship as follows: η = 0.17Hos = 0.24Horms

(1.16)

with scatter that suggests η/Hos ranging approximately from 0.05 to 0.50 for individual experiments. Holman and Sallenger28 conducted a field experiment for measuring wave setup as well as other surf zone parameters at the U.S. Army Corps of Engineers field research pier in Duck, NC, USA. Data on water level at the shoreline were collected using longshore looking time lapse photography from Super-8 movie cameras mounted on the research pier scaffolding. The beach at the experiment site had a very steep foreshore (∼ 1 on 10) while the offshore profile slope is much milder (∼ 1 on 100). Beach material was bimodal in size with a median sand size of 0.25 mm and a coarse fraction of 0.75 mm. Results of the experiments showed considerable scatter and dependence on tide level. Regression lines were fit to the data (segmented by tide levels) with results as follows for high tide and mid-tide data: η = 0.35ξ0 + 0.14 (high tide) , (1.17) Hs η = 0.46ξ0 + 0.06 (mid-tide) . (1.18) Hs As most of the data fell in a range of ξ0 = 1 to 2, the maximum setup was noted to be of the same order as the significant wave height in many of the experiments, much higher than theoretically suggested values. Note that in terms of Horms (based on consideration of monochromatic theory results) the setup would be much higher than most other studies show or suggest. Although Holman and Sallenger28 conclude from their experiments that the setup is dependent on the Iribarren number, it is not entirely clear from their data, especially for higher waves (i.e., see Fig. 1.4, Ref. 28). An additional problem that must be considered when computing the Iribarren number for real beaches and irregular waves is how to define beach slope. It should be noted that video camera (visual) approaches estimate setup via the measurement of the water surface elevation on the beach (similar to the Guza and Thornton measurements) rather than an actual vertically fluctuating water level. The anomaly between dependence of

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setup on Iribarren number as noted by Holman and Sallenger28 is likely due to the aforementioned relationship between the wave-breaking coefficient, κ, and beach slope. Nielsen31 and Davis and Nielsen32 conducted a novel setup experiment on Dee Why Beach, in New South Wales, Australia using a set of manometer tubes as shown in Figs. 1.7 and 1.8 from Davis and Nielsen.32 The tubes were deployed throughout the beach face and surf zone. A total of 120 setup profiles were measured in 11 days. Wave heights Horms ranged from 0.6 to 2.6 m in height and significant wave periods (Ts ) ranged from 5.8 to 12.1 seconds. A shoreline setup of about 40% of Horms was found although Davis and Nielsen32 point out that there is reason to believe that the surf zone characteristics influence the relationship between wave height and setup magnitude, and also note a problem of defining beach slope via the Iribarren number. Nielsen31 and Davis and Nielsen32 also observe that a major portion of the setup occurred on the beach face as shown in Fig. 1.9. Nielsen31 points out that previous field investigations have typically measured the mean water level elevation on the beach as opposed to the average fluctuating mean water level in the vertical plane (i.e., the wave setup as usually defined), and that the two measurements are often different in part due to the beach permeability, which in turn is related to beach material size. The issue of extracting wave setup from runup and rundown on the beach is illustrated in Fig. 1.10. King et al.33 collected wave setup data at Woolacombe Beach in North Devon, U.K. which faces the North Atlantic Ocean. The beach face slope varied between 1 on 40 at high tide and 1 on 70 at mid-tide level with a tidal range of 3 m at neap and as much as 9 m at springs. Beach face material consisted of fine sand with

Fig. 1.7.

Manometer setup of Davis and Nielsen32 for measuring setup.

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Fig. 1.8.

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Schematic diagram of apparatus (from Ref. 32).

Fig. 1.9. Dimensionless setup versus total depth where much of the setup occurs on the beach face (from Ref. 32). In this figure, B and D are equal to η and h as used in this chapter, respectively.

90% in the 0.125 mm to 0.25 mm size range. Pressure transducers were utilized to collect wave and setup information at various stations across the beach and also in a longshore direction to assess the spatial variability of the mean setup. Both tripod mounted and buried pressure transducers were utilized. The buried pressure transducers were 50 to 80 cm below the beach surface and were protected by a porous cover. Instruments collected pressure data which were then transformed to water level data over 4096 second intervals. Data did not include sampling in very shallow water and the maximum wave setup was estimated by extrapolating the

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Fig. 1.10. Illustration of differences between mean water level (MWL) shoreline and mean water line on beach.

water surface from the most shoreward water stations. Wave setup estimated from the data showed the wave setup to be roughly: η = 0.10H os = 0.14H orms

(1.19)

with most of the values of η/H orms between 0.11 and 0.15. The authors do not speculate as to why such low values of setup (compared to analytical results) were found in this measurement program. Yanagishima and Katoh34 discuss field measurements of mean water level near the shoreline on the Pacific Coast of Japan as measured by an ultrasonic wave gage mounted on a pier where the mean depth of water was ∼ 0.4 m. The setup was determined via a multiple regression approach on 1305 sets of (20 minute records) data taking into account astronomical tide, wind setup, and atmospheric pressure head components of mean water level. Their data included 91 records in which the offshore wave height was above 3 m. Yanagishima and Katoh’s34 regression analysis suggested the following relationship:  0.2 Los η = 0.0520H os , (1.20) Hos which can be formulated in terms of Iribarren number for their beach slope (1 on 60) to the following: η = 0.27H os (ξ0 )0.4 = 0.38H orms (ξ0 )0.4 .

(1.21)

Yanagishima and Katoh34 noted reasonable agreement with the theory of Goda35 (to be discussed later). Even higher values of setup would be expected on the beach face in accord with theory and findings of other researchers. Greenwood and Osborne36 conducted field measurements on a Georgian Bay Beach, in Lake Huron, Ontario, Canada. Lake Huron has no measurable tide and the beach profile at the site had a slope of 0.015 with a steeper sloped (0.031 to 0.047) inshore bar. Setup was measured using surface piercing resistance wire wave staffs with the shoreward most gage being in approximately 0.4 m of water depth. Measured setup values were found as follows: η = 0.19H os = 0.27H orms .

(1.22)

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It is again noted that even higher values of setup would be expected on the beach face in accord with theory and experience of other researchers. Further work by Hanslow and Nielsen37,38 utilized the manometer tube deployment shown in Fig. 1.7 on three additional beaches (Seven Mile, Palm, and Brunswick) in New South Wales, Australia. With beach face slopes ranging from 0.03 to 0.16 and mean grain sizes of swash zone beach material ranging from 0.18 to 0.5 mm, shoreline beach setup was measured using 20 minute record averages. Using the data from these three beaches as well as earlier measurements at Dee Why Beach (see Refs. 31 and 32), linear least square relationships were fit to the data as follows: η = 0.27H os = 0.38H orms

with R = 0.65

(1.23)

  η = 0.040 Hos L0 = 0.048 Horms L0

with R = 0.77 ,

(1.24)

or

where a somewhat higher value of explained regression was noted using wave height and wave period. Data and regression lines for these two relationships are shown in Figs. 1.11 and 1.12. The improvement in fit due to inclusion of the deep water wavelength is not evident visually. A significant finding of these studies was that a major portion of the setup occurred on the beach face (see Fig. 1.9). Further measurements on wave setup at two river entrances is also discussed in Hanslow and Nielsen37 and Dunn et al.39 with the result that the wave setup at river entrances was found to be (somewhat surprisingly) negligible. Lentz and Raubenheimer40 report on a field experiment at the U.S. Army Field Research Pier in Duck, NC, USA where 11 pressure sensor gages and 10 sonar altimeters extended across the surf zone from 2 to 8 m of water depth. Close agreement with Longuet-Higgins radiation stress theory for wave setup was noted

Fig. 1.11.

Empirical relationship between setup and wave height (from Ref. 38).

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Fig. 1.12.

Empirical relationship between setup and wave parameters (from Ref. 38).

although the lack of setup measurements in shallow water (< 2 m) did not allow conclusions regarding the maximum setup that might be expected on the beach. Raubenheimer et al.41 report on a second field experiment at the U.S. Army Field Research Pier in Duck, NC, USA where 12 buried pressure sensor gages were employed across the surf zone from the shoreline to 5 m of water depth. Again good agreement with Longuet-Higgins and Stewart42 radiation stress theory was noted by integration of the cross-shore momentum equation to estimate the wave setup for water depths greater than 1 m but the theory was found to under-predict wave setup in shallow water (h < 1 m). The lack of setup measurements on the beach face did not allow conclusions regarding the maximum setup that might be expected on a beach although an empirical equation was provided to estimate wave setup at the SWL line as follows: η SWL −1/3 = 0.019 + 0.003βf , Hos

(1.25)

where βf is the average slope across the surf zone. Raubenheimer et al.41 suggest that theory under-predicts the setup by a factor of 2 for water depths less than 1 m. Stockdon et al.43 using video shoreline water level time series determined wave setup and wave runup results during 10 diverse field experiments (four from Duck, NC, USA; four from West Coast beaches in California/Oregon, USA; and two from Terschelling, The Netherlands). These wave setup results were analyzed to provide empirical parameterizations for wave setup under many natural beach conditions as follows:  η = 0.385βf H0 L0 , (1.26) which, assuming that tan βf ≈ βf can be expressed in terms of the Iribarren number as η = 0.385ξ0 H0

(1.27)

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and for extremely dissipative beaches η 0.043 = , H0 H0 /L0

(1.28)

where H0 is the effective deep water wave height, L0 is the deep water wave length associated with the peak spectral period and βf is the average slope over a depth range defined in terms of the standard deviation of the water surface displacement. It should again be noted that the video camera (visual) approach estimates setup via the mean of measurements of the water surface elevation on the beach rather than the mean of fluctuating water levels at one location. Results from nine of the field experiments presented here have been analyzed to determine the average ratio of wave setup at the still water line to significant wave height and its associated standard deviation. The ratios at the still water line were determined to be 0.191±0.100. Several caveats apply to these results. In cases where the beach slope and/or the deep water wave steepness was incorporated into the expression presented, these were taken as 0.01 and 0.04, respectively. Some of the published expressions were in terms of the breaking wave height and some in terms of the deep water wave height and no attempt was made to differentiate between breaking and deep water wave heights. The Holman and Sallenger results were not included in these results as they appeared to be anomalously high. Finally, the wave setup ratio at the intersection of the mean water line intersection with the beach profile would be greater than the average ratio (0.191) above. Also, although not examined in detail here, the dynamic wave setup which increases with energetic narrow spectra, would also contribute to the total wave setup. It is relevant to note that results from field measurements are often not consistent, possibly due to: (1) Limited measurement distances across the nearshore. (2) Use of many different approaches to measure/evaluate setup (i.e., videos, pressure sensors, runup gages, manometers, etc.). (3) Inherent difficulties in obtaining a consistent datum for nearshore measurements and offshore measurements. (4) A clear definition of setup on the beach face is lacking due to the nature of the permeable beach and the difficulty of sub-aerial setup measurements. 1.6. Published Guidance on Wave Setup for Engineering Applications Several sources of wave setup recommendations are available; two are reviewed here. The U.S. Army Corps of Engineers 1984 Shore Protection Manual (SPM) presents a graphical method to calculate wave setup at mid-depth of the surf zone. This method, developed for irregular waves, is presented in Fig. 1.13 in which the normalized setup has been multiplied by a factor of 2 to transfer approximately the results to the still water shoreline. The effect of beach slope and deep water wave steepness in Fig. 1.13 are evident.

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Fig. 1.13. Nondimensional wave setup versus deep water wave steepness and profile slope by the 1984 Shore Protection Manual recommendations as incorporated in Appendix D of FEMA44 Guidelines. Note that the normalized setup has been multiplied by a factor of 2 to transfer the setup from the mid-depth of the surf zone as it appears in SPM to the approximate still water level contour. Note: S in this figure is equal to η in this chapter.

Fig. 1.14. Nondimensional wave setup by Goda versus deep water wave steepness and relative water depth within the surf zone. Profile slope = 1:100.

Goda35 has presented guidance for static and dynamic wave setups due to irregular waves. The guidance for static setup and a profile slope of 1:100 is shown in Fig. 1.14. The effects of various deep water wave steepness values are illustrated in Fig. 1.14.

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Table 1.2. Comparison of nondimensional wave setup by SPM and Goda methods for irregular waves. η/H0 H0 /L0

SPM

Goda

0.005 0.01 0.02 0.04 0.08

0.154 0.135 0.120 0.103 0.097

0.122 0.102 0.083 0.065 0.049

Note: Values in SPM method have been multiplied by 2.0 to transfer from surf zone mid-depth to still water line.

Table 1.2 presents a comparison of ratios of nondimensional wave setup values at the still water shoreline as recommended by SPM and Goda. In examining the results in Table 1.2, recall that an additional wave setup occurs from the still water line to the location where the maximum setup intersects the beach profile.

1.7. Summary and Recommendations The reviews of theory, laboratory and field data, and published guidance for engineering applications presented here have identified static and dynamic wave setup components as contributing to the deviation from still water level in the surf zone and their relevance to engineering design. Examination of the static wave setup has reinforced the effect of beach slope on wave setup. The theory presented here does not account for the onshore bottom stress acting on the water column due to undertow. The available field measurement results exhibit a wide range of wave setup to wave height ratios. Some of this variability is undoubtedly due to the effect of profile slope, which is not accounted for explicitly in some of the analyses and part is due to the effect of wave-breaking in depths greater than the shallow water limit. Design methodology should account for the static and dynamic wave setup components. In determining the wave setup to include in design, the characteristics of the particular application of interest should be compared with those of the various field and laboratory experiments available including those referenced here. The dominant role of beach slope should be recognized. The preliminary results presented here of η/Hs = 0.191 ± 0.100 may serve as a useful guide for the static wave setup component. It is hoped that further research with improved instrumentation, modern surveying techniques, and more diverse field site studies will help to clarify both the static and dynamic wave setup components for future design applications.

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References 1. A. T. Ippen, Estuary and Coastline Hydrodynamics (McGraw-Hill Book Company, 1966), 744 pp. 2. M. S. Longuet-Higgins, Radiation stress and mass transport in gravity waves with application to “surf beats”, J. Fluid Mech. 13(4), 481–504 (1962). 3. M. S. Longuet-Higgins and R. W. Stewart, A note on wave set-up, J. Marine Res. 21, 4–10 (1963). 4. M. S. Longuet-Higgins and R. W. Stewart, Radiation stresses in water waves; a physical discussion, with applications, Deep-Sea Res. 11, 529–562 (1964). 5. R. A. Holman, Wave set-up, Handbook of Coastal and Ocean Engineering, Vol. 1, ed. J. Herbich (Gulf Publishing, 1990), Chapter 11, pp. 635–646. 6. van der Meer, TAW Technical Report: Wave Run-up and Wave Overtopping at Dikes, Technical Advisory Committee for Flood Defence, Delft (2003). 7. M. S. Longuet-Higgins, On the wave-induced difference in mean sea level between the two sides of a submerged breakwater, J. Marine Res. 25(2), 148–153 (1967). 8. W. R. Dally, R. G. Dean and R. A. Dalrymple, Wave height variation across beaches of arbitrary profile, J. Geophys. Res. 90(C6), 11917–11927 (1985). 9. R. G. Dean, Stream function representation of nonlinear ocean waves, J. Geophys Res. 79(30), 4489–4504 (1965). 10. R. G. Dean, Evaluation and Development of Water Wave Theories for Engineering Application; Volume I: Presentation of Research Results; Volume II: Tabulation of Dimensionless Stream Function Variables, Special Report No. 1, Published by U.S. Army Corps of Engineers, Coastal Engineering Research Center (1974). 11. R. A. Dalrymple, A finite amplitude wave on a linear shear current, J. Geophys. Res. 87(C1), 483–491 (1974). 12. R. G. Dean and C. J. Bender, Static wave setup with emphasis on damping effects by vegetation and bottom friction, Coast. Eng. 13, 149–156 (2006). 13. W. R. Munk, Surf beats, Trans. Am. Geophys. Union 30, 849–854 (1949). 14. M. J. Tucker, Surf beats: Sea waves of 1 to 5 minutes period, Proc. Roy. Soc. A 202, 565–573 (1950). 15. G. Symonds, D. A. Huntley and A. J. Bowen, Two-dimensional surf beat: Long wave generation by a time-varying breakpoint, J. Geophys. Res. 87(C1), 492–498 (1982). 16. G. Symonds and A. J. Bowen, Interaction of nearshore bars with wave groups, J. Geophys. Res. 89(C2), 1953–1959 (1984). 17. H. A. Schaffer and I. A. Svendsen, Surf beat generation on a mild-slope beach, Proc. ASCE Int. Conf. Coastal Engineering (1988), pp. 1058–1072. 18. J. K. Kostense, Measurements of surf beat and set-down beneath wave groups, Proc. ASCE Int. Conf. Coastal Engineering (1984), pp. 724–740. 19. A. J. Bowen, D. L. Inman and V. P. Simmons, Wave set-down and set-up, J. Geophys. Res. 73, 2569–2577 (1968). 20. J. A. Battjes, Set-Up Due to Irregular Waves, Report No. 72–2, Communications on Hydraulics, Delft University of Technology, Department of Civil Engineering (1972), 13 pp. (Paper also presented at the 13th Int. Conf. Coastal Engineering, Vancouver, B.C.). 21. I. A. Svendsen, Wave heights and set-up in a surf zone, Coast. Eng. 8, 302–329 (1984). 22. M. J. F. Stive and H. G. Wind, A Study of Radiation Stress and Set-Up in the Nearshore Zone, Publication No. 267, Waterlopkundig Laboratorium, Delft Hydraulics Laboratory (1982), 25 pp.

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23. T. S. Hedges and H. Mase, Modified Hunt’s equation incorporating wave setup, J. Waterway Port Coast. Ocean Eng. 130(3), 109–113 (2004). 24. H. Mase, Random wave runup height on gentle slopes, J. Waterway Port Coast. Ocean Div. 115(WW2), 649–661 (1989). 25. T. L. Walton, Interim guidance for prediction of wave run-up on beaches, Ocean Eng. 19(2), 199–207 (1999). 26. T. Saville, Experimental determination of wave set-up, Proc. 2nd Technical Conf. Hurricanes, Miami Beach, FL, National Hurricane Research Project Report No. 50, U.S. Department of Commerce Washington, D.C. (1961), pp. 242–252. 27. R. Dorrestein, Wave set-up on a beach, Proc. 2nd Technical Conf. Hurricanes, Miami Beach, FL, National Hurricane Research Project Report No. 50, U.S. Department of Commerce, Washington, D.C. (1961), pp. 230–241. 28. R. A. Holman and A. H. Sallenger, Setup and swash on a natural beach, J. Geophys. Res. 90(C1), 945–953 (1985). 29. U. A. Hansen, Wave setup and design water level, J. Waterway Port Coast. Ocean Div. 104(WW2), 227–240 (1978). 30. R. T. Guza and E. B. Thornton, Wave set-up on a natural beach, J. Geophys. Res. 96(C2), 4133–4137 (1981). 31. P. Nielsen, Wave setup: A field study, J. Geophys. Res. 93(C12), 15643–15652 (1988). 32. G. A. Davis and P. Nielsen, Field measurement of wave set-up, ASCE Int. Conf. Coastal Engineering, Malaga, Spain (1988), Chapter 38, pp. 539–552. 33. B. A. King, M. W. L. Blackley, A. P. Carr and P. J. Hardcastle, Observations of wave induced setup on a natural beach, J. Geophys. Res. 95(C12), 22289–22297 (1990). 34. S. Yanagishima and K. Katoh, Field observation on wave setup near the shoreline, Proc. 22nd Int. Conf. Coastal Engineering, Vol. 1, ASCE, New York, N.Y. (1990), Chapter 7, pp. 95–108. 35. Y. Goda, Random Seas and Design of Maritime Structures (World Scientific Publishing Co., 2000), 443 pp. 36. B. Greenwood and P. D. Osborne, Vertical and horizontal structure in cross-shore flows: An example of undertow and wave setup on a barred beach, Coast. Eng. 14, 543–580 (1990). 37. D. J. Hanslow and P. Nielsen, Wave setup on beaches and in river entrances, 23rd Int. Conf. Coastal Engineering, Venice, Italy (1992), pp. 240–252. 38. D. J. Hanslow and P. Nielsen, Shoreline set-up on natural beaches, J. Coast. Res. SI15, 1–10 (1993). 39. S. L. Dunn, P. Nielsen, P. A. Madsen and P. Evans, Wave setup in river entrances, Proc. 27th Int. Conf. Coastal Engineering, ASCE, New York, Sydney, Australia (2000), pp. 3432–3445. 40. S. Lentz and B. Raubenheimer, Field observations of wave setup, J. Geophys. Res. 104(C11), 25867–25875 (1999). 41. B. Raubenheimer, R. T. Guza and S. Elgar, Field observations of wave-driven setdown and setup, J. Geophys. Res. 106(C3), 4629–4638 (2001). 42. M. S. Longuet-Higgins and R. W. Stewart, Radiation stresses in water waves: A physical discussion with applications, Deep Sea Res. 11(4), 529–562 (1964). 43. H. F. Stockdon, R. A. Holman, P. A. Howd and A. H. Sallenger, Jr., Empirical parameterization of setup, swash, and runup, Coast. Eng. 53, 573–588 (2006). 44. FEMA, Guidelines and Specifications for Flood Hazard Mapping Partners, Appendix D: Guidance for Coastal Flooding Analysis and Mapping, Map Modernization Program, Washington, D.C. (2003). Also available at: www.fema.gov/fhm/ dl cgs.shtm.

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Chapter 2

Wavemaker Theories Robert T. Hudspeth School of Civil and Construction Engineering Oregon State University, Corvallis, OR 97331, USA [email protected] Ronald B. Guenther Department of Mathematics Oregon State University, Corvallis, OR 97331, USA [email protected] The fundamental solutions to the wavemaker boundary value problem (WMBVP) are given for 2D channels, 3D basins, and circular basins. The solutions are given in algebraic equations that replace integral and differential calculus. The solutions are generic and apply to both full- and partial-draft piston and hinged wavemakers; to double-articulated wavemakers, and to directional wave basins. The WMBVP is solved by conformal mapping and by domain mapping. The loads on a wavemaker are connected to the radiation boundary value problem for semiimmersed bodies and demonstrate the connection of these loads to the added mass and radiation damping coefficients required to compute the dynamic response of large Lagrangian solid bodies.

2.1. Introduction Wavemaker theories play several important roles in coastal and ocean engineering. The most important role is the application to laboratory wavemakers for both wavemaker designs and wave experiments. A second role for wavemaker theories is to compute a scalar radiated wave potential to compute the wave-induced loads on large solid bodies applying potential wave theory. The displacements and rotations of a semi-immersed six degrees-of-freedom large Lagrangian solid body are related to the displacements and rotations of wavemakers. The boundary between a planar wavemaker and an ideal fluid requires special care because the fluid motion

25

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is an Eulerian field with time and space as the independent variables, and the planar wavemaker is a Lagrangian solid body with time and the wavemaker as the independent variables. Consequently, the kinematic boundary condition will be different from the free surface boundary that separates two Eulerian fluid fields of air and water. The boundary between the fluid and wavemaker separates an Eulerian field (the fluid) from a Lagrangian body (the wavemaker), and the wavemaker kinematic boundary condition (WMKBC) must convert the Lagrangian wavemaker motion to a Eulerian field motion in order that the independent variables for both dependent motion variables are equivalent. This may be accomplished by multiplying the Lagrangian motion of the wavemaker by the unit normal to the boundary. Because the unit normal is a function of space and the Lagrangian wavemaker motion is a function of time, the product will produce a motion that is a function of both space and time that are the independent variables of the Eulerian fluid field. Although this fact is not central to the WMBVP, it is an important connection between the WMBVP and the radiation potential boundary value problem for semi-immersed large Lagrangian solid bodies.1 The formulae for computing the two fundamental fluid unknowns for an incompressible fluid of the velocity q(x, z, t) and the pressure p(x, z, t) from a scalar velocity potential Φ(x, z, t) are given first. The classical linear WMBVP for dimensionless 2D planar wavemaker is reviewed for two types of double-articulated planar wavemakers. The sway X1 (t) displacement of a full-draft piston wavemaker and the roll Θ5 (t) rotation of a hinged wavemaker are connected directly to the sway displacement and the roll rotation of a semi-immersed large Lagrangian solid body. In this review, integral calculus formulae for computing the integrals that are required to compute the coefficients of the eigenseries for the fluid motion, to compute the loads on the wavemaker and the average power required to generate the propagating waves are replaced by generic algebraic formulae. For example, an integral equation that is required to compute the nth eigenseries coefficient Cn for the nth eigenfunction Ψn (Kn , z/h) from a wavemaker shape function χ(z/h) may be computed symbolically and expressed by a dimensionless algebraic formula In (α, β, b, d, Kn ), that is given by
Cn = h

0

−1

χ(z/h)d(z/h)Ψn (z/h)d(z/h) = In (α, β, b, d, Kn ).

(2.1)

The coefficient in (2.1) may then be computed very efficiently by substitution into algebraic formulae in all subsequent applications. Next a dimensionless theory for both amplitude-modulated (AM) and phase-modulated (PM) circular wavemakers is reviewed. Then, a dimensionless theory for double-actuated wavemakers is reviewed. Following that, a dimensionless directional wavemaker theory for large wave basins based on a WKBJ approximation1 is reviewed. Next, a theory for sloshing waves due to transverse motions of a segmented wavemaker in a narrow wave channel is reviewed. Then, 2D planar wavemakers are mapped to a unit circle by conformal mapping and to a fixed rectangular domain by domain mapping; and both the linear and nonlinear wavemaker solutions are computed numerically.

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2.2. Planar Wavemaker in a 2D Channel Two generic planar wavemaker configurations are shown in Figs. 2.1(a) and 2.1(b). The fluid motion may be obtained from the negative gradient of a dimensional scalar velocity potential Φ(x, z, t) according to q(x, z, t) = u(x, z, t)ex + w(x, z, t)ez = −∇2 Φ(x, z, t), where the 2D gradient operator in (2.2a) is given by ∇2 (•) =

∂(•) ∂(•) ex + ez . ∂x ∂z

Fig. 2.1(a).

Definition sketch for a Type I planar wavemaker.

Fig. 2.1(b).

Definition sketch for a Type II planar wavemaker.

(2.2a)

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The total pressure field P (x, z, t) may be computed from the unsteady Bernoulli equation according to 

 ∂Φ(x, z, t) 1 2 P (x, z, t) = p(x, z, t) + pS (z) = ρ − |∇Φ(x, z, t)| + Q(t) − ρgz, ∂t 2 (2.2b) where Q(t) = the Bernoulli constant; and the free surface elevation η(x, t) for zero atmospheric pressure according to 1 η(x, t) = g



 ∂Φ(x, η, t) 1 2 − |∇Φ(x, η, t)| + Q(t) ; ∂t 2

x ≥ ξ(η, t);

z = η(x, t). (2.2c)

The scalar velocity potential must be a solution to the Laplace equation ∇22 Φ = 0;

x ≥ ξ(z, t);

−h ≤ z ≤ η(x, t),

(2.3a)

with the following boundary conditions: Kinematic Bottom Boundary Condition (KBBC): ∂Φ = 0; ∂z

x ≥ ξ(−h, t);

z = −h.

(2.3b)

Combined Kinematic and Dynamic Free Surface Boundary Condition (CKDFSBC):   ∂ 2Φ 1 ∂ ∂Φ dQ − − ∇Φ · ∇ |∇Φ|2 + = 0; + g ∂t2 ∂z ∂t 2 dt

x ≥ ξ(η, t);

z = η(x, t). (2.3c)

Kinematic WaveMaker Boundary Condition (KWMBC): A Stokes material surface for planar wavemaker is W (x, z, t) = x − ξ(z, t), and the Stokes material derivative gives the KWMBC from ∂Φ ∂ξ ∂Φ ∂ξ DW = + − = 0; Dt ∂x ∂t ∂z ∂z

x = ξ(z, t);

−h ≤ z ≤ η(t).

(2.3d)

Kinematic Radiation Boundary Condition (KRBC): A KRBC is required as x → +∞ for uniqueness to insure that propagating waves are only right progressing or that evanescent eigenmodes are bounded. For a temporal dependence proportional to exp ±iωt, the KRBC may be expressed by  lim

x→+∞

 ∂ ± iKn Φ(x, z, t) = 0. ∂x

(2.3e)

A velocity potential ϕ(x, z) may be defined by the real part of Φ(x, z, t) = Re{ϕ(x, z) exp −i(ωt + ν)},

(2.4)

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where Re{•} means the real part of {•}; and ν = arbitrary phase angle. The linearized WMBVP for kh = O(1) is1 ∇22 ϕ(x, z) = 0;

0 ≤ x < +∞;

∂ϕ(x, z) = 0; ∂z

0 ≤ x + ∞;

∂ϕ(x, z) − k0 ϕ(x, z) = 0; ∂z  lim

x→+∞

η(x, t) = Re

z = −h,

(2.5b)

z = 0,

 ∂ − iKn ϕ(x, z) = 0, ∂x x = 0;

 −iω ϕ(x, 0) exp −i(ωt + ν) ; g

p(x, z, t) = ρ

(2.5a)

0 ≤ x < +∞;

∂ϕ(x, z) ∂ξ(z, t) exp −i(ωt + ν) = − ; ∂x ∂t 

−h ≤ z ≤ 0,

∂Φ(x, z, t) ; ∂t

(2.5d)

−h ≤ z ≤ 0, x ≥ 0;

0 ≤ x < +∞;

(2.5c)

z = 0,

z = 0,

(2.5e)

(2.5f)

(2.5g)

where k0 = ω 2 /g. Because the boundary conditions defined by (2.5b)–(2.5e) are prescribed on boundaries with constant values of the independent variables x and z, a solution by the method of separation of (independent) variables may be computed.1 The instantaneous wavemaker displacement ξ(z, t) from its mean position x = 0 is assumed to be strictly periodic in time with period T = 2π/ω, and may be expressed by     S χ(z/h) exp −i(ωt + ν) ξ(z/h, t) = Re i (∆/h)   S = χ(z/h) sin(ωt + ν). (2.6) (∆/h) The specified shape function χ(z/h) for the Type I wavemaker shown in Fig. 2.1(a) is valid for either a double-articulated piston or hinged wavemaker of variable draft and is given by the following dimensionless equation for a straight line2 : χ(z/h) = [α(z/h) + β][U (z/h + 1 − d/h) − U (z/h + b/h)],

(2.7a)

where α, β = dimensionless constants; U (•) = the Heaviside step function with two boundary conditions given by [S/(∆/h)]χ(z/h = −1 + d/h + ∆b /h + ∆/h) = S,

(2.7b)

[S/(∆/h)]χ(z/h = −1 + d/h + ∆b /h) = Sb ,

(2.7c)

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that may be solved simultaneously for the dimensionless coefficients α, β to obtain α = (1 − Sb /S);

β = ∆/h + α(1 − d/h − ∆b /h − ∆/h).

(2.7d,e)

The coefficients α and β for the specified shape function χ(z/h) in (2.7a) may be obtained for the Type II wavemaker shown in Fig. 2.1(b) by substituting ˆ S = S¯ + S;

¯ Sb = S;

∆=h−b−d

into the following boundary conditions2 in (2.7b) and (2.7c):   ˆ (S¯ + S) ˆ χ(z/h = −b/h) = S¯ + S, 1 − b/h − d/h 

 ˆ (S¯ + S) ¯ χ(z/h = −1 + d/h) = S, 1 − b/h − d/h

(2.8a)

(2.8b)

that may be solved simultaneously for the constant coefficients α, β to obtain  ¯ d b S Sˆ ; β =1− − . (2.8c,d) α= ¯ ˆ ¯ ˆ h S+S S+S h 2.2.1. Eigenfunction solution to the WMBVP Because all of the boundary conditions defined by (2.5b)–(2.5e) are now prescribed for constant values of the independent variables (x, z) and the dimensionless parameter kh = O(1), a solution by separation of independent variables1 is suggested according to ϕ(x, z) = X(x) • Z(z).

(2.9)

The eigenseries solution may be written compactly as1,3,4

Φ(x, z, t; Kn ) = Cn cosh Kn (z + h) exp +i(Knx − ωt + ν),

(2.10a)

n=1

where Kn = k for n = 1 and Kn = +iκn for n ≥ 2 provided that ko h − kh tanh kh = ko h + κn h tan κn h = 0;

n > 2.

(2.10b)

The eigenseries (2.10a) may be separated into a propagating Φp (x, z, t; k) and evanescent eigenmodes Φe (x, z, t; κn) or “local ” wave components3 according to Φ(x, z, t; Kn ) = Φp (x, z, t; k) + Φe (x, z, t; κn ) 

Cn cos κn (z + h) exp +i(Kn x − ωt + ν). = C1 cosh k(z + h) + n=2

(2.10c)

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The wave number k = 2π/λ where λ = wavelength. Because the numerical value of kh must be computed from an eigenvalue problem in the vertical z coordinate, equivalence of the eigenvalue k to the wave number 2π/λ requires a pseudo-horizontal boundary condition of periodicity given by k = 2π/λ and ϕ(x + λ, z) = ϕ(x, z). It is computationally efficient to normalize the eigenseries in (2.10a) according to cosh Kn h(1 + z/h) ; n = 1, 2, . . . , Nn where the nondimensional normalizing constant Nn is  2kh + sinh 2kh  
0 ; n = 1,  4kh Nn2 = cosh2 Kn h(1 + z/h)d(z/h) =  2κn h + sin 2κn h −1   ; n ≥ 2. 4κn h Ψn (Kn , z/h) =

(2.11a)

(2.11b) (2.11c)

The eigenseries in (2.10a) may be written as an orthonormal eigenseries by Φ(x, z, t; Kn ) =

∞

Cn Ψn(Kn , z/h) exp i(Knx − ωt − ν),

(2.12)

n=1

where the orthonormal eigenfunction Ψn (•,•) is dimensionless. 2.2.2. Evaluation of Cn by WM vertical displacement χ(z/h) The following dimensionless coefficient computed from (2.5e) will replace integral calculus with algebraic substitution for the coefficients Cn in the eigenseries (2.12):
−b/h [α(z/h) + β]Ψn (Kn , z/h)d(z/h) In (α, β, b, d, Kn ) = −1+d/h

    d b     sinh K K h 1 − d − K b sinh K h 1 −   n n n  n h h  α =   (Kn h)2 Nn  b     − cosh Kn h 1 − + cosh Kn d   h    b β (2.13) + sinh Kn h) 1 − − sinh Kn d (Kn h)Nn h that is dimensionless when α and β are given by (2.7d) and (2.7e) or (2.8c) and (2.8d). The coefficients Cn may be computed algebraically by (2.13) from the KWMBC (2.5e) to obtain Sωh In (α, β, b, d, Kn ), Kn ∆ and the orthonormal eigenseries (2.12) is given by Cn = i

(2.14)

Φ(x, z, t; Kn ) =

∞

iSωh In (α, β, b, d, Kn )Ψn (Kn , z/h) exp i(Kn x − ωt − ν). Kn ∆ n=1

(2.15)

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2.2.3. Decay distance of evanescent eigenmodes n ≥ 2 Numerical solutions and experimental measurements of ocean and coastal designs require that the KRBC (2.5d) be applied far enough away so that only the propagating eigenmode for n = 1 in (2.12) is measurable. The evanescent eigenseries in (2.12) for n ≥ 2 will decay spatially at least as fast as the smallest evanescent eigenvalue κ2 . This eigenvalue must be κ2 h > (n − 3/2)π = π/2. If the smallest value for κ2 h > π/2, then κ2 > π/2h and ϕ(x, z) ∝ exp −(πx/2h). For the values of the evanescent eigenseries to be less than 1% of their values at the wavemaker, ϕ(x, z) ∝ exp −(πxd /2h) = 0.01 and πxd /(2h) = 4.6 ≈ 3π/2, and the minimum decay distance is xd ≥ 3h. 2.2.4. Transfer function for wave amplitude from wavemaker stroke The average rate of work or power done by a wavemaker of width B is1
˙ τ = Pτ = B W

τ +1




0

h

p(x, z, τ )u(x, z, τ )d(z/h)dτ,

τ

(2.16a)

−1

where the temporal averaging operator is defined by
•τ =

τ +1

(•)dτ,

(2.16b)

τ

and  ˙ τ = Pτ = W

ρω 3 S 2 Bh4 ∆2 2kh

 I12 (α, β, b, d, k),

(2.16c)

so that all of the average power from a wavemaker is transferred to only the propagating eigenmode. The average energy flux in a linear wave is given by1  ˙ τ = E

ρgBA2 2

CG ,

(2.16d)

where the group velocity CG is given by1 CG =

  2kh C 1+ . 2 sinh 2kh

(2.16e)

Equating (2.16c) to (2.16d) gives the following transfer function for a planar wavemaker:  A ko h = Ψ1 (k, 0)I1 (α, β, b, d, k). (2.16f) S kh

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2.2.5. Hydrodynamic pressure loads (added mass and radiation damping) The wave loads on a planar wavemaker may be estimated by integrating the total pressure over the wetted surface of the wavemaker, i.e., 

 F n = dS, (2.17a,b) P M r×n 0

S

where the outward pointing unit normal n points from the wavemaker into the fluid, and the pseudo-unit normal n for the rotational modes is given by n = r × n = (z + h − d)nx ey = ny ey .

(2.17c)

Force. For the Type I piston wavemaker of total width B, the horizontal component of the pressure force on the fluid side only may be computed from the real part of 
−b/h

Cn Ψn (Kn , z/h)d(z/h) exp −i(ωt + ν) F1 (t) = Re iρωBh n=1

−1+d/h

= −F1 cos(ωt + ν − α1 ),

(2.18a)

where the static component of the pressure force on the fluid side only is Fs = −

ρgBh2 [1 − 2(d/h) + (d/h)2 − (b/h)2 ]. 2

(2.18b)

The hydrodynamic component of F1 (t) may be separated linearly into a propagating and an evanescent component that are related to the piston wavemaker translational velocity and acceleration, respectively, from the real part of F1 (t) = −Re{[λ11 (Sω) + µ11 (−iSω 2 )] exp −i(ωt + ν)} = −µ11 (−Sω 2 sin(ωt + ν)) − λ11 (Sω cos(ωt + ν))

(2.18c)

¨ 1 (t) − λ11 X˙ 1 (t)}, = Re{−µ11X

(2.18d)

where the added mass coefficient µ11 may be computed from the evanescent eigenmodes only, and the radiation damping coefficient λ11 may be computed from the propagating eigenmode only. The average power may be computed from λ11 (Sω)2 . −F1 X˙ 1 t = 2

(2.19a)

Equating (2.19a) to (2.16d) yields  λ11 =

A1 S1

2

ρBh CG , ko h

(2.19b)

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that relates the radiation damping coefficient to the square of the ratio of the radiated wave amplitude to the amplitude of the wavemaker displacement. Moment. For the Type I wavemaker of width B, the dynamic pressure moment on one side only of the wavemaker may be computed from the real part of  
 −b/h 

d z 2 Cn M5 (t) = Re iρωBh 1+ − h h −1+d/h n=1   z  × Ψn (Kn , z/h)d exp −(ωt + ν) h = −M5 cos(ωt + ν − α5 ),

(2.20a)

and the static component of the pressure moment on the fluid side only is       2   3  2  3 2 d d b ρgBh3 b d b −3 . 1− 1− +2 +3 − Ms = 6 h h h h h h (2.20b) The pressure moment M5 (t) in (2.20a) may be separated linearly into a propagating and an evanescent component that are related to the rotational velocity and acceleration from the real part of1   Sω 2 M5 (t) = −Re µ55 −i ∆(1 + ∆b /∆)    Sω + λ55 −i exp −i(ωt + ν) , ∆(1 + ∆b /∆) ¨ 5 (t) − λ55 Θ ˙ 5 (t), M5 (t) = −µ55 Θ

(2.21a)

where µ55 = ρBh4

I 2 (α, β, b, d, κn ) n , κn h n=2

λ55 = ρωBh4

I12 (α, β, b, d, k) . kh

(2.21b)

(2.21c)

2.3. Circular Wavemaker Havelock5 applied Fourier integrals to develop a theory for surface gravity waves forced by circular wavemakers in water of both infinite and finite depth. The fluid motion may be obtained from the negative gradient of a scalar velocity potential Φ(r, θ, z, t) according to q(r, θ, z, t) = −∇Φ(r, θ, z, t),

(2.22a)

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where the 3D gradient operator ∇(•) in polar coordinates is  ∇(•) =

∂ er + ∂r

  1 ∂ ∂ eθ + e3 (•). r ∂θ ∂z

(2.22b)

The total pressure field P (r, θ, z, t) may be computed from the unsteady Bernoulli equation in polar coordinates according to P (r, θ, z, t) = p(r, θ, z, t) + pS (z)   ∂Φ(r, θ, z, t) 1 − |∇Φ(r, θ, z, t)|2 + Q(t) − ρgz, =ρ ∂t 2

(2.22c)

where Q(t) = the Bernoulli constant, and the free surface elevation η(r, θ, t) for zero atmospheric pressure according to 1 η(r, θ, t) = g



Q(t) ∂Φ(r, θ, η, t) 1 − |∇Φ(r, θ, η, t)|2 + ∂t 2 ρ

r ≥ b + ξ(θ, η, t);

 ;

z = η(r, θ, t).

(2.22d)

The scalar velocity potential Φ(r, θ, z, t) must be a solution to the continuity equation ∇2 Φ =

1 ∂ r ∂r

 ∂Φ 1 ∂2Φ ∂2Φ r + 2 + = 0, ∂r r ∂θ2 ∂z 2

r ≥ b + ξ(θ, z, t);

0 ≤ θ ≤ 2π;

−h ≤ z ≤ η(r, θ, t),

(2.23a)

with the following boundary conditions: Kinematic Bottom Boundary Condition (KBBC): ∂Φ = 0; ∂z

r ≥ b + ξ(θ, −h, t);

0 ≤ θ ≤ 2π;

z = −h.

(2.23b)

Combined Kinematic and Dynamic Free Surface Boundary Condition (CKDFSBC):   ∂Φ dQ ∂ 1 ∂ 2Φ + g − − ∇Φ · ∇ |∇Φ|2 + = 0; ∂t2 ∂z ∂t 2 dt r ≥ b + ξ(θ, η, t);

0 ≤ θ ≤ 2π;

z = η(r, θ, t).

(2.23c)

Kinematic WaveMaker Boundary Condition (KWMBC): ∂Φ ∂ξ 1 ∂Φ ∂ξ ∂Φ ∂ξ + − − = 0; ∂r ∂t r2 ∂θ ∂θ ∂z ∂z

r = ξ(θ, z, t);

−h ≤ z ≤ η(b, θ, t).

(2.23d)

Two types of circular cylindrical wavemaker displacements ξ(θ, z, t) may be analyzed, viz., amplitude-modulated (AM) and phase-modulated (PM) wavemakers.

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The distinction between these two types is in the azimuthal θ dependency of the wavemaker displacement ξ(θ, z, t) from its mean position r = b, given by   cos mθ sin(ωt + ν) mS (2.23e) χ(z/h) . ξ(θ, z, t) = (∆/h) sin(ωt + ν + mθ) (2.23f) Kinematic Radiation Boundary Condition (KRBC):   ∂ ± iKn Φ(r, θ, z, t) = 0; r → ∞. √ lim |Kn r|→+∞ ∂r

(2.23g)

Finally, physically realizable solutions to (2.23a) must be periodic in θ; i.e., Φ(r, θ, z, t) = Φ(r, θ + 2π, z, t).

(2.23h)

1

The dimensional WMBVP may be scaled and linearized by expanding the variables in perturbation series with a dimensionless perturbation parameter ε = kA. A scalar radiated velocity potential ϕ(r, θ, z) may be defined by the real part of Φ(r, θ, z, t) = Re{ϕ(r, θ, z) exp −i(ωt + ν)}.

(2.24)

A linearized WMBVP may be obtained by setting the dimensionless parameter kA = ε = 0 and by requiring that kh = O(1). This linearized WMBVP is ∇2 ϕ(r, θ, z) = 0;

b ≤ r < +∞; 0 ≤ θ ≤ 2π; −h ≤ z ≤ 0,

∂ϕ(r, θ, z) = 0; ∂z

b ≤ r < +∞; 0 ≤ θ ≤ 2π; z = −h,

∂ϕ(r, θ, z) − ko ϕ(r, θ, z) = 0; ∂z  √

lim

|Kn r|→+∞

b ≤ r < +∞; 0 ≤ θ ≤ 2π; z = 0,  ∂ − iKn ϕ(r, θ, z) = 0, ∂r

∂ϕ(r, θ, z) ∂ξ(θ, z, t) exp −i(ωt + ν) = − ; ∂r ∂t  η(r, θ, t) = Re

(2.25b)

(2.25c)

(2.25d)

r = b; 0 ≤ θ ≤ 2π; −h ≤ z ≤ 0,

 −iωϕ(r, θ, z) exp −i(ωt + ν) ; g

(2.25e) b ≤ r < ∞; 0 ≤ θ ≤ 2π; z = 0,

P (r, θ, z, t) = {p(r, θ, z, t)} + ps (z) = Re{−iωρϕ(r, θ, z) exp −i(ωt + ν)} − ρgz, ϕ(r, θ, z) = ϕ(r + λ, θ, z);

(2.25a)

ϕ(r, θ, z) = ϕ(r, θ + 2π, z).

(2.25f) (2.25g) (2.25h,i)

The specified wavemaker shape function χ(z/h) is valid for either a doublearticulated piston or hinged circular AM or PM wavemaker of variable draft that

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37

Definition sketch for circular wavemaker.

is shown in Fig. 2.2 is identical to (2.7) for a 2D planar wavemaker with the dimension b replaced with a and the stroke S replaced with the azimuthal stroke m S. The solution to the WMBVP (2.25) may be compactly expressed by the following orthonormal eigenseries: m ϕ(r, θ, z)

=

∞

(1) Cmn Ψn (Kn , z/h)Hm (Kn r)MA(P ) (mθ),

(2.26a,b)

n=1

where the azimuthal mode function is  cos mθ ; MA(P ) (mθ) = exp −imθ

m ≥ 0 and integer,

(2.26c,d)

and where (2.26a) represents an AM wavemaker; (2.26b) represents a PM wavemaker; Ψn (Kn , z/h) = the orthonormal eigenseries defined in (2.11); (1) Hm (Kn r) = the Hankle function of the first kind. When K1 = k and Kn = iκn for n ≥ 2 and integer, (1) Hm (iκn r) = Jm (iκn r) + iYm (iκn r) =

2 −(m+1) i Km (κn r), π

(2.26e)

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where Km (•) = the Modified Bessel (or Kelvin) function of the second kind of order m. The coefficients Cmn may be computed by expanding the KWMBC in an eigenseries following the procedure in (2.14) and obtaining Cmn = −

m Sj hω

In (α, β, a, d, Kn )

(1) Kn ∆ Ln (Hm (Kn b))

Ln (Zm (ζn )) =

;

n ≥ 1 and integer,

(2.26f)

dZm (ζn ) 1 = {Zm−1 (ζn ) − Zm+1 (ζn )}; dζn 2

(1) (ζn ). Zm (ζn ) = Jm (ζn ), Ym (ζn ), Km (ζn ), Hm

(2.26g)

The solution to (2.25) is given by the real part of the following eigenseries expansion: m Φj (r, θ, z, t)

[m Sj hω]

=

m Φpj (r, θ, z, t)

+ m Φej (r, θ, z, t) [m Sj hω]

  (1)   I1 (α, β, a, d, k) Ψ1 (k, z/h)Hm (kr)    (1)  k∆   L1 (Hm (kb))   = −Re  ∞ 

In (α, β, a, d, κn ) Ψn (κn , z/h)Km (κn r)       +  κn ∆ Ln (Km (κn b)) n=2       −i(ωt+ν) . (2.27a,b) × MA(P ) (mθ)e      Because the asymptotic behavior of the evanescent eigenseries Km (κn r) depends on the mode m(1) , it is not possible to specify a minimum distance from the wavemaker equilibrium boundary at r = b where the evanescent eigenvalues are less than 1% of their value at the circular wavemaker boundary. The wave field must be computed far away from the wavemaker, and it is understood that far away must be computed uniquely for each radial mode m for either an AM or PM circular wavemaker. The evaluation of the power, forces, and moments, and added mass and radiation damping coefficients for both AM and PM circular wavemakers are given by Hudspeth.1 2.4. Directional Wavemakers Directional wavemakers are vertically segmented wavemakers that undulate sinuously and, consequently, are also called snake wavemakers. Segmented directional wavemakers may be driven either in the middle of each vertical segment or at the joint between vertical segments. Because of these two methods of wave generation, parasitic waves are formed along the wavemaker due to either the discontinuity of

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the wavemaker surface (middle segment driven) or of the derivative of the wavemaker surface (joint driven). A dimensional scalar spatial velocity potential ϕ(x, y, z) may be defined by the real part of Φ(x, y, z, t) = Re{ϕ(x, y, z) exp −i(ωt + ν)}.

(2.28a)

The dimensional linear fluid dynamic pressure field p(x, y, z, t) and 3D fluid velocity vector field may be computed from p(x, y, z, t) = ρ

∂Φ(x, y, z, t) , ∂t

q(x, y, z, t) = −∇3 Φ(x, y, z, t), ∇3 (•) =

∂(•) ∂(•) ∂(•) ex + ey + ez . ∂x ∂y ∂z

(2.28b) (2.28c) (2.28d)

The dimensional WMBVP for directional waves is given by ∇23 ϕ(x, y, z) = 0;

x ≥ 0;

B ≤ y ≤ +B;

−h(x, y) ≤ z ≤ 0,

(2.29a)

∂ϕ(x, y, 0) − ko ϕ = 0; x ≥ 0; −B ≤ y ≤ +B; z = 0, ∂z

(2.29b)

∂ϕ = 0; x ≥ 0; y = ±B; −h(x, y) ≤ z ≤ 0, ∂y  x = 0, ∂Φ(x, y, z, t) ∂ξ(y, z, t)  =− ; −B ≤ y ≤ +B,  ∂x ∂t −h(0, y) ≤ z ≤ 0,  lim

x→+∞

 ∂ − iKn ϕ(x, y, z) = 0, ∂x

∂ϕ(x, y, z) = −∇2 ϕ(x, y, z) · ∇2 h(x, y); z = −h(x, y), ∂z   2 ∂ ∂2 (•), ∇22 (•) = , ∂x2 ∂y 2  ˆ z) U(y, ξ(y, z, t) = Re i [∆U (y, a)][∆U (z, b, d)] exp −i(ωt + ν) , ω  ˆ (y, z) = U

Sω ∆/ho

(2.29c)

(2.29d)

(2.29e)

(2.29f) (2.29g)

(2.29h)

Γ(y)χ(z/ho ),

∆U (y, a) = U (y + a− ) − U (y − a+ ), ∆U (z, b, d) = U (z + h − d) − U (z + b),

(2.29i)

(2.29j)

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Fig. 2.3.

Definition sketch for rectangular directional wave basin.

where ko = ω 2 /g and where a± denotes the (possibly nonsymmetric) transverse ends of the directional wavemaker in the transverse y-direction in Fig. 2.3. The solution to the WMBVP in (2.29) is given by the following set of orthonormal eigenfunctions: ϕ(x, y, z) = i

g

ζn (x, y)Υn (Kn , z/h), ω n=1

  Ψ1 (k, z/h) ; Ψ1 (k, 0) Υn (Kn , z/h) =  Ψ1 (κn , z/h); ko h = Kn h tanh Kn h = 0;

(2.30a)

n = 1,

(2.30b)

n ≥ 2,

(2.30c)

n = 1, 2, 3, . . . ,

(2.30d)

where K1 = k and Kn = +iκn for n ≥ 2 and Ψn (•) is defined in (2.11). The orthonormal eigenfunctions (2.30b) and (2.30c) are applicable strictly only for constant depth wave basins; however, they may be applied to slowly varying depth wave basins if (2.30b) and (2.30c) are considered to be evaluated only locally over relatively small horizontal length scales (e.g., several wavelengths λ), where the depth may be considered to be locally equal to a constant by a Taylor series expansion of the depth.1 Substituting (2.30a) into (2.29a) yields the following 2D Helmholtz equation: ∇22 ζ(x, y) + Kn2 ζ(x, y) = 0;

x ≥ 0;

−B ≤ y ≤ +B.

(2.31)

Alternatively, for wave basins with mildly sloping bottoms, the mild slope equation may be applied according to1 ∇2 • (CCG ∇2 ζ(x, y)) + Kn2 CCG ζ(x, y) = 0,

(2.32)

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where the wave group velocity CG is given by (2.16e). If the product CCG is a constant, (2.32) reduces to the 2D Helmholtz equation (2.31). Applying the WKBJ approximation1 for the x-dependent solution in the method of separation of variables to (2.32) yields the following solution1 :   g Φ(x, y, z, t) = Re i ζ(x, y)Υn (Kn , z/h) exp −i(ωt + ν) ω    M   g

[C(x)CG (x)]x=0     Ξm (µm , y/B) Amn i  ω m=0 n=1 C(x)CG (x) = Re 
x        × Υn(Kn , z/h) exp i Qmn dξ exp − i(ωt + ν)  (2.33a)

Ξm (µm , y/B) =

cos µm B(y/B − 1) ; Mm 

Qmn =

Mm = 1 + δm0 ;

Kn2 − µ2m ;

m > 0;

µm =

mπ 2B

K1 = k > µ2m .

(2.33b–d)

(2.33e)

The coefficients Am may be computed from (2.29d) by expanding the wavemaker shape function in orthonormal eigenfunctions1 and are given by k

Amn

S Qmn (∆/hx0 )

= −Ψn (Kn , 0)In (α, β, b, d, k)
×

+

a+ /B

− a /B −

Γj (qj , y/B)Ξm (µm , y/B)d(y/B).

(2.34)

2.4.1. Full-draft piston wavemaker The prescribed transverse y-component of the snake displacement of a full-draft (b = d = 0) piston (α = 0 and β = 1) wavemaker may be expressed as Γj (qj , y/B) =

+∞

c˜j exp i[qj B(y/B + νy )],

(2.35a)

j=−∞

where the coefficients c˜j may be computed from the integral in (2.34) by
cmj = =

+

a+ /B

− a /B −

Γj (qj , y/B)Ξm (µm , y/B)d(y/B)

Ra+ ,a− + iIa+ ,a− . mB(qj2 − µ2m )(1 + δm0 )

(2.35b)

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If the full-draft piston snake wavemaker spans the entire width of the wave basin so that a± = ±B, then (2.35b) reduces to the integral in (2.34) and  4qj B cmj =

sin[qj B(νj − 1)] + (−1)m sin[qj B(νj + 1)] −i{cos[qj B(νj − 1)] + (−1)m cos[qj B(νj + 1)]} ((qj B − (mπ)2 )2 )(1 + δm0 )

.

(2.35c)

Values for cmj for (possibly nonsymmetric) values for a± are given by Hudspeth.1

2.5. Sloshing Waves in a 2D Wave Channel A long rectangular wave channel with a horizontal flat bottom, two rigid vertical sidewalls, and a wavemaker may generate either 2D, long-crested progressive waves or two types of transverse waves, viz., (1) sloshing waves that are excited directly by transverse motion of the wavemaker or (2) cross waves that are excited parametrically by the progressive waves at a sub-harmonic of the wavemaker frequency. The WMBVP for 3D sloshing waves is identical to (2.5) for planar 2D wavemakers except for the KWMBC at x = 0 and an additional kinematic boundary condition on the sidewalls of the 2D wave channel at y = ±B in Fig. 2.4. The kinematic and dynamic wave fields may be computed from a dimensional 3D scalar

Fig. 2.4.

Definition sketch for a sloshing wave channel.

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velocity potential Φ(x, y, z, t). The fluid velocity q(x, y, z, t) may be computed from the negative directional derivative of a scalar velocity potential by q(x, y, z, t) = −∇Φ(x, y, z, t). (2.36a) The dimensional fluid dynamic pressure field p(x, y, z, t) may be computed from p(x, y, z, t) = ρ

∂Φ(x, y, z, t) . ∂t

(2.36b)

A spatial velocity potential ϕ(x, y, z) may be defined by the real part of Φ(x, y, z, t) = Re{ϕ(x, y, z) exp −i(ωt + ν)}.

(2.36c)

The WMBVP for sloshing waves is given by the following: ∇2 ϕ(x, y, z) = 0;

x ≥ 0; −B ≤ y ≤ +B; −h ≤ z ≤ 0,

(2.37a)

∂ϕ(x, y, −h) = 0; x ≥ 0; −B ≤ y ≤ +B; z = −h, ∂z

(2.37b)

∂ϕ(x, y, 0) − ko ϕ = 0; x ≥ 0; −B ≤ y ≤ +B; z = 0, ∂z

(2.37c)

∂Φ(x, y, z, t) ∂ξ(y, z, t) =− ; x = 0; −B ≤ y ≤ +B; −h ≤ z ≤ 0, ∂x ∂t ∂ϕ = 0; x ≥ 0; y = ±B; −h ≤ z ≤ 0, ∂y  ∂ lim − iKn ϕ(x, y, z) = 0, x→+∞ ∂x     x ≥ 0, −iω −B ≤ y ≤ +B, Φ(x, y, 0, t) ; η(x, y, t) = Re  g z = 0.

(2.37d)

(2.37e)



A solution to (2.37) is given by the following eigenfunction expansions1 : 

˜ Φ(x, y, z, t) = Re ψn (x, y)Ψn (Kn , z/h) exp −i(ωt + ν) ,

(2.37f)

(2.37g)

(2.38a)

n=1

ϕ(x, y, z) = ψ˜n (x, y)Ψn (Kn , z/h), 

η(x, y, t) = Re ζn (x, y) exp −i(ωt + ν) ,

(2.38b) (2.38c)

n=1

ω ζn (x, y) = −i ψ˜n (x, y)Ψn (Kn , 0), g

(2.38d)

g ζn (x, y) , ψ˜n (x, y) = i ω Ψn (Kn , 0)

(2.38e)

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where ζn (x, y) is sometimes referred to as a displacement potential. The scalar potential (2.38a) may be expressed from (2.38d) and (2.38e) as 

g Ψn (Kn , z/h) Φ(x, y, z, t) = Re exp −i(ωt + ν) . (2.39) i ζn (x, y) ω Ψn (Kn , 0) n=1 The WMBVP may be expressed in terms of a displacement potential ζn (x, y) by   x ≥ 0, 2 2 ∂ ζn (x, y) ∂ ζn (x, y) 2 −B ≤ y ≤ +B, (2.40a) + + K ζ (x, y) = 0; n n  ∂x2 ∂y 2 −h ≤ z ≤ 0.   x = 0,

∂ζn (x, y) Ψn (Kn , z/h) ω (2.40b) = i U (y, z) B ≤ y ≤ +B,  ∂x Ψn (Kn , 0) g n=1 −h ≤ z ≤ 0.   ∂ (2.40c) lim − iKn ζn (x, y) = 0. x→+∞ ∂x ∂ζn (x, y) = 0; x ≥ 0; y = ±B; −h ≤ z ≤ 0, ∂y

(2.40d)

where (2.40a) is the 2D Helmholtz equation.9,10 Because the boundary conditions are prescribed on boundaries that are constant values of (y,z), a solution to the WMBVP (2.40) may be computed by the method of separation of variables and is given by1 Φ(x, y, z, t)                   g = Re i  ω                

 M

 Ψ1 (k, z/h)   Cm1 Υm (y/B)  exp iPm1 x   Ψ1 (k, 0)  m=0    

 Ψ1 (k, z/h)   exp + Cm1 Υm (y/B)    Ψ (k, 0)                 

                 

             1 m=M+1 exp −i(ωt + ν) ,   − Ξm1 x          

  Ψn(κn , z/h)    exp + Cmn Υm (y/B)       Ψ (κ , 0) n n  m=0 n=2     − Qmn x (2.41a)

where Υm (µm , y/B) =

cos µm B(y/B − 1) ; Mm µm =

mπ , 2B

2 Mm = 1 + δm0 ;

(2.41b)

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Pm1 =

K12 − µ2m = Ξm1 =



k 2 − µ2m ;

 µ2m − k 2 ;

Q2mn = µ2m − Kn2 ; n ≥ 1: K1 = k > µm : Qm1 = i

k < µm

45

k > µm ; m ≤ M,

k < µm ; m ≥ 0;

(2.41d)

n ≥ 1,

(2.41e) m ≤ M,

n = 1: k = µm : Qm1 = 0  : Qm1 = µ2m − k 2 = Ξm1 > 0; m > M, 

(2.41c)

m > M,

 k 2 − µ2m = iPm ;

n ≥ 2 : Kn = iκn : Qmn =

FA

µ2m + κ2n > 0,

(2.41f)

(2.41g) (2.41h)

where M is the maximum integer value for m in order for µm < k. Substituting (2.41a) into (2.40b) yields the following solutions for Cmn 1 :

  ω Ψ1 (Kn , 0) +1  y  0  z   z  z y d d U y, Ψ1 K n , Υ m µm ; Cm1 = g Pm1 B −1 h h h B −1

Cm1 = −i

Cmn = −i

ω Ψ1 (Kn , 0) g Ξm1 ω Ψn (Kn , 0) g Qmn




+1

d

y B

−1




+1

d

y

−1

B




m≤M (2.42a) 0   z   z  z y U y, Ψ1 K n , Υm µm , ; d h h h B −1




m>M +1 (2.42b) z   z  0   z y U y, Ψn K n , Υ m µm , ; d h h h B −1 m ≥ 0, n ≥ 2.

(2.42c)

The first three transverse eigenmodes are illustrated in Fig. 2.5. 2.6. Conformal Mappings Conformal and domain mappings are applications of complex variables to solve 2D boundary value problems. Conformal mapping is an angle preserving transformation that will compute exact nonlinear solutions for surface gravity waves of constant

Fig. 2.5.

First three transverse eigenmodes in a 2D wave channel.

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form that may be treated as a steady flow following a Galilean transformation from a fixed inertial coordinate system to a noninertial moving coordinate system. Domain mapping is a transformation of the wavemaker geometry into a fixed computational domain where a solution may be computed efficiently. 2.6.1. Conformal mapping1 Conformal mapping of the WMBVP provides a global solution that accurately accounts for the singular behavior at all irregular points. The irregular points in the physical wavemaker domain are transformed into both weak and strong singular kernels in a Fredholm integral equation. The two irregular points on the WMBVP boundary are located at (1) the intersection between the free-surface and the wavemaker boundary and (2) the intersection between the horizontal bottom and the wavemaker boundary. These two irregular points exhibit integrable weakly singular kernels. The far-field radiation boundary exhibits a strongly singular kernel and significantly affects the solution. The irregular frequencies3,4 are included in the global solution by the Fredholm alternative. A theory for the planar WMBVP computes a global solution by a conformal mapping of the physical wavemaker boundary to a unit disk that includes the motion of an inviscid fluid near irregular points that are illustrated in Fig. 2.6. A numerical solution to Laplace’s equation in a transformed unit disk may be computed from a Fredholm integral equation. The WMBVP defined by (2.5) is transformed into complex-valued analytical functions where the complex-valued coordinates are defined as z = x + iy. The coordinates for the semi-infinite wave channel in Fig. 2.1(a) must be transformed to complexvalued coordinates z. The fluid velocity q(x, y, t) and dynamic pressure p(x, y, t) may be computed from a scalar velocity potential Φ(x, y, t) according to q(x, y, t) = −∇Φ(x, y, t);

p(x, y, t) = ρ

∂Φ(x, y, t) . ∂t

(2.43a,b)

The WMBVP and Type I wavemaker shape function are given by (2.5)–(2.7). There are both Irregular (I) and Regular (R) points at the intersections between the Smooth (S) and Critical (C) boundaries B1 and B2 in the WMBVP as illustrated in Fig. 2.6 where these two boundary intersection points are identified as P1 and P2 . The classification of the boundary points P1 and P2 in Fig. 2.6 depends on (1) the boundary conditions ϕi (Pj ) and (2) the continuity of the boundaries Bm and their derivatives where i, j, and m = 1 or 2. A conformal mapping of the semi-infinite wave channel strip in the physical plane will yield a Fredholm integral equation,6,7 where these critical points may be transformed to singular points that are integrable over a smooth continuous mapped boundary. 2.6.1.1. Conformal mapping to the unit disk 1 Two conformal mappings are: (i) the physical z-plane to a semi-circle in the Z-plane shown in Fig. 2.7; and (ii) a semi-circle in the upper Z-plane to a unit disk in the

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Fig. 2.6. Combinations of Irregular (I) and Regular (R) boundary points P1 and P2 between Smooth (S) and Critical (C) boundaries B1 and B2 intersections in the WMBVP.8

Fig. 2.7. Mapping of the semi-infinite strip in the lower half x–y-plane in the physical z-plane to the upper half X–Y -plane in the Z-plane.8

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Fig. 2.8.

Mapping of the upper half-plane in the Z-plane to the unit disk in the Q-plane.

Q-plane shown in Fig. 2.8. The Schwarz–Christoffel transformation dz C1 √ =√ dZ Z +1 Z −1

(2.44a)

may be integrated to obtain z = x + iy  h = − Log[−Z − Z 2 − 1], π

(2.44b)

where the Log[•] denotes the principal value of Log[•] and the argument of the Log[•] is −π ≤ arg < π. Inverting (2.44b) yields Z = X + iY = − cosh(πz/h),

(2.44c)

where ! πy " cos , h h ! πx " ! πy " Y = − sinh sin . h h

X = − cosh

! πx "

(2.44d) (2.44e)

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The radiation boundary in the z-plane transforms to a semi-circle in the Z-plane by     1 2πy 2πx cos + cosh , (2.44f) R2 = X 2 + Y 2 = 2 h h  πx   πy  Y tan θ = = tanh tan . (2.44g) X h h Details of the transformation of the WMBVP are given by Hudspeth.1 2.6.1.2. Mapping Z-plane to a unit disk The upper-half-plane of the Z-plane may be mapped into a unit disk shown in Fig. 2.8 by the following bilinear transformation:   i−Z Q = ξ + iζ = exp(iθ0 ) , (2.45a) i+Z that may be inverted to obtain  1−Q , Z = X + iY = i 1+Q 

(2.45b)

and the mapping function coordinates are ξ=

1 − X2 − Y 2 , X 2 + (Y + 1)2

ζ=

2X , X 2 + (Y = 1)2

(2.45c,d)

that may be transformed into the cylindrical coordinates for the unit disk in Fig. 2.8 by   2X (X 2 + Y 2 − 1)2 − 4X 2 ζ . = arctan r2 = ξ 2 +ζ 2 = ;  = arctan (X 2 + (Y + 1)2 )2 ξ 1 − X2 − Y 2 (2.45e,f) Details of the transformation are given by Hudspeth.1 A numerical solution to the transformed WMBVP may be computed from the following Fredholm integral equation1 :   
+π  ˆ r , ) ∂G(r, r ˆ , , ) ˆ ∂ Φ(ˆ ˆ 2π ˆ ˆ r , ) ˆ + G(r, rˆ, , ) ˆ rˆd, ˆ π Φ(r, ) = − −π Φ(ˆ ∂ˆ r ∂ˆ r (2.46a) where G(r, rˆ, , ) ˆ = − ln[ρ(r, rˆ, , )]; ˆ

ρ2 (r, rˆ, , ) ˆ = r2 − 2rˆ r cos( ˆ − ) + rˆ2 . (2.46b,c)

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Fig. 2.9. Nodal points on the unit disk in the Q-plane and the corresponding nodal points on the wavemaker in the physical z-plane.8

Numerical solutions to (2.46a) may be computed by discretizing the unit disk boundary as shown in Fig. 2.9. The numerical details regarding the evaluation (2.46a) at the two weakly singular irregular points at B and D in the physical z-plane in Fig. 2.7 and the strongly singular point at ±π that is the vertical radiation boundary A–E at +∞ in the physical z-plane in Fig. 2.7 are tedious.8 Global numerical solutions may be computed for both the linear and the nonlinear WMBVPs. 2.6.1.3. Conformal mapping to the unit disk 2 The wavemaker geometry shown in Fig. 2.10 is mapped to the unit disk by two transformations. The WMBVP is given by (2.5) and the WM shape function is χ(y/h) = [α(y/h) + β][U (y/h + 1 − b0 /h) − U (y/h + a0 /h)].

(2.47)

In order to transform the wavemaker geometry to a Jacobian elliptic function, it must be rotated and translated as shown in Fig. 2.11. The 90◦ rotation to the z  -plane is given by z  = x + iy  = iz = −y + ix.

(2.48a)

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Fig. 2.10.

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51

WMBVP11 with the six critical boundary points at a–a0 –b–b0 –c–d.

Fig. 2.11. Rotation and translation of the physical wavemaker rectangular strip in the z-plane to the w-plane.11

The horizontal shift to the left in the z  -plane is given by z  = x + iy  = w − h/2 = −y − h/2 + ix.

(2.48b)

In order to map the WM geometry in the z-plane to the semi-circle in the Z-plane shown in Fig. 2.12 as a Jacobian elliptic function, the rotated and translated strip in the z  -plane must have the dimensions of −K ≤ ξ ≤ +K and 0 ≤ ζ ≤ K  , where K = h/2 and K  = 3h = 6K. This requires a coordinate amplification given by 2K  (x + iy  ) h  h 2K −y − + ix . = h 2

w=

(2.48c)

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Fig. 2.12.

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Mapping of the wavemaker semi-circle in the Z-plane to the unit disk in the Q-plane.11

The Schwarz–Christoffel transformation from the w-plane to the Z-plane is dw = 

ˆ CkdZ (a − Z)(b − Z)(c − Z)(d − Z)

.

(2.48d)

The following change of variables: ˆ dZ = adZˆ : κ = a/c; Cˆ = c, Z = aZ; modifies (2.48d) to the following Jacobian elliptic function:
Z dZˆ = sn−1 [Z, κ], w= 1 [(1 − Zˆ 2 )(1 − κ2 Zˆ 2 )] 2 0

(2.48e)

where sn[Z, κ] = the Jacobian elliptic function of modulus κ or sine amplitude function.9 Define kˆ = sn−1 [1, κ],

(2.48f)

and the mapping of the rectangle {x1 , x2 ; y1 , y2 } = {0, 3h; 0, h} is given by Z = X + iY "   # $ % & ! ˆ  sn − 2K  y + h2 , κ dn 2Kx h h ,k " ! =  1 − dn2 #− 2K $y + h % , κ& sn2 2Kx , kˆ  h 2 h ! " ! "  # $ % & # $ % & h 2K h 2Kx ˆ 2Kx ˆ   cn − 2K y + , κ dn − y + , κ , sn , k cn , k h 2 h 2 h h " ! , +i # 2K $ % &   h 2Kx ˆ 2 2 1 − dn − h y + 2 , κ sn h ,k (2.48g) where sn[•, •] in the copolar half-period trio in (2.48g) is defined in (2.48e), and cn[•, •] and dn[•, •] are defined by cn 2 [•, •] = 1−sn 2 [•, •], dn 2 [•, •] = 1−κ2 sn2 [•, •]. The mapping from the semi-circle in the Z-plane to the unit disk in the Q-plane

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Fig. 2.13.

53

Transformed boundary conditions mapped to arcs on the perimeter of the unit disk.11

is shown in Fig. 2.12; and the mapping to the unit disk in the Q-plane is shown in Fig. 2.13. The mapping of the Z-plane to the Q-plane is given by Q=

ˆ )2 + 2iZ(1 − α ˆ )2 i−Z −α ˆ (i + Z) (1 − α ˆ )2 − Z 2 (1 + α , = i+Z −α ˆ (i − Z) (1 − α ˆ 2 ) + Z 2 (1 + α) ˆ 2

(2.49a)

where −1 < α ˆ < +1. Changing variables to circular cylindrical coordinates by (1 − α ˆ 2 ) − 2Y (1 − α ˆ 2 ) + (X 2 + Y 2 )(1 + α ˆ )2 , (1 − α ˆ )2 − 2Y (1 − α ˆ 2 ) + (X 2 + Y 2 )(1 + α ˆ2 )   2X(1 − α ˆ2) , θ(X, Y ) = arctan (1 − α) ˆ 2 − (X 2 + Y 2 )(1 + α) ˆ 2

R2 (X, Y ) =

(2.49b) (2.49c)

the unit disk may be transformed into functions of the copolar trio of Jacobian elliptic functions. The transformed WMBVP in circular cylindrical coordinates is given by Hudspeth.1 A general solution to the transformed WMBVP may be written as10 ϕ(R, θ) =

N

n=0

R

n



 a ˆn ˆ cos nθ + bn sin nθ , 1 + δn0

(2.50)

where δij is the Kronecker delta function. Substituting (2.50) into the generic boundary conditions on each of the six arcs on the perimeter of the unit disk illustrated in Fig. 2.13, multiplying each of these six boundary conditions by a member of the set of the orthogonal series in (2.50), integrating over the interval of orthogonality −π ≤ θ ≤ +π yields the following matrix equation for each of the coefficients a ˆn and ˆbn AB = H.

(2.51)

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Fig. 2.14.

Physical fluid domain.12

2.7. Domain Mapping Domain mapping of the WMBVP12 follows the theory by Joseph.13 The physical fluid domain shown in Fig. 2.14 for the fully nonlinear WMBVP is mapped to a fixed computational fluid domain, and the discretized coupled free-surface boundary conditions are computed by an implicit Crank–Nicholson (C–N) method.14,15 At each iteration of the C–N method, the potential field is computed by the conjugate gradient method.15 The wavemaker motion Ξ(y/h, t) is assumed to be periodic with period T = 2π/ω, and the WMBVP with the surface tension Tˆ is given by  0 ≤ y ≤ Γ(x, t), 2 ∇ Φ(x, y, t) = ∆Φ(x, y, t) = 0; (2.52a) Ξ(y/h, t) ≤ x ≤ L. Tˆ ∂ 2 Γ(x, t) ∂Φ(x, y, t) 1 ρ ∂x2 + |∇Φ(x, y, t)|2 −  + gΓ(x, t) = 0.  2 3/2 ∂t 2 ∂Γ(x, t) 1+ ∂x

(2.52b)

∂Φ(x, y, t) ∂Γ(x, t) ∂Φ(x, y, t) ∂Γ(x, t) − + = 0; ∂y ∂x ∂x ∂t Ξ(Γ(x, t), t) ≤ x ≤ L; ∂Φ(x, y, t) = 0; ∂y ∂Φ(x, y, t) = 0; ∂x

y = Γ(x, t).

Ξ(Γ(x, t), t) ≤ x ≤ L;

x = L;

(2.52c) y = 0.

0 ≤ y ≤ Γ(L, t).

∂Φ(x, y, t) ∂Ξ(y/h, t) ∂Ξ(y/h, t) ∂Φ(x, y, t) =− + ; ∂x ∂t ∂y ∂y



x = Ξ(y/h, t), 0 ≤ y ≤ Γ(0, t).

(2.52d)

(2.52e)

(2.52f)

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The initial conditions for t = 0 are Γ(x, 0) = H(x); ∂Γ(x, t) = 0;

(2.52g)

 Ξ(x, 0) ≤ x ≤ L.

Φ(x, Γ(x, 0), 0) = 0;

(2.52h) (2.52i)

The physical fluid domain shown in Fig. 2.14 may be mapped into a dimensionless fixed rectangle of dimensions 0 ≤ ξ ≤ 1 by 0 ≤ ζ ≤ 1 by the transforms ξ=

x ; L

ζ=

y ; Γ(x, t)

τ = ωt;

Γ(x, t) , h

(2.53a–d)

P (x, y, t) , ρA2 ω 2

(2.53e,f)

γ(ξ, τ ) =

and dimensionless variables by q(ξ, ζ, τ ) = −

ϕ(ξ, ζ, τ ) =

∇Φ(x, y, t) ; Aω

Φ(x, y, t) ; Ahω

w(ζ, τ ) =

p(ξ, ζ, τ ) =

Ξ(y/h, t) ; S

ˆ= T

˜ T . ρALhω 2

(2.53g–i)

Because ζ is a function of both x and y in (2.53b), transforming partial derivatives with respect to x must be done with some care.12 Details of these lengthy transformations and the transformed WMBVP in the fixed mapped domain are given by Hudspeth.1 References 1. R. T. Hudspeth, Waves and Wave Forces on Coastal and Ocean Structures (World Scientific, Singapore, 2006). 2. R. T. Hudspeth, J. M. Grassa, J. R. Medina and J. Lozano, J. Hydraulic Res. 32, 387 (1994). 3. F. John, Commun. Pure Appl. Math. 2, 13 (1949). 4. F. John, Commun. Pure Appl. Math. 3, 45 (1950). 5. T. H. Havelock, Phil. Mag. 8, 569 (1929). 6. P. R. Garabedian, Partial Differential Equations (Wiley, Inc., New York, 1964). 7. P. M. Morse and H. Feshbach, Methods of Theoretical Physics (McGraw-Hill Book Company, New York, 1953). 8. Y. Tanaka, Irregular points in wavemaker boundary value problem, PhD thesis, Oregon State University (1988). 9. G. F. Carrier, M. Krook and C. E. Pearson, Functions of a Complex Variable, Theory and Technique (McGraw-Hill Book Co. Inc., New York, 1966). 10. R. B. Guenther and J. Lee, Partial Differential Equations of Mathematical Physics and Integral Equations (Dover Publications, Inc., New York, 1996). 11. P. J. Averbeck, The boundary value problem for the rectangular wavemaker, MS thesis, Oregon State University (1993). 12. S. J. DeSilva, R. B. Guenther and R. T. Hudspeth, Appl. Ocean Res. 18, 293 (1996).

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13. D. D. Joseph, Arch. Rational Mech. Anal. 51, 295 (1973). 14. B. Carnahan, H. A. Luther and J. O. Wilkes, Applied Numerical Methods (John Wiley and Sons, New York, 1965). 15. R. Glowinski, Numerical Methods for Nonlinear Variational Problems (SpringerVerlag, 1984).

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Chapter 3

Analyses by the Melnikov Method of Damped Parametrically Excited Cross Waves Ronald B. Guenther Department of Mathematics Oregon State University, Corvallis, OR 97331, USA [email protected] Robert T. Hudspeth School of Civil and Construction Engineering Oregon State University, Corvallis, OR 97331, USA [email protected] The Wiggins–Holmes extension of the generalized Melnikov method (GMM) to higher dimensions and the extension of the Generalized Herglotz Algorithm (GHA) to nonautonomous systems are applied to weakly damped parametrically excited cross waves with surface tension in order to demonstrate that cross waves are chaotic. The GMM is a global perturbation analysis about a manifold of fixed points that are connected by separatrices for higher dimensional nonlinear dynamical systems. The Luke Lagrangian, density function for surface gravity waves with surface tension and dissipation is expressed in three generalized coordinates that are the time-dependent components of three velocity potentials that represent three standing waves. The Hamiltonian for these cross waves is homomorphic to the Hamiltonian for a parametrically excited pendulum that is an example of a Floquet oscillator that may be approximated by the Mathieu equation. Neutral stability curves measured from wave tank data motivated the inclusion of dissipation in the Luke Lagrangian density function for cross waves. An integral containing a generalized dissipation function that is proportional to the Stokes material derivative of the free surface is added to the Luke Lagrangian integral so that dissipation is correctly incorporated into the dynamic free surface boundary condition. The generalized momenta are computed from the Lagrangian; and the Hamiltonian is computed from a Legendre transform of the Lagrangian. This Hamiltonian contains nonautonomous components and is transformed by three canonical transformations in order to obtain a suspended system for the application of the Kolmogorov–Arnold–Moser (KAM) averaging

57

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58

operation and the GMM. The system of nonlinear nonautonomous evolution equations determined from Hamilton’s equations of motion of the second kind must be averaged in order to obtain an autonomous system that may be analyzed by the GMM. Hyperbolic saddle points that are connected by heteroclinic orbits are computed from the unperturbed autonomous system. The nondissipative perturbed Hamiltonian system with surface tension satisfies the KAM nondegeneracy requirements; and the Melnikov integral is calculated to demonstrate that the motion is chaotic. For the perturbed dissipative system with surface tension, the only hyperbolic fixed point that survives the averaged equations is a fixed point of weak chaos that is not connected by a homoclinic orbit; and, consequently, the Melnikov integral is identically zero. The chaotic motion for the perturbed dissipative system with surface tension is demonstrated by numerical computation of positive Liapunov characteristic exponents. A chaos diagram of the largest Liapunov exponent demonstrates regions in the Floquet forcing parameter space of possible chaotic motions; and the range of values of the Floquet parametric forcing parameter ε and of the wavemaker dissipation parameter α in the α−ε space where chaotic motions may exist.1,3

3.1. Introduction The cross waves shown in Fig. 3.1 are excited parametrically by progressive wavemaker waves at a sub-harmonic of the wavemaker frequency.1 Parametrically excited standing cross waves that oscillate in a direction transverse to the wavemaker forcing with crests perpendicular to the wavemaker are analyzed by the generalized Melnikov method (GMM) and by the Generalized Herglotz Algorithm (GHA) extended to nonautonomous systems. The cross wave wavelengths and wave modes

Fig. 3.1.

Mode 2 cross wave in a 2D wave channel.1–3

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possible are determined by the channel width L
c = 2(channel width)/n, where n = the mode number and is equal to the number of half-wavelengths across the channel. Floquet instabilities occur when the cross wave wavelengths also satisfy the frequency dispersion equation for surface gravity waves. Energy is transferred from the progressive wavemaker waves to the parametrically forced cross waves through the spatial mean motion of the free surface and their growth is due to the rate of working of the transverse stresses of the progressive wavemaker waves. The cross wave instability is shown to be homomorphic to the Floquet oscillator instability2 by constructing the neutral Floquet stability diagram shown in Fig. 3.2. The simultaneous generation of a primary resonance (ωp : ωc = 2 : 1) and a secondary resonance (ωp : ωc = 1 : 1) may be observed in the cross wave data. The criteria for horseshoe maps and for homoclinic/heteroclinic orbits are selected to test whether or not cross waves are a chaotic dynamical system. Specifically, the GMM provides local criterion for the transverse intersection of stable and unstable manifolds of the perturbed system and for the resulting chaotic motion near the unperturbed (undamped and unforced) homoclinic/heteroclinic orbits. In order to apply the GMM to a suspended dynamical system that will survive the KAM averaging operation, the nonautonomous Hamiltonian is transformed by three separate canonical transformations by applying the GHA for nonautonomous transformations. Three canonical transformations are applied in order to (1) eliminate cross product terms by a rotation of axes; (2) to eliminate two degrees of freedom, and (3) to suspend the nonautonomous terms in the wavemaker forcing component of the Hamiltonian by applying the Hamilton–Jacobian transformation. The Liapunov characteristic exponents represent an alternative criterion for diagnosing the chaotic behavior of a dynamical system by measuring the mean rate of exponential divergence of nearby trajectories, and is computed numerically for the perturbed dissipative Hamiltonian when the GMM fails to predict chaos.

Fig. 3.2. Neutral Floquet stability diagram for mode 2 cross waves (o = mode 2 cross waves and
= no cross waves).1,2

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Fig. 3.3.

Definition sketch of fluid domain.

3.2. Hamilton’s Principle for Cross Waves The fluid is assumed to be incompressible and inviscid and the flow to be irrotational. The dimensional fluid particle velocities u
and the dimensional total pressure in the fluid P
are computed from ¯
ϕ
; u
= − ∇

P
1 ¯

2 = −g
z
+ ϕ
t
− |∇ ϕ| ; ρ
2

(3.1a,b)

where ϕ
= a velocity potential; ρ
= the fluid mass density; and g
= the gravitational acceleration. The fluid domain is the 3D rectangular wave channel shown in Fig. 3.3. 3.2.1. Variational principle The Lagrangian L
with kinematic surface tension T
is given by L
=




V
(t
)


(ζ
− 1)
1 ¯

2 dSη
; |∇ ϕ | − ϕ
t
+ g
z
dV
+ T
2 ζ
S



(3.2a)

η

where η
= the free surface profile and where dSη

= ζ
dx
dy
;

¯
ϕ
|2 ]1/2 . ζ
= [1 + |∇

(3.2b,c)

Generalized Hamilton’s principle with dissipation.4 The first variational of (3.2) is
δ

t
2 t
1

L
dt
+




t
2

t
1

F




Dη
Dt




dt
= 0;

(3.3a)

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where the generalized dissipation function per unit mass density F
(Dη
/Dt
) is F






Dη
Dt






= −α

g
κ



Sη



Dη
δη

dSη
; Dt
ζ


(3.3b)

where α = a dimensionless damping parameter and κ
= the cross wave wavenumber. The dimensional boundary-value-problem may be obtained by requiring that the variation of φ
and η
vanish at the arbitrary values t
1 and t
2 , and is given by δL
+

d dt









 1

2 |∇ ϕ | −ϕ
t
+g
η
−T
∇
(∇
η
/ζ
) δη
dx
dy


−b
χη 2 z =η


2



ˆ s
) − (∇ ϕ )δϕ dV + (ϕn
+ Us
· n V
(t
) S








ηn (3.4) × δϕ dSs
+ T
δη /ζ dSη
,


δϕ
dV
=

(t )

b


Sη



where n
denotes a normal derivative. The Lagrangian in (3.2) reduces to1  1

ˆ s
ϕ
dS
ϕ ∇ ϕ dV + ϕ
+ Us
· n 2 n V
S





b


1 b + g
η
2 dx
dy
+ T
(ζ
− 1)dx
dy
. 2 −b
χη
−b
χ



1 L =− 2








2










(3.5)

η

The velocity potential ϕ
is the field variable, and the free surface η
may be expressed in terms of ϕ
by the free surface boundary conditions and the contact line conditions at the vertical sidewalls.1,3 The velocity potential ϕ
and the free surface displacement η
are assumed to be linear combinations of the progressive wavemaker wave and the parametrically excited cross wave given by the following: ϕ
(x
, y
, z
, t
) = ϕ
p (x
, z
, t
) + ϕ
c (y
, z
, t
),











η (x , y , t ) =

ηp
(x
, t
)

+

ηc
(y
, t
).

(3.6a) (3.6b)

The dimensional variables may be scaled by the following scales1,5: x
=

x ; k


y
=



h=κh; ωp
2 = g
k
τλ ; τ12 = 1 + τ ;

y ; κ


z
=

z ; κ






t ; g
κ


(3.7a–g)





ξ=k;

b=κb; τ = T
κ
2 /g
;

τλ = 1 + (τ /λ4r ); λ2r = κ
/k
;

t
= √

ω
c = g
κ
τ12 ; 2

(3.7h–m)

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 ϕ
p

=

ϕp a
p

ηc


=

g
; k


a
c ηc ;

 ϕ
c

=




χ =

ϕc a
c

a
w χ;

g
; κ


ηp
= a
p ηp ;

La
2 g
L =
c
. kκ

(3.7n–s)




Scaling by (3.7) yields the following dimensionless scaling parameters1,3: ε = κ
a
c ;

β = ωc
/ωp
;

γ = k
a
w ;

Γ = a
p /a
c ;

(3.8a–d)

and the following dimensionless field variables: ϕ(x, y, z, t) = ϕc (y, z, t) + Γλr ϕp (x, z, t); η(x, y, t) = ηc (y, t) + Γηp (x, t).

(3.9a,b)

The independent variation of ϕ
and η
vanish at arbitrary temporal values t
1 and t
2 in Hamilton’s principle (3.3) yielding the following scaled boundary value problem: 1 ϕxx + ϕyy + ϕzz = 0; γχ ≤ x ≤ ξ, y ≤ |b|, −h ≤ z ≤ εη, (3.10a) λ4r   1 z = εη, γχ ≤ x ≤ ξ, y ≤ |b|, ϕ η + ϕ η ϕz = −τ1 ηt + ε x x y y ; λ4r (3.10b)     ε 1 2 1 ϕ + ϕ2y + ϕ2z − τ1 ϕt + η = τ ηxx + ηyy − αϕz ; 2 λ4r x λ4r  γχ ≤ x ≤ ξ, (3.10c) y ≤ |b|, z = εη  ˆ = 0; ∇ϕ · n

y = |b|,

−h ≤ z ≤ εη,

γχ ≤ x ≤ ξ

z = −h,

y ≤ |b|,

γχ ≤ x ≤ ξ,

γ ϕx = − λ4r τ1 χt + γλ4r ϕz χz ; ε

x = γχ,

y ≤ |b|,

−h ≤ z ≤ εη,

(3.10d,e)

(3.10f)

plus an appropriate radiation condition at x = ξ. The free surface curvature requires a dynamical constraint that is given either by the contact line condition ηn

= 0 or by the edge constraint boundary condition η
= 0.1,3 The wavemaker perturbation forcing parameter γ is smaller than the Floquet parametric forcing parameter ε because experiments demonstrate that the standing cross wave amplitude becomes larger than the wavemaker forcing amplitude as t → ∞. The parameter ordering is 0 < γ 2 < εγ < ε2 < γ < ε < 1 or

0<

γ2 γ < γ < < 1. ε ε

(3.11a,b)

The higher order terms that will be neglected are O(ε2 ), O(εγ), and O(γ 2 ); but terms O(γ/ε) will be retained. Neglecting these higher order terms, the nondimensional Lagrangian integrals may be approximated by the following linear

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combination1,3 : Lχ = −

γτ1 2ε
b

+ −b







b −b

0

−h

[(ϕc + Γλr ϕp )χt ]x=0 dzdy

[εχt (ηc + Γηp )(ϕc + Γλr ϕp )]x,z=0 dy,


b
0 Γ [ϕpx (ϕc + Γλr ϕp )]x=ξ dzdy Lξ = − 3 2λr −b −h
b + [εϕpx (ηc + Γηp )(ϕc + Γλr ϕp )]x=ξ,z=0 dy, Lη =

1 2




−b b

−b




ξ

(3.12a)

(3.12b)

{τ1 (ηc + Γηp )t [(ϕc + Γλr ϕp )

0

+ ε(ηc + Γηp )(ϕc + Γλr ϕp )r ] + (ηc + Γηp )2 }z=0 dxdy
γ b − {χ[τ1 (ϕc + Γλr ϕp )(ηc + Γηp )t + (ηc + Γηp )2 ]}x,z=0 dy, (3.12c) 2 −b  


b   2 τ b ξ Γ2 2 Γ 2 2 2 Lτ = η + ηcy dxdy − η + ηcy dy. γχ 2 −b 0 λ4r px λ4r px −b x,z=0 (3.12d) The wavemaker forcing χ is specified to be t χ = f (z) sin β



f (z) = 1 + f (z) = 1,

z , for a full draft hinge, h for a full draft piston;

(3.13a,b)

and the deepwater cross wave and wavemaker wave are approximated by ϕc (y, z; t) = q(t) cos(y − b) exp[z], ϕp (x, z; t) = [Q1 (t) cos x +

Q2 (t) sin x] exp[z/λ2r ],

(3.14a) (3.14b)

where the variables q(t), Q1 (t), and Q2 (t) are the generalized coordinates. The freesurface displacement η is a solution to the linearized inhomogeneous boundary value problem given by L η η = [Lpη ηp − fp ] + [Lcη ηc − fc ] ≡ 0; Lpη ηp − fp = 0; ηpx = 0;

x = 0,

(3.15a)

Lcη ηc − fc = 0;

(3.15b,c)

ηcy = 0;

(3.15d,e)

y = |b|,

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where (3.15d and 3.15e) are the contact line conditions and where Lpη ηp =

τ ηp − ηp , λ4r xx

fp = αλr ϕpz − τ1 λr ϕpt ;

Lcη ηc = τ ηcyy − ηc ,

fc = αϕcz − τ1 ϕct ;

z = 0,

z = 0.

(3.15f,g) (3.15h,i)

Substituting (3.14a) and (3.15h,i) into (3.15c) yields ηc =

1 (−qα + τ1 q) ˙ cos(y − b). τ12

Substituting (3.14b) and (3.15f,g) into (3.15b) yields   −Q2 α + λ2r τ1 Q˙ 2 −Q1 α + λ2r τ1 Q˙ 1 cos x + ηp = λr τλ λr τλ  

√ λ2p τ . × sin x + 2 exp − √ x λp τ

(3.16)

(3.17)

The Lagrangian may be decomposed into the following components: L = L0 + Lε + Lγ + O(ε2 , εγ, γ 2 );

(3.18a)

where   bΓ2  2    (t) + Q22 (t)) + Q1 (t)Q2 (t)] [ξ(Q −   1   2         2 2 bτ Γ 1 4 2 2 ˙ ˙ ˙ ˙ L0 = ; + [ξλ τ ( Q (t) + Q (t)) + 4τ Q (t) Q (t)] 1 2 r λ 1 2   2λ2r τλ2             bξ 2 2   − (q (t) − q˙ (t)) 2 εbΓτ1 τ εbΓ Lε = − 3 q 2 Q˙ 2 − q qQ ˙ 2; 2λr τλ 2τ1 λ3r

(3.18b)

(3.18c)

   √ t q q˙ Γλr τ1 Q1 Γ2 τ12 2 ˙ ˙ Q1 (λr Q1 + τ Q2 ) + f1 + Lγ = −γb cos β εβ βτλ 2β    √ t Γ2 τ 2 −q˙2 + q2 Γ2 Q21 − 2 12 (λ4r Q˙ 21 + λ2r τ Q˙ 1 Q˙ 2 + τ Q˙ 22 ) + , + γb sin β λr τλ 2 (3.18d) where f1 is an integral of (3.13a or 3.13b), L0 represents the free oscillations, Lε represents the Floquet parametric forcing of the cross wave by the progressive wave, and Lγ represents the completely nonautonomous perturbation. The analogy to the nonautonomous Hamiltonian for a Floquet oscillator follows.

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3.2.2. Legendre transform The Legendre transform of the Lagrangian yields the following Hamiltonian: ˙ t), H(q, p, t) = pq˙ + P1 Q˙ 1 + P2 Q˙ 2 − L(q, q,

(3.19)

where p = (p, P1 , P2 ) are the conjugate momenta corresponding to the generalized coordinates q = (q, Q1 , Q2 ). The conjugate momenta are computed from pi =

∂(L0 + Lε ) ; ∂ q˙i

p1 = p = bξ q˙ −

i = 1, 2, 3, εbΓ qQ2 ; 2τ1 λ3r

  2τ ξ Q˙ 1 + 4 Q˙ 2 ; λr τλ   2 2 2 εbτ1 Γτ 2 bτ1 Γ λr 2τ ˙ ˙ ξ Q2 + 4 Q1 . p3 = P2 = − 3 q + 2λr τλ τλ λr τλ p2 = P1 =

bτ12 Γ2 λ2r τλ

(3.20a) (3.20b) (3.20c) (3.20d)

Inverting (3.20b–3.20d) yields q˙1 = Q˙ 1 =

p εΓqQ2 ; + bξ 2τ1 ξλ3r

λ4r τλ P1 −

Q˙ 2 =

2τ ξ P2 2 bτ1 Γ2 λ6r ξ

λ4r τλ P2 −

−

2τ ξ P1 2 2 bξτ1 Γ λ6r

(3.21a)

ετ 2 q 2 ; Γλ9r ξ 2 τ1 τλ

(3.21b)

ετ q 2 . 2Γξλ5r τ1

(3.21c)

+

Substituting (3.18), (3.20), and (3.21) into (3.19) yields the following Hamiltonian: H(q, p, t) = H0 (q, p) + Hε (q, p) + Hγ (q, p, t) + O(ε2 , εγ, γ 2 ),

(3.22a)

where the free oscillation component is H0 =

bΓ2 1 p2 (ξ(Q21 + Q22 ) + Q1 Q2 ) + bξq 2 + 2 2 2bξ +

2τ P1 P2 (P12 + P22 ) − 2 2 6 2; 2bβ 2 Γ2 ξ bτ1 Γ λr ξ

(3.22b)

the Floquet parametric forcing component is Hε =

εΓ ετ ετ 2 2 qpQ + q P − q 2 P1 ; 2 2 2τ1 ξλ3r 2τ1 Γξλ5r τ1 Γξ 2 λ9r τλ

(3.22c)

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and the nonautonomous perturbed component from the wavemaker forcing is   √   Q1 (λ2r P1 + τ P2 ) bΓλr τ1 Q1     +    f1      εβ βξλ2r   t       γ cos   √     2 β   2τ Q1 (λr P2 + τ P1 ) pq − + . Hγ = 2 6 βξ λr τλ 2βξ            √      (λ2r P1 + τ P2 )2 1 2 p2 t   2 2  −γ sin  + bq − bΓ Q1 − 2 2 2 6 β bτ1 Γ ξ λr 2 2bξ 2

(3.22d)

Although the nonautonomous Hamiltonian in (3.22d) is homomorphic to a nonautonomous Floquet Hamiltonian, a subtle but important distinction exists. The nonautonomous component in a Floquet Hamiltonian is due to parametric forcing; while, in contrast, the nonautonomous component in (3.22d) is due to external wavemaker boundary forcing that generates the progressive wave that then parametrically forces the cross waves. This sequence of external forcing to parametric forcing distinguishes cross wave parametric forcing from the simple pendulum Floquet parametric forcing. Damping forces. The scaled Hamilton’s principle (3.3) is given by1,3
δ

t2

t1


L dt =

t2

t1

3 


Di δqi dt = −

i=1

t2




t1

b −b




ξ

(αϕz δη)dxdydt,

(3.23a)

0

where D = (D1 , D2 , D3 ) is a set of generalized components of the damping force corresponding to the set of generalized coordinates q = (q, Q1 , Q2 ) and where

δη =

3   ∂η i=1

∂qi

δqi +

 ∂η δ q˙i . ∂ q˙i

(3.23b)

Integrating by parts, the last term1,3 reduces D to the canonical variables q, p   ∂ ∂η ∂η + Di = α −ϕz dxdy; i = 1, 2, 3, ϕz ∂qi ∂t ∂ q˙i −b 0      2  αεbΓ α α bξ p+ qQ2 + q, D1 = τ1 2τ12 λ3r τ12         2 α 2ατ αεbΓτ 2 α bξΓ2 2 D2 = P1 − P2 − q + Q1 , λ2r τ1 ξλ6r τ1 τλ λ9r ξτλ λ2r τλ    2  2     α αεbΓτ 2α bτ Γ2 α bξΓ2 2 P2 + q + Q1 + Q2 . D3 = τ1 λ2r 2λ5r τλ λ6r τλ2 λ2r τλ


b




ξ



(3.24a) (3.24b) (3.24c) (3.24d)

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3.3. Hamilton’s Dynamic Equations and Canonical Transformations Hamilton’s equations of motion are given by q˙i =

∂H ; ∂pi

p˙ i = −

∂H − Di ; ∂qi

i = 1, 2, . . . , N.

(3.25a,b)

If the canonical transformation Q = Q(q, p, t), P = P(q, p, t) is computed from a generating function F (u, U, t) computed from the GHA-Type I (Appendix A), then the Hamiltonian K for the new set of variables is given by K(Q, P, t) = H[q(Q, P, t), p(Q, P, t), t] +

∂F (u, U, t) ; ∂t

(3.26a)

and the transformed Hamilton’s equations of motion are N

∂K  ˜ ∂qj Q˙ i = + ; Dj ∂Pi j=1 ∂Pi

i = 1, 2, . . . , N,

N  ∂K ˜ j ∂qj ; P˙i = − − D ∂Qi j=1 ∂Qi

i = 1, 2, . . . , N,

˜ i (Q, P, t) = Di [q(Q, P, t), p(Q, P, t)]; D

i = 1, 2, . . . , N.

(3.26b)

(3.26c) (3.26d)

3.3.1. Three canonical transformations and analyses by the GMM In order to apply the GMM to the dynamical system in (3.26), canonical transformations are required first to simplify terms in H0 ; second to simplify terms in Hε ; and third to suspend nonautonomous terms in Hγ . The GHA will be applied to two of the following three canonical transformations, because the rotation of axes transformation is well known so that the GHA will not be applied for that transformation even though it is still applicable. (1) Rotation of axes. The cross product terms in (3.22b) may be eliminated by q = q˜,

p = p˜,

(3.27a,b)

˜ 1 cos θ + Q ˜ 2 sin θ, Q1 = Q

P1 = P˜1 cos θ + P˜2 sin θ,

(3.27c,d)

˜ 2 cos θ − Q ˜ 1 sin θ, Q2 = Q

P2 = P˜2 cos θ − P˜1 sin θ,

(3.27e,f)

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that satisfy the Poisson bracket requirements in Appendix A. The rotation angle θ = π/4 eliminates the cross product terms, and the transformed Hamiltonian is given in component form by ˜ε + H ˜ γ (t) + O(ε2 , εγ, γ 2 ), ˜ =H ˜0 + H H→H where

   2   bξ 2 p˜2 P˜12 bΓ ξ ˜ 2     q˜ + + Q1 +    2 2   2 2bξ 2 2bβ Γ ξ            

    2 2 ˜2 P ξ bΓ 2 ˜ ˜ , + Q2 + H0 =   2 2bβ 2 Γ2 ξ          

    2 2 2 ˜ ˜   τ ( P − P ) bΓ   1 2 2 2  ˜  ˜ + (Q2 − Q1 ) +    2 4 bτ1 Γ2 λ6r ξ 2 

(3.28a)

(3.28b)

4 ˜2 − Q ˜ 1 ) − ετ√(2τ + ξλr τλ ) q˜2 P˜1 ˜ ε = √ εΓ q˜p˜(Q H 2 2τ1 ξλ3r 2 2τ1 Γξ 2 λ9r τλ

+

ετ (−2τ + ξλ4r τλ ) 2 ˜ √ q˜ P2 , 2 2τ1 Γξ 2 λ9r τλ

(3.28c)

   ˜1 + Q ˜ 2) bΓλr τ1 (Q     √ f   1      2εβ               √ √     2 2  ˜ 2 )((λr − τ )P˜1 + (λr + τ )P˜2 )  ˜1 + Q t ( Q       γ cos     +   2     β 2βξλ   r           √ √   2 2 ˜ ˜ ˜ ˜ ˜ τ (Q1 + Q2 )((λr + τ )P2 + (−λr + τ )P1 ) p˜q˜ . Hγ = − +   2 λ6 τ   βξ 2βξ   r λ         √ √     2 2 2 2 ˜ ˜   ((λ − τ ) P + (λ + τ ) P ) bΓ   1 2 r r 2   ˜ ˜    + Q ) −  ( Q 1 2   2 2 2 6   2 t   2bτ1 Γ ξ λr       −γ sin     2   β 1 p ˜   2   + b˜ q − 2 2bξ 2 (3.28d) The first three energy square brackets [•] in (3.28b) are the action variables in the next canonical transformation. (2) Action/Angle Transformation. The three new canonical variables are given by pˆ =

bξ 2 p˜2 q˜ + , 2 2bξ

bΓ2 ξ ˜ 2 P˜12 Pˆ1 = , Q1 + 2 2bβ 2 Γ2 ξ

bΓ2 ξ ˜ 2 P˜22 , Q2 + Pˆ2 = 2 2bβ 2 Γ2 ξ

(3.29a–c)

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that satisfy the Poisson brackets. The Herglotz auxiliary functions Xi are the ratios p; X1 = q˜/˜

˜ 1 /P˜1 ; X2 = Q

˜ 2 /P˜2 . X3 = Q

(3.30a–c)

The angle variables are qˆi =

3 

q˜j

j=1

∂F ∂ p˜j + ; ∂ pˆi ∂ pˆi

i = 1, 2, 3,

(3.31a)

where qˆ = tan−1 (bξX1 );

ˆ 1 = β tan−1 (bβξΓ2 X2 ); Q

ˆ 2 = β tan−1 (bβξΓ2 X3 ). Q (3.31b–d)

The canonical transformation to action/angle variables is given by the following:  q˜ =

 ˜1 = Q

 ˜2 = Q

2Pˆ1 sin bξΓ2

2Pˆ2 sin bξΓ2





ˆ1 Q β

ˆ2 Q β

2ˆ p sin qˆ, bξ

p˜ =





2bξ pˆ cos qˆ,



 ,

P˜1 = Γβ

,

P˜2 = Γβ



2bξ Pˆ1 cos



 2bξ Pˆ2 cos

(3.32a,b)

ˆ1 Q β

ˆ2 Q β

,

(3.32c,d)

,

(3.32e,f)



that satisfy the Poisson brackets.1,3 The transformed Hamiltonian is ˜ ⇒H ˆ =H ˆ0 + H ˆε + H ˆ γ (t) + O(ε2 , εγ, γ 2 ), H     ˆ1 2β 2 τ Q 2β 2 τ 2 ˆ P1 cos − λ6r ξτ12 β λ6r ξτ12      ˆ2 ˆ1 ˆ2 ˆ1 ˆ2 Q Q Q P P 2 2 2 × Pˆ2 cos − sin + sin , β 2ξ β 2ξ β

(3.33a)



ˆ 0 = [ˆ p + Pˆ1 + Pˆ2 ] + H

(3.33b)

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    

ˆ2 ˆ2   Q Q   ˆ   P2 cos 2ˆ − cos 2ˆ q+ q−     β β   εˆ p ˆ Hε =   

   4 bξ 3 λ3r τ1    ˆ1 ˆ1   Q Q   ˆ   − cos 2ˆ q+ − P1 cos 2ˆ q−   β β εβτ pˆ −  2 bξ 5 λ9r τ1 τλ     

ˆ1 ˆ1   Q 1 Q   4   (2τ + ξλr τλ ) Pˆ1 cos − cos 2ˆ q+       β 2 β          

       ˆ2  ˆ1 Q 1 Q 4 ˆ , × q− + (2τ − ξλr τλ ) P2 cos − cos 2ˆ  2 β β                   ˆ ˆ   Q 1 Q 1   2 2   cos 2ˆ q + − cos 2ˆ q − −     2 β 2 β (3.33c)    √     ˆ1 ˆ2 bλr f1 τ1  Q Q     √ + Pˆ2 sin Pˆ1 sin       β β βε ξ                    √      2 4   ˆ   + τ )(−2τ + ξλ τ ) (λ Q   λ 1 r r   ˆ     P + cos 1     6 2     λ ξ τ β λ   t r     γ cos           β   √      2 4   ˆ + τ )(−2τ + ξλ τ )  (λ  Q λ 2   r r   ˆ   P + cos   2   6 ξ2τ  λ β    λ  r                  pˆ sin(2ˆ q)   + ˆ . Hγ = 2βξ    

        ˆ   √ −β 2   2   ˆ1 cos Q1 P (λ − τ )     r   6τ 2   λ β     r 1               2        ˆ √   Q   2   2   ˆ  γ  + τ ) cos + (λ P   2 t   r     β   − sin       ξ β       

  2         ˆ ˆ     Q Q 1 2   ˆ1 sin ˆ2 sin     + + P P        β β            2 2 + pˆ(sin qˆ − cos qˆ) (3.33d)

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The nonautonomous perturbed component in (3.33d) will not survive the KAM averaging theorem, and some of the nonautonomous terms may be suspended by applying the following Hamilton–Jacobian nonautonomous canonical transformation. (3) Hamilton–Jacobian transformation. The following nonautonomous transformations: qˆ =

t + q + Q1 , 2β

ˆ 1 = t + 2βQ1 , Q

ˆ 2 = 3t + 2β(q + Q1 + Q2 ) Q

(3.34a–c)

satisfy the Poisson brackets. The Herglotz auxiliary functions Xi = Xi (p, q) satisfy the nonzero Jacobian condition given by ∂(ˆ q, X) 4β 2 = 0. = ∂(p, q) qQ1 Q2

(3.35)

Solving (3.35) for qi = qi (p, X), substituting qi (p, X) into (3.34), and inverting yields p=

X1 ˆ 1 ); P1 = −X2 (t − Q ˆ 1 ); (2β qˆ − Q 2β 2β −X3 ˆ 2 ). P2 = (2t + 2β qˆ − Q 2β

(3.36a–c)

The generating function F (•, •, •) for the nonautonomous canonical transformation may be computed from the following indefinite integral: F (ˆ q, p(ˆ q, X, t), t)  3
Xi   ∂P1 ∂P2 ∂p
=− q
+ Q1 ∂X
+ Q2 ∂X
dXi ∂X i i i i=1 =

−1 ˆ 1 )2 + X2 (t − Q ˆ 1 )2 + X3 (2t + 2β qˆ − Q ˆ 2 )2 ]. [X1 (2β qˆ − Q 4β 2

(3.37)

Computing the three remaining conjugate momenta from pˆi = −

3  j=1

qj

∂F ∂pj − ; ∂ qˆi ∂ qˆi

i = 1, 2, 3

(3.38a)

1 Pˆ2 = P2 . 2β

(3.38b–d)

gives pˆ = p − P2 ,

1 Pˆ1 = (P1 − p), 2β

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The transformations in (3.34) and (3.38) satisfy the Poisson brackets, and the first square bracket term [•] in (3.33b) may be transformed to [ˆ p + Pˆ1 + Pˆ2 ] =

  1 P1 1− (p − P2 ) + . 2β 2β

(3.39)

The primary Floquet resonance condition is, approximately, ωc
− ωp
≈ O(ε) so that a detuning parameter Ω may be defined by     ωp
1 1− = 1 − = εΩ; 2ωc
2β

(3.40)

and the primary parametric Floquet resonance is ωp
: ωc
= 2:1. A generating function is now given by   1 ˆ 1 ˆ 1 ) + 1 P2 (2t + 2β qˆ − Q ˆ 2 ); F (ˆ q, p, t) = p −ˆ q+ P1 (t − Q Q1 + 2β 2β 2β

(3.41)

and the transformed Hamiltonian H(p, q, t) with autonomous and nonautonomous terms by ˆ − ∂F ⇒ H = H0 + H0 (t) + Hε + Hε (t) H ∂t + Hγ + Hγ (t) + O(ε2 , εγ, γ 2 ),

(3.42a)

where     βτ 1 βτ 1 1 (P P2 , + + − − − p) + − 1 8βξ 2ξλ6r τ12 β 8βξ 2ξλ6r τ12 √ ε(p − P2 ) P1 − p Hε = εΩ(p − P2 ) +  [(2βτ 2 − ξλ4r τλ (λ2r − βτ )) cos(2q)], 4 2bβξ 5 λ9r τ1 τλ  √   γ bf1 λr τ1  2β − 1  P1 − p sin(2Q1 ) + γ Hγ = 4βξ 2ε 2ξβ 3 H0 =

× (P2 − p) sin 2(q + Q1 ),

(3.42b) (3.42c)

(3.42d)

and where the nonautonomous terms are given by Fadel.5 (4) Transformation to original variable (qorig , porig ) ⇒ (q, p). In order to compute the generalized damping forces in (3.24), the original canonical variables (qorig , porig ) must be expressed as functions of the final transformed canonical

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variables (q, p) by

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  2 t , p − P2 sin q + Q1 + bξ 2β     t , porig = 2bξ p − P2 cos q + Q1 + 2β     1 t Q1orig = √ P1 − p sin 2Q1 + β Γ 2bβξ    3t + P2 sin 2(q + Q1 + Q2 ) + , β     bβξ  t P1orig = Γ P1 − p cos 2Q1 + 2 β    3t , + P2 cos 2(q + Q1 + Q2 ) + β     1 t √ Q2orig = − P1 − p sin 2Q1 + β Γ 2bβξ    3t + P2 sin 2(q + Q1 + Q2 ) + , β      bβξ t − P1 − p cos 2Q1 + P2orig = Γ 2 β    3t + P2 cos 2(q + Q1 + Q2 ) + . β qorig =

(3.43a) (3.43b)

(3.43c)

(3.43d)

(3.43e)

(3.43f)

3.3.2. Transformed damping forces The transformed damping forces are1,3    √ t 2bβξα      p − P2 cos q + Q1 +     τ1 2β         √       2    t 2bξα     + p − P sin q + Q +   2 1 2   τ 2β   1           εα t , D1 = √ + p − P sin q + Q + 2 1  2β  2λ3r ξ βτ12                  3t     P sin 2(q + Q + Q ) +   2 1 2     β         ×              t     − P1 − p sin 2Q1 + β

(3.44a)

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    √ t bβαΓ(2τ + λ4r ξτλ )      √ P − p cos 2Q + 1 1   6 2ξ   β τ τ λ   1 λ r       √         4    3t bβαΓ(−2τ + λ ξτ ) λ   r   √ + P cos 2(q + Q + Q ) +   2 1 2   6 2ξ β   τ τ λ 1 λ r             2 2Γεατ t − 9 2 2 (p − P2 ) sin 2(q + Q1 ) + , D2 =   λr ξ τλ 2β         √     2    Γ bξ α t     √   + P − p sin 2Q + 1 1   2τ   β λ 2β   r λ       √     2      Γ bξ α 3t     + √ P sin 2(q + Q + Q ) + 2 1 2 2 β λr τλ 2β (3.44b)        t     − P − p cos 2Q + 1 1  √      β   αΓ bβξ      √          2τ      2λ   3t r 1     + P cos 2(q + Q + Q ) +   2 1 2   β                 εαΓτ t   + ) sin 2(q + Q ) + (p − P 2 1 5 D3 = , λ ξτ 2β r λ          √       t α2 Γ b(2τ − λ4r ξτλ )       √  P − p sin 2Q + + 1 1   2 6τ   β λ 2βξ   r λ          √     2 4    3t Γ b(2τ + λ ξτ ) α   λ r     √ P sin 2(q + Q + Q ) + + 2 1 2  2 6 β  λr τλ 2βξ (3.44c) that are completely autonomous and will survive the KAM averaging operation. Averaging H(q, p, t) over the dimensionless cross wave period 2π yields

H = H0 (p, P1 , P2 ) + Hε (q, p, P1 , P2 ) + Hγ (q, p, P1 , P2 , Q1 )     βτ 1 βτ 1 1 (P P2 + + − = − − p) + − 1 8βξ 2ξλ6r τ12 β 8βξ 2ξλ6r τ12

 √ (p − P2 ) P1 − p 2 4 2 + ε Ω(p − P2 ) +  [(2βτ − ξλr τλ (λr − βτ )) cos(2q)] 4 2bβξ 5 λ9r τ1 τλ 

 √   2β − 1 bf1 λr τ1   (P2 − p) sin 2(q + Q1 ) . P1 − p sin(2Q1 ) + +γ 4βξ 2ε 2ξβ 3 (3.44d)

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The averaged system of autonomous equations is1,3   2P1 + P2 − 3p     √ −a cos[2q] + εΩ + εa 1 3     P − p   1         sin[2Q1 ] 2 − γc2 sin[2(q + Q1 )] + α d1 , − γc1 √ q˙ = ε P1 − p               d (P − p) + d (P − p)   2 1 3 2     √ sin[2q] − αε P1 − p      4εα3 P1 − p(p − P2 ) sin[2q] − 2γc2 (P2 − p) cos[2(q + Q1 )]   p˙ = ,  α     + αd4 P2 − p + 2αεd2 P1 − p(p − P2 ) cos[2q] τ1    4γ     − P − p cos[2Q ] − 2γc (P − p) cos[2(q + Q )] c 1 1 2 2 1   ε 1 ˙ , P1 = 2  (4λr − 1)(P2 − p) − P1       +α − αεd P − p(p − P ) cos[2q] 5 1 2 2λ2r τ1   τ − λ4r ξτλ P˙ 2 = αP2 , λ6r ξτ1 τλ   (p − P2 ) sin[2Q1 ]     √ √ cos[2q] + γc + εa a 3 1   1 P1 − p ε P1 − p ˙ , Q1 =   P2 − p     sin[2q] + α2 d6 + αεd3 √ P1 − p     a2 − εΩ − 2εa3 P1 − p cos[2q] + γc2 sin[2(q + Q1 )]  , Q˙ 2 =    + α2 d7 + αεd2 P1 − p sin[2q]

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(3.45a)

(3.45b)

(3.45c)

(3.45d)

(3.45e)

(3.45f)

where ai , ci , and di are tabulated in Appendix B. 3.4. Application of the GMM and Largest Liapunov Characteristic Exponent The GMM determines the existence of transverse homoclinic/heteroclinic points that are transverse intersections between the stable and unstable manifolds to any invariant sets of the perturbed system when a homoclinic/heteroclinic orbit exists to a hyperbolic invariant manifold in the unperturbed (undamped α = 0 and unperturbed γ = 0) system. The unperturbed vector field may be computed by setting the perturbation (wavemaker forcing) parameter γ = 0 and the dissipation parameter α = 0 in (3.45), and are q˙ = −a1 + εΩ + εa3

2P1 + P2 − 3p √ cos(2q), P1 − p

(3.46a)

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p˙ = 4εa3



P1 − p(p − P2 ) sin(2q),

P˙1 = P˙2 = 0,

(3.46b) (3.46c,d)

(p − P2 ) Q˙ 1 = a1 + εa3 √ cos(2q), P1 − p  Q˙ 2 = a2 − εΩ − 2εa3 P1 − p cos(2q),

(3.46e) (3.46f)

where the coefficients a1 , a2 , and a3 are defined in Appendix B. The unperturbed vector field α = γ = 0 in (3.46) is a three degrees of freedom Hamiltonian system with (q, p, P1 , P2 , Q1 , Q2 ) ∈ T1 × R1 × R2 × T2 . An important consequence of the Hamilton–Jacobian canonical transformation is that the unperturbed Floquet Hamiltonian H(γ = 0) = H0 (p, P1 , P2 ) + Hε (q, p, P1 , P2 ) is independent of Q1 and Q2 . Hyperbolic saddle points. For every (P1 , P2 ) ∈ R2 , the (q − p) components of the unperturbed vector field α = γ = 0 in (3.46) possess a hyperbolic saddle point that varies smoothly with P1 and P2 and is given by p0 (P2 ) = P2 ;

2q0 = nπ;

n = 0, 1, . . . ,

  a1 − εΩ 1 √ , cos−1 2 2εa3 P1 − P2  2 a1 − εΩ P1 ≥ P2 + . 2εa3

q0 (P1 , P2 ) =

(3.47a,b)

(3.47c)

(3.47d)

The fixed point (q0 , p0 ) is a hyperbolic saddle point if the determinant satisfies # # # ∂ q˙ ∂ q˙ # # # # ∂q ∂p # # # = (a1 − εΩ)2 − 4(P1 − P2 )ε2 a23 < 0. (3.48) # ∂ p˙ ∂ p˙ # # # # ∂q ∂p # (q=q0 ,p=p0 ) The symmetry properties of Hamiltonian systems imply that the stable and the unstable manifolds of the hyperbolic saddle point (q0 , p0 ) have equal dimensions in the full 6D phase space (q, p, P1 , P2 , Q1 , Q2 ) ∈ T1 × R1 × R2 × T2 . The unperturbed system α = γ = 0 has a 4D (R2 × T2 ) normally hyperbolic invariant manifold given by the union of the hyperbolic saddle points (q0 , p0 ) according to M = {(q, p, P1 , P2 , Q1 , Q2 ) ∈ T1 × R1 × R2 × T2 : q = q0 (P1 , P2 ); p = p0 (P2 )}. (3.49) The normally hyperbolic invariant manifold M has 5D (R1 ×R2 ×T2 ) stable manifold W s (M) and unstable manifold W u (M) that coincide along the 5D heteroclinic manifold H = W s (M) ∩ W u (M) − M, where W s (M) and W u (M) are the set of

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initial conditions that approach the hyperbolic saddle points as t → ±∞ under the action of the unperturbed flow. Dynamics on M. The unperturbed vector field α = γ = 0 restricted to the normally hyperbolic invariant manifold M may be determined from1,3 P˙1 = 0;

P˙2 = 0;

Q˙ 1 = a1 ;

Q˙ 2 = a2 − a1 .

(3.50a–d)

The unperturbed vector field α = γ = 0 restricted to M has the form of two degrees of freedom, 4D phase space (T1 × R1 × T2 ), completely integrable Hamiltonian system with the Hamiltonian given by the level energy surfaces H(P1 , P2 ) = H(q = q0 , p = p0 ; γ = 0) = a1 P1 + (a2 − a1 )P2 = E,

(3.51)

where E is a constant energy that allows the phase space motion to be reduced from four dimensions (R2 × T2 ) to three dimensions. Because P1 = P2 = 0, the constant energy surfaces may be reduced to a 1D surface in the 3D constant energy space. On this 1D surface, the angular motion is parameterized by the two frequencies σ1 = a1 , σ2 = a2 − a1 .

(3.52a,b)

The angular components of the motion on the normally hyperbolic invariant manifold M are given by Q1 (t) = a1 t + Q1 (0);

Q2 (t) = (a2 − a1 )t + Q2 (0).

(3.53a,b)

The normally hyperbolic invariant manifold M is a two-parameter (P1 , P2 ) family of 2D tori shown in Fig. 3.4. A 2D torus on M may be defined by  $ (q, p, P1 , P2 , Q1 , Q2 ) ∈ T1 × R1 × R2 × T2 : q = q0 (P¯1 , P¯2 ); ¯ ¯ Υ(P1 , P2 ) = , p = p0 (P¯2 ); P1 = P¯1 ; P2 = P¯2 (3.54) where each 2D torus is invariant. The level energy surface is a family of concentric tori as shown in Fig. 3.5. Because the frequencies of motion σ1 and σ2 in (3.52) are independent of P1 or P2 , they do not change from one concentric torus to another concentric torus. The motion on the surface of the invariant torus Υ ⊂ M is quasiperiodic. When the frequency ratio σ1 /σ2 = a1 /(a2 − a1 ) is an irrational number, the motion on the surface of the 2D invariant nonresonant torus may no longer be periodic; i.e., trajectories wind densely on the surface of the torus and never close on themselves. This 2D torus has a 3D (R1 × T1 ) stable manifold W S and unstable manifold W U that coincide along the 3D (R1 × T1 ) heteroclinic manifold H defined for fixed values of P1 and P2 , as shown in Fig. 3.4. On a constant level energy surface, neither the nonresonant invariant tori nor the stable and unstable manifolds may be isolated. In addition, the 2D nonresonant invariant torus has a 2D center manifold corresponding to nonexponentially expanding or contracting directions tangent to the normally hyperbolic invariant manifold M.

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Fig. 3.4.

3D unperturbed heteroclinic manifold.1

Fig. 3.5. Motion of a phase space point for an integrable Hamiltonian system with two degrees of freedom. (a) Invariant tori in a 3D constant energy space E; and (b) the flow on a 2D torus.1,3

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Heteroclinic orbits. For fixed values of (P1∗ , P2∗ ) ∈ R2 , the (q, p) components of the unperturbed vector field α = γ = 0 in (3.46) possess a 1D heteroclinic orbit connecting the hyperbolic saddle points (q0 , p0 ). The heteroclinic orbit lies on each of the level energy surfaces defined by (3.51) and are solutions to

H(γ = 0) − (a1 P1∗ + (a2 − a1 )P2∗ ) = 0.

(3.55)

Values for q on the heteroclinic orbit are given by & %  1 q = cos−1 (a1 − εΩ)/2εa3 P1∗ − p . 2

(3.56)

In the full 6D phase space (q, p, P1 , P2 , Q1 , Q2 ) ∈ T1 ×R1 ×R2 ×T2 , the heteroclinic manifold H may be determined by substituting   2   a1 − εΩ a1 − εΩ √ √ , sin[2q] = 1 − , (3.57a,b) cos[2q] = 2εa3 P1 − p 2εa3 P1 − p into the unperturbed vector field α = γ = 0 in (3.46), and integrating to obtain1,3 √ A ph (t) = P2 + sech2 [ At] + p(0), (3.58a) B 

√ √ 1 A −1 (3.58b) − qh (t) = tan tanh( At) + q(0), 2 a1 − εΩ P1h (t) = P1 (0), P2h (t) = P2 (0),

(3.58c,d)

Q1h (t) = a1 t − qh (t) + Q1 (0) + q(0) 

√ √ 1 A −1 tanh( At) + Q1 (0), = a1 t − tan − 2 a1 − εΩ Q2h (t) = (a2 − a1 )t + Q2 (0), A = [−(a1 − εΩ)2 + (P1 − P2 )B] > 0,

(3.58e) (3.58f)

B = 4ε2 a23 ,

(3.58g,h)

π q = qn (0) = (2n + 1) ; n = 0, 1, 2, . . . . (3.58i,j) 2 The trajectories of the unperturbed system α = γ = 0 along the 5D phase space (R1 × R2 × T2 ) heteroclinic manifold H may be expressed as p(0) = 0,

Ψ(P1 ,P2 ) = {qh (t), ph (t), P1 (0), P2 (0), Q1h (t), Q2h (t)}.

(3.59)

The 6D phase space (q, p, P1 , P2 , Q1 , Q2 ) ∈ T × R × R × T is a direct product between a 4D space with coordinates (q, p, P1 , P2 ) and a 2D torus with angular 1

1

2

2

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Fig. 3.6.

Unperturbed 4D phase space.

coordinates (Q1 , Q2 ). Because P1 and P2 are constants, the motion in the 6D phase space is reduced to the four dimensions (q, p, Q1 , Q2 ) shown in Fig. 3.6. Small perturbations are anticipated to break up the geometric structure of the unperturbed system α = γ = 0 and to separate the manifolds. The behavior of the perturbed systems γ > 0 and α > 0 near the unperturbed heteroclinic manifold H is required in order to apply the GMM. The distance between the stable and unstable manifolds of any surviving invariant set in the perturbed system must be computed at a point P on the unperturbed heteroclinic manifold H. Two perturbed systems are evaluated: (1) γ > 0, α = 0 and (2) γ > 0 and α > 0. 3.4.1. Geometric structure for γ > 0 and α = 0 The perturbed vector field for nonzero wavemaker forcing γ > 0 and no dissipation α = 0 is given by q˙ = −a1 + εΩ + εa3

2P1 + P2 − 3p √ cos[2q] P1 − p

sin[2Q1 ] − γc1 √ − γc2 sin[2(q + Q1 )], ε P1 − p

    4εa3 P1 − p(p − P2 ) sin[2q] − 2γc2 (P2 − p) cos[2(q + Q1 )] ,  p˙ =  +αd4 P2 − α p + 2αεd2 P1 − p(p − P2 ) cos[2q] τ1  4γ P˙1 = − c1 P1 − p cos[2Q1 ] − 2γc2 (P2 − p) cos[2(q + Q1 )], ε

(3.60a)

(3.60b)

(3.60c)

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P˙2 = 0, (p − P2 ) sin[2Q1 ] Q˙ 1 = a1 + εa3 √ cos[2q] + γc1 √ , P1 − p ε P1 − p  Q˙ 2 = a2 − εΩ − 2εa3 P1 − p cos[2q] + γc2 sin[2(q + Q1 )],

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(3.60d) (3.60e) (3.60f)

where (q, p, P1 , P2 , Q1 , Q2 ) ∈ T1 × R1 × R2 × T2 . The perturbed vector field is computed from 5D level energy surfaces defined by

H = H0 (p, P1 , P2 ) + Hε (q, p, P1 , P2 ) + Hγ (q, p, P1 , P2 , Q1 ).

(3.61)

Because the perturbation is Hamiltonian, the 3D level energy surfaces are preserved. In the 4D normally hyperbolic invariant manifold of the unperturbed space, the locally stable and unstable manifolds and the flow describe the geometric structure of the perturbed phase space given by the perturbed normally hyperbolic locally invariant manifold, the locally stable and unstable manifolds, and the persistence of the 2D nonresonant invariant tori Υγ (P1 , P2 ). Persistence of M. The perturbed system γ > 0 and α = 0 possesses a 4D normally hyperbolic locally invariant manifold Mγ given by   (q, p, P1 , P2 , Q1 , Q2 ) ∈ T1 × R1 × R2 × T2 :     Mγ = q = q˜0 (P1 , P2 , Q1 , Q2 ; γ) = q0 (P1 , P2 ) + O(γ); .     p = p˜0 (P1 , P2 , Q1 , Q2 ; γ) = p0 (P2 ) + O(γ)

(3.62)

On Mγ there are locally stable and unstable manifolds that are of equal dimensions and are close to the unperturbed locally stable and unstable manifolds. The perturbed normally hyperbolic locally invariant manifold Mγ intersects each of the 5D level energy surfaces in a 3D set of which most of the two-parameter family of 2D nonresonant invariant tori persist by the KAM theorem.1,3 The Melnikov integral may be computed to determine if the stable and unstable manifolds of the KAM tori intersect transversely. KAM Theorem.1,3 The KAM theorem determines whether the recurrent motions occur on the perturbed normally hyperbolic locally invariant manifold Mγ and whether any of the two parameter families of 2D nonresonant invariant tori survive the Hamiltonian perturbation. The unperturbed Floquet Hamiltonian H(γ = 0) = H0 (p, P1 , P2 ) + Hε (q, p, P1 , P2 ) satisfies the following nondegeneracy (or nonresonance) condition: # 2 # ∂ H # # ∂P 2 # 1 # 2 # ∂ H # # ∂P ∂P 2

1

# # # # (a1 − εΩ)2 # =− < 0. # 4(P1 − P2 )2 ∂ 2 H ## ∂P22 #(q=q0 ,p=p0 ;γ=0)

∂ 2 H ∂P1 ∂P2

(3.63)

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Most of the 2D nonresonant invariant tori Υ(P1 , P2 )) that persist are only slightly deformed on the perturbed normally hyperbolic locally invariant manifold Mγ and are KAM tori. In the phase space of the perturbed system γ > 0 and α = 0, there are invariant tori that are densely filled with winding trajectories that are conditionally periodic with two independent frequencies σ1 and σ2 . The resulting conditionally-periodic motions of the perturbed system are smooth functions of the perturbation γ. A generalization of the KAM theorem states that the KAM tori have both stable and unstable manifolds by the invariance of manifolds.1,3 In order to determine if chaos exists, two measurements are required in order to determine whether or not W s (Υγ ) and W u (Υγ ) intersect transversely. Melnikov integral. The distance between W s (Υγ ) and W u (Υγ ) at any point P ∈ H may be computed from


βn

∂H γ (P1 ,P2 ) (Ψ )dt, ∂Q1

(3.64a)

2c1  P1 − p sin(2Q1 ) + c2 (P2 − p) sin[2(q + Q1 )], ε

(3.64b)

M(Q1 (0)) = −

lim

αn ,βn →∞

−αn

where Hγ =

where Hγ = γH γ in (3.61), and where1,3 M(Q1 (0)) = cos[2(Q1 (0) + q(0))]     4c1 
βn   P1 − ph (cos(2qh − 2a1 t)) −  dt  ε lim   αn ,βn →∞ −αn − 2c2 (P2 − ph ) cos(2a1 t) + sin[2(Q1 (0) + q(0))]     4c1 
βn   P1 − ph (sin(2qh − 2a1 t)) −  dt ,  ε lim   αn ,βn →∞ −αn + 2c2 (P2 − ph ) sin(2a1 t) (3.64c) that after retaining only even integrands reduces to1,3 M(Q1 (0)) = cos[2(Q1 (0) + q(0))][I1 + I2 + I3 ],  $
−2c1 (a1 − εΩ) βn I1 = lim cos(2a1 t)dt , n→∞ ε 2 a3 −αn ' (
√ 2c2 A ∞ cos(2a1 t)sech2 ( At)dt , I2 = B −∞  $ √
∞ √ −2c1 A I3 = sin(2a1 t)tanh( At)dt . ε 2 a3 −∞

(3.65a) (3.65b)

(3.65c)

(3.65d)

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The Melnikov integral M(Q1 (0)) = 0 when ¯ 1n (0) = (2n + 1) Q1 (0) = Q

π − q(0); 4

n = 0, 1, 2, . . . ,

¯ 1n (0)) ∂M(Q ¯ 1n (0) + q(0))][I1 + I2 + I3 ] = 0, = −2 sin[2(Q ∂Q1 (0)

(3.66a)

n = 0, 1, 2, . . . . (3.66b)

Consequently, the stable W s (Υγ (P1 , P2 )) and unstable W u (Υγ (P1 , P2 )) manifolds of the KAM tori Υγ (P1 , P2 ) intersect transversely yielding Smale horseshoes1,3 on the appropriate 5D level energy surfaces. This implies multiple transverse intersections and the corresponding existence of chaotic dynamics in the perturbed system γ > 0 and α = 0. 3.4.2. Geometric structure for γ > 0 and α > 0 The perturbed dissipative system α > 0 and γ > 0 possesses a 4D normally hyperbolic locally invariant manifold Mγα that is given by

Mαγ

  (q, p, P1 , P2 , Q1 , Q2 ) ∈ T1 × R1 × R2 × T2 :       = q = q˜0 (P1 , P2 , Q1 , Q2 ; γ) = q0 (P1 , P2 ) + O(γ), .       p = p˜0 (P1 , P2 , Q1 , Q2 ; γ) = p0 (P2 ) + O(γ)

(3.67)

The manifold Mγα has locally stable and unstable manifolds that are close to the unperturbed locally stable and unstable manifolds; and if these manifolds intersect transversely, then the Smale–Birkhoff theorem predicts the existence of horseshoes and their chaotic dynamics in the perturbed dissipative system. A 2D hyperbolic invariant torus Υγα (P1 , P2 ) may be located on Mγα by averaging the perturbed dissipative vector field γ > 0 and α > 0 restricted to Mγα over the angular variables Q1 and Q2 . The averaged equations have a unique stable hyperbolic fixed point (P1 , P2 ) = (0, 0) with two negative eigenvalues provided that the determinant1,3 # # # ∂ P˙1  ∂ P˙ 1  # # # # ∂P ∂P2 ## ν 2 γ 2 (λ4r ξτλ − τ ) 1 # > 0. (3.68) # #= # ∂ P˙  ∂ P˙  # 2λ8r ξτ12 τλ 2 2 # # # # ∂P1 ∂P2 The Melnikov method fails to predict chaos for this dissipative system when α > 0.1,3 Liapunov characteristic exponents (LCE). Dissipative systems are characterized by the attraction of all trajectories passing through a certain domain toward an invariant surface or an attractor of lower dimensionality than the original space.

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For the 6D phase space (q, p, P1 , P2 , Q1 , Q2 ) ∈ T1 × R1 × R2 × T2 , there are six real exponents that may be ordered as µ1 ≥ µ2 ≥ µ3 ≥ µ4 ≥ µ5 ≥ µ6 , where µ1 is the largest Liapunov characteristic exponent (LCE). The LCE is calculated from the first variation of the perturbed dissipative vector field γ > 0 and α > 0 according to  ∂ q˙  ∂q   ∂ p˙     δ q˙  ∂q     δ p˙   ˙    ∂ P1  ˙     δ P1   ∂q    δ P˙  =  ∂ P˙2  2      δ Q˙   ∂q  1    ∂ Q˙ 1  δ Q˙ 2  ∂q    ∂ Q˙ 2 ∂q

∂ q˙ ∂p

∂ q˙ ∂P1

∂ q˙ ∂P2

∂ q˙ ∂Q1

∂ p˙ ∂p

∂ p˙ ∂P1

∂ p˙ ∂P2

∂ p˙ ∂Q1

∂ P˙ 1 ∂p

∂ P˙1 ∂P1

∂ P˙ 1 ∂P2

∂ P˙1 ∂Q1

∂ P˙ 2 ∂p

∂ P˙2 ∂P1

∂ P˙ 2 ∂P2

∂ P˙2 ∂Q1

∂ Q˙ 1 ∂p

∂ Q˙ 1 ∂P1

∂ Q˙ 1 ∂P2

∂ Q˙ 1 ∂Q1

∂ Q˙ 2 ∂p

∂ Q˙ 2 ∂P1

∂ Q˙ 2 ∂P2

∂ Q˙ 2 ∂Q1

∂ q˙  ∂Q2   ∂ p˙    ∂Q2    ∂ P˙ 1   ∂Q2    ∂ P˙ 2   ∂Q2    ˙ ∂ Q1   ∂Q2    ˙ ∂Q 



δq



   δp       δP1     δP  .  2    δQ   1 δQ2

(3.69)

2

∂Q2

For 20 different values of the dimensionless damping parameter and of the dimensionless Floquet parametric forcing parameter, the largest LCEs were calculated and a chaos diagram for the positive values of the largest Liapunov exponents identified the parameter space in which chaotic motions exist.1,3

Appendix A The extension of the Herglotz algorithm to nonautonomous dynamical systems (GHA) significantly reduces the effort required to suspend the nonautonomous Hamiltonian component in (3.33d).

Generalized Herglotz algorithm (GHA) The Herglotz algorithm for autonomous dynamical systems4 may be generalized by: (1) including time t; and (2) defining a generating function with a nonzero determinant of second derivatives. The GHA transforms a set of 2N variables (u, v) to a set of 2N new variables (U, V) by choosing N new variables Ui and then computing the remaining N new variables Vi uniquely from the chosen Ui so that the transformation (u, v) → (U, V) is canonical as shown by satisfying the Poisson bracket conditions.

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GHA:Type I Equate N new variables to N old variables Ui = Ui (u, v, t) such that the Poisson brackets satisfy [[Ui , Uj ]]uv = 0 and the determinant # ∂U # 1 # # ∂v1 # # # · # # # # ∂UN # ∂v1

·

·

·

∂U1 ∂vN

·

∂Ui ∂vj

·

·

·

·

·

∂UN ∂vN

# # # # # # # = 0. # # # # #

(3.A.1)

Equate N Herglotz auxiliary functions Xi (u, v) to the absolute value of the ratios of the old variables in either of the following forms:  |ui /vi | Xi = Xi (u, v) = ; i = 1, 2, . . . , N, (3.A.2a,b) |vi /ui | such that the Jacobian of the new variables U and the Herglotz auxiliary functions X is nonzero ∂(U, X)/∂(u, v) = 0. Solve (3.A.2) for vi = vi (u, X); substitute vi (u, X) into Ui (u, v, t) and invert to obtain ui = ui (U, X, t). Compute the generating function F (u, U, t) from dF (u(U, X, t), U, t) =

N 

(vi dui − Vi dUi ),

(3.A.3)

i=1

 N   ∂F ∂F dXi + dUi ∂Xi ∂Ui i=1   N N N    ∂uj ∂uj  vj dXi + vj dUi − Vi dUi  . = ∂X ∂U i i i=1 j=1 j=1

(3.A.4)

Equate the coefficients of like differentials and compute: (1) the generating function F (u, U, t) =

N
 i=1

N Xi  j=1

vj

∂uj dXi
+ C(U, t), ∂Xi


(3.A.5)

and (2) the new N variables from Vi (u, v, t) =

N  j=1

vj

∂F ∂uj − ; ∂Ui ∂Ui

i = 1, 2, . . . , N.

(3.A.6)

In order to compute the transformed Hamiltonian in terms of the new variables (U, V), the inverse canonical transformation (u(U, V, t), v(U, V, t)) must be computed,

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and the new Hamiltonian K(Q, P, t) is given by K(Q, P, t) = H(q(Q, P, t), p(Q, P, t), t) +

∂F (u, U, t) . ∂t

(3.A.7)

The GHA may also be applied to equate N old variables ui = ui (U, V, t) to obtain N new variables Ui = Ui (u, v, t).1,3 Appendix B

τ1 =

√ 1+τ

κτ = κ(1 + τ )

τλ = 1 + (τ /λ4r )

    h  2  λ 1 − exp − ; for a full draft piston,   
0  r λ2r z f1 = f (z) exp 2 ds =     λr h −h  λ4r h  ; for a full draft hinge − 1 + exp − 2  h λ2r λr a1 = −

1 βτ + −8βξ 2ξλ6r τ12

c1 =

√ bf λ τ 1 r 1 4 2ξβ 3

d1 =

a2 = −

1 βτ 1 + − β 8βξ 2ξλ6r τ12

c2 =

2β − 1 4βξ

1 d2 =  8 2bβξ 3 λ3r τ12

2βτ 2 − ξλ4r τλ (λ2r − βτ )  8 2bβξ 5 λ9r τ1 τλ

d3 =

τ (2τ + λ4r ξτλr )  8 2bβξ 5 λ9r τλ2

d4 =

a3 =

τ τ 2 − λ2r τλ d5 =  1 2 2bβξ 3 λ5r τλ τ12

−τ + λ4r ξτλ d6 = 4βλ6r ξτλ2

1 4



τ − λ4r ξτλr 2 2 + τ1 βλ6r ξτλ2



τ + λ4r ξτλ (λ2r − 1) λ6r ξτ1 τλ

1 d7 = 4



−2 τ + λ4r ξτλ + τ12 βλ6r ξτλ2



References 1. R. T. Hudspeth, Waves and Wave Forces on Coastal and Ocean Structures (World Scientific, Singapore, 2006). 2. C. M. Bowline, R. T. Hudspeth and R. B. Guenther, Applicable Anal. 72, 287 (1999). 3. R. T. Hudspeth, R. B. Guenther and S. Fadel, Acta Mechanica 175, 139 (2005). 4. R. B. Guenther, H. Schwerdtfeger and G. Herglotz, Vorlesungen u ¨ber die Mechanik der Kontinua (1985). 5. S. Fadel, Application of the generalized Melnikov method to weakly damped parametrically excited cross waves with surface tension, PhD dissertation, Oregon State University, USA (1998).

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Chapter 4

Random Wave Breaking and Nonlinearity Evolution Across the Surf Zone Yoshimi Goda ECOH CORPORATION, 2-6-4 Kita-Ueno Taito-ku, Tokyo 110-0014, Japan [email protected] A review is made on the statistical features of breaking waves in the nearshore waters. Inherent variability of the breaker index for regular waves is examined with the revised Goda’s formula. The incipient breaking height of significant wave is about 30% lower than that of regular waves. Nonlinearity of random waves is strongest at the outer edge of surf zone, but it is destroyed by wave-breaking process inside the surf zone. The wave height distribution is the narrowest in the middle of the surf zone, but it returns to the Rayleigh near the shoreline. Large differences among various wave models are noted for prediction of wave heights in the surf zone.

4.1. Introduction Breaking waves exert strong actions on maritime and coastal structures, while wave dissipation through breaking plays a major role in the generation of nearshore currents. Without good understanding of wave-breaking process, we cannot pursue any study for coastal engineering works. Nevertheless, wave breaking is an elusive phenomenon. Not many people spend enough time to observe wonderful pictures of wave deformation by breaking and regeneration after breaking. One needs some kind of a pier at a beach to have a good look of waves that break and rush toward the shore. Otherwise, one should work for some hours in a laboratory to measure waves at various locations along a wave flume. Our knowledge on wave breaking mostly comes from various literature based on previous research works. Quite a number of people use the formulas, diagrams, and other information listed there without examining the credibility of the information. For example, many people regard the breaker index, or the ratio of wave height to water depth at breaking, as a deterministic value without paying consideration to the fact that the breaker height exhibits large fluctuations even for a given wave 87

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condition and that the breaker index has been obtained by drawing a mean curve among scattered data. Nowadays, many researchers are developing various numerical models of random wave transformations. They have to adopt some kind of wave-breaking criteria and energy dissipation mechanism so that the model can reproduce wave deformation in the nearshore waters. However, modelers seem to pick up whatever is readily available without deliberation on the physical features of the wave-breaking process and the appropriateness of the breaking model. The present chapter is a revised version of the author’s paper1 presented at the 4th International Conference on Asian and Pacific Conference 2007, in Nanjin, China. It aims at providing coastal engineers and researchers with the most advanced knowledge on the statistics of wave breaking in the nearshore waters so that they can make a correct approach to the problems related to wave-breaking processes.

4.2. Physical Definitions of Wave Breaking Waves are defined as breaking when the crest starts to contain foams or when water particles jump out from the wave crest. It is a disruption of smooth water surface, and breaking waves are often classified into three types: spilling, plunging, and collapsing breakers. In the sea, there is no pure spilling breaker, because it is always accompanied with a small portion of plunging water. For example, we can observe from a window of an airplane such a scene of wave breaking in deep water that a crest of large wave makes plunging, leaves a patch of white foam behind, and moves forward with blue color. The following wave grows in height and breaks. It is a manifestation of wave energy being transported by the group velocity, which is one-half the phase velocity in deep water. Theoretically, three criteria can be cited. The first is the condition that the horizontal velocity of the water particle at the wave crest becomes equal to or greater than the phase speed of wave profile. The second is the upward vertical acceleration of the water particle at the wave crest to be equal to or to exceed the gravitational acceleration. The third is the vertical gradient of the total pressure at the wave crest to be zero or negative. The first criterion has been employed by mathematicians to find out the limiting waves of permanent form on a horizontal bed. Conventional perturbation techniques are ineffective to derive the limiting waves, and the approach specific to the limiting form needs to be employed. These limiting waves are characterized with the angular crest having the angle of 120◦ . Yamada and Shiotani2 have produced the most reliable computation results so far, which are summarized by Goda3 as reproduced in Table 4.1. The symbol H denotes the wave height, h is the water depth, L is the wavelength, L0 is the deepwater wavelength, C is the wave celerity, and ηc is the crest elevation. The subscripts “b” and “A” refer to the quantities at breaking and those of the small amplitude waves (Airy’s wave), respectively.

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Table 4.1. Characteristics of breaking waves of permanent type (after Goda3 based on Yamada and Shiotani2 ). Hb /L0

hb /LA a

hb /Lb

Cb /CA a

Hb /Lb

Hb /hb

ηc /Hb

infinity 0.935 0.471 0.286 0.1856 0.1117 0.0763 0.0474 0.0284 0.01669 0.01095 0.00575 0.00239 0.001144 0.000437 0

infinity 0.935 0.474 0.300 0.216 0.1510 0.1198 0.0915 0.0694 0.0525 0.0422 0.0306 0.01953 0.01351 0.00833 0

infinity 0.7686 0.4011 0.2597 0.1885 0.1331 0.1050 0.07915 0.05909 0.04398 0.03499 0.2483 0.01570 0.01075 0.00660 0

1.193 1.189 1.181 1.154 1.143 1.134 1.141 1.156 1.174 1.193 1.207 1.231 1.244 1.257 1.263 1.285b

0.1412 0.1409 0.1386 0.1277 0.1115 0.08997 0.07410 0.05771 0.04430 0.03371 0.02720 0.01962 0.01260 0.00871 0.00538 0

0 0.1791 0.3456 0.4919 0.5912 0.6683 0.7059 0.7293 0.7496 0.7666 0.7774 0.7904 0.8028 0.8099 0.8160 0.8261b

0.6706 0.6765 0.6908 0.7165 0.7619 0.7939 0.8392 0.8766 0.9061 0.9242 0.9453 0.9649 0.9757 0.9849 1.0000b

Notes: a LA and CA denote the wavelength and celerity of small amplitude waves. b These values are those of solitary wave computed by Yamada et al.4

The limiting height of solitary wave is (H/h)b = 0.8261 by Yamada et al.,4 instead of the often-quoted value of (H/h)b = 0.78 by McCowan.5 The limiting wave steepness of (H/L)b = 0.142 by Miche6 is calculated for waves of nonlinear waves, and it becomes Hb /(L0 )A = 0.1684 when the linear deepwater wavelength (L0 )A is used. The second criterion is for the breaking of standing waves, but no theoretical computation of limiting standing waves has been made. The third criterion has been proposed by Nadaoka et al.,7 for defining wave breaking in numerical time-domain computation. Zero vertical gradient of total pressure implies no presence of water below the wave crest, i.e., wave breaking.

4.3. Parameters Governing Breaker Index Because wave breaking attracts attention of many researchers, there have been proposed a number of formulas to describe the ratio of wave height to water depth. Kaminsky and Kraus8 called this ratio as the breaker height-to-depth index, but the present chapter employs the term of breaker index for the ratio of wave height to water depth at breaking for the sake of simplicity. Kamphuis9 has compared 11 formulas for testing of goodness-of-fit with his 225 sets of hydraulic model tests. Rattanapitikon and Shibayama10 have collected 574 data points from 24 papers/reports and calculated the root-mean-square errors of 24 breaking index formulas. Both authors have proposed their own formulas by modifying some of the previous ones.

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Rattanapitikon et al.11 have further added 121 data points in large-scale flume tests and proposed revised formulas. Similar to the approach by Rattanapitikon et al.,11 the breaker index formulas can be categorized into the following four functional forms: Hb /hb = f1 (0) = constant ,

(4.1)

Hb /hb = f2 (hb /L0 or hb /Lb ) ,

(4.2)

Hb /hb = f3 (s) ,

(4.3)

Hb /hb = f4 (s, hb /L0 or hb /Lb ) ,

(4.4)

where s denotes the bed slope. The wavelength L0 is the deepwater wavelength given by L0 = g/(2π)T 2 , where T is the wave period. Because the relative water depth hb /L0 is easily converted to hb /Lb through the dispersion relationship, the two relative depths hb /L0 and hb /Lb are interchangeable. The formula of Hb /hb = 0.78 is a typical example of Eq. (4.1). There are some other formulas using the parameter of deepwater wave steepness H0 /L0 . They can be useful for predicting the breaker height of regular waves. Nevertheless, they cannot be applied for breaking of random waves, because individual zero-crossing waves in a train of random waves are unrelated to individual waves in deepwaters. Thus, there is little room for the parameter H0 /L0 to function in the breaker index for random waves. Performance of a breaker index formula can be judged by the magnitude of the bias of the predicted breaker height from the observed heights. It should also be examined with either the root-mean-square error of predicted breaker heights or the correlation coefficient between prediction and observation. The root-mean-square error analysis by Rattanapitikon and Shibayama10 is not conclusive in differentiating the merits of four functional forms, but they recommend a certain modification of the slope effect in the function f4 (s, hb /L0 ), apparently indicating their preference of this functional form. Kamphuis9 calculated the correlation coefficients between 11 formulas and his laboratory data. By assigning the best-fitting value to the proportionality coefficient of each formula, he obtained the determination coefficient R2 = 0.69 to f1 (0), R2 = 0.67 to f2 (hb /L0 ), R2 = 0.84 to f3 (s), and R2 = 0.88 to f4 (s, hb /L0 ). His result clearly suggests the necessity to include both the parameters of bed slope and relative water depth in the breaker index formula.

4.4. Breaker Index for Regular Waves and Its Scatter 4.4.1. Scatter of regular breaking waves In 1970, Goda3 presented a diagram of breaker index curves of regular waves for four bed slopes, based on the laboratory data from eight sources, which included his own large-scale tests with Hb = 0.43 to 0.93 m; Rattanapitikon et al.11 did not

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analyze this data set. Later, Goda12,13 approximated the breaker index curves with the following empirical formula:
  Hb hb  A 4/3 1 − exp −1.5π 1 + 15s : A = 0.17 . (4.5) = hb hb /L0 L0 Equation (4.5) was derived by a graphical curve-fitting technique without direct comparison with the original laboratory data. Rattanapitikon and Shibayama10 have recommended a modification of the slope effect term of (1 + 15s4/3 ) into (1.033 + 4.71s − 10.46s2 ). Upon reexamination of the original laboratory data, the author has also recognized a necessity of modifying the slope effect term. The revised formula is as follows:
  hb  Hb A 1 − exp −1.5π 1 + 11 s4/3 : A = 0.17 . (4.6) = hb hb /L0 L0 Muttray and Oumeraci14 found the best-fitting coefficient of Eq. (4.5) being 0.167 instead of 0.170 for the slope of 1/30. When Eq. (4.6) is applied to their data, the coefficient would have the value of 0.173. Comparison of the laboratory data of breaker index with Eq. (4.6) is shown in Fig. 4.1 for five groups of bed slopes, i.e., 1/7 to 1/12, 1/20, 1/30, 1/50, and 1/200 to 1/200. (See Goda3 for description of the laboratory data employed here.) Recent data sets by Li et al.15 for s = 1/30 and s = 1/50, Li et al.16 for s = 1/200, and Lara et al.17 for s = 1/20 are also added. It is clear in Fig. 4.1 that the value of the breaker index increases as the bed slope becomes steep. Thus, it is absolutely necessary to incorporate the slope effect into the breaker index formula. Because the experimental data are scattered around the index curves, the upper and lower bound curves with the range of 87%–115% of the value by Eq. (4.6) are drawn in Fig. 4.1. A quantitative evaluation of the degree of the scatter is made by means of the relative error of the breaker index, i.e., E = (1 − γmeas/γest ), where γmeas is the measured value of Hb /hb and γest is the predicted value by the breaker index formula of Eq. (4.6). The mean and the standard deviation of the relative error are calculated for each group of the bed slope. The mean Emean indicates a bias of the breaker index and the standard deviation of E represents the degree of scatter of the breaker index. Because E is defined as the relative error, the standard deviation σ(E) is equivalent to the coefficient of variation (CoV). A positive bias indicates a tendency of overestimate, while a negative bias shows an underestimate. Table 4.2 lists the bias and CoV of the breaker index of Eq. (4.6) for the data of various bed slopes. The slope data of 1/9 and 1/12 are excluded from the analysis because of their small sample sizes. The bias varies from −2.9% to +6.2% depending on the bed slope, but the formula of Eq. (4.6) can be regarded as yielding reasonable estimates of the breaker heights. The scatter of the data as expressed by CoV is about 5% to 7% for the bed slope of 1/200 to 1/50; it increases as the slope becomes steep, and it takes the value of 14% for the bed slope of 1/10. Such scatter of data represents an inherent stochastic nature of wave-breaking phenomenon. It resides in the data set itself, being independent of the breaker index formula being applied.

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2.0

Breaker Index, γ = Hb /hb

Slope = 1/9 to 1/12

1.0 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.001

Kishi-Iohara 1/9 Iversen 1/10 Goda et al. 1/10 Galvin 1/10 Bowen et al. 1/12 Index curve 1/10 Upper 115% Lower 87%

0.01

0.1

Relative water depth, hb /L0

(a) Breaker index data for the slope of 1/9−1/12. 1.5 Breaker Index, γ = Hb /hb

Slope = 1/20 1.0 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.001

Iversen 1/20 Galvin 1/20 Toyoshima 1/20 Lara 1/20 Index curve 1/20 Upper 115% Lower 87%

0.01 Relative water depth, hb /L0

0.1

(b) Breaker index data for the slope of 1/20.

Breaker Index, γ = Hb /hb

1.2

Slope = 1/30

1.0 0.9 0.8 0.7 0.6 0.5 0.4

0.3 0.001

Iversen 1/30 Mitsuyasu 1/30 Toyoshima 1/30 Li Y.C. et al. 1/30 Index curve 1/30 Upper 115% Lower 87%

0.01 Relative water depth, hb /L0

0.1

(c) Breaker index data for the slope of 1/30. Fig. 4.1.

Comparison of breaker index formula Eq. (4.6) with laboratory data of regular waves.

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Breaker Index, γ = Hb /hb

1.0 0.9 0.8 0.7

Slope = 1/50

0.6

Iversen 1/50 Mitsuyasu 1/50 Li Y.C. et al. 1/50 Index curve 1/50 Upper 115% Lower 87%

0.5 0.4

0.3 0.001

0.01 Relative water depth, hb /L0

0.1

(d) Breaker index data for the slope of 1/50.

Breaker Index, γ = Hb /hb

1.0 0.9 0.8 0.7 0.6

Slope = 1/100 to 1/200

0.5 0.4

0.3 0.001

Goda 1/100 Li Y.C. et al. 1/200 Index curve 1/100 Index curve 1/200 Upper 115% Lower 87%

0.01 Relative water depth, hb /L0

0.1

(e) Breaker index data for the slope of 1/100–1/200. Fig. 4.1.

Table 4.2.

(Continued )

Bias and CoV of the breaker index formula of Eq. (4.6).

Bed slope

No. of data

Bias = Emean (%)

CoV = σ(E) (%)

1/10 1/20 1/30 1/50 1/100 1/200

29 47 73 28 32 19

−0.5 +3.9 +1.4 −2.9 +6.2 +3.5

14.0 11.3 8.6 4.8 5.5 7.4

Sources Iversen, Goda et al. Iversen, Galvin, Toyoshima et al., Lara et al. Iversen, Mitsuyasu, Toyoshima et al., Li et al. Iversen, Mitsuyasu, Li et al. Goda Li et al.

Even under a well-controlled laboratory test, the breaking point fluctuates over some distance and the breaker height varies from wave to wave. Smith and Kraus18 reported on their regular wave tests that “despite care in conducting the tests and use of the average value of the given quantity (i.e., over 10 waves), wide scatter appeared in some quantities and must be considered inherent to the breaking process of realistic waves.” One cause of the data scatter is the presence of small-amplitude, long-period oscillations of water level in a laboratory flume, but the breaking process itself is triggered by many small factors beyond the control of experimenters. We should regard the wave-breaking phenomenon as stochastic one and accept a certain range of natural fluctuation. As listed in Table 4.2, the coefficient of

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variation is large for steep slope and it becomes small for gentle slope. In any research work involving wave-breaking phenomena, due consideration should be given to such stochastic nature of breaker heights. 4.4.2. Scatter of random breaking wave heights The first effort to identify the breaker heights of individual waves among trains of irregular waves was undertaken by Kimura and Seyama19 and Seyama and Kimura,20 who analyzed about 1000 individual breaking waves for each slope of 1/10, 1/20, 1/30, and 1/50 with the aid of a video camera and wave gauges. The breaker index of individual waves varied over a wide range, which was equivalent to CoV of 18% to 23% (the present author’s visual inspection of scatter diagrams). In order to reduce the scatter of data, they proposed to employ an artificial water depth below the mid-level between the wave crest and trough of individual breaking waves and succeeded in reducing CoV to the values between 8% and 11%. Because Eq. (4.5) yielded the breaker index larger than most of the observed value, they proposed its modified version. When the revised breaker index formula of Eq. (4.6) is employed, however, the center line of the scatted data appears to be at the level of 85% (s = 1/10) to 95% (for other slopes) of the predicted value. Black and Rosenberg21 made observation of individual breaking waves on a natural beach with a depth of 1.0–1.5 m at Apollo Bay in southern Australia. The median value of the breaker index was about 84% and 87% of those by Eqs. (4.5) and (4.6), respectively. Another observation was made by Kriebel22 in a large wave flume with a bed slope of 1/50 for waves with a significant height of 0.46 m and peak period of 2.9 s. When he applied Eq. (4.5) to individual breaker heights, he found that the value of the proportionality coefficient A fitted to the data varied from about 0.09 to 0.18 (against the original value of A = 0.17) with the mean of 0.142 (84% of Eq. (4.5)) and the standard deviation of 0.017 (equivalent to CoV of 12%). Li et al.23 also reported the result of their measurements of random breaking waves on the slope of 1/50 and 1/200, recommending the coefficient value of A = 0.150 with the standard deviation of 0.031. It should be recalled that the random wave breaking model by Goda24 in 1975 had already incorporated the variability of breaker heights by assigning a variable probability of individual wave breaking. The probability was assumed to increase linearly from 0 to 1 in the range of the wave height from 71% to 106% of the height predicted by Eq. (4.5), corresponding to A = 0.12 to 0.18. Therefore, it is expected that the median value of individual breaker heights would be smaller than those predicted by Eq. (4.5) or (4.6). 4.5. Breaker Index for Random Waves 4.5.1. Incipient breaking index of significant wave Equations (4.5) and (4.6) are examples of the breaker index for regular waves. There are some people who try to apply such breaker index formulas to coastal

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waves or random waves, but such application does not yield correct answers. The breaker index for regular waves may be utilized for the highest wave in an irregular wave train, but it cannot be applied for the significant wave, the root-mean-square wave, or any other characteristic wave. When random waves approach the shore, breaking of individual waves occurs gradually with large waves first at the far distance, medium waves next at some distance from the shore, and small waves near the shoreline. The variation of significant wave height from the offshore toward the shore is so gradual that we cannot employ the concept of wave-breaking line, which is so obvious in the case of regular waves. Against such difficulty of defining the breaking point of significant wave, Kamphuis9 introduced the definition of “incipient wave breaking.” He measured cross-shore variations of significant wave height beyond and across the surf zone, drew a curve of wave shoaling trend in the outside of the surf zone and a curve of wave height decay within the surf zone, and called the condition at the cross-point of the two curves as the incipient wave breaking. By using the data of the significant wave height at incipient breaking, he calibrated 11 breaker index formulas and determined the best-fitting proportionality coefficient. For the formula of Eq. (4.5), he obtained the proportionality coefficient of A = 0.12 for significant wave height. Li et al.23 have also presented a data of the breaker index of (H1/3 /h)b on the slope of 1/200, which is fitted to Eq. (4.5) with a modified constant value of 0.12 for the initial stage of breaking. Their breaking condition was some observation of large individual breaking waves in an irregular wave train. Goda24 has prepared a set of diagrams depicting variations of significant wave height across the surf zone (reproduced as Figs. 3.29 to 3.32 in Ref. 25). The boundary lines of 2% decay in these diagrams approximately correspond to the breaker index with A = 0.11, and the water depth (h1/3 )peak at which the significant wave takes a peak value within the surf zone (Fig. 3.34 in Ref. 25) corresponds to A = 0.11–0.13. Therefore, the incipient breaker index of the significant wave can be expressed with the following formula: 
   H1/3, b 0.12 (hb )incipient  4/3 = 1 − exp −1.5π 1 + 11 s . (4.7) hb hb /L0 L0 incipient Thus, the incipient breaker index of significant wave is about 30% lower than that of regular waves. The incipient breaking of significant wave corresponds to the condition that the high waves of upper several percent among individual waves have begun to break. 4.5.2. Laboratory data of breaker index of random waves After incipient breaking, the percentage of wave breaking increases as waves proceed across the surf zone. The ratio of the significant wave height to the water depth gradually increases toward the shoreline. Ting26,27 made detailed laboratory investigations of random wave deformations on a uniform slope of 1/35, using frequency spectra of broad- and narrow-band with the peak enhancement factor of 3.3 and 100, respectively. Waves had the significant height of Hs = 0.15 m and the spectral

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Breaker indices, H1/3 /h & Hrms /h

1.0

0.6 0.4 0.3 0.2

0.10

Fig. 4.2.

Slope = 1/35

0.8

Broad spec. H1/3 /h Narrow spec. H1/3 /h Breaker envlp. H1/3 /h Broad spec. Hrms /h Narrow spec. Hrms /h Breaker envlp. Hrms /h

0.01

0.1 0.03 Relative depth, h/L0

0.3

Breaker indices for H1/3 and Hrms on s = 1/35 with the data by Ting.26,27

peak period of Tp = 2.0 s. He recorded wave profiles at an offshore station with the depth of 0.457 m and at six stations on the slope with the depth of 0.27–0.0625 m. Waves at the six stations on the slope had the percentage of breaking ranging from 5% to 94% (the case of broad-band spectrum). Wave records were analyzed by the zero-downcrossing method, and calculated results of characteristic wave heights and periods are presented in tabular forms. From these results, the ratios of H1/3 and Hrms to the local depth (inclusive of mean water level change) are calculated and plotted against the relative water depth h/L0 , as shown in Fig. 4.2. The curves denoted as breaker envelopes are calculated by Eq. (4.6) for s = 1/35 with the proportionality coefficient of A = 0.145 (85% of regular waves) for the significant height H1/3 and to A = 0.111 (65%) for the root-mean-square height Hrms . Because the percentage of breaking waves is high in these data, an A value higher than that for incipient breaking fits to the data. The breaker index data for Hrms by Tick is higher than the value proposed by Sallenger and Holman,28 who gave an expression of Hrms /h = 3.2s + 0.32 without inclusion of the relative depth (h/L0 ) term. They converted the orbital velocity spectra to the surface wave spectra with the transfer function based on the linear theory, and estimated the energy-based Hrms , which must have been smaller than the statistical Hrms value based on direct measurement data of surface profiles. 4.5.3. Description of field data employed for analysis In the field wave observation at a fixed station, it is not feasible to judge whether individual waves are at the stage of breaking or not, unless simultaneous measurements with video cameras are taken. However, we may find out an upper limit of significant wave height for a given water depth by taking an envelope of many data at different relative water depths. For this type of analysis, stationary wave

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Summary of stationary coastal wave data employed in the present analysis.

Observation station Rumoi Port Yamase-Domari Port Tomakomai Port

Kanazawa Port Caldera Port, Costa Rica Sakata Port

97

Type of wave gauge

Water depth (m)

Sampling interval ∆t (s)

Significant height H1/3 (m)

Significant period T1/3 (s)

No. of data

Step-resistance ”

−11.5 −12.7

0.5 0.5

2.2–7.1 1.9–6.2

5.9–11.7 7.7–15.6

44 9

” ” Ultrasonic ” ”

−10.8 −13.8 −20 −20 −18

1.0 0.5 0.5 1.0 0.5

2.9–5.8 2.6–2.8 2.4–2.5 1.0–6.8 1.5–3.6

7.7–10.9 6.7 – 7.5 6.9–7.4 1.0–6.8 14.2–18.4

9 2 2 13 50

Pressure

−14.5 −10.5

0.5 0.5

1.7–9.7 1.7–6.1

6.3–13.4 6.5–15.0

123 123

(Source: Goda and Nagai,29 Goda24,30 ).

observation data analyzed by Goda and Nagai29 and the data of long-traveled swell recorded with an ultrasonic wave sensor reported by Goda30 were utilized. Table 4.3 lists the characteristics of these field data. Waves were recorded by means of either step-resistance gauges or ultrasonic wave sensors so that they were reliable registration of surface wave profiles. The data at Tomakomai and Kanazawa as well as Caldera were measured with ultrasonic wave sensors. They are not analyzed for breaker limits but for wave nonlinearity effects to be described in Sec. 4.6. Table 4.3 also lists the wave records at Sakata Port measured by means of pressure gauges, which were utilized by Goda24 for calibration of his random wavebreaking model. Although there remains a problem of pressure conversion to surface profiles, the conversion error would have been small because of the relatively shallow water depth at Sakata stations (10.5 and 14.5 m). They were included in the present analysis to increase the size of database. Other sources of nearshore waves are the photogrametric measurement data by Hotta and Mizuguchi31,32 as well as by Ebersole and Hughes.33 Hotta and Mizuguchi mobilized 11 motion-picture cameras set on top of a coastal observation pier at Ajigaura Beach, Ibaragi, Japan. They took film pictures of instantaneous water surfaces simultaneously at some 60 surveyor’s poles erected in the nearshore waters on a line perpendicular to the shoreline stretched over a distance of about 120 m. Films of surface wave records were taken every 0.2 s for an effective duration of 760 s. The beach profile in September 1978 had a trough at the distance of 25 m from the shoreline and the slope of about 1/60 offshore of the trough. The beach profile in December 1978 was somewhat uniform without any bar or trough, and the slope was about 1/70. The water depth inclusive of tides at the poles varied from 0.1 to 2.7 m. Photogrametric measurements of nearshore waves were also executed by Ebersole and Hughes33 during the DUCK85 campaign in Duck, North Carolina, USA with the cooperation of Dr. Hotta who brought twelve cameras with him and took charge of filming. They referred to this technique as “the photopole method”; this terminology is employed in the present chapter. Over the distance of 64 m,

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Summary of photopole measurements.

Date

No. of data

h (m)

H0 (m)

T1/3 (s)

H0 /L0

1978/09/05 1978/12/13–14 1985/09/04–05 1986/09/11–19

54 175 99 140

0.6–2.7 0.1–1.8 0.5–2.4 0.4–3.7

0.7 0.5–0.7 0.3–0.5 0.5–1.1

8.4 7.2–7.9 10.3–11.1 5.4–11.5

0.0064 0.0059–0.0081 0.0017–0.0028 0.0027–0.0254

12 poles were erected in the initial depth ranging from 0.4 to 1.9 m. Measurements were taken for nine runs during the fourth and fifth days of September 1985. With the variation of the tide level, the actual water depth varied from 0.5 to 2.4 m. The effective duration of wave recording was about 650 s, judging from the number of waves and average periods listed by Ebersole and Hughes.33 The beach profile was nearly flat for about 25 m from the shoreline with the depth of about 0.5 m below the mean sea level, and it had the slope of about 1/30 beyond that. Another series of photopole measurements were carried out during the SUPERDUCK campaign in 1986. Dr Hughes kindly supplied the author with the data files of measured wave statistics. The number of poles was increased to 20 and the water depth inclusive of tides varied from 0.4 to 3.7 m. The beach profile during SUPERDUCK is not known, but it would have been nearly the same as DUCK85 because of the same season. All the photopole measurement data were analyzed by the zero-downcrossing method, and various statistical wave characteristics were calculated. Table 4.4 lists the summary of the photopole wave measurement conditions. The significant wave period T1/3 has been converted from the pole-averaged values of either the mean period Tmean or the spectral period Tp by assumption of T1/3 = 1.05Tmean or T1/3 = 0.95Tp, which would be appropriate for swell of very low steepness. The offshore wave height H0 was converted from the significant height H1/3 measured at the most offshore pole using the shoaling coefficient; no refraction effect was taken into consideration as no information of wave direction was available. All the waves were swell of very low steepness ranging from 0.0017 to 0.0081, except one case of SUPERDUCK with H0 /L0 = 0.0254. 4.5.4. Field data of breaker index for energy-based significant waves Coastal surface wave data, pressure-converted wave data, and three sets of photopole data are plotted together in Fig. 4.3 in the form of Hm0 /h versus h/L0 . The energybased significant wave height Hm0 defined by Eq. (4.8) is employed here instead of the zero-crossing height H1/3 , because the latter is greatly enhanced over Hm0 by strong effects of wave nonlinearity and it is not representative of breaking-dissipated wave energy level; this aspect is discussed in Sec. 4.6. √ (4.8) Hm0 = 4.0ηrms = 4.0 m0 , where m0 denotes the zeroth moment of frequency spectrum.

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Step-resistance ( h=11-14m) Presssure gage (h=10-14m) Photopole Ajigaura (h=0.1-2.7 m) Photopole DUCK85 (h=0.5-2.4m) Photopole SUPERDUCK (h=0.5-3.2 m) Breaker envelope (80% limit)

Wave height ratio, Hm0 /h

0.8 0.6 0.4

0.2

0.1 0.001

Fig. 4.3.

0.01 0.1 Relative depth, h/L0

1

Breaker index for Hm0 based on the field wave data.

Coastal wave data, on the other hand, were not much affected by wave nonlinearity effects, and the zero-crossing significant height H1/3 was almost the same as Hm0 . Many data points in Fig. 4.3 belong to nonbreaking condition, but what interests us is the upper envelope which provides an estimate of the upper limit of breaking wave height. The curve of dash-dot line in Fig. 4.3 has been calculated by Eq. (4.6) for the slope of s = 0.0143 (1/70) with the coefficient being reduced to A = 0.136 (80%). Similar with the laboratory data shown in Fig. 4.2, the wave height ratio Hm0 /h is higher than the incipient breaker index of significant wave expressed by Eq. (4.7). It is because the breaker index increases inside the surf zone as the percentage of breaking waves increase. It is seen that the energy-based significant wave height Hm0 on gentle slopes does not exceed 0.7 times the local water depth except for the low-steepness swell in very shallow water. For the range of h/L0 > 0.03, the upper limit of significant wave height is about 0.6 times the local water depth. Some data points above the dash-dot curve are those of DUCK85 and SUPERDUCK, which were conducted on the beach steeper than the beach in Ajigaura.

4.6. Evolution of Wave Nonlinearity Across Surf Zone 4.6.1. Variations of skewness and kurtosis across the surf zone Ocean waves are characterized with almost linear property, as evidenced by the Gaussian distribution of surface elevation. Wave linearity is the basis of spectral

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representation and analysis. Deviation from the linearity is measured with the values of the skewness and kurtosis of the surface elevation with reference to the mean water level during a wave record. The skewness is zero when a distribution is symmetric with respect to the mean, and takes a positive value when a distribution is asymmetric with a long tail toward the right side (large value). The kurtosis takes a value of 3.0 for the Gaussian distribution. When the mode of distribution has a sharp peak and the distribution has long tails in both the left and right sides, the value of kurtosis becomes much larger than 3.0. The degree of positive skewness and the deviation of kurtosis from 3.0 are the measure of the strength of wave nonlinearity. The skewness of ocean waves is less than 0.5, and the kurtosis is below 4.0 for most cases, and thus the nonlinearity of waves in deepwater is weak. The variations of the skewness and kurtosis of field waves are examined with coastal surface waves listed in Table 4.3 (excluding the pressure sensor data of Sakata Port), and the photopole data in Ajigaura, DUCK85, and SUPERDUCK listed in Table 4.4. The data of skewness and kurtosis of September 5th in Ajigaura were kindly provided by Dr Hotta. The data of the photopole measurements of SUPERDUCK provided by Dr Hughes had the data of skewness only. As waves approach the shore, wave nonlinearity is enhanced and both the skewness and kurtosis increase significantly. Figure 4.4(a) exhibits the increase of the skewness with the nonlinearity parameter Π1/3 , which was introduced by Goda34 with the following definition: Π1/3 =

H1/3 2πh coth3 , LA LA

(4.9)

where LA denotes the wavelength calculated by small amplitude wave theory or Airy’s theory. The data are grouped by the range of the offshore wave steepness H0 /L0 : the first group for 0.001 < H0 /L0 < 0.0029, the second group for 0.003 < H0 /L0 < 0.0049, the third group for 0.0050 < H0 /L0 < 0.0099, the fourth group for 0.010 < H0 /L0 < 0.029, and the fifth group for 0.030 < H0 /L0 < 0.049 (Legends are shown with abbreviated figures). The data shown in the left diagram of Fig. 4.4 are those outside the surf zone. Because the boundary of surf zone is difficult to be set for random waves, an arbitrary boundary of h/H0 = 2.5 is employed here to separate the wave data outside and inside the surf zone. As shown in the left diagram, the skewness outside the surf zone shows a clear correlation with the wave nonlinearity parameter. The skewness begins from the value of zero at Π1/3 = 0, increases almost linearly with Π1/3 , and attains the value of 2.0 around Π1/3 = 4. The dashed line represents a semitheoretical relationship, which is based on the analysis of finite amplitude regular wave profiles by Goda34 with the consideration of the probability of individual wave heights according to the Rayleigh distribution. Variation of the skewness inside the surf zone (h/H0 < 2.5) is shown in Fig. 4.4(b). The ordinate is the ratio of offshore wave height to water depth, H0 /h, which increases rapidly as waves approach the shore. There is a clear trend of skewness decreasing toward β1 = 0 with the increase of the height-to-depth ratio

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Random Wave Breaking and Nonlinearity Evolution Across the Surf Zone

2.5

2.5 0.001
h/H0 < 2.5

h/H0 > 2.5 2.0

1.5

Skewness, β 1

Skewness, β 1

2.0

1.0

1.0

0.5

0.0

0.0

0.1 1 Nonlinearity parameter, Π1/3

0.6

0.8

1

1.2

1.4

Height-to-depht ratio, H0 /h

(a) O utside the surf zone (h/H0 > 2.5) Fig. 4.4.

-0.5 0.4

10

0.001
1.5

0.5

-0.5 0.01

(b) Inside the surf zone (h/H0 < 2.5)

Variation of the skewness of surface elevation outside and inside the surf zone.

10.5

10.5

7.5

h/H0 > 2.5

h/H0 < 2.5 9.0

Kurtosis, β 2

0.001
9.0 Kurtosis, β 2

101

6.0 4.5 3.0

0.001
7.5 6.0 4.5 3.0

1.5 0.01

0.1 1 Nonlinearity parameter, Π1/3

(a) Outside the surf zone (h/H0 > 2.5) Fig. 4.5.

10

1.5 0.4

0.6

0.8 1.2 1.0 Height-to-depht ratio, H0 /h

1.4

(b) Inside the surf zone (h/H0 < 2.5)

Variation of kurtosis of surface elevation outside and inside the surf zone.

H0 /h. Use of the height-to-depth ratio in Fig. 4.4(b) is to provide a kind of contrast of the increase and decrease of skewness in the outside and inside of the surf zone, respectively. While the increase of skewness outside the surf zone seems indifferent to the wave steepness, the value of skewness inside the surf zone is much affected by the wave steepness; waves of low steepness maintain large values of skewness, while waves of high steepness have small values only. Variation of the kurtosis of surface elevation is shown in Fig. 4.5; Fig. 4.5(a) shows kurtosis outside the surf zone and Fig. 4.5(b) shows kurtosis inside the surf zone. The pattern of variation is the same as that of skewness, though the available number of kurtosis data is smaller than the skewness data. The kurtosis starts from the value of 3.0 at Π1/3 = 0, increases as the nonlinearity parameter increases, and attains the value of 9 around Π1/3 = 4. In Fig. 4.5(b), the kurtosis inside the surf

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zone decreases toward β2 = 3.0 as waves approach the shoreline (i.e., increase of H0 /h). Waves with the steepness larger than 0.01 exhibit the kurtosis value much smaller than swell with very low steepness. It is because waves with large steepness cannot experience the full process of nonlinear shoaling owing to the early start of wave breaking. 4.6.2. Variations of zero-crossing wave heights across the surf zone Wave nonlinearity is also reflected in individual wave heights defined by the zerocrossing method. In the offshore, individual wave heights almost follow the Rayleigh distribution. Close examination of wave height distribution has revealed a distribution slightly narrower than the Rayleigh with the result of the mean ratio of H1/3 /Hm0 being 0.95. The difference is due to the energy spread over frequency as represented by wave spectrum, the degree of which can be measured with the spectral shape parameter as discussed by Goda and Kudaka.35 As waves propagate toward the shore, however, waves undergo nonlinear shoaling. In this process, wave profiles become skewed with high and sharp crests and low and flat troughs. Because of skewed wave profile, the potential energy contained in such a profile is smaller than the energy of sinusoidal wave with the same height. In other words, nonlinear waves can have the height much larger than the linear (sinusoidal) waves for the same potential energy. Owing to the nonlinear shoaling, individual wave heights increase rapidly with the rate greater than that of linear shoaling. Significant wave heights and other characteristic wave heights also grow rapidly. The nonlinearity effect becomes most conspicuous around the outer edge of the surf zone. After waves enter the surf zone and begin to be attenuated through breaking process, the wave nonlinearity is gradually lessened. The degree of nonlinear shoaling effect may be judged by a departure of the statistical wave height from the theoretical prediction by the Rayleigh distribution, which is expressed as below: H1/10 = 1.27Hm0 ,

H1/3 = Hm0 ,

Hrms = 0.707 Hm0 .

(4.10)

Figures 4.6–4.8 show the evolutions of the wave height ratios H1/10 /Hm0 , H1/3 /Hm0 , and Hrms /Hm0 , respectively. The difference between H1/3 and Hm0 in the nearshore waters has been pointed by several researchers such as Thompson and Vincent36 and Ebersole and Hughes.33 In Figs. 4.6–4.8, the (a) parts are for those outside the surf zone and the (b) parts are for those inside the surf zone, similar to Figs. 4.4 and 4.5. The boundary of the surf zone is subjectively set at h/H0 = 2.5 in Figs. 4.6 and 4.7 and h/H0 = 1.5 in Fig. 4.8. In case of the root-mean-square wave height Hrms , the surf zone should be defined in a shallower area than for the significant wave, because Hrms is calculated with all individual waves. Thus, the boundary of Hrms was set at h/H0 = 1.5. The dashed lines in the left diagrams are semi-theoretical predictions based on the potential energy calculation of finite amplitude waves and the Rayleigh distribution of wave heights, by referring to the methodology employed by Longuet-Higgins.37

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2.2

Wave height ratio, H1/10 /Hm0

1.8

h/H0 > 2.5

h/H0 < 2.5 2.0 Wave height ratio, H1/10 /Hm0

0.001
2.0

1.6

1.4

1.2

0.001
1.8

1.6

1.4

1.2

1.0 0.01

1.0 0.1 1 Nonlinearity parameter, Π1/3

10

0.4

(a) Outside the surf zone (h/H0 > 2.5)

4 1 Height-to-depht ratio, H0 /h

10

(b) Inside the surf zone (h/H0 < 2.5)

Variation of wave height ratio H1/10 /Hm0 outside and inside the surf zone.

Fig. 4.6.

1.75

1.75 0.001
1.50

h/H0 > 2.5

h/H0 < 2.5 Wave height ratio, H1/3/Hm0

Wave height ratio, H1/3 /Hm0

103

1.25

1.00

0.75 0.01

0.1

1

Nonlinearity parameter, Π1/3

(a) Outside the surf zone (h/H0 > 2.5) Fig. 4.7.

10

1.50

0.001
1.25

1.00

0.75

0.4

1

4

10

Height-to-depht ratio, H0 /h

(b) Inside the surf zone (h/H0 < 2.5)

Variation of wave height ratio H1/3 /Hm0 outside and inside the surf zone.

Outside the surf zone shown in the left diagrams of Figs. 4.6–4.8, the wave height ratios H1/10 /Hm0 , H1/3 /Hm0 , and Hrms /Hm0 increase with the increase of the wave nonlinearity parameter. Although there is much scatter of data, they follow the trend of semi-theoretical curves. The one-tenth highest wave height H1/10 in Fig. 4.6 is given the initial value of 1.20Hm0 based on the trend of the data in consideration of the spectral width effect. The maximum values of the ratios of the statistical wave heights to the energybased significant wave height for swell of very low steepness are around 2.1 for H1/10 /Hm0 , 1.6 for H1/3 /Hm0 , and 1.15 for Hrms /Hm0 . With reference to the wave height ratios at the weak nonlinearity state, the statistical wave heights are

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1.2

1.2 h/H0 > 1.5

h/H0 < 1.5

1.1 Wave height ratio, Hrms /Hm0

0.001
1.1 Wave height ratio, Hrms /Hm0

FA

1.0

0.9 0.8

0.7

0.001
1.0

0.9

0.8

0.7

0.6 0.01

0.6 0.1 1 Nonlinearity parameter, Π1/3

(a) Outside the surf zone (h/H0 > 1.5) Fig. 4.8.

10

0.4

1 4 Height-to-depht ratio, H0 /h

10

(b) I nside the surf zone (h/H0 < 1.5)

Variation of wave height ratio Hrms /Hm0 outside and inside the surf zone.

enhanced by 1.75 times for H1/10 , 1.68 times for H1/3 , and 1.64 times for Hrms . Such enhancement of statistical wave heights are apparent ones without real increase of wave energy as discussed earlier. Inside the surf zone shown in the right diagrams of Figs. 4.6–4.8, the data exhibit large scatter but they indicate a clear tendency of decrease with the height-to-depth ratio H0 /h; the wave height ratios H1/10 /Hm0 , H1/3 /Hm0 , and Hrms /Hm0 converge to 1.27, 1.0, and 0.70, respectively, as H0 /h becomes larger than 2. As for the effect of wave steepness, waves with low steepness in the range of 0.001 to 0.0049 have the mean value of H1/3 /Hm0 = 1.4 around H0 /h = 0.4, while waves with the medium steepness of 0.005 to 0.009 starts from the mean value of H1/3 /Hm0 = 1.2. Waves with a high steepness of 0.010 to 0.049 have the H1/3 /Hm0 value less than 1.1. Decrease of the wave height ratio with decreasing water depth has been pointed out by Thompson and Vincent,36 who suggested earlier start of decrease for waves of high steepness compared with waves of low steepness. Such effects of wave steepness on the nonlinear features of waves inside the surf zone are originated from the fact that waves of low steepness experience a high degree of nonlinear wave shoaling before they are attenuated by breaking, while waves of high steepness are attenuated much earlier before they experience strong nonlinear shoaling. It is seen in the left diagrams of Figs. 4.6–4.8 that waves with the steepness of 0.010–0.049 have the nonlinearity parameter up to Π1/3 = 0.3 only and waves are transferred into the right diagrams which represent waves inside the surf zone. Waves with the steepness of 0.005 to 0.0099 have the nonlinearity parameter up to Π1/3 = 1.2 and moves into the group of those inside the surf zone. Waves with the steepness below 0.0029 can have the nonlinearity parameter up to Π1/3 = 4 before they enter into the surf zone. Thus, the evolution of wave nonlinearity expressed in terms of the skewness β1 , kurtosis β2 , and wave height ratios H1/10 /Hm0 , H1/3 /Hm0 , and Hrms /Hm0 is summarized as follows. It is weak in the offshore, increases gradually as waves propagate shoreward, becomes strongest in the outer half of the surf zone, begins

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to decrease as breaking process of random waves become intensive, and is nearly lost at the shoreline.

4.7. Changes of Wave Height Distribution and Probability Density Function Across Surf Zone 4.7.1. Histogram of wave height distribution The first formulation of random wave-breaking process in a numerical model was made by Collins38 by modifying the probability density function (pdf ) of the Rayleigh distribution. He assumed that all waves beyond the breaking limit will reduce their heights to the breaker height Hb , i.e., the pdf is truncated at H = Hb , and it has an infinite intensity of delta function there. An early work of Battjes39 adopted the same approach, while Kuo and Kuo40 redistributed the portion of broken waves over the range of 0 < H ≤ Hb in proportion to the remaining probability density. On the other hand, Goda24 proposed to make a gradual cut over an upper one-third range of wave height below the breaker height as explained in Sec. 4.4.2. Goda’s proposal was based on the field data of wave height histograms, which was analyzed by himself. Figure 4.9 is a replotting of the data of surface wave records of Rumoi, Yamase-Domari, Tomakomai, and Kanazawa Ports listed in Table 4.3. The data is presented in four groups according to the value of relative water height H1/3 /h. The group with H1/3 /h < 0.4 demonstrates the wave heights being in good agreement with the Rayleigh distribution. Though not discernible in the figure, there is a small number of waves in the class of H/Hmean = 3.25–3.50, corresponding to the probability density of the Rayleigh. The next group with 0.4 ≤ H1/3 /h < 0.5 does not have any wave in the class of H/Hmean > 3.25, and the histogram indicates a slight leftward shift. As the relative wave height increases, disappearance of large individual waves becomes clearer and so is the deviation from the Rayleigh distribution. However, there is no concentration of wave heights at the upper end as assumed by Collins38 and Battjes.39 Changes of the pdf of wave heights within the surf zone have been observed in laboratory data such as Goda,24 Baldock et al.,41 and Ting.42,43 Figure 4.10 is an example of the cumulative distribution of wave heights in the surf zone, which has been prepared for the data presented in Fig. 4.5 of Ting.42 The abscissa of Fig. 4.10 is the relative wave height H/H1/3 and the ordinate is the nonexceedance probability. When wave heights follow the Rayleigh distribution, their cumulative distribution is aligned on the heavy straight line of Fig. 4.10. The data at the relative depth of h/H0 = 1.90 deviates only slightly from the straight line of the Rayleigh. As the relative depth decreases, the data of cumulative distribution shift leftward with the largest relative height being Hmax /H1/3 = 1.3 or less. At the relative depth of h/H0 = 0.69 and 0.48, however, the cumulative distribution exhibits a tendency of returning to the Rayleigh with Hmax /H1/3 = 1.5. By referring to the original diagram of Fig. 4.5 of Tick,42 it is observed that deviation from the Rayleigh distribution is largest around h/H0 = 1.0. Closer to the

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1.0

1.0

H1/3/h = 0.400 - 0.499 17 records with 1398 waves

0.8

Probability density, p (x)

Probability density, p (x)

H1/3/h = 0 - 0.399 52 records with 5111 waves

Observation Rayleigh

0.6

0.4

0.2

0.8 Observation Rayleigh

0.6

0.4

0.2

0

0 0

1.0 1.5 2.0 2.5 3.0 0.5 Normalized wave height, x = H /Hmean

0

3.5

0.5

1.0

1.5

2.0

2.5

3.0

3.5

Normalized wave height, x = H /Hmean

1.0

1.0

H1/3/h = 0.500 - 0.599 7 records with 534 waves Observation Rayleigh

0.6

H1/3/h = 0.600 - 0.630 3 records with 188 waves

0.8 Probability density, p (x)

0.8 Probability density, p (x)

FA

0.4

0.2

Observation Rayleigh

0.6

0.4

0.2

0

0 0

0.5

1.0

1.5

2.0

2.5

3.0

Normalized wave height, x = H /Hmean

Fig. 4.9.

3.5

0

0.5

1.0

1.5

2.0

2.5

3.0

3.5

Normalized wave height, x = H /Hmean

Histograms of field wave data in four groups of relative water depth (after Goda24 ).

shore, the range of wave height begins to spread owing to regeneration of waves after breaking, and the wave heights show a tendency to approach the Rayleigh again. Some researchers such as Battjes and Groenendijk44 have tried to apply certain statistical distributions to the wave heights in the surf zone and to empirically formulate the parameter values. As demonstrated in Figs. 4.9 and 4.10, however, the wave height distribution continues to vary throughout the surf zone, and any effort to describe it with a combination of multiple distribution functions will be futile. 4.7.2. Variation of wave height ratios across the surf zone Change in the pdf of wave heights also yields variation of the ratios among characteristic wave heights. General tendency in the surf zone is that the values of Hmax /H1/3 , H1/20 /H1/3 , and H1/10 /H1/3 decrease as waves enter the surf zone and take the minimum values around at its middle, while the values of Hrms /H1/3 and Hmean /H1/3 increase. However, the wave height ratios come back to the values predicted by the Rayleigh distribution as waves approach the shoreline. Goda45 has made a prediction of such variation of wave height ratios by means of a numerical model based on the Parabolic Equation with a Gradational Breaker

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2.5 0.99

Rayleigh h/H0'=1.90 h/H0'=1.60 h/H0'=1.28 h/H0'=0.97 h/H0'=0.69 h/H0'=0.48

2.0

1.5

0.95 0.90 0.75

1.0 0.50 0.5

0.25

0

0.10 0.05 0.01 0

0.5

1.0

1.5

Nonexceedance Probability, P

Reduced variate, SQRT [ -ln (1 - P)]

Ting's Lab. Data

2.0

Relative wave height, H/H1/3 Fig. 4.10. Evolution of cumulative distribution of wave heights in the surf zone based on the data by Ting.42

Index for Spectral waves (PEGBIS). It has been demonstrated that the difference among characteristic wave heights gradually decreases as the relative water depth becomes shallow down to around h/H0 = 1, but the range of wave heights expands again as the water becomes much shallower toward the shoreline. Laboratory tests by Ting42,43 on the wave breaking on a uniform slope of 1/35 show a variation of the wave height ratio Hrms /H1/3, following the general trend described above. Similar variations of the wave height ratios H1/10 /H1/3 and Hrms /H1/3 are found in the photopole data at Ajigaura, DUCK85, and SUPERDUCK. Figure 4.11 presents a compilation of the laboratory and field data concerning the wave height ratio changes. Also included in Fig. 4.11 is the prediction of the wave height ratios across the surf zone by means of the PEGBIS model applied for the Ajigaura beach condition. Although there are large scatters of data, the ratio H1/10 /H1/3 takes the minimum value of about 1.15 around h/H0 = 1.4 for the field data, which indicate slightly larger values than the PEGBIS prediction, probably owing to the wave nonlinearity effect. The ratio Hrms /H1/3 takes the maximum value of about 0.81 around h/H0 = 1.0 for both the laboratory and field data. The DUCK85 data exhibit the Hrms /H1/3 being closer to 1 than the Ajigaura data. It might be due to lower wave steepness of the swell at DUCK85. Such changes of the wave height ratios across the surf zone are the result of the evolution of the pdf of wave heights discussed in Sec. 4.7.1. Narrowing of the wave height distribution at the middle of the surf zone provides a larger safety margin in the design of breakwaters and other structures than the case of applying the Rayleigh distribution, because the design wave height is specified with either Hmax (composite breakwaters) or H1/20 (mound breakwaters) for a prescribed offshore

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H1/10 /H1/3: Ajigaura H1/10 /H1/3: DUCK85 H1/10 /H1/3: SUPERDUCK H1/10 /H1/3: PEGBIS Hrms /H1/3: Ajigaura

Hrms /H1/3: DUCK85 Hrms /H1/3: SUPERDUCK Hrms /H1/3: Ting's Lab. Hrms /H1/3: PEGBIS

Wave height ratios, H1/10 /H1/3 & Hrms /H1/3

1.50

1.25

1.00

0.75

0.50 0.0

1.0

2.0

3.0

4.0

5.0

6.0

Relative depth, h/H0 Fig. 4.11. Variations of wave height ratios H1/10 /H1/3 and Hrms /H1/3 by photopole field data and Hrms /H1/3 in laboratory tests by Ting together with prediction by PEGBIS model.

significant wave height. Therefore, reliable information on the evolution of characteristic wave heights, not just Hm0 or Hrms , needs to be provided by numerical models of wave transformation with random breaking.

4.8. Incorporation of Wave Breaking Process in Numerical Wave Transformation Models For engineering applications, numerical wave transformation models are indispensable tools, and there have been many models, proposed by various researchers and institutions. The models must be capable of dealing with random waves and the process of individual wave breaking, because the majority of engineering problems are concerned with the situations within the surf zone. For the simplest case of random waves on a uniform slope, Goda46 compared performance of seven models listed in Table 4.5, which include those by Battjes,39 Kuo and Kuo,40 Goda,24,45 Battjes and Janssen,47 Thornton and Guza,48 and Larson and Kraus,49 though there are many more models for random wave breaking. All the models introduced the breaker index Hb /hb , but the degree of inclusion of the parameters of the relative depth hb /L0 and the slope s differs among them. The inherent variability of breaker heights is taken into account by the models by

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109

Main characteristics of random wave-breaking models for comparative study. Factors affecting breaking wave height

Models Battjes39

Energy dissipation mechanism

PDF of broken wave

pdf Delta function deformation at breaker height Kuo and ” Remove and Kuo40 adjust the remainder Goda24 ” ” Bore model Delta function Battjes at breaker and height Janssen47 Thornton ” Adjust with and weight Guza48 function Larson and Dally Delta function Kraus49 model at breaker height ” Remove and Goda45 (PEGBIS) adjust the remainder

Breaker variability

Waves spectrum

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Goda24,45 only. The energy dissipation is dealt directly with either the bore analogy or the model by Dally et al.50 or indirectly through modification of the pdf of wave heights. Figure 4.12 is an example of differences in the variations of the root-mean-square wave height in the nearshore waters predicted by the seven models on the slope of s = 0.02 (Goda46 ). Differences are very large beyond expectation and Goda46 shows other examples on the slope of s = 1/10, 1/30, and 1/100. Because the generation of longshore currents is governed by the spatial gradient of the radiation stresses, these models produce much different cross-shore profiles of longshore current velocities as demonstrated by Goda.46 Smith51 has also demonstrated different performances of five wave-breaking models with different breaker indices against field measurement of spectral waves. Because of the large difference in the prediction of nearshore wave heights, one should be careful in selecting a numerical model for engineering applications. Development of numerical models for time-domain wave transformation has been tried by many researchers, but random wave-breaking process has not been well reproduced in these models. One of the exceptions is the Boussinesq-type model developed by Hirayama et al.52 and Hirayama and Hiraishi,53 who employed the breaking criterion of the vertical pressure gradient by Nadaoka et al.7 They raised the threshold gradient from 0 to 0.5 to compensate the insufficiency in numerical accuracy due to the features of weak nonlinearity inherent to the Boussinesq equation. They have succeeded in reproducing the pdf variation across the surf zone.

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Depth, h (m) Fig. 4.12. Variations of wave heights Hrms in the surf zone on planar beaches computed by seven random wave-breaking models for (H1/3 )0 = 2.0 m, T = 8.004 s, and θ0 = 30◦ (Goda46 ).

Good reproduction of random wave-breaking process having the features discussed in the present chapter is a key factor for future development of reliable and practical numerical models for engineering applications.

4.9. Summary Wave-breaking phenomenon in the nearshore waters is characterized by the following features: (1) The breaker index formula by Goda13 for regular waves is revised by reducing the bed slope effect for better agreement with laboratory data. (2) Depth-limited breaker heights have inherent variability with the coefficient of variation of about 6% for the slope of 1/100 and 14% for the slope of 1/10, which increases as the bed slope becomes steep. (3) Breaker index or the ratio of breaker height to water depth is governed by both the relative water depth hb /L0 and the bed slope. (4) Incipient breaker index for the significant height of random waves is smaller than that for regular waves by about 30%, and the significant wave height has an upper limit of about 0.6 times the water depth on gentle slopes. (5) Wave nonlinearity expressed in terms of skewness, kurtosis, and the wave ratios H1/10 /Hm0 , H1/3 /Hm0 , and Hrms /Hm0 increases with wave propagation toward the shore and is most enhanced just outside the surf zone. For swell of very low steepness, the skewness, kurtosis, and wave height ratio H1/3 /Hm0 may go up to 2.0, 9.0, and 1.6, respectively, at the wave nonlinearity parameter around 4.0.

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(6) For wind waves and swell of relatively large steepness, random wave breaking begins at large depths compared with swell of small steepness. Thus, the former waves do not exhibit conspicuous wave nonlinearity. (7) Wave nonlinearity is weakened by wave breaking inside the surf zone and is eventually lost near the shoreline with return to linear wave features. (8) Probability density function (pdf ) of wave heights gradually deviates from the Rayleigh as waves propagate toward the shore, with the largest deviation taking place at the middle of the surf zone. Near the shoreline, the wave height distribution approaches the Rayleigh again. (9) Differences between characteristic wave heights such as H1/10 , H1/3 , and Hrms decrease in the middle of the surf zone, but they return to those by the Rayleigh near the shoreline. The PEGBIS model can simulate such changes of characteristic wave heights. (10) Various existing models of random wave breaking produce quite different prediction of wave heights in the surf zone. Careful selection of a random wave-breaking model should be made so that the model will yield reliable information for engineering applications.

Acknowledgment The author much appreciates kind cooperation by Dr Steven Hughes and Dr Shintaro Hotta for providing him with their valuable wave data.

References 1. Y. Goda, How much do we know about wave breaking in the nearshore waters, Proc. 4th Int. Conf. Asian and Pacific Coast (APAC 2007), Nanjin, China, CD-RM (2007). 2. H. Yamada and T. Shiotani, On the highest water waves of permanent type, Bull. Disaster Prevention Res. Inst., Kyoto Univ. 18–2(135), 1–22 (1968). 3. Y. Goda, A synthesis of breaker indices, Trans. Japan Soc. Civil Eng. 2(2), 39–49 (1970). 4. H. Yamada, G. Kimura and J. Okabe, Precise determination of the solitary waves of extreme height on water of a uniform depth, Rep. Res. Inst. Appl. Mech., Kyushu Univ. XVI(52), 15–32 (1968). 5. J. McCowan, On the highest waves in water, Phil. Mag. Ser. 5, 36, 351–358 (1894). 6. R. Miche, Mouvements ondulatoires de lamer en profondeur ou d´ecroissante, Annales de Ponts et Chauss´ees 19, 370–406 (1944). 7. K. Nadaoka, O. Ono and H. Kurihara, Near-crest pressure gradient of irregular water waves approaching to break, Proc. Coastal Dynamics ’97 (1997), pp. 255–264. 8. G. M. Kaminsky and N. C. Kraus, Evaluation of depth-limited wave breaking criteria, Second Int. Symp. Ocean Waves Measurement and Analysis, WAVES 1995 (1995), pp. 180–193. 9. J. W. Kamphuis, Incipient wave breaking, Coastal Eng. 15, 185–203 (1991). 10. W. Rattanapitikon and T. Shibayama, Verification and modification of breaker height formulas, Coastal Eng. J. 42(4), 389–406 (2000).

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11. W. Rattanapitikon, T. Vivattanasirisak and T. Shibayama, A proposal of new breaker height formula, Coastal Eng. J. 45(1), 29–48 (2003). 12. Y. Goda, A new method of wave pressure calculation for the design of composite breakwaters, Rept. Port Harbour Res. Inst. 14(3), 59–106 (1973) (in Japanese). 13. Y. Goda, New wave pressure formulae for composite breakwaters, Proc. 14th Int. Conf. Coastal Eng., Copenhagen, ASCE (1974) pp. 1702–1720. 14. M. Muttray and H. Oumeraci, Wave transformation on the foreshore of coastal structure, Coastal Eng. 2000 (Proc. ICCE) (2000) pp. 2178–1191. 15. Y. C. Li, G. H. Dong and B. Teng, Wave breaker indices in finite water depth, China Ocean Eng. 5(1), 51–64 (1991). 16. Y. C. Li, Y. Yu, L. F. Cui and G. H. Dong, Experimental study of wave breaking on gentle slope, China Ocean Eng. 14(1), 59–67 (2000). 17. J. L. Lara, I. J. Losada and P. L.-F. Liu, Breaking waves over a mild gravel slope: Experimental and numerical analysis, J. Geophys. Res. 111(C11019), 1–26 (2006). 18. J. W. Smith and N. C. Kraus, Laboratory study of wave-breaking over bars and artificial reefs, J. Waterway, Port, Coastal, Ocean Eng. ASCE, 117(4), 307–325 (1991). 19. A. Kimura and A. Seyama, Breaking limit of irregular waves on slopes, Proc. 33rd Japanese Coastal Eng. Conf. (1986), pp. 174–178 (in Japanese). 20. A. Seyama and A. Kimura, The measured properties of irregular wave breaking and wave height change after breaking on the slope, Proc. 21st Int. Conf. Coastal Eng. (1988), pp. 419–497. 21. K. P. Black and M. A. Rosenberg, Semi-empirical treatment of wave transformation outside and inside the breaker line, Coastal Eng. 16, 313–345 (1992). 22. D. Kriebel, Breaking waves in intermediate-depths with and without current, Coastal Eng. 2000 (Proc. ICCE), (2000) pp. 203–215. 23. Y. C. Li, Y. Yu, L. F. Cui and G. H. Dong, Transformation and breaking of irregular waves on very gentle slope, China Ocean Eng. 14(3), 261–278 (2000). 24. Y. Goda, Irregular wave deformation in the surf zone, Coastal Eng. Jpn, JSCE 18, 13–26 (1975). 25. Y. Goda, Random Seas and Design of Maritime Structures, 2nd Edn. (World Scientific, Singapore, 2000). 26. F. C. K. Ting, Laboratory study of wave and turbulence velocities in a broad-banded irregular wave surf zone, Coastal Eng. 43, 183–208 (2001). 27. F. C. K. Ting, Laboratory study of wave and turbulence characteristics in a narrowbanded irregular breaking waves, Coastal Eng. 46, 291–313 (2002). 28. A. H. Sallenger and R. A. Holman, Wave energy saturation on a natural beach of variable slope, J. Geophys. Res. 90(C6), 11939–11944 (1985). 29. Y. Goda and K. Nagai, Investigation of the statistical properties of sea waves with field and simulation data, Rept. Port Harbour Res. Inst. 13(1), 3–37 (1974) (in Japanese). 30. Y. Goda, Analysis of wave grouping and spectra of long-travelled swell, Rept. Port Harbour Res. Inst. 22(1), 3–41 (1983). 31. S. Hotta and M. Mizuguchi, A field study of waves in the surf zone, Coastal Eng. Jpn, JSCE 23, 59–79 (1980). 32. S. Hotta and M. Mizuguchi, Statistical properties of field waves in the surf zone, Proc. 33rd Jpn Coastal Eng. Conf. (1986) pp. 154–157 (in Japanese). 33. B. A. Ebersole and S. A. Hughes, DUCK85 photopole experiment, US Army Corps of Engrs., WES, Misc. Paper, CERC-87-18, (1987) pp. 1–165. 34. Y. Goda, A unified nonlinearity parameter of water waves, Rept. Port Harbour Res. Inst. 22(3), 3–30 (1983).

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35. Y. Goda and M. Kudaka, On the role of spectral width and shape parameters in control of individual wave height distribution, Coastal Eng. J. 49(3), 311–335 (2007). 36. E. F. Thompson and C. L. Vincent, Significant wave height for shallow water design, J. Waterway, Port, Coastal, Ocean Eng. ASCE 111(5), 828–841 (1985). 37. M. S. Longuet-Higgins, On the distribution of the heights of sea waves: Some effects of nonlinearity and finite band effects, J. Geophys. Res. 85(C3), 1519–1523 (1980). 38. J. I. Collins, Probabilities of breaking wave characteristics, Proc. 12th Int. Coastal Eng. Conf., Washington, DC, ASCE (1970) pp. 399–414. 39. J. A. Battjes, Setup due to irregular wave, Proc. 13th Int. Conf. Coastal Eng., Vancouver, ASCE (1972) pp. 1993–2004. 40. C. T. Kuo and S. T. Kuo, Effect of wave breaking on statistical distribution of wave heights, Proc. Civil Eng. Ocean, ASCE (1974) pp. 1211–1231. 41. T. E. Baldock, P. Holmes, S. Bunker and P. van Weert, Cross-shore hydrodynamics with an unsaturated surf zone, Coastal Eng. 34(3–4), 173–196 (1998). 42. F. C. K. Ting, Laboratory study of wave and turbulence velocities in a broad-banded irregular wave surf zone, Coastal Eng. 43, 183–208 (2001). 43. F. C. K. Ting, Laboratory study of wave and turbulence characteristics in a narrowbanded irregular breaking waves, Coastal Eng. 46, 291–313 (2002). 44. J. A. Battjes and H. W. Groenendijk, Wave height distributions on shallow foreshores, Coastal Eng. 40, 161–182 (2000). 45. Y. Goda, A 2-D random wave transformation model with gradational breaker index, Coastal Eng. J. 46(1), 1–38 (2004). 46. Y. Goda, Examination of the influence of several factors on longshore current computation with random waves, Coastal Eng. 53, 156–170 (2006). 47. J. A. Battjes and J. P. F. M. Janssen, Energy loss and set-up due to breaking of random waves, Proc. 16th Int. Conf. Coastal Eng., Hamburg, ASCE (1978), pp. 1–19. 48. E. B. Thornton and R. T. Guza, Transformation of wave height distribution, J. Geophys. Res. 88(C10), 5925–5938 (1983). 49. M. Larson and N. C. Kraus, Numerical model of longshore current for bar and trough beaches, J. Waterway, Port, Coastal, Ocean Eng., ASCE 117(4), 326–347 (1991). 50. W. R. Dally, R. G. Dean and R. A. Darlymple, Wave height variation across beaches of arbitrary profile, J. Geophys. Res. 90(C6), 11917–11927 (1985). 51. J. M. Smith, Breaking in a spectral wave model, Ocean Wave Measurement and Analysis (Proc. WAVES 2001), ASCE (2001) pp. 1022–1031. 52. K. Hirayama, N. Hara and T. Hiraishi, Application of bore model to non-linear wave transformation, Proc. 13th Int. Offshore Polar Eng. Conf. (2003), pp. 796–801. 53. K. Hirayama and T. Hiraishi, A Boussinesq model for wave breaking and run-up in a coastal zone, 1D, Fifth Int. Symp. Ocean Waves Measurement Analysis, WAVES 2005, CD-ROM, No. 151 (2005), pp. 1–10.

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Chapter 5

Aeration and Bubbles in the Surf Zone Nobuhito Mori Disaster Prevention Research Institute Kyoto University, Gokasho, Uji, Kyoto 611-0011, Japan [email protected] Shohachi Kakuno Osaka City University 3-3-138 Sugimoto, Sumiyoshi-ku, Osaka 558-8585, Japan [email protected] Daniel T. Cox Oregon State University, Corvallis, OR 97331-4501, USA [email protected] This chapter presents a brief summary of aeration in the surf zone, beginning with a review of air–water characteristics in surf zone waves. Second, measurements techniques of the bulk of air and bubbles induced by breaking waves in the surf zone are described, and third, the bulk of air and bubble characteristics are summarized based on the in situ and visualization laboratory measurements. Finally, the gas transfer in the surf zone is described and related to the wave characteristics.

5.1. Introduction Ocean surface gravity wave propagation from offshore to shoreline is often regarded as a single phase flow using potential flow theories or solving the Navier–Stokes equation. Generally, the single phase flow approach to ocean waves is successful for simulating wave transformation in the coastal area. However, this assumption breaks down because waves become steep and start to break owing to the bottom bathymetric effects in the near shore. The wave-breaking process creates dense bubble plumes, dissipates energy and generates turbulence in the surf zone. Figure 5.1 shows an example of the crest of wave breaking on a gentle slope. Most of the waves 115

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Fig. 5.1.

Image of breaking wave crest at an initiation of breaking in the surf zone.

can be regarded as a single (liquid) phase flow, and only the near-crest region is aerated by the jet attachment to the water surface. An accurate estimation of the bulk of the void fraction, bubble size, and population distributions in the surf zone are important for understanding the two-phase flow characteristics, solving engineering problems, and elucidating environmental mechanisms of the coastal area.1 For example, Peregrine2 summarized that the entrapped air of breaking wave reduces the wave impact pressure owing to its greater compressibility compared with the single-phase approach. There is also a significant difference between salt water and fresh water experiments on the wave impact pressure (bubbles generated by turbulence in saltwater are smaller than fresh water), and the entrapped air reduced the pressure by approximately 10%.3 The compressibility owing to air– water mixture decreases the velocity of sound and is being used to estimate largescale prototype impacts, since the usual Froude scaling is unlikely to be correct for engineering problems. Therefore, the connection between air-mixture, bubble distribution, and wave-breaking-induced turbulence is essential to understand gas– liquid interactions in the surf zone. The air bubble plume is always present in the surf zone breaking waves. Experimental data show that the entrained air is proportional to the energy dissipation.4 For these breaking waves, entrained air enhances the gas-exchange at the water surface. For example, the dissolved oxygen is associated with the breaking waves in the coastal waters,5,6 although the carbon dioxide gas transfer can be regarded as a function of wind speed. Understanding air entrainment and bubble distribution for the depth-limited breaking of surf zone waves is limited in comparison with the wind-wave breaking. The air bubble distributions under the wind-wave breaking in the upper layer of the ocean surface has been summarized in detail by Thorpe.7 The wind-wave bubble formation and its dependence on the turbulent dissipation rate of fluids have been discussed,8 and it is proposed a −2 power-law scaling with bubble diameter. The −10/3 power-law scaling is alternatively proposed based on the discussion of bubble fragmentation due to strong turbulent shear flow. These bubble size scaling laws are still under development.9 However, the knowledge of wind-wave breaking induced

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bubbles and associated gas transfer has become well-known issue over the last decade.10 Despite the fruitful knowledge of air entrainment of wind-wave breaking, few studies of air entrainment for surf zone wave breaking exist. It has been found that there is little difference in the bubble populations beneath the mechanically generated surface waves in saltwater and fresh water.11 On the other hand, it has been reported that the void fraction depends on the turbulent intensity in the bore region of surf zone waves.12 The large-scale bubbles entrained by breaking waves in the surf zone are split by strong local turbulent shear at the bubble scale.13 However, neither qualitative nor quantitative bubble characteristics in the surf zone and connections between bubble characteristics and wave breaking are well known owing to the lack of detailed observations. This detailed information of two-phase flow characteristics is required for mathematical modeling to solve both basic scientific interests and applied engineering problems.

5.2. Measurements Techniques There are several measurement methods for bubble population and size distributions, i.e., in situ measurements,14,15 video or photographic measurements,16 laser measurements,17 and acoustic measurements.18 The bubble size measurements using lasers probe such as a phase Doppler anemometer show high accuracy in small-scale bubble measurements smaller than 500 µm but is of limited use in the presence of high void fractions and large-scale bubble measurements due to fundamental instrument limitations. Acoustic measurements of bubbles are useful in deepwater but have limitation for very shallow water region owing to the multi-reflection of sound beams. Therefore, conventional optical or resistivity type void probes are useful for the surf zone breaking waves. The electrical conductivity probe (so-called void probe) has been generally used for the high void fractions.19 This type of void probe is robust and easy to handle for both wave flume experiments and field observations. Therefore, the void probe measurements have been widely used for the twophase flow measurement in the surf zone.12,14,20 However, the in situ measurement has clear limitation for inhomogeneous surf zone breaking phenomena. The photographic or imaging technique has been advanced to in situ measurements which can cover the spatial and temporal variations of the two-phase flow characteristics in the surf zone. Recent photographic studies have illustrated the disintegration of entrapped air cavities divided into bubbles.13,21 Several imaging techniques have been proposed to measure the bubble characteristics in the surf zone. One is an application of Particle Image Velocimetry (PIV) method with forward and backward lighting.22,23 This method basically correlates the scattered light intensity and bubble plume intensity, and it can be used for high void fraction region over 20%, although it cannot measure the bubble shape itself. The other imaging technique is a bubble shape tracking method with backlighting.24 This method is a so-called shadow graph method because the bubble shape is enhanced by a backlight source toward a camera. The shadow graph method can measure both bubble shape and velocity simultaneously. However, the density of the bubbles has to be within a certain limit to separate individual bubbles from the image by the shadow graph method.

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In summary, there is no ideal method that can be used for universal purpose of bubble measurements and one can choose appropriate method(s) for a particular purpose.

5.3. Bulk Aeration Characteristics in the Surf Zone Figure 5.2 shows successive images of breaking wave taken by high-speed video camera in the surf zone. Figure 5.2(a) is just after attachment of jet on the water surface which corresponds to the initial stage of bubble generation. Figures 5.2(b) and 5.2(c) are 42 ms and 108 ms after image (a), respectively. The lifetime of the large volume of entrained air can be categorized into two phases. The first phase is the initial stage of breaking waves [Figs. 5.2(a) and 5.2(b)]. The newly injected air column produces pluses of sound and, therefore, this phase is regarded as the acoustically active phase.21 In the active phase, the trapped volume of air between the overturning jet and water surface has fragmented and injected into the deeper region, and the air pocket is fragmented into small size bubbles. This phase occurs within 1 s of the initial stage of wave breaking. The second phase is the acoustically silent phase. Most of the bubbles are advected by the surface roller of the carrier waves with buoyancy, and some of them are trapped by oblique descending eddy in this phase [Figs. 5.2(c)]. In the active phase, the void fraction ratio is relatively high and there is no clear formation of bubbles in that region. Therefore, the only possible approach to this phase is to account for the total amount of air, which is the summation of the air pocket and bubbles. The silent phase has a lower void fraction and is more dispersive, and it is possible to discriminate individual bubbles. These injected air penetrate into a deeper region during the active phase. After that, the volume of air and/or bubbles are advected by the wave motion due to their buoyancy. Therefore, the gas phase characteristics in the surf zone, such as the void fraction, are time–space variables and are inhomogeneous in both time and space. The next part of this chapter discusses the total amount of air, bubble size distributions, and gas-transfer in the surf zone, respectively. 5.3.1. Temporal characteristics The aeration process by breaking wave in the surf zone is obviously time-dependent. Therefore, the temporal transition is important in understanding the phenomena in the beginning. Figure 5.3 shows an example of phase-averaged time series of surface elevation and turbulence components of the mechanically generated periodic wave breaking on a slope.20 The notations z and Hb denote the vertical axis positive upward from the mean water level and wave height at the breaking point, respectively. The initial wave height and period are 16.3 cm and 1.6 s, respectively. Using the phase-averaged method, the velocity data were divided into the mean, periodic, 2 and turbulence components. The notation of σu2 , σv2 , and σw are the turbulent stresses for the velocity components (u, v, w). The peak of the turbulent stress is located at the front of the crest and decreases linearly as time passes. This is a typical time series of turbulence characteristics of depth-limited breaking waves.

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(a)

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Fig. 5.2. Formation of bubble plume under the breaking wave in the surf zone. (a) Attachment of jet on the surface (initiation of bubble plume generation); (b) Active phase of bubble plume generation [42 ms after (a)]; (c) Bubble plume in silent phase [108 ms after (a)].

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η and turbulence components 1.0 m from B.P. in Case 1. (a) Free surface η (solid) and standard deviation η ± ση ; (b)–(d) temporal variation of σu (thick solid), σv (thin solid), and σw (dotted) and (e) the phase-averaged void fraction
α.20

The highest void fraction occurs at the wave front, and the void fraction is decreased rapidly in time. The temporal variation of the void fraction depends on the air supply from the surface, advection, and buoyancy, and these effects are difficult to discriminate from each other. Although the process is complex, the normalized temporal characteristics show a universal behavior. Similar laboratory experiments were observed by several researchers.12,25,26 Furthermore, Cox and Shin12 found the self-similar exponential decay of the temporal series of void fraction normalized by the averaged value in different types of breaking waves. In addition, they reported that the maximum void fraction in the time series was three to four times higher than its average value. 5.3.2. Vertical distribution of void fraction The vertical distribution of void fraction of the surf zone breaking waves was first systematically measured by Hoque and Aoki.27 They showed a simple description of vertical void fraction distribution as follows. If a void fraction α simply follows a

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Vertical distribution of void fraction for plunging breaker.27

diffusion equation in steady state, we have ∂ ∂(αu) = ∂z ∂z


 ∂α D , ∂z

(5.1)

where z is the vertical axis, u is the water velocity, and D is the turbulent diffusivity of air bubble plume. Assuming a uniform velocity distribution and D independent in the vertical direction, we have ∂ 2α u0 ∂α , = 2 ∂z D ∂z

(5.2)

where u0 denotes the uniform velocity. The solution of the above equation is α(z) = α0 exp(kz),

(5.3)

where k = u0 /D is the newly introduced decay parameter of the vertical void fraction distribution and α0 denotes the void fraction at the mean water level (z = 0). Figure 5.4 shows the comparison between experimental data and Eq. (5.3) for plunging breakers on a gentle slope. The experimental data shows an exponential decay of the void fraction in the vertical direction. Furthermore, introducing k0 = kh, where h is the local depth, k0 can be regarded as a constant value ranging approximately from 3.75 to 4.0, independent of breaker type. Similar results were measured by a visualization technique23 after that. In addition, the linear decay of the maximum water depth of penetrated bubble shoreward is also observed as shown in Fig. 5.4.27 5.3.3. Horizontal distribution of void fraction The characteristics of the volume of air changes both horizontally, vertically and temporally. The void fraction and total kinetic energy show similar distributions, and Cox and Shin12 also reported the dependence of void fraction on turbulent intensity in the bore region of surf zone waves. The entrapped air induced by wave breaking generates strong turbulence near the free surface. Therefore, the relationship between void fraction and turbulence should be connected to each other.

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×10

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xb/Hb Fig. 5.5. Horizontal relationship between time-averaged void fraction α (solid line) and timeaveraged TKE k (dotted line).20

Figure 5.5 shows experimental results of void fraction and total kinetic energy (TKE) variations from the breaking point xb to the shore line.20 The spatial variations of the time-averaged void fraction α and time-averaged TKE, k show similar tendency, although the peak of α is located approximately 0.2 × Hb ∼ 2 × Hb seaward of the peak of k, as shown in Fig. 5.5. The short spatial lag between α and k might be associated with the spatial difference of entrapped air and associated turbulence. The spatial variations of the turbulence and void fraction are highly correlated near the mean water level. The entrained bulk of air and bubbles generate turbulence, and turbulence splits the coarse bubbles into fine bubbles in this area. Moreover, the TKE and void fraction increased by an order of magnitude from the trough level (z/Hb = −0.013) to above the mean water level (z/Hb = 0.033); thus, the two-phase flow characteristics are remarkably strong near the wave crest. Neglecting the short spatial lag between the void fraction and TKE, the direct comparison between the time-averaged void fraction α and normalized timeaveraged TKE k of all the measuring locations is shown in Fig. 5.6. For small values of α or k, the relationship between the two shows a high correlation and becomes scattered at high void fraction or TKE region. Overall, the relationship between time-averaged void fraction and time-averaged TKE shows high correlation12 as does the incident wave-dependence on the relationship between α and k.

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0.14

0.12

0.1

α

0.08

0.06

0.04

Case1 Case2 Case3

0.02

0

0

0.1

0.2 ( k/gh )

0.3

0.4

1/2

Fig. 5.6. Relationship between time-averaged TKE k and time-averaged void fraction α. (•: Case 1, +: Case 2,
: Case 3).

The relation between the void fraction and TKE, the scale-dependence of void fraction, and further theoretical considerations will be required to find the universal relationship between the turbulence characteristics of depth-limited breaking waves and air entrainment.

5.4. Bubble Characteristics in the Surf Zone Bubble generation is a small-scale process of aeration in the surf zone.28 There is a large body of literature dealing with the distribution of bubbles in wind-wave breaking.7,8,29,30 From the knowledge of wind-wave breaking, there are two modes of bubble generation.31 One is entrapment of air under the falling water jet by the wave overturning and another is air entrainment at the point where the falling water jet meets the vortex formed at the rear of the splash-up. Figure 5.7 shows the time series of bubble diameter distribution just beneath the mean water level near the breaking point under the same wave condition as in Fig. 5.5.24 The bubble diameter here means the averaged value of the major and minor diameter. The horizontal axis is the characteristic bubble diameter, the vertical axis is the temporal change, and the contour indicates number of bubbles per unit area. The plunging wave breaking not only entrains the bulk of air but also generates a large number of bubbles in comparison with the spilling breaker. As discussed earlier, the void fraction decreases exponentially in time. The time series

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Time series of bubble size distribution.24 (a) Spilling breaker; (b) Plunging breaker.

of the bubble number density distribution (Fig. 5.7) shows a similar tendency, but the small-scale bubbles survive longer than large-scale bubbles. Generally, entrapped air by breaking waves is split into small bubbles owing to turbulence shear of surrounding flows. The early work of this problem for a general condition is discussed by Kolmogorov32 and Hinze33 in the middle of the last century, and a large bibliography has been generated.34 Longuet-Higgins35 proposed a crushing air cavity model for the bubble distributions after the formation of a bubble plume and shows good agreement of bubble aeration in general condition, qualitatively. Figure 5.8 shows the instantaneous and time-averaged bubble size spectra with the estimated power law measured by an imaging technique.24 The instantaneous bubble size spectra corresponds to the time of the maximum void fraction observed. There is no significant difference of bubble size distribution

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Power law of bubble size distribution.

between instantaneous and time-averaged spectra at small scale (diameter < 1 mm). On the other hand, two scaling laws can be seen in the bubble size distribution. The small and large size bubble power scaling laws are d−1 and d−3–−4 , respectively. Garrett et al.9 proposed semi-empirical −10/3 power-law scaling for the bubble which is larger than the Hinze scale, based on the discussion of bubble fragmentation in the strong turbulence flow below the trough level. On the other hand, Deane and Stokes21 empirically proposed a −3/2 power-law scaling, smaller than the Hinze scale in the acoustically active phase near the crest. The node point between the two power scaling corresponds to the Hinze scale of the bubble splitting theory.33 The measured power-law scalings of the small size bubble shown in Fig. 5.8 are close to −3/2 and −10/3 power laws, respectively. These power-law scalings can be seen in different conditions in mechanically generated waves in the flume.13,20,21 The Hinze scale dH 21 of bubble splitting can be given by
 ρw ε−2/5 , dH = 2−8/5 c (5.4) γ where ρw is the water density, γ is the surface tension coefficient, ε is turbulence energy dissipation at the bubble diameter, and c = (W ec )3/5 , where W ec is the critical Weber number. The Weber number W e is defined as ρw 2 δu d, We = (5.5) γ where δu is turbulence velocity at the bubble scale d. The turbulence velocity δu at the bubble scale in the inertial range can be given by δu = 2ε2/3 d2/3 . If the energy dissipation rate ε can be estimated by wave energy dissipation, the Hinze scale can be obtained. For example, if ε = 12 W/kg, the Hinze scale is 2 mm in diameter. These theoretical predictions agree fairly well with the experimental data,20,21 although the Hinze scale is formulated under the assumption of isotropic and homogeneous turbulence flow.

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5.5. Gas Exchange in the Surf Zone For moderately solvable gases such as carbon dioxide and oxygen, the gas transfer across the air–sea interface is controlled by the aqueous condition, and is parameterized as a function of a gas transfer velocity kL kL = αSc−1/2 u∗ ,

(5.6)

where Sc = νκ is the Schmidt number, α is the empirical coefficient, and u∗ is the friction velocity, respectively.36 The role of bubbles in the exchange of gas in the surf zone is poorly understood in comparison to the wind-wave breaking. An analogy of wind-wave breaking to surf zone breaking waves is similar, but surface renewal process and the bubble-mediated gas transfer are different from each other. For example, the surface renewal by the wind stress is important for the windwave scenario, but the turbulence generated by the wave overturning is dominant in the surf zone. Moreover, the bubble-mediated gas transfer process depends on the bubble population and size distribution in the surf zone. For the wind-wave case, the large-scale bubbles with a diameter greater than approximately 1 mm play an important role in air–sea gas exchange.37,38 However, there are a few confident prediction models that exist for macro or microscopic bubble-mediated gas transfer for surf zone breaking waves. The oxygen gas transfer is important for fish and marine mammals near the coastal region. Therefore, oxygen gas transfer, rather than carbon dioxide, plays a vital role in aquatic environment in the surf zone.39 There are significant difference between the wind-waves and surf zone waves generally. For example, the spatial homogeneity can be assumed in wind-waves, but there is a strong spatial dependence for surf zone waves. Therefore, a different approach to predict the gas transfer is required. One may assume that the gas transfer is related to the total water mass volume of surf zone (Fig. 5.9). The general expression of the gas transfer in an aeration tank of constant volume can be formulated: ∂C A (5.7) = kL ∆C = k2 ∆C, ∂t V where C is the gas concentration, ∆C is the concentration difference between liquid and gas phase, A is the aeration (surface) area, kL is the gas transfer velocity,

Fig. 5.9.

Illustration of gas transfer in the surf zone.

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and k2 is the reaeration coefficient. Eckenfelder40 proposed the bubble-mediated gas transfer velocity kL for aeration tank as kL =

αh2/3 Qa , A dm Sc1/2

(5.8)

where α is the empirical coefficient, h is the water depth of aeration tank, Qa is the total amount of air flow rate, dm is mean bubble diameter. Multiplying the surf zone area A by Eq. (5.8), it can be formulated for aeration by surf zone breaking waves as41 2/3

kL A =

αHb Vg 1/2

dm Sc T

,

(5.9)

where Hb is the wave height at breaking point, Vg is the total entrained air volume per wave period, and T is the wave period (Fig. 5.10). In this formulation, the gas transfer in the surf zone linearly depends on the total entrained air volume. Now, we introduce the gas transfer flow rate kL A instead of the gas transfer velocity kL in Eq. (5.9). This is because the aeration area A is difficult to be estimated in the surf zone, but the total water volume V is easily obtained. Assuming a uniform bottom slope, Kakuno41 empirically obtained a relationship between Vg and surf zone parameter as Vg ∝ Hb hb / tan θ, where hb is the water depth at the breaking point, and θ is the bottom slope. Substituting this expression into Eq. (5.9), we have the gas transfer flow rate coefficient kL A for slope beach41 : 5/3

kL A =

α2 Hb hb 1/2

T dm Sc

tan θ

.

(5.10)

Kakuno compared Eq. (5.10) with the fresh water experimental data and found that the empirical coefficient α2 is approximately 2.3 × 10−3 ∼ 2.8 × 10−3 as shown in Fig. 5.11.41 Equation (5.10) expresses the gas transfer flow rate as a function of breaking wave height Hb , water depth at breaking point hb , wave period T , mean bubble diameter dm , bottom slope θ, and Schmidt number, although salinity and scaling effects are neglected. Furthermore, Kakuno41 obtained an empirical relation between the gas transfer velocity and wave energy dissipation as kL ∝ ε1/3 . wave1

wave2

wave3

Vg XTi Xb Fig. 5.10. Total bubble volume distribution in the surf zone (Xb : length of surf zone, Vg : bubble volume per wave period).

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Fig. 5.11. Gas transfer flow rate and bottom bathymetry in the surf zone (•: 1/10 slope, ◦: 1/20 slope, : 1/30 slope).

Research to estimate the gas transfer velocity kL from the wave characteristics through the wave energy dissipation is ongoing.

5.6. Conclusions and Future Study In this chapter, we summarize our general review of gas–water two-phase flow characteristics in the surf zone. There are several measurement methods for the aeration in the surf zone, such as in situ measurements, video or photographic measurements, laser measurements, and acoustic measurements. There is no one appropriate method to measure the volume of air and bubbles in the surf zone. The spatial and temporal characteristics of void fraction are summarized, and the bubble size distribution is described with simple theories. Furthermore, the gas transfer prediction method is presented as a function of the bottom slope and wave characteristics for the surf zone waves. There are many other problems such as salinity effect, scale effects, and more mathematical modeling should be investigated in the future.

References 1. M. Donelan, W. Drennan, E. Saltzman and R. Wanninkhof, Gas Transfer at Water Surfaces (AGU, Washington, DC, 2002). 2. D. Peregrine, Water wave impact on walls, Ann. Rev. Fluid Mechanics 35, 23–43 (2003). 3. B. G. N. A. Crawford, P. Hewson and P. Bird, Experiments on full-scale wave impact pressures, Coastal Eng. 8, 331–346 (1999). 4. S. Longo, M. Petti and I. J. Losada, Turbulence in the swash and surf zones: A review, Coastal Eng. 45, 129–147 (2002).

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5. D. Farmer, C. McNeil and B. Johnson, Evidence for the importance of bubbles in increasing air-sea gas flux, Nature 361, 620–623 (1993). 6. D. Wallace and C. Wirick, Large air-sea fluxes associated with breaking waves, Nature 356, 145–147 (1992). 7. S. Thorpe, On the clouds of bubbles formed by breaking wind waves in deep water, and their role in air-sea gas transfer, Phil. Trans. R. Soc. London A 304, 155–210 (1982). 8. S. Baldy, A generation-dispersion model of ambient and transient bubbles in the close vicinity of breaking waves, J. Geophys. Res. C10, 18277–18293 (1993). 9. C. Garrett, M. Li and D. Farmer, The connection between bubble size spectra and energy dissipation rates in the upper ocean, J. Phys. Oceanogr. 30, 2163–2171 (2000). 10. B. J¨ ahne and H. Haußecker, Air-water gas exchange, Ann. Rev. Fluid Mechanics 30, 443–468 (1998). 11. M. Loewen, M. O’Dor and M. Skafel, Bubbles entrained by mechanically generated breaking waves, J. Geophys. Res. 101, 20759–20769 (1996). 12. D. Cox and S. Shin, Laboratory measurements of void fraction and turbulence in the bore region of surf zone waves, J. Eng. Mechanics, ASCE 129(10), 1197–1205 (2003). 13. G. B. Deane and M. D. Stokes, Air entrainment processes and bubble size distributions in the surf zone, J. Phys. Oceanogr. 29, 1393–1403 (1999). 14. H. Chanson, S. Aoki and M. Maruyama, Unsteady air bubble entrainment and detrainment at plunging breaker: Dominant time scales and similarity of water level variation, Coastal Eng. 46, 139–157 (2002). 15. H. Chanson, Comments on fiber optic refectometer for velocity and fraction ratio measurements in multiphase flows, Rev. Sci. Instrum. 75(1), 284–285 (2003). 16. I. Leifer and G. de Leeuw, Bubble measurements in breaking-wave generated bubble plumes during the LUMINY wind-wave experiment, in Gas Transfer at Water Surfaces, eds. M. Donelan, W. Drennan, E. Saltzman and R. Wanninkhof (AGU, Washington, DC, 2002), pp. 303–310. 17. N. Mori, Experimental study of air bubble distributions induced by wind wave breaking, in Proc. Asia Pacific Coast 2003, JSCE, Makuhari (2003), CD–ROM. 18. G. B. Deane, Sound generation and air entrainment by breaking wave in the surf zone, J. Acoustic Soc. Am. 102, 2671–2689 (1997). 19. S. Vagle and D. Farmer, A comparison of four methods of bubble measurement, IEEE Oceanic Eng. 23(3), 211–222 (1998). 20. N. Mori, T. Suzuki and S. Kakuno, Experimental study of air bubbles and turbulence characteristics in the surf zone, J. Geophys. Res. 112(C05014) (2007), doi: 10.1029/2006JC003647. 21. G. B. Deane and M. D. Stokes, Scale dependence of bubble creation mechanisms in breaking waves, Nature 418, 839–844 (2002). 22. Y. Ryu, K.-A. Chang and H. J. Lim, User of bubble image velocimetry for measurement of plunging wave impinging on structure and associated greenwater, Meas. Sci. Technol. 16, 1945–1953 (2002). 23. O. Kimmoun and H. Branger, A particle image velocimetry investigation on laboratory surf-zone breaking waves over a sloping beach, J. Fluid Mech. 588, 353–397 (2007), doi: 10.1017/S0022112007007641. 24. N. Mori and S. Kakuno, Aeration and bubble measurements of the coastal breaking waves, Fluid Dynamics Res. 40(7–8), 616–626 (2008). 25. F. Ting and J. Kirby, Observation of undertow and turbulence in a laboratory surf zone, Coastal Eng. 24, 51–80 (1994).

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26. D. Cox, N. Kobayashi and A. Okayasu, Vertical variations of fluid velocities and shear stress in surf zones, Proc. 24th Int. Conf. on Coastal Eng., ASCE (1994), pp. 98–112. 27. A. Hoque and S. Aoki, Distributions of void fraction under breaking waves in the surf zone, Ocean Eng. 32, 1829–1840 (2005). 28. M. Loewen, Inside whitecaps, Nature 418, 830 (2002). 29. R. Cipriano and D. Blanchard, Bubble and aerosol spectra produced by a laboratory breaking waves, J. Geophys. Res. C86, 8085–8092 (1981). 30. C. Garrett, M. Li and D. Farmer, The connection between bubble size and energy dissipation rates in the upper ocean, J. Phys. Oceanogr. 30, 2163–2171 (2000). 31. P. Bonmarin, Geometric properties of deep-water breaking waves, J. Fluid Mech. 209, 405–433 (1989). 32. A. Kolmogorov, On the breakage of drops in a turbulent flow, Dokl. Akad. Navk. SSSR 66, 825–828 (1949). 33. J. Hinze, Fundamentals of the hydrodynamic mechanism of splitting in dispersion processes, Am. Inst. Chem. Eng. J. 1, 289–295 (1955). 34. C. Mart´ınez-Baz´ an, J. Montan´es and J. Lasheras, On the breakup of air bubble injected into a fully developed turbulent flow. Part I: Breakup frequency, J. Fluid Mech. 401, 157–182 (1999). 35. M. Longuet-Higgins, The crushing of air cavities in a liquid, Proc. Roy. Soc. London A 439, 611–626 (1992). 36. W. Melville, The role of surface-wave breaking in air-sea interaction, Ann. Rev. Fluid Mechanics 28, 279–321 (1996). 37. L. Memery and L. Marlivat, Modeling of gas flux through bubbles at the air-water interface, Tellus 37B, 272–285 (1985). 38. R. Keeling, On the role of large bubbles in air-sea gas exchange and supersaturation in the ocean, J. Marine Res. 51, 237–271 (1993). 39. C. Moutzouris and E. Daniil, Water oxygenation in the vicinity of coastal structures due to wave breaking, Proc. 24th Int. Conf. Coastal Eng., ASCE (1994), pp. 3167–3177. 40. W. Eckenfelder Jr., Absorption of oxygen from air bubbles in water, J. Sanitary Eng. Div. Proc. ASCE, 89–99 (1959). 41. S. Kakuno, D. B. Moog, T. Tatekawa, K. Takemura and T. Yamagishi, The effect of bubble on air-water oxygen transfer in the breaker zone, in Gas Transfer at Water Surfaces, eds. M. Donelan, W. Drennan, E. Saltzman and R. Wanninkhof (AGU, Washington, DC, 2002), pp. 265–277.

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Chapter 6

Freak Wave Nobuhito Mori Disaster Prevention Research Institute Kyoto University, Gokasho, Uji, Kyoto 611-0011, Japan [email protected] In this chapter, we describe mechanisms that lead to the occurrence of freak waves in the ocean. It is now generally recognized that freak waves can be generated by any one of four possible mechanisms. The traditional mechanism is a simple linear superposition of waves, the theory of which is reviewed here. The newest mechanism attempts to include the third-order nonlinearities that depart from linear wave theory. Therefore, we present here a state-of-the-art review that is based on nonlinear wave dynamics.

6.1. Introduction The history contains many reports of encounters with monstrous waves.1 For example, the captain of the cargo vessel Junior reported a wave estimated to be 100 ft high and the famous reliable report was that of the wave encountered by the U.S.S. Ramapo in the North Pacific in 1993; that wave was estimated to be 112 ft high. There are many more reports of encounter of similar waves in the history of the seas. Over the last decade, more evidence has accumulated for such extreme waves. This may be because of an increase in the number of measurement locations together with the development of more accurate field measurements, for example, several extreme wave events, the so-called freak waves, recorded in the North Sea.2 A sample wave profile is shown in Fig. 6.1, which was extracted from the data provided by wave radar shooting downward from the platform to the sea surface in the North Sea. The data were collected over 12–15 hours at a depth of 40 m. Since the early 1990s such an event has come to be called a freak wave or a rogue wave. More evidence has been obtained by analyzing field data from the North Sea,3,4 the Sea of Japan,5,6 and the Gulf of Mexico.7 The first clear definition of a freak wave was proposed by Klinting and Sand2 ; in its original form, the definition required a wave to satisfy three conditions to be

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Fig. 6.1. Freak wave recorded on the 24th November, 1981 at the Gorm Field in the Danish Sector of the North Sea.8

considered “freak wave”: (1) It has a wave height higher than twice the significant wave height. (2) Its wave height is larger than two times of the foregoing and the following wave heights. (3) Its wave crest height is larger than 65% of of its wave height. Subsequently, these conditions were found to be overly stringent, and today the first condition is generally considered to be sufficient to identify a freak wave.8 Freak wave research has two basic objectives. The first is an understanding of the mechanisms that create freak waves; several mechanisms have been proposed to explain why extreme-wave events occur in the ocean. The second is to establish a reliable statistical model for the occurrence of freak waves. However, the problem is difficult because a freak wave is only one wave, or perhaps just a few waves, in a wave train. Therefore, its occurrence exhibits statistical sensitivity. Moreover, an extreme wave event is transient: it forms and disappears quickly in both time and space. To address these problems, we should study wave populations that contain freak waves, rather than concentrate on the features of an individual wave profile. (This is not to say, however, that observations of single events are unimportant.) Once such wave populations can be characterized, then it should be possible to estimate how often waves of any given size will occur. Over the last decades, researchers in coastal and ocean engineering, and the oceanography have dedicated much efforts to the study of extreme wave events. Detail on the state of the art are available in the conference proceedings of Rogue Waves 2004.9 In this chapter, we provide an overview of freak waves. First, we review the four mechanisms that are now thought to generate freak waves in Sec. 6.2. Next, we describe the probability distribution for the populations of linear waves in Sec. 6.3. Third, we show how that probability distribution changes when nonlinear effects are taken into account in Sec. 6.4. Fourth, we discuss the accuracy with which extreme wave events can be predicted in Sec. 6.5. Finally, we make some brief suggestions about future work. 6.2. Mechanisms for Generating Freak Waves It is now accepted that at least four mechanisms can explain the formation of freak waves. A problem common to these four is whether a freak wave obeys an ordinary

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statistical law or whether other physical mechanisms come into play. That is, if we are to be able to predict freak waves, then any underlying mechanism must lead to reliable statistical model for extreme wave events. First Mechanism. The first mechanism is simply that a freak wave can be caused by a linear superposition of waves. In this case, the probability distribution for wave height in the limit of the narrow-band approximation obeys a Rayleigh distribution10 ; corrections due to finite spectral band width have been obtained.11–13 Wave crest statistics can be obtained by using a second-order theory.14 The probability distribution for wave crests has been found in a narrow-band approximation15 and for finite band width.16 This mechanism is theoretically well established and relatively easy to follow; the basic theory will be reviewed in Sec. 6.3. Second mechanism. The second mechanism is that freak wave can result when waves interact with currents. When this situation occurs, ray theory can be used to explain the formation of extreme waves. However, it is not yet known how the statistical properties of surface are related to the properties of currents. From the studies of freak waves generated over a random current, it has been found that wave focusing with 1 m/s velocity fluctuations occurs in the uniform current.17 Third Mechanism. A third mechanism has its basis in crossing-sea states (i.e., two-sea systems). An example is a swell and a wind sea in which the two have different directions coexisting over some region of the ocean, or diffraction of waves behind an island. Usually, both the wave-current mechanism and the two-sea state mechanism occur in specific locations, such as along the Angus current; therefore, freak waves caused by either of these mechanisms are relatively predictable. Fourth Mechanism. In the fourth mechanism, extreme wave events result from the high-order nonlinear interactions; specifically, the underlying mechanism is fourwave interactions.18,19 Numerical and experimental studies have demonstrated that freak-like waves can be generated frequently in a two-dimensional wave flume, even in the absence of a current, or wave refraction, or wave diffraction18,20–22 ; see Fig. 6.2. Further, numerical studies clearly indicate that a freak wave having a single, steep crest can be generated in deep-water by the third-order nonlinear interactions.18 Freak wave generation is sometimes discussed in the context of the Benjamin– Feir instability in deep-water waves because of the similarity in the steep wave

Fig. 6.2.

Freak wave simulated by the high-order spectrum method.18

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profile.18,22 Over the last two decades, the Benjamin–Feir instability of deepwater gravity waves has been studied using the nonlinear Schr¨ odinger-type equations,23–25 mode-coupling equations,26 pseudo-spectral methods,27 and experiments.28 However, the Benjamin–Feir instability vanishes when the wave spectrum is sufficiently broad. This means, there is a discrepancy between the nonlinear behaviors of periodic waves and random waves. Janssen19 investigated freak waves that occur as a consequence of four-wave interactions, including the effects of nonresonant four-wave interactions. Using Monte Carlo simulations of the Zakharov equation, he found that homogeneous nonlinear interactions give rise to deviations from the Gaussian distribution for the surface elevation. The fourth mechanism for freak wave generation requires additional knowledge before predictions can be made. Relationship between kurtosis and freak wave events has been studied using laboratory data,29 numerical results,30 and field data.7 North Sea data shows that the maximum crest, normalized by its wave heights, correlates with skewness, while the wave heights, normalized by significant wave height, correlate with the kurtosis.7 This is because the kurtosis is the fourth moment of the probability density function; hence, it measures the relevance of tails in a distribution. These results explain the general characteristics of a freak wave. (a) A secondorder nonlinearity causes wave asymmetry over the short-time evolution, O(1/ε2 ), of a wave train (Here ε is a small parameter that corresponds to the steepness of deep-water waves). (b) But the kurtosis is associated with the third-order nonlinear interactions.14 Therefore, it may be that third-order nonlinear interactions are the fundamental factors contributing to the occurrence of freak waves. However, the timescales for such interactions are longer than those for second-order interactions, O(1/ε4 ), and they are two orders of magnitude longer than those for second-order nonlinearities. The formal relation between the kurtosis and the exceedance probability for wave height have been discussed (Mori and Janssen31 ; see also Tayfun and Lo32 ; Mori and Yasuda33 ). The kurtosis enters into the distribution function as a correction to the Rayleigh distribution: when the kurtosis tends to three, the expected Gaussian values, and then the distribution function tends to the Rayleigh distribution. In this context, changes in the kurtosis can be evaluated once the evolution of the wave spectrum is known.19 The wave height exceedance probability has been derived as a function of the number of waves.31 This relation will be explained in a later part in this chapter. In all these studies, an ultimate goal is the determination of the shape of the exceedance probability function P (H). Once an analytical form for P (H) is found for all sea conditions (for example, as a function of the wave spectrum), then the probability of a wave exceeding a certain height can be estimated directly. One difficulty in studying extreme wave events is the verification of theory. Statistical theories are evaluated under the assumptions of stationarity; however, the real seas change states both spatially and temporally. Therefore, statistical theories are difficult to verify.34 In addition, freak wave prediction requires us to predict the maximum wave height distribution. However, although study of the maximum wave height distribution is necessary to verify a theory for freak wave

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Surface Elevation (m)

8 6 4 2 0 −2 −4 45000

45100

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Wavelet Spectrum of Yura Y88121401 Surface Waves

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Surface Elevation (m)

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Wavelet Spectrum of Yura Y88010901 Surface Waves

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0.6 0.5 0.4 0.3 0.2 0.1

10

Fig. 6.3.

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30

40

50

60

70

Freak wave recorded in 1988 in the Sea of Japan.34

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prediction, verification of wave height distribution is not sufficient to completely verify a theory.3,31 Samples of freak wave profiles and their wavelet spectrum, recorded in the Sea of Japan, are shown in Fig. 6.3.34 The wave profile shown in Fig. 6.3(a) is a 10 min time-series segment that contains a freak wave along with the corresponding contour plot for the wavelet spectrum. It appears that for the well-defined freak wave shown in Fig. 6.3(a), the occurrence of the freak wave can be readily identified from the wavelet spectrum: the strong energy density in the spectrum instantly increases at the onset of the freak wave, at the same time, that increase in energy appears to carry over into high frequency components. However, for another freak wave time series shown in Fig. 6.3(b), no corresponding focused surge in energy appears in the wavelet spectrum. So in this case, it is uncertain whether a freak wave is begun, generated by linear or nonlinear wave focusing. Nevertheless, at the onset of the event in Fig. 6.3(a), the freak wave profile appears rather asymmetric with respect to the mean level, whereas the freak wave profile in Fig. 6.3(b) is generally symmetric. So, these differences in wavelet spectra might result from differences in the processes that generated these freak waves. In any case, time series alone may or may not be reliable guides for identifying freak wave events. These observations suggest that linear and nonlinear focusing effects are coupled (that is, linear and nonlinear processes do not belong to totally different populations), and this coupling must be understood if we are to be able to predict freak waves. 6.3. Linear Wave The surface gravity waves’ celerity depends on the wave length owing to dispersive effects in deep-water. The nonlinear effects slightly modify the dispersion relation in the order of wave steepness.35 The order of nonlinear effects, both dispersion and wave profile, are square or cubic of the wave steepness if there is no resonant interactions in the system. Therefore, in general, the ocean waves can be regarded as a linear, ergodic, homogeneous, stationary, Gaussian processes. Because of these assumptions, the probability density function for water surface elevation of unidirectional waves can be expressed as the Gaussian distribution
 1 η2 p(η) = √ exp − , (6.1) m0 2πm0  ∞ 2 2 S(ω)dω , (6.2) σ = ηrms = m0 = √

0

and η is the surface elevation while σ = m0 is its variance. Let η(t) be the sea surface elevation at time t and let ζ(t) be an auxiliary variable such that η(t) and ζ(t) are not correlated. Assuming both η(t) and ζ(t) are real zero-mean functions with variance σ, we have Z(t) = η(t) + iζ(t) = A(t)eiφ(t) ,  A(t) = η 2 (t) + ζ 2 (t) ,

(6.3) (6.4)

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 ,

(6.5)

where A is the envelope of the wave train and φ is the phase. As there is no correlation between η and ζ, the joint probability density function of η and ζ becomes   1 1 p(η, ζ) = exp − (η2 + ζ 2 ) . (6.6) 2π 2 √ All variables will be normalized by the variance of the surface elevation σ = m0 . The probability density function (PDF) for the envelope A can now be obtained by integrating the joint probability distribution p(A, φ) = A p(η, ζ) ,

(6.7)

over φ; hence, 

2π

p(A) =

dφ p(A, φ).

(6.8)

0

The results are the usual Rayleigh distributions for wave amplitude p(A) = Ae−(1/2)A

2

(6.9)

and wave height p(H) =

2 1 He−(1/8)H . 4

(6.10)

In the narrow-band approximation, the wave height H = 2A. The exceedance probability PH (H) for wave height now follows by integrating (6.10) from H to ∞: 2

PH (H) = e−(1/8)H .

(6.11)

√ In an exact, linear narrow-band random wave theory, H1/3 = 4.004 m0 , but for √ simplicity we will assume H1/3 = 4 m0 . In this notation, the freak wave condition √ H/H1/3 > 2 corresponds to H/ m0 > 8. According to Eq. (6.11), the probability of occurrence for a freak wave in a wave train is P (8) = e−8 = 3.35 × 10−4 . This is equivalent to a freak wave occurring once in every 2981 waves. Assuming a characteristic wave period of 10 s, this corresponds to a storm having a duration of 8.3 h. When this linear wave theory is extended to 3D space, the appearance probability for fully directional conditions (Smax = 10) increases by 45% over that for the unidirectional condition.36 The maximum wave height distribution becomes important when we examine statistical properties of freak waves over a series of recorded data. This is because a freak wave prediction is equivalent to estimating the maximum wave height and the exceedance probability (that is, Eq. (6.11) is insufficient for prediction). The PDF for the maximum wave height pm in wave trains can be obtained once the

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PDF of wave height p(H) and exceedance probability of the wave height P (H) are known.37 Thus, pm (Hmax )dHmax = N [1 − P (Hmax )]N −1 p(Hmax )dHmax ,

(6.12)

where N is the number of waves in the wave train. For sufficiently large N , we may use the approximation N

N

N →∞

N →∞

lim [1 − P (Hmax )]N  lim exp [−N P (Hmax )] .

(6.13)

Substituting Eqs. (6.11) and (6.13) into Eq. (6.12), gives the PDF for the maximum wave height pm ,   2 2 N (6.14) pm (Hmax )dHmax = Hmax e−Hmax /8 exp −N e−Hmax /8 dHmax , 4 and the exceedance probability of maximum wave height Pm ,   2 (6.15) Pm (Hmax ) = 1 − exp −N e−Hmax /8 . Note that Eqs. (6.14) and (6.15) are evaluated as functions of N number of waves √ in the train. The freak wave condition in this study becomes Hmax / m0 > 8, and we obtain from Eq. (6.15) the following simple formula for probability of a freak wave occurring as a function of N ,

(6.16) Pfreak = 1 − exp −e−8 N . This is a general expression for the expected probability of freak waves in unidirectional linear random waves. Therefore, in the linear wave case, the occurrence probability for a freak wave depends only on the number of waves N . For example, if the number of waves equals 2981, then Eq. (6.16) gives Pfreak = 0.632, that is, the expected probability of a freak wave is 63.2% when we randomly sample from a population of 2981 waves. For N = 120 waves, the expected probability of a freak wave is 3.9% and for N = 10,000 waves, it is 96.5%. These results show that the number of waves in a wave train (or, equivalently, the duration of a storm) is important for predicting freak waves.

6.4. Nonlinear Interactions 6.4.1. Role of nonlinear interactions Since the beginning of freak wave research, high-order nonlinear effects have been considered.18,30 As a result, the nonlinear occurrence of freak waves, including their mechanisms and detailed dynamic properties, are becoming better understood.21,22,34,38–40 The state of the art has been summarized at the Rogue Wave Conference, held in 2000 and 2004.9,41 The consensus from those conferences is that third-order nonlinear interactions not only enhance the appearance of freak wave, but they are also the primary cause of freak waves. The only exceptions occur during strong wave–current interactions or wave diffraction behind an island.

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6.4.2. Quasi-resonant four-wave interactions and high-order moments Investigations have been carried out on the creation of freak waves in response to four-wave interactions, including the effects of nonresonant four-wave interactions.19,31 Those studies culminated in the formulation of a probability for the occurrence of freak waves in nonlinear wave fields. However, the modulational instability is a quasi-resonant interaction process, i.e., wave numbers and frequencies satisfy the following conditions: k1 + k2 − k3 − k4 = 0 , ω(k1 ) + ω(k2 ) − ω(k3 ) − ω(k4 ) ≤ 2 ,

(6.17) (6.18)

where  is a small parameter that corresponds to the steepness of deep-water waves. The modulational instability takes place when two wave numbers are the same, k1 = k2 , while k3 and k4 are side bands separated from k1 by ∆k, which should be small in order to satisfy the conditions Eqs. (6.17) and (6.18). The standard kinetic equation that describes the evolution of the wave spectrum in time42 is formally valid only for exact resonances and large times, O(−4 ω0−1). It has been extended to include quasi-resonant interactions,19 where a kinetic equation, which should also be valid on the timescale of the modulational instability, has been derived. If we then consider the evolution of higher order moments, such as the kurtosis µ4 , we find that the quasi-resonant interactions are responsible for deviations from the Gaussian distribution. The explicit relation between nonlinear interactions of free waves and the fourth-order moment of the surface elevation η(x, t) in deep-water has been investigated.19 The result can be expressed in terms of κ40 , which is the fourth-order cumulant of the surface elevation (κ40 = µ4 − 3), η 4  −3 m20  √ 12 dk1,2,3,4 T1,2,3,4 ω1 ω2 ω3 ω4 = 2 2 g m0 × δ1+2−3−4 Rr (∆ω, t)N1 N2 N3 .

κ40 =

√

(6.19)

Here, m0 is the variance of the surface elevation η, g is the acceleration of gravity, k is the wave number, ω is the angular frequency, N is the wave action spectral density, and T1,2,3,4 is the coupling coefficient in the Zakharov equation (see Krasitskii43 for its analytical form), δ1+2−3−4 = δ(k1 + k2 − k3 − k4 ), dk1,2,3,4 = dk1 dk2 dk3 dk4 . The resonant function is Rr =

1 − cos(∆ωt) , ∆ω

(6.20)

with ∆ω = ω1 + ω2 − ω3 − ω4 .

(6.21)

In the limit of large times, Rr → P/∆ω, where P denotes the principle value of the integral to avoid singularity in the integral.

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In the narrow-band approximation, we assume that the spectrum E(ω) has a Gaussian shape, E(ω) =

2 m0 √ e−(1/2)ν , σω 2π

(6.22)

where ν = (ω − ω0 )/σω and σω is the spectral band width; the integral in Eq. (6.19) for large times becomes  2 2 2 242 dν1,2,3 e−(1/2)[ν1 +ν2 +ν3 ] P . (6.23) κ40 = ∆2 (2π)3/2 (ν1 + ν2 − ν3 )2 − ν12 − ν22 + ν32 √ Here,  = k0 m0 is the steepness parameter and ∆ = σω /ω0 is the relative spectral band width. The integral can be evaluated analytically to obtain π κ40 = √ BF I 2 , 3

(6.24)

where BFI is defined as by Janssen19: BFI =

√ 2. ∆

(6.25)

Equation (6.24) is the simplified predictive equation for the kurtosis of the surface elevation assuming a narrow-band, unidirectional wave train, but a full description requires evaluation of six-dimensional integral in wave number space Eq. (6.19). 6.4.3. Wave height distributions in a nonlinear wave field To include nonlinear effects in the wave height distribution function, thereby possibly obtaining deviations from Rayleigh statistics, the standard approach is to use the Edgeworth series.44 The theory for wave height has been described,31–33 and the resulting distribution has been named the Modified Edgeworth Rayleigh (MER) distribution. The MER wave height and exceedance wave height distributions are given by Mori and Janssen31 : p(H)dH =

1 −(1/8)H 2 He [1 + κ40 AH (H)] dH , 4

(6.26)

2

(6.27) PH (H) = e−(1/8)H [1 + κ40 BH (H)] . √ Here, H is the wave height normalized by ηrms = m0 , κ40 is defined in Eq. (6.23), while AH (H) and BH (H) are polynomials defined as 1 (H 4 − 32H 2 + 128) , 384 1 BH (H) = H 2 (H 2 − 16) . 384 AH (H) =

(6.28) (6.29)

Note that the distributions describe the deviations from linear statistics under the hypothesis of a narrow-band, weakly nonlinear wave train. Further, in the case of linear waves (κ40 = 0), these distributions are equivalent to Eqs. (6.10) and (6.11).

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Second-order contributions, which are important for the distribution of wave crests, can be included by using an approach like that used by Tayfun.15,45 The probability distribution function, pm , and the exceedance probability, Pm , of the maximum wave height can also be expressed as a function of the fourth cumulant of the surface elevation κ40 and the number of waves N in the wave train, pm (Hmax ) =

2 N Hmax e−Hmax /8 [1 + κ40 AH (Hmax )] 4   2 × exp −N e−Hmax /8 [1 + κ40 BH (Hmax )] ,

  2 Pm (Hmax ) = 1 − exp −N e−Hmax /8 [1 + κ40 BH (Hmax )] ,

(6.30) (6.31)

where Hmax is the maximum wave height normalized by ηrms . Qualitative agreement has been found between the theoretical wave height distribution, given by Eq. (6.30), and field data.31 Equations (6.30) and (6.31) are evaluated as functions of N and κ40 (or µ4 ). When κ40 = 0, Eqs. (6.30) and (6.31) reduce to Eqs. (6.14) and (6.15), which are provided by the Rayleigh distribution. The occurrence probability of a freak wave as a function of N and κ40 , taking into account nonlinear effects, now becomes  (6.32) Pfreak = 1 − exp −e−8 N (1 + 8κ40 ) . Using Eq. (6.31) we see that the effect of kurtosis becomes of the same order as the linear theory for κ40 = 1/8. This corresponds to µ4 = 3.125, which is not a strong nonlinear condition. Hence, both the effects of finite kurtosis and the number of waves N are important for determining the probability of maximum wave height in the nonlinear wave train. It should be emphasized here that the above discussion does not require any empirical, ad hoc parameters. A different approach, also based on the dynamics of a cubic nonlinearity, has been used for finding the exceedance probability for wave height.46 That approach involves an optimization problem, requiring that at a specific space and time, the Fourier phases are all equal, under the constraints that the conservation laws of the Zakharov equation are satisfied. To summarize, quasi-resonant four-wave interactions introduce deviations from linear statistics for surface elevation; in particular, for weakly nonlinear, narrowbanded and long-crested wave trains, the kurtosis evolves according to Eq. (6.19). In the narrow-band approximation, the kurtosis is related to the BFI [defined in Eq. (6.25)]; the tail of the wave height distribution depends on the kurtosis/BFI and it increases as the kurtosis increases. Finally, the maximum wave height distribution depends on the number of waves in the wave train (record length) and on the kurtosis of the nonlinear wave field, as shown in Eq. (6.30). 6.5. Accuracy of Freak Wave Prediction Figure 6.4 compares the linear (Rayleigh) theory with those from the present theory for the occurrence probability of a freak wave, Pfreak ; the plot shows how Pfreak

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0

Pfreak

10

−1

10

Rayleigh µ4 = 3.1 µ4 = 3.2 µ4 = 3.3 µ4 = 3.4 µ4 = 3.5 −2

10

2

3

10

10 N

4

10

Fig. 6.4. Occurrence probability for freak waves as a function of number of waves N and kurtosis µ4 .31

responds to changes in the number of waves N and in the strength of nonlinearities, as measured by the kurtosis µ4 . For N = 100 waves, the occurrence probability of a freak wave is predicted to be 3.3% by the linear theory, while it is 15.4% according to Eq. (6.30) with µ4 = 3.5. For N = 1000 waves, the occurrence probability of a freak wave is 28.5% according to linear theory, while it is 81.3% according to Eq. (6.31) with µ4 = 3.5. The number of waves N = 1000 corresponds to a duration of about 3 hours, assuming T1/3 = 10 s, which is not unusual in stormy conditions. Alternatively, defining the threshold value for the occurrence probability of a freak wave as 50%, the expected number of waves that would include at least one freak wave is predicted to be 2000 waves from linear theory, but only 500 waves when Eq. (6.31) is used with µ4 = 3.5. Thus, in a strong nonlinear wave field, freak waves can occur several times more frequently than in a linear field. Figure 6.5 shows freak wave occurrence probability as a ratio Rfreak of the value predicted by the present approach to that predicted by the Rayleigh theory (κ40 = 0); the ratio depends on the kurtosis µ4 , Rnonlinear =

Pfreak −1. Pfreak |κ40 =0

(6.33)

For a small number of waves N ≤ 250, the ratio Rfreak depends linearly on µ4 . If µ4 = 3.1 and N ≤ 500, the occurrence probability of freak waves is 50% more from the nonlinear theory than from the linear theory. However, the rate of increase in Rfreak decreases as the number of waves increases. This is because, for very large number of waves, the maximum wave height almost always exceeds 2H1/3 , even in linear wave trains.

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350 300

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N=100 N=250 N=500 N=1000 N=5000

Rfreak [%]

250 200 150 100 50 0 3

Fig. 6.5.

3.1

3.2

µ4

3.3

3.4

3.5

Ratio of freak wave occurrence predicted by Eq. (6.31) to the Rayleigh theory.31

The consequences of the maximum wave height distribution [Eq. (6.30)] are generally hard to verify, not only because Eq. (6.30) depends on the number of waves but also because large amount of data are required to characterize nonlinear effects. In the following discussion, we compare the above theories with a set of long time-series experimental data47 ; these data appear to be sufficiently long to make useful comparisons. Figure 6.6 shows the probability of occurrence of freak waves Hmax /ηrms ≥ 8 for short time records, as a function of kurtosis. We assumed 100 waves per record and counted the number of freak waves. In the experimental data, the occurrence probability for freak waves shows a clear dependence on the kurtosis (this effect of course is not described by the Rayleigh distribution). But the experimental data appear to depend quadratically on the kurtosis, while the nonlinear theory Eq. (6.31) predicts a linear dependence. This is probably because the nonlinear theory includes only the lowest-order correction for nonlinearity: it excludes high-order cumulants. Figure 6.7 compares the maximum wave height distribution from the experimental data with linear, Eq. (6.14), and nonlinear theories, Eq. (6.30), for N = 150. In Fig. 6.7(a), the comparison between theory and experiment is shown for a nearby linear wave condition (µ4 = 3.06). The peak in the observed maximum wave height distribution is larger than the Rayleigh Hmax distribution and the MER Hmax distribution with µ4 = 3.06, but the observed distribution is not as wide (we ascribe this difference to the effects of the finite width of the spectrum). The maximum wave height distribution from the laboratory data departs from the Rayleigh distribution [Figs. 6.7(a)–6.7(c)] at some distance from the wavemaker, and the peak of the observed distribution shifts to larger wave heights. While the Rayleigh Hmax distribution, Eq. (6.14), is independent of the kurtosis, the MER Hmax distribution,

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N=11900 0.45 Exp. MER Rayleigh

0.4

Probability of Freak Wave [%]

0.35 0.3 0.25 0.2 0.15 0.1 0.05 0 3

3.5

4

4.5

µ4 Fig. 6.6. Occurrence probability for freak waves in long recorded wave data: N = 11, 900 (•: Experimental data, solid line: MER distribution from Eq. (6.31), dashed line: Rayleigh distribution).

Eq. (6.30), follows the behavior of the experimental data. The MER Hmax distribution seems to reproduce the experimental data reasonably well; for very small kurtosis, it overestimates the experimental data, and for large kurtosis it slightly underestimates the data. This is consistent with what we have seen in Fig. 6.6 where the occurrence probability of freak waves is plotted as a function of the kurtosis. Similar results have been obtained in other studies.31 We also consider the general behavior of the probability density function of maximum wave height in a nonlinear wave field. To do so, we plot in Fig. 6.8 the expected value of the maximum wave height, indicated by angle brackets, as a function of µ4 and N . In the figure, we show the numerically integrated value of Eq. (6.30) together with the ensemble average of the experimental data. The quantity Hmax /H1/3  in Fig. 6.8 from the Rayleigh theory corresponds to √ H/ m0 > 8. The dependence of Hmax /H1/3  on µ4 and N is clear in both the experimental data and Eq. (6.30). Overall, Hmax /H1/3  from the experimental data is smaller than that from the MER Hmax distribution; nevertheless, the theoretical and experimental data show similar trends. The above evidence shows that freak waves can be predicted for given strengths of nonlinearities, as measured by the kurtosis. The remaining problems, therefore,

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0.8 Exp. MER Rayleigh

(a) µ4 =3.06 0.7

0.5

P(H

max

/η

rms

)

0.6

0.4 0.3 0.2 0.1 0 4

5

6

7

8

H

9

10

11

12

/η

max

rms

0.8 Exp. MER Rayleigh

(b) µ4 =3.43 0.7

rms

0.4

P(H

/η

0.5

max

)

0.6

0.3 0.2 0.1 0 4

5

6

7

8

H

9

10

11

12

/η

max

rms

0.8

Exp. MER Rayleigh

(c) µ4 =4.10 0.7

P(Hmax/ηrms)

0.6 0.5 0.4 0.3 0.2 0.1 0 4

5

6

7

8

9

10

11

12

H max/ηrms

Fig. 6.7. Comparison of maximum wave height distribution Hmax /ηrms with N = 150 (•: experimental data, solid line: MER Hmax distribution, dashed line: Rayleigh Hmax distribution).

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2.2



2.1 2

1.9 1.8 1.7 1.6 1.5

1.4 1000 800 600 400 200 N

3

3.2

3.4

3.6

3.8

4

µ4

2.2



2.1 2

1.9 1.8 1.7 1.6 1.5

1.4 1000 800 600 400 N

Fig. 6.8.

200

3

3.2

3.4

3.6

3.8

4

µ4

Comparison of expected value of
Hmax /H1/3 .48

are developing accurate methods for estimating the kurtosis and learning how to take directional effects into account for the application in the ocean. 6.6. Conclusion and Future Study This chapter summarized generally accepted mechanisms that cause freak waves and we have described linear and nonlinear mechanisms in more detail. Four-wave

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quasi-resonant interactions have been shown to play an important role in determining the statistical properties of surface elevation. This discovery, which is fairly new, is of some relevance to extreme wave forecasting. Previously, deviations from Gaussian behavior have been attributed only to bound modes. However, even in the most severe sea states, the kurtosis obtained from second-order theory rarely reaches values above 3.15. Moreover, contributions to wave height distribution from second-order theory are practically negligible (in the narrow-band approximation that are exactly zero). If waves are long-crested and sufficiently steep, the dynamics of the free waves can cause very strong departures from Gaussian behavior. Large values of the kurtosis can substantially change the tails of the probability density function for wave height and can increase the occurrence of freak waves. Although nonlinear effects have been found to be important, some issues remain unresolved. First, we need an evaluation of the accuracy with which the kurtosis is estimated from spectra. BFI originally describes the asymptotic behavior, given an initial value, but in reality, there is no initial condition in the ocean. This should be verified in the near future. Second, we need to account for wave directionality. In real sea states, directional effects are important. Directional contributions to the evolution of the kurtosis, using a modified nonlinear Schr¨ odinger equation, have been described,49 and the effects of four-wave interactions only appear close to longcrested waves conditions.50 A quantitative discussion of directional effects should be presented in the near future.

References 1. L. Draper, “Freak” ocean waves, Marine Observer 35, 193–195 (1965). 2. P. Klinting and S. Sand, Analysis of prototype freak waves, in Coastal Hydrodynamics, ed. R. Darlymple (ASCE, 1987), pp. 618–632. 3. P. Stansell, J. Wolfram and S. Zachary, Horizontal asymmetry and steepness distributions for wind-driven ocean waves, Appl. Ocean Res. 25, 137–155 (2003). 4. G. Guedes Soares, Z. Cherneva and E. Ant¨ ao, Characteristics of abnormal waves in North Sea storm sea states, Appl. Ocean Res. 25, 337–344 (2003). 5. T. Yasuda and N. Mori, Occurrence properties of giant freak waves the sea area around Japan, J. Waterway, Port, Coastal Ocean Eng. 123(4), 209–213 (1997). 6. T. Yasuda, N. Mori and S. Nakayama, Characteristics of giant freak waves observed in the Sea of Japan, in Waves97 (ASCE, 1997), pp. 482–495. 7. G. Guedes Soares, Z. Cherneva and E. Antao, Abnormal waves during hurricane Camille, J. Geophys. Res. Ocean 109, C08008 (2004). 8. S. Sand, N. O. Hansen, P. Klinting, O. Gudmestad and M. Sterndorff, Freak Wave Kinematics (Kluwer Academic Pub., 1990), pp. 535–549. 9. M. Olagnon (ed.), Rogue Waves 2004 (IFREMER, 2004). 10. M. Longuet-Higgins, On the statistical distribution of the heights of sea waves, J. Marine Res. 11, 245–266 (1952). 11. A. Næss, On the distribution of crest-to-trough wave heights, Ocean Eng. 12(3), 221–234 (1985). 12. P. Boccotti, On Mechanics of Irregular Gravity Waves, Chap. VIII (Atti Accademia Nazionale dei Lincei, Memorie, 1989), pp. 111–170.

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13. M. A. Tayfun, Distribution of crest-to-trough wave heights, J. Waterway, Port, Coastal Ocean Eng. 107(3), 149–158 (1981). 14. M. Longuet-Higgins, The effect of non-linearities on statistical distirbutions in the theory of sea waves, J. Fluid Mechanics 17, 459–480 (1963). 15. M. A. Tayfun, Narrow band nonlinear sea waves, J. Geophys. Res. Ocean 85(C3), 1548–1552 (1980). 16. F. Fedele and F. Arena, Weakly nonlinear statistics of high non-linear random waves, Phys. Fluids 17, 026601 (2005). 17. B. White and B. Fornberg, On the chance of freak wave at sea, J. Fluid Mechanics 355, 113–138 (1998). 18. T. Yasuda, N. Mori and K. Ito, Freak waves in a unidirectional wave train and their kinematics, Proc. 23rd Int. Conf. Coastal Eng., Vo. 1, ASCE, Venice (1992), pp. 751–764. 19. P. Janssen, Nonlinear four-wave interactions and freak waves, J. Phys. Oceanogr. 33(4), 863–884 (2003). 20. C. Stansberg, Extreme waves in laboratory generated irregular wave trains, in Water Wave Kinematics, eds. A. Tørum and O. Gudmestad (Kluwer Academic Pub., 1990), pp. 573–590. 21. K. Trulsen and K. Dysthe, Freak waves — A three-dimensional wave simulation, Proc. 21st Symp. Naval Hydrodynamics (National Academy Press, 1997), pp. 550–558. 22. M. Onorato, A. Osborne, M. Serio and S. Bertone, Freak waves in random oceanic sea states, Phys. Rev. Lett. 86(25), 5831–5834 (2001). 23. H. C. Yuen and B. M. Lake, Nonlinear dynamics of deep-water gravity waves, Adv. Appl. Mech. 22, 67–229 (1982). 24. E. Caponi, P. Saffman and H. Yuen, Instability and confined chaos in a nonlinear dispersive wave system, Phys. Fluids 25(12), 2159–2166 (1982). 25. K. Dysthe, Note on modification to the nonlinear Schr¨ odinger equation for application to deep water waves, Proc. Roy. Soc. Lond. A 369, 105–114 (1979). 26. M. Stiassnie and L. Shemer, Energy computations for evolution of class I and II instabilities of Stokes waves, J. Fluid Mechanics 174, 299–312 (1987). 27. T. Yasuda and N. Mori, Roles of sideband instability and mode coupling in forming a water wave chaos, Wave Motion 26(2), 163–185 (1997). 28. M. Su, Three-dimensional deep-water waves. Part 1. Experimental measurement of skew and symmetric patterns, J. Fluid Mechanics 124, 73–108 (1982). 29. N. Mori, T. Yasuda and K. Kawaguchi, Breaking effects on waves statistics for deepwater random wave train, in Waves97 (ASCE, 1997), pp. 316–328. 30. T. Yasuda and N. Mori, High order nonlinear effects on deep-water random wave trains, Int. Symp. Waves-Physical and Numerical Modelling, Vol. 2, Vancouver (1993), pp. 823–832. 31. N. Mori and P. Janssen, On kurtosis and occurrence probability of freak waves, J. Phys. Oceanogr. 36(7), 1471–1483 (2006). 32. M. A. Tayfun and J.-M. Lo, Nonlinear effects on wave envelope and phase, J. Waterway, Port, Coastal Ocean Eng. 116(1), 79–100 (1990). 33. N. Mori and T. Yasuda, A weakly non-Gaussian model of wave height distribution for random wave train, Ocean Eng. 29(10), 1219–1231 (2002). 34. N. Mori, P. Liu and T. Yasuda, Analysis of freak wave measurements in the Sea of Japan, Ocean Eng. 29(11), 1399–1414 (2002). 35. M. Longuet-Higgins and O. Phillips, Phase velocity effects internary wave interactions, J. Fluid Mechanics 12, 133–135 (1962).

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36. A. Kimura, Probability of the freak wave appearance in a 3-dimensional sea condition, Proc. 24th Int. Conf. Coastal Eng., Vol. 1, ASCE (1994), pp. 356–369. 37. Y. Goda, Random Seas and Design of Maritime Structures, 2nd Edn. (World Scientific, 2000). 38. I. Lavrenov, The wave energy concentration at the Agulhas current off South Africa, Natural Hazard 17, 117–127 (1998). 39. A. Osborne, M. Onorato, M. Serio and S. Bertone, The nonlinear dynamics of rogue waves and holes in deep water gravity wave trains, Phys. Lett. A 275(5–6), 386–393 (2000). 40. S. Haver, Evidences of the existence of freak waves, in Rogue Waves, eds. M. Olagnon and G. Atanassoulis (IFREMER, 2001), pp. 129–140. 41. M. Olagnon and G. Athanassoulis (eds.), Rogue Waves 2000 (IFREMER, 2000). 42. K. Hasselmann, On the nonlinear energy transfer in gravity-wave spectrum. I. General theory, J. Fluid Mechanics 12, 481–500 (1962). 43. V. Krasitskii, On reduced equations in the Hamiltonian theory of weakly nonlinear surface waves, J. Fluid Mechanics 272, 1–20 (1994). 44. F. Edgeworth, On the representation of statistical ferquency by a series, J. Roy. Statist. Soc. 70, 102–106 (1907). 45. M. A. Tayfun, Statistics of nonlinear wave crests and groups, Ocean Eng. 33(11–12), 1589–1622 (2006). 46. F. Fedele and M. A. Tayfun, Explaining freak waves by a stochastic theory of wave groups, 25th Int. Conf. Offshore Mechanics and Arctic Engineering 1 (2006). 47. M. Onorato, A. R. Osborne, M. Serio, L. Cavaleri, C. Brandini and C. T. Stansberg, Extreme waves, modulational instability and second order theory: Wave flume experiments on irregular waves, Eur. J. Mech. B/Fluids 25(5), 586–601 (2006). 48. N. Mori, M. Onorato, P. A. E. M. Janssen, A. R. Osborne and M. Serio, On the extreme statistics of long crested deep water waves: Theory and experiments, J. Geophys. Res., Ocean 112, C09011 (2007), doi: 10.1029/2006JC004024. 49. H. Socquet-Juglard, K. Dysthe, K. Trulsen, H. Krogstad and J. Liu, Probability distribution of surface gravity waves during spectral changes, J. Fluid Mechanics 542, 195–216 (2005). 50. O. Gramstad and K. Krulsen, Influence of crest and group length on the occurrence of freak waves, J. Fluid Mechanics 582, 463–472 (2007).

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Chapter 7

Short-Term Wave Statistics Akira Kimura Department of Social Systems Engineering Faculty of Engineering, Tottori University, 680-8552 Koyama Minami 4-101, Tottori, Japan [email protected] This chapter deals with the short-term statistical properties of irregular sea waves. It begins with a description of an early study by Rice [Bell Syst. Tech. J. 23, 282–332; 24, 46–156 (1944, 1945)] about the relation between wave spectra and statistics of wave amplitudes. Next, an explanation is given of studies which deal with the effect of spectrum width and wave nonlinearity on wave statistics as well as with joint statistical properties between wave heights and periods. Finally, the chapter closes with a look at the statistics of wave direction, length of wave crest, and spatial maximum amplitude (which are the 3D wave’s properties) as well as the time series of wave height.

7.1. Introduction Waves in sea conditions have certain irregular properties. These comprise properties such as wave heights and wave periods. For many years, engineers have known that it is not sufficient to know only statistically representative values such as the mean value. Larger wave heights can induce damage on coastal structures while longer wave periods may induce resonance phenomena. To understand their overall statistical properties, engineers and researchers have tried to clarify the statistical characteristics of populations for these properties with the intention of finding out their probability density function (pdf). In particular, their pdf have been studied in relation to the wave spectrum. The study of this development was initiated primarily by Rice.20 The application of this idea in coastal engineering was started notably by Longuet-Higgins.15 He developed pdf for surface elevation and gradient, maxima and minima of water surface, number of zero-crossings along a line, etc. Of course, sailors have carried out visual observations of waves since ancient times. From the theoretical and empirical knowledge of wave statistics, two types of statistical study have appeared: shortterm and long-term wave statistics. Since waves must be stationary stochastic to 151

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study short-term statistics, the duration of concern is 1 or 2 days at most. On the other hand, long-term statistics cover waves for time periods of at least dozens of years. In this type, the time series of water surface elevation is not used. Instead, representative wave properties such as significant wave height and period are utilized. The wave height/period is estimated for coastal structures within their lifetime from long-term statistics, for example. The effectiveness of results from long-term statistics is heavily dependent on short-term statistics. Both are inseparable for the efficient use of wave statistics in coastal engineering.

7.2. Rice Model Rice20 proposed a model of the Fourier sum for irregular signals. The irregular signal I(t) is expressed by the summation of trigonometric functions as I(t) =

N


(an cos σn t + bn sin σn t) =

n=1

N


cn cos(σn t + εn ),

(7.1)

n=1

in which σn is the angular frequency and εn (= tan−1 (−bn /an )) is the phase of the nth component wave, respectively. an , bn , and cn are amplitudes of nth component waves which are distributed normally with zero mean. Their variances are  2 cn 2 2
an  =
bn  = = S(σ)dσ , (7.2) 2 where
 means the ensemble mean, S(σ) is the energy spectrum of I(t). Rice defined the envelope function for I(t) as I(t) =

N


cn cos(σn t + εn ) = Ic (t) cos σt − Is (t) sin σt ,

(7.3)

n=1

where σ is the mean frequency of S(σ), and Ic (t) and Is (t) are Ic (t) =

N


cn cos(σn t − σt + εn ),

n=1

Is (t) =

N


cn sin(σn t − σt + εn ) .

(7.4)

n=1

The envelope function R(t) is R(t) = {Ic2 (t) + Is2 (t)}1/2 .

(7.5)

The nth component wave in Eq. (7.4), cn cos(σn t − σt + εn ) or cn sin (σn t − σt + εn ), has a U-shape pdf of finite width.14 From the central limit theorem, the infinite sum (N → ∞) of the trigonometric functions, Ic (t) and Is (t), are distributed normally. Their joint pdf is   2 1 Ic + Is2 p(Ic , Is )dIc dIs = dIc dIs , exp − (7.6) 2πϕ0 2ϕ0

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since Ic (t) and Is (t) are independent (
Ic (t) · Is (t) = 0). ϕ0 is the variance of Ic (t), Is (t), and I(t), respectively:
Ic2  =
Is2  =
I 2  = ϕ0 .

(7.7)

The variables R(t), I(t), Ic (t), and Is (t) are represented as R, I, Ic , and Is , respectively. It is convenient to introduce the variable θ, where Ic = R cos θ,

Is = R sin θ .

By substituting Eq. (7.8) into Eq. (7.6), the joint pdf of R and θ becomes   R R2 dRdθ . p(R, θ)dRdθ = exp − 2πϕ0 2ϕ0

(7.8)

(7.9)

Since R and θ are independent and θ is distributed uniformly over the range 0 to 2π, the pdf of R is obtained as   R R2 p(R)dR = exp − dR . (7.10) ϕ0 2ϕ0 Figure 7.1(a) shows the relation between I and R. The spectrum S(σ) for I is the following normalized Wallops type (r = 10)8 :    −4   −r r σ σ exp . (7.11) · 1− S(σ) = σp 4 σp The maxima or minima of I almost coincide with R within a “zero-up/downcrossing” interval of I. If I takes extremes (∂I/∂t = 0) at t = t∗1 , t∗2 , . . . , wave amplitude can be approximated by R|t∗i , (i = 1, 2, . . .) [Fig. 7.1(b)]. The intervals between t∗1 , t∗2 , . . . are almost constant when the spectrum is very narrow. The pdf for R|t∗i is also the Rayleigh distribution approximately, since the local maxima/ minima is equal to the wave amplitude. Equation (7.10) becomes the pdf for wave amplitude14 approximately.

7.3. Pdf of Wave Heights 7.3.1. Nonlinear effect on the pdf Since the study by Longuet-Higgins,14 the pdf for wave amplitudes has been used extensively as pdf for wave heights. Many measurements and numerical simulations have supported the Rayleigh distribution for the wave height distribution. The standard form of the widely used Rayleigh distribution is   π H π H2 p(H)dH = exp − dH , (7.12) 2 H2 4 H2 in which H is the mean wave height. When the wave spectrum is very narrow, wave height distribution can be the Rayleigh distribution, since wave height can be approximated as H ≈ 2R .

(7.13)

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Fig. 7.1. (a) Wave profile I(t) (solid line) and its envelope R(t) (dotted line). (b) Wave profile I(t) (solid line) and envelope R(t) (dotted line) (small discrepancies between I(t) and R(t) at t = t∗1 , t∗2 , · · · ).

However, the zero-crossing wave height is defined by H = Il max − Il min , where Il max and Il min are the consecutive local maximum and minimum of the wave profile, respectively. The pdf for H may not be the Rayleigh distribution if the pdf for Il max and Il min are the Rayleigh distribution. Field measurements have shown that there are small discrepancies between the Rayleigh distribution p(H) and frequency distributions of measured data F (H): F (H) < p(H) for H < Hp and F (H) > p(H) for H > Hp , where p(Hp ) is a peak of p(H). In particular, as nonlinearity on the wave profile becomes prominent, this characteristic may bring a considerable effect on the pdf. Forristall3 used the following Weibull distribution to improve the goodness of fit:

α −ς , (7.14) P (ς) = exp β where P (ς) is the Weibull distribution function and α = 2.126, β = 8.42, and ς=

H∗ 1/2

m0

,

(7.15)

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in which H ∗ is the difference between the heights of consecutive wave crest and trough within a zero-crossing interval of the wave profile. Longuet-Higgins17 pointed out that the discrepancy should be improved using rms (root mean square) wave amplitude. The ith moment of a spectrum is determined as ∞ mi = σ i S(σ)dσ (i = 1, 2, . . .) . (7.16) 0

From the Khintchine theorem, the 0th moment is also defined in another way as 2 , m0 = ηrms

(7.17)

where η is the wave profile and ηrms is its rms. When the spectrum is narrow and the nonlinearity on the wave profile is negligible, (arms )2 , (7.18) 2 in which arms is the rms of wave amplitude a. The Rayleigh distribution function given by Longuet-Higgins is

 a2 P (a) = exp − 2 . (7.19) arms m0 =

a2rms can be replaced by 2m0 if the nonlinearity is negligible. He discussed the nonlinearity effect on the wave profile as follows. Applying the third-order Stokes wave theory, the 2nd moment of the periodic wave profile is derived from Eq. (7.17) as

 1 1 2 19 3077 2 4 6 (ak) − · · · , (7.20) m0 = a 1 − (ak) − (ak) − 2 2 12 720 in which k is the wave number. m0 for irregular waves is calculated by ∞ m0 = 2V (a)dp(a) ,

(7.21)

0

in which V (a) denotes the potential energy (2V = η 2 = m0 ) of the zero-crossing wave with an amplitude a given as Eq. (7.20), p(a) is the pdf of a. Applying Eq. (7.19), m0 becomes

  amax a2 2V (a)d exp − 2 m0 = +δ, (7.22) arms 0 in which amax is the maximum wave amplitude of nonbreaking waves. He used the same V for the waves (ak)2 ≥ 0.196. δ is the remainder and is exponentially small for the large value of a2max /a2rms . The new m0 is

 1 19 3077 (arms k)6 − · · · . (7.23) m0 = a2rms 1 − (arms k)2 − (arms k)4 − 2 2 30 Since arms k for irregular waves distributes roughly 0.158 < arms /k < 0.203,13 0.935 <

2m0 < 0.968 . a2rms

(7.24)

If wave profiles have a nonlinearity, rms wave amplitude between 2.066m0 < a2rms < 2.140m0 may give a correction for the Rayleigh distribution of wave amplitudes.

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7.3.2. Effect of spectrum width Since the zero-crossing method for irregular waves does not have a physical basis, a mathematical treatment is difficult except for the very narrow band spectrum. Cartwright and Longuet-Higgins2 derived a spectrum width parameter as

 m22 ε= 1− : 0 < ε < 1, (7.25) m0 m4 in which m0 , m2 , and m4 are determined by Eq. (7.16). Pdf of amplitudes (local maxima) is the Rayleigh distribution when ε → 0. If ε = 0, however, waves must be periodic. Longuet-Higgins17 investigated the effect of the spectrum width on pdf of wave amplitudes. He used a periodic wave profile with small disturbance given as
y = b cos(σt + ε) + an cos(σn t + εn ) , (7.26) n

in which an , σn , and εn are the amplitude, angular frequency, and phase of the disturbance, respectively. Their amplitudes are b >> an . The spectrum of the disturbances S  (σ) is defined as σn +dσ
σn

The rms of y is

2
yrms  = b2rms + 0

∞

1 2 a = S  (σn )dσ . 2 n

(7.27)



πσ  σ 2  sin2 S  (σ)dσ , + 2σ σ

(7.28)

and the 0th moment of the spectrum of y is ∞ ∞ b2rms  {S(σ) + S (σ)}dσ = S  (σ)dσ , + m0 = 2 0 0

(7.29)

where S(σ) is the spectrum of the periodic wave and brms is the rms of the periodic wave profile. From Eqs. (7.28) and (7.29), yrms is  ∞  2

 σ 2 2 πσ  = 2m0 + − 1 − cos (7.30)
yrms S  (σ)dσ. 2σ σ2 0 Using the relations   2   σ 2

σ σ    −1 + −1 −1 =2   σ σ σ2  

πσ  1 − cos π(σ/σ − 1) 2 − 1) π(σ/σ 2 cos = = + ···  2 4  2σ  ∞   i   (σ − σ) S (σ)dσ , µi = 

(7.31)

0

the next relation is obtained. 2
yrms (π 2 /8 − 1/2)µ2 µ1  − =1+ . 2m0 m0 σ m0 σ 2

(7.32)

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If the spectrum is symmetry for σ ¯, 2 
yrms =1− 2m0



1 π2 − 8 2

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157

 ν2 ,

(7.33)

µ2 m0 m2 − m21 = , m21 m0 σ 2

(7.34)

where ν2 =

in which ν is also the spectrum width parameter.18 Since b  an , yrms ≈ brms , Eq. (7.33) may give a correction for the Rayleigh distribution of wave amplitudes when the spectrum is not a narrow band. Tayfun25 also investigated the effect of spectrum width. A better definition for the zero-crossing wave height than Eq. (7.13) may be given as Hi = Rmi + Rmi+1 ,

(7.35)

where Rmi (i = 1, 2, . . .) are the amplitudes of the wave envelope when the wave profile I takes the local extreme values (t = t∗1 , t∗2 , . . .) [Fig. 1(b)]. The joint pdf for the consecutive amplitudes Rmi , Rmi+1 is     πκξ1 ξ2 π(ξ12 + ξ22 ) π2 ξ1 ξ2 p(ξ1 , ξ2 ) = exp − I , (7.36) 0 4(1 − κ2 ) 4(1 − κ2 ) 2(1 − κ2 ) in which I0 [ ] is the 0th order modified Bessel function, ξi = Rmi /(πm0 /2)1/2 , ξi+1 = Rmi +1 /(πm0 /2)1/2 . They are replaced as ξi = ξ1 and ξi+1 = ξ2 , respectively for simplicity. κ is the correlation parameter given by  1/2 , (7.37) k = ρ2 + λ2 in which ρ=

1 m0



∞ 0

τ S(σ) cos(σ − σ) dσ , 2

−1 ∞ τ λ= S(σ) sin(σ − σ) dσ , m0 0 2 m1 τ = , m0 where τ is the mean wave period. Normalized wave height is given as H ξ1 + ξ2 . ς= = Hm 2

(7.38)

Substituting the relation ξ2 = 2ς − ξ1 into Eq. (7.36) and the integration in terms of ξ1 from 0 to 2ς gives the pdf for ς: 2ς p(ξ1 , 2ς − ξ1 )dξ1 . (7.39) p(ς) = 0

Figure 7.2 shows p(ς) for two different wave spectra [Wallops-type, Eq. (7.11): r = 5 (dotted line), r = 10 (broken line)] and the Rayleigh distribution (solid line). Kimura et al.10 showed that the discrepancy in Fig. 7.2 comes from the difference between I and R at (t = t∗1 , t∗2 , . . .) [Fig. 7.1(b)].

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1.2 1.0

r=5 r = 10

p (ζ)

0.8

Rayleigh 0.6 0.4 0.2 0.0 0.0

0.5

1.0

Fig. 7.2.

1.5 ζ

2.0

2.5

3.0

Pdf of wave height.

7.4. Maximum Wave Height Zero-crossing wave amplitude in Eq. (7.19) extends the range from 0 to infinity if the number of waves is infinity. When the population of waves is finite, such as data from 20 min measurements every 2 h, maximum wave amplitude may be finite. If a time series of wave amplitudes a1 , a2 , . . . , aN , aN +1 , . . . , a2N , a2N +1 , . . . , a3N , a3N +1 . . . , a4N ,             Group No. 1

2

3

4

is divided into groups with N waves, the maximum values (amax ) from individual groups show irregular properties. The pdf of the maximum wave amplitude is derived by Longuet-Higgins.14 The probability that one wave amplitude a is greater than a0 is P (a0 ) [Eq. (7.19)]. When the size of the group is N , the probability that all the wave amplitudes are a < a0 is {1 − P (a0 )}N . The probability that one amplitude, at least, exceeds a0 is 1 − {1 − P (a0 )}N . Therefore, the probability that maximum wave amplitude lies in the range amax ∼ amax + da is 1 − {1 − P (amax )}N − 1 + {1 − P (amax + da)}N = −d[1 − {1 − P (amax )}N ] dP (amax ) = −N {1 − P (amax )}N −1 da . da The pdf for the maximum wave amplitude is     2  2  N −1 amax −amax amax exp p(amax ) = 2N 1 − exp − . arms arms a2rms

(7.40)

(7.41)

Its mean value is derived as E(amax ) = (ln N )1/2 + γ(ln N )−1/2 + · · · arms

(7.42)

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with an approximation   2 N −θ ≈ e−e , 1 − e−(amax /arms )

(7.43)

in which θ =



amax arms

2 − ln N ,

(7.44)

and γ is the Euler constant (γ = 0.5772 . . .). Maximum wave height E(Hmax )/Hrms is usually compared also with Eq. (7.42).

7.5. Zero-Crossing Wave Period The problem of deriving the pdf of zero-crossing wave periods is very difficult. Rice derived a pdf for the zero-crossing interval. He treated the interval where I(t) [Eq. (7.1)] crosses the 0 line upward at t = 0 and crosses the 0 line downward between τ and τ + dτ . 1 τ 1 τ τ p(τ ) = p0 (τ ) − p1 (r, τ )dr + p2 (r, s, τ )drds 1! 0 2! 0 0 1 τ τ τ p3 (r, s, u, τ )drdsdu + · · · , (7.45) − 3! 0 0 0 where p0 (τ ) is the probability that I(t) = 0 at t = 0 and I(t) = 0 lies between τ < t < τ + dτ , p1 (r, τ ) is the probability that I(t) = 0 at t = 0 and I(t) = 0 lies between r < t < r + dr and τ < t < τ + dτ (r < τ ), p2 (r, s, τ ) is the probability that I(t) = 0 at t = 0, and I(t) = 0 lies between s < t < s + ds, r < t < r + dr, and τ < t < τ + dτ (s < r < τ ), and so on. Since this series of integrations is inexcusable, Rice21 showed p(τ ) only applying the first term of Eq. (7.45) as p(τ ) =

dτ 2π

1/2

 −3/2 M23  2 ψ0 −  ψ0 − ψτ2 {1 + H cot−1 (−H)} , ψ0 H

(7.46)

with  2  2 −1/2 − M23 , H = M23 M22 where M22 and M23 are the cofactors of elements µ22 and µ23 of the matrix, 

µ11  µ21 M =  µ31 µ41

µ12 µ22 µ32 µ42

µ13 µ23 µ33 µ43

  µ14 ψ0  0 µ24  = µ34   ψτ ψτ µ44

0 −ψ0 −ψτ −ψτ

ψτ −ψτ −ψ0 0

 ψτ −ψτ   0  ψ0

(7.47)

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and 1 ψτ = 2π ψτ = − ψτ = −



∞

S(σ) cos στ dσ ,

0

1 2π 1 2π



∞

σS(σ) sin στ dσ ,

0

(7.48) ∞

σ 2 S (σ) cos στ dσ ,

0

ψ0 = ψτ |τ =0 ,

ψ0 = ψτ |τ =0 .

Equation (7.46) has a peak around the mean value of τ . The second and third peaks appear for larger τ . This is because the higher terms in Eq. (7.45) are neglected. No successful analytical solution for the zero-crossing wave period has been derived at the present time. Instead of wave periods, Longuet-Higgins16 derived the pdf for wave lengths. The wave profile of unidirectional irregular waves is expressed as ς = R cos {χ} ,

(7.49)

where χ = ux + φ,

χx = u + φx ,

(7.50)

in which R is the amplitude of the wave envelope, φ is the phase, u is the mean wave number in x-direction, and subscript x means a differentiation in terms of x. Joint pdf for χ and χx was derived by Longuet-Higgins15 as p (χ, χx ) = with m∗n



∞

=

µ∗2 = u=

(m∗0 /µ∗2 )

1/2

&3/2 , % 2 4π 1 + (χx − u) · m∗0 /µ∗2

(7.51)

E (u) un du ,

(7.52)

(n = 0, 1, . . .)

0



∞

2

E (u) (u − u) du ,

0 m∗1 /m∗0

,

(7.53) (7.54)

in which u is the wave number in the x-direction and E (u) is its spectrum. The event ς = 0 is equivalent to χ = (2r − 1/2) π or χ = (2r + 1/2) π (r: integer). The probability that χ = (2r − 1/2) π between x and x + dx is 1/2  ∞ 1 m∗2 H(χ)dx = p(χ, χx )χ=(2r−1/2)π |χx | dxdχx = dx . (7.55) 2π m∗0 −∞ Since the probability that χ = (2r + 1/2)π between x and x + dx is given by the same equation, Eq. (7.55), the probability that ς = 0 between x and x + dx is 2H(χ)dx. The average zero-crossing interval is  ∗ 1/2 1/2 m0 π 1 + ν ∗2 =π = , (7.56) ∗ m2 u

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with ν ∗2 =

m∗0 m∗2 − m∗2 1 = m∗2 1



µ∗2 u2 µ∗0

 .

(7.57)

If ν ∗ is small, ≈

π . u

(7.58)

The zero-up-crossing event of ς takes place when χ = (2r − 1/2)π with ∂χ/∂x > 0 or χ = (2r + 1/2)π with ∂χ/∂x < 0. The probability of the latter condition is

∞

−∞

p (χ, χx )χ=(2r+1/2)π |χx | dxdχx 1 = 4



m∗2 m∗0

1/2  −1/2   1 − 1 + ν ∗2 dx .

(7.59)

This probability is negligible for small values of ν ∗ . Between any two successive zero-crossing points x1 and x2 , χ must increase by π. Hence,   1 π = χ(x2 ) − χ(x1 ) = χx + 2 χxx + · · · . 2 The terms on the left of this equation, Taylor series expansion of χ(x2 ) around x = x1 is used. Since χxx = O(ν 2 ), χxxx = O(ν 3 ), . . . and they are very small, neglecting these terms, ≈

π . χx

(7.60)

Applying the same method by Rice,20 a mean zero-crossing number of ς per unit length, the conditional pdf of χx is p(χx )χ =

1 p(χ, χx )|χx | = H(χ) 2



m∗0 m∗2

1/2

|χx |(m∗0 /µ∗2 )1/2 , {1 + (χx − u)2 (m∗0 /µ∗2 )}3/2

(7.61)

in which |χx |/H(χ) is the probability that χ falls between x and x + dx. Since Eq. (7.61) does not involve χ, p(χx )χ directly becomes the pdf for χx . Substituting Eqs. (7.57), (7.58), and (7.60) into Eq. (7.61), p(ξ) =

1 , 2ν{1 + (ξ/ν ∗ )2 }3/2

(7.62)

in which ξ = ( − )/. If the frequency spectrum S(σ) is used instead of the wave number spectrum E(u), the pdf for the zero-crossing interval (half-wave period) is derived following the same method.

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7.6. Joint Pdf for Wave Height and Period Longuet-Higgins18 derived the joint pdf for wave amplitude and period. He used the following expression for the wave profile: ς = Re {A exp (iσt)} ,

(7.63)

where Re means a real part and A = a exp{iφ} ,

(7.64)

in which a is the wave amplitude, φ is the phase of A, and σ is the mean angular frequency given by m1 σ= , (7.65) m0 where m0 and m1 are calculated by Eq. (7.16). The phase of ς is χ = σt + φ. Differentiation of the phase with t becomes χt = φt + σ .

(7.66)

When the spectrum is very narrow, the envelope of ς changes very slowly and φt  σ. The wave period can be approximated as 2π 2π τ= = . (7.67) χt σt + φt Longuet-Higgins derived a joint pdf for a and φt : 



2 a2 a (1/µ0 + φ2t /µ2 ) , p(a, φt ) = exp − 2 (2πµ20 µ2 )1/2

(7.68)

where µ0 and µ2 are calculated with Eq. (7.31), and ν is calculated by Eq. (7.34). With new variables τ a , T = , R= τ (2m0 )1/2 Eq. (7.68) is expressed as ' ' 

2 2 2 ' ∂(a, φt ) ' ' = 2R L exp − R [1 + (1 − 1/T ) ] , p(R, T ) = p(a, φt ) '' ∂(R, T ) ' π 1/2 νT 2 ν2

(7.69)

in which L is the correction factor to neglect the p(R, T ) in the region T < 0, L=

2 . 1 + (1 + ν 2 )−1/2

(7.70)

Figure 7.3 shows the contour map of p(R, T ). Solid curves are contours for p(R, T )/pmax = 0.99, 0.90, 0.70, 0.50, 0.30, 0.10 from inside to outside, respectively. The marginal distribution for normalized wave amplitudes and periods are R/ν 2L p(R) = 1/2 R exp(−R2 ) exp(−β 2 )dβ, (7.71) π −∞

−3/2 (1 − 1/T )2 L . (7.72) 1+ p(T ) = 2νT 2 ν2

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Joint pdf for wave amplitude and period Eq. (7.69) (after Longuet-Higgins18 ).

Fig. 7.4.

Pdf for wave amplitude p(R) Eq. (7.71) (after Longuet-Higgins18 ).

Figures 7.4 and 7.5 show p(R) and p(T ), respectively. If ν = 0, p(R) is not the Rayleigh distribution. Peak of the p(R) slightly shifts to the larger side of R with increasing spectrum width ν. When ν = 0, p(T ) becomes a straight line (broken line, T : constant). Figure 7.6 shows the comparison of data and Eq. (7.69) by Goda.8 The number of waves within a grid of ∆H = 0.2H and ∆T = 0.2T are shown. H and T are the mean values of wave height and period, respectively. Many data sets, their correlation coefficients between H and T distributed from 0.19 to 0.25, are used. The spectrum width parameter in Eq. (7.69) is v = 0.26.

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Fig. 7.5.

Fig. 7.6.

Pdf for wave period p(T ) Eq. (7.72) (after Longuet-Higgins18 ).

Joint distribution of wave height and period (after Goda5 ).

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7.7. Statistical Properties of 3D Irregular Waves Statistical properties of waves, such as length and direction of wave crest, are important in the design of coastal structures. Wave height and period are determined from a record of point measurement in the ordinary method. In contrast, a 3D wave profile is necessary for the definition of wave crest length, direction, etc. As a result, very few studies have examined their properties. 7.7.1. Pdf for wave directions Kimura12 introduced a theoretical pdf for the wave direction. Figure 7.7 shows axes and wave crest lines (dotted line). Two wave gauges are installed along the y-axis. One is on the origin y1 and the other is on y2 (0, y0 ). Figure 7.8 shows the wave profiles measured at these points. ∆t1 , ∆t2 , ∆t3 , . . . are the time lags between corresponding wave crests. The phase lag of individual waves is determined as   ∆ti εi = 2π , (i = 1, 2, . . .) , (7.73) Ti in which Ti is the zero-crossing wave period. The angle between the crest line and y-axis (Fig. 7.7) is

 εi Li /2π −1 θi = sin , (i = 1, 2, . . .) , (7.74) y0 where Li is the wave length corresponding to the zero-crossing wave period Ti and θi is the direction of wave propagation measured from the x-axis.

Fig. 7.7.

Wave crest lines (dotted line) and axes.

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Fig. 7.8.

Wave profiles (solid line) at y1 and y2 .

Assuming y0 u < 1 in which u is the mean wave number in the x direction, wave profiles at these points are η1 (t) =

∞


ci cos(σi t + δi ) = ηc1 cos σt − ηs1 sin σt ,

(7.75)

ci cos(σi t + δi + vi y0 ) =ηc2 cos σt − ηs2 sin σt ,

(7.76)

i=1

η2 (t) =

∞
i=1

in which ci , σi , δi , and vi are the amplitude, the angular frequency, the initial phase, and the wave number in the y-direction of the ith component wave, respectively. σ is the mean angular frequency. ηc1 , ηs1 , ηc2 , and ηs2 are derived in the same way as Eq. (7.4). Their pdfs are the normal distribution with 0 mean, respectively. Their joint pdf is 1 p(ηc1 , ηs1 , ηc2 , ηs2 ) = (2π)2 (m20 − µ23 − µ24 )

1 2 − 2µ3 ηc1 ηc2 − 2µ4 ηc1 ηs2 × exp − (m0 ηc1 2(m20 − µ23 − µ24 )  2 2 2 + m0 ηs1 + 2µ4 ηs1 ηc2 − 2µ3 ηs1 ηs2 + m0 ηc2 + m0 ηs2 ) , (7.77)

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with 2 2 2 2  =
ηs1  =
ηc2  =
ηs2 , m0 =
ηc1

µ3 =
ηc1 ηc2  =
ηs1 ηs2  , µ4 =
ηc1 ηs2  = −
ηs1 ηc2  ,

(7.78)


ηc1 ηs1  =
ηs2 ηc2  = 0 . From the translations ηc1 = R1 cos(σt + φ1 ) = R1 cos(χ1 ) , ηs1 = R1 sin(σt + φ1 ) = R1 sin(χ1 ) , ηc2 = R2 cos(σt + φ2 ) = R2 cos(χ2 ) ,

(7.79)

ηs2 = R2 sin(σt + φ2 ) = R2 sin(χ2 ) , the joint pdf for R1 , R2 , χ1 , and χ2 is 1 p(R1 , R2 , χ1 , χ2 ) = (2π)2 (m20 − µ23 − µ24 )

1 (m0 (R12 + R22 ) × exp − 2(m20 − µ23 − µ24 )

 − 2R1 R2 [µ3 cos(χ1 − χ2 ) − µ4 sin(χ1 − χ2 )]) ,

(7.80)

in which Ri and χi are the amplitude and phase of ηi (i = 1, 2), respectively.15 The crest of the wave profile passes the origin of axes when χ1 = 2nπ (n = 0, 1, 2, . . .). The probability of the event that χ1 = 2nπ between t and t + dt is H(χ1 )dt = m1 /m0 dt. The conditional pdf for R1 , R2 , and χ2 when χ1 = 2nπ is ' σp(R1 , R2 , χ1 , χ2 ) '' p(R1 , R2 , χ2 | χ1 = 2nπ) = ' H(χ1 ) χ1 =2nπ

=

R1 R2 exp 2πm20 (1 − κ2 )



−

1 2 m20 (1

− κ2 )

[m0 (R12 + R22 )

 − 2R1 R2 (µ3 cos(2nπ − χ2 ) − µ4 sin(2nπ − χ2 ))] , (7.81) in which κ2 =

µ23 + µ24 ≈ 1 − y02 u2 γ∗2 m20

(7.82)

and γ∗ is the long-crestedness parameter.15 Putting χ2 = 2nπ + ε, pdf for ε is p(R1 , R2 , ε|χ1 = 2nπ) p(R1 , R2 )  ( 

R1 R2 κ R1 R2 κ cos(ε − β) 2πI0 , = exp m0 (1 − κ2 ) m0 (1 − κ2 )

p(ε|R1 , R2 ) =

(7.83)

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where I0 [ ] is the 0th order modified Bessel function and β = tan−1 (µ4 /µ3 ). β is an angle of the dominant wave direction from x-axis. From Eq. (7.74), the pdf for the wave direction from x-axis becomes 2π y0 cos θ L  (  

2πy0 R1 R2 ) R 1 R2 ) 2 cos 2 , 1 − y 1 − y sin θ − β 2πI × exp 0 ∗ ∗ m0 y∗2 m0 y∗2 L (7.84)

p(θ|R1 , R2 ) =

in which L is the wave length corresponding to σ, and * ) µ2 + µ2 y∗ ≈ 1 − κ2 = 1 − 3 2 4 . m0

(7.85)

If the interval between two wave gauges is infinitesimally small, i.e., y∗ → 0, R1 = R2 → R. Putting β = 0 and applying the relation z → ∞ ⇒ I0 (z) →

exp(z) , 2πz

(7.86)

Eq. (7.84) reduces to p(θ|R)dθ =

 1 R2 R cos θ √ exp − 2 sin2 θ dθ . 2γ∗ m0 γ∗ 2πm0

(7.87)

Figure 7.9 shows p(θ|R) for several R. R is the mean of R. The 3D spectrum is the Bretschneider–Mitsuyasu type with Mitsuyasu-directional function modified by Goda.8 The peakedness parameter for the directional function is Smax = 10. The pdf for larger R shows a narrower width. 7.7.2. Mean length of 3D wave crest The length of 3D irregular wave crests may affect the total wave force on coastal structures with finite width such as caisson-type breakwaters, for example. Since a definition of the wave crest length is difficult, there are very few studies about its statistical properties. Goda6 studied its statistics numerically. He calculated a spatial wave profile η(x, y) of 3D irregular waves. The black and white pattern in Fig. 7.10 shows  η(x, y) > ηth , Black area , (7.88) η(x, y) < ηth , Other , in which ηth is the threshold value (ηth = 0.1H1/3 in Fig. 7.10). The 3D wave spectrum is the Bretschneider–Mitsuyasu type (H1/3 = 2.0 m, T1/3 = 8.01 s) with Mitsuyasu-directional function modified by Goda, and Smax = 25. The pattern recognition method is applied to determine the wave crest lengths. Four threshold values ηth = 0.2 m, 0.5 m, 1.0 m, and 2.0 m were used. The calculated

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Pdf for wave directions: Eq. (7.87).

crest lengths were arranged in descending order. The top 0.4%, 1%, 5%, and 10% of total data for each ηth were used in the analysis, respectively. Figure 7.11 shows the distribution of the mean length of wave crests. The four lines in the figure show the results from the above four different data. 7.7.3. Pdf of the local maximum of 3D wave amplitudes Figure 7.12 shows a short crested spatial wave profile and a wave gauge. The fixedtype wave gauges usually fail to record the peak wave profile in the neighborhood. Wave height changes along a large object such as a ship, caisson-type breakwater, etc. If a large wave hits any part of these objects, it is recognized that the large wave attacked it regardless of its position. Since local damage on the caisson-type breakwater, for example, may lead to the whole failure, the size effect on the pdf becomes important. Kimura et al.11 showed the pdf of the maximum wave amplitudes within a finite interval. x , y  , and vertical z  -axes are taken in the 3D wave field. The vertical plane H is installed parallel to x -axis. Figure 7.13 schematically shows the envelope R(x ) (solid line) for the cross section of the 3D wave profile on H. A wave gauge is installed at x = 0. The width of the object is 2∆L (−∆L < x < ∆L). R∗ is the local maximum amplitude of R(x ) (−∆L < x < ∆L), and R(0) is the amplitude of the envelope at x = 0. ∆R is the difference between R∗ and R(0). ∆R is approximately

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Fig. 7.10.

Fig. 7.11.

Wave crests (after Goda6,7 ).

Mean length of wave crest (after Goda6,7 ).

given as '  ' ' R (0) ' ' ' ' R (0) ' ≤ ∆L : ' '  ' R (0) ' ' ' ' R (0) ' > ∆L

∆R = −

{R (0)}2 , 2R (0)

(7.89)

R (0)(∆L)2 , 2

R (0) ≥ 0 :

∆R = R (0)∆L +

R (0) < 0 :

∆R = −R (0)∆L +

R (0)(∆L)2 , 2

(7.90) (7.91)

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Fig. 7.12.

Fig. 7.13.

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3D wave profile and the vertical plane H.

Wave envelope (solid line) for 3D wave profile.

in which R (0) and R (0) are the first and second derivatives of R(x ) at x = 0, respectively. The probability that R∗ falls between c < R∗ < c + dR on condition R(0) is p[R (0), R (0) : R(0)]dS , (7.92) Prob(c < R∗ < c + dR)|R(0) = S

in which p[R (0), R (0) : R(0)] is a conditional joint pdf of R (0) and R (0) on condition R(0), and S is the region in which c < R∗ < c + dR in the R ∼ R plane. The probability that R∗ falls between c < R∗ < c + dR regardless of R(0) is Prob(c < R∗ < c + dR) = 0

c

{Prob(c < R∗ < c + dR)|R(0) }p[R(0)]dR(0) ,

(7.93)

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Fig. 7.14. Pdf of local maximum wave amplitudes (dotted lines) within three finite width ∆L = 0 (Rayleigh dist.), 0.05L1/3 , and 0.10L1/3 .

where p[R(0)] is the pdf of R at x = 0 (Rayleigh distribution). Connecting Eqs. (7.92) and (7.93), the pdf of the local maximum amplitudes within the finite width is given as R∗ p(R∗ )dR∗ = p[R(0), R (0), R (0)]dGdR(0) , (7.94) 0 

G



in which p[R(0), R (0), R (0)] is the joint pdf of R(0), R (0), and R (0), and G is the region in which the local maximum amplitude falls within R∗ and R∗ + dR∗ on condition R(0). Rice20 formulated p[R(0), R (0), R (0)]. Figure 7.14 shows p(R∗ ) (dotted lines). The spectrum is the Bretschneider– Mitsuyasu-type with H1/3 = 5.5 m and T1/3 = 10 s and Goda’s directional function with Smax = 10, the water depth is h/L1/3 = 1.0, and the half-object width is ∆L = 0.050L1/3 and ∆L = 0.10L1/3, respectively. The Rayleigh distribution (solid line ∆L = 0) is also shown in the figure. Broken lines are the Weibull distribution with m ≈ 2.5 (shape factor) shows good agreement when ∆L/L1/3 > 0.05 (h/L1/3 = 1.0, Smax = 10). 7.8. Statistics of the Time Series of Wave Heights Sawhney23 and Goda4 started to investigate the statistical properties of the time series of irregular wave heights, . . . , Hi−2 , Hi−1 , Hi , Hi+1 , Hi+2 , . . . ,

(7.95)

since this property may affect wave overtopping, slow drift oscillation of floating bodies, etc. This property is analyzed by applying the concept of “run.” Figure 7.15 shows a time series of wave heights. The run of high waves is defined by waves, which are Hi > H∗ consecutively, in which H∗ is the threshold wave height. In the case shown in Fig. 7.15, the length of the high wave run is 3.

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Fig. 7.15.

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Run of wave heights (high waves, low waves, and total run).

Goda4 gave the pdf for the run of high waves as p(j1 ) = pj1 −1 q ,

(7.96)

in which j1 is the length of the high wave run, p is the probability that the event H > H∗ takes place and q = 1 − p. Since the pdf for wave heights is the Rayleigh distribution, p and q are given as 

H2 , p = exp − ∗ 8m0 (7.97) q = 1−p. The mean and standard deviation of j1 are j1 =

1 , q

√

σ(j1 ) =

p , q

(7.98)

respectively. The recurrence interval (total run) of high waves is defined as the sum of the high wave run and the consecutive low wave run. The run of low waves is defined by waves, which are Hi < H∗ , consecutively. The total run in Fig. 7.15 is jt = 3 (j1 : high wave run) + 4 (j2 : low wave run) = 7. The pdf for the total run is pq p(jt ) = (pjt −1 − q jt −1 ) . (7.99) p−q The mean and standard deviation of jt are jt =

1 1 + , p q

σ(jt ) =

+

q p + 2, 2 q p

(7.100)

respectively. Equations from (7.96) to (7.100) are derived assuming that the wave height is independent: γ(Hm Hm+n ) = 0,

m, n = 1, 2, 3, . . . ,

(7.101)

where γ( ) is the correlation coefficient of wave heights. From observation, Rye22 showed that there is a correlation between consecutive wave

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heights γ(Hm Hm+1 ) = 0. From numerical calculations, Kimura9 showed that γ(Hm Hm+1 ) = 0 but γ(Hm Hm+n ) ≈ 0 (n ≥ 2). He assumed the time series of wave heights from a Markov chain9 and introduced the pdf for the high wave run: p(j1 ) = pj221 −1 (1 − p22 ) .

(7.102)

p22 is the transition probability from the event Hi > H∗ to Hi+1 > H∗ . Its mean and standard deviation are √ p22 1 j1 = , σ(j1 ) = . (7.103) 1 − p22 1 − p22 The pdf for the total run is p(jt ) =

(1 − p22 )(1 − p11 ) jt −1 (p22 − pj11t −1 ) . p22 − p11

(7.104)

p11 is the transition probability from the event Hi < H∗ to Hi+1 < H∗ . The mean and standard deviation of jt are 1 1 jt = + , 1 − p22 1 − p11 (7.105) 1/2

p11 p22 σ(jt ) = + , 2 (1 − p11 ) (1 − p22 )2 where p11 and p22 are given by H∗ H∗ p(H1 , H2 )dH1 dH2 / p11 = 0



p22 =

0 ∞ H∗



p(H1 )dH1 ,

0

∞ H∗

p(H1 , H2 )dH1 dH2 /

with p(H1 , H2 ) =

H∗



H12 + H22 2 (κ2 − 1)Hrms 

2H1 H2 . p(H1 ) = 2 exp − 21 Hrms Hrms 4H1 H2 exp 4 (1 − κ2 )Hrms

∞

H∗



 I0

(7.106) p(H1 )dH1 ,

2κH1 H2 2 (1 − κ2 )Hrms

 ,

(7.107) (7.108)

Equations (7.107) and (7.108) are the 2D Rayleigh and Rayleigh distributions, respectively. k is the correlation parameter. The relation between κ and the correlation coefficient γ(Hm Hm+1 ) is

 1 π 1 2 γ(Hm Hm+1 ) = E(κ) − (1 − κ )K(κ) − , (7.109) 1 − π/4 2 4 in which E( ) and K( ) are complete elliptic integrals of the first and second kind, respectively. Battjes et al.1 and Longuet-Higgins19 showed κ and the relation between κ and γ(Hm Hm+1 ) as   2  ∞ 2  ∞ 1 2 S (σ) cos στ0 dσ + S (σ) sin στ0 dσ (7.110) κ = m0 0 0

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Fig. 7.16. (a) Mean length of high wave runs (solid line: Eq. (7.103); circles: after Goda8 ). (b) Mean length of total runs (solid line: Eq. (7.105); circles: Goda used j2 for the total run, after Goda8 ).

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and κ2 ≈ γ(Hm Hm+1 ) κ2 < 0.6 , γ(Hm Hm+1 ) ≈ 1 −

1 − κ2 4−π

κ: very close to 1 .

Figures 7.16(a) and 7.16(b) show the comparison between j1 and Eq. (7.103), and jt and Eq. (7.105), respectively. The threshold wave height is H1/3 .

7.9. Remarks Statistical studies have shown us the entire structures of wave irregularities in sea conditions. A knowledge about their pdf has brought us an advance not only for designing coastal structures but also for understanding nearshore hydraulics of the wave and current system. However, we have to keep several things in mind when we use these wave statistics. The Rayleigh distribution for wave height is derived on the condition that the wave spectrum is narrow. The problem still remains to clarify a good correspondence between measured distributions and the Rayleigh distribution regardless of the spectrum. The assumption of the narrow band spectrum gives a considerable effect on the pdf of wave periods. Although the theoretical models have the spectrum parameter in the equation, we have to be careful about the stretch use for the spectrum with finite band width. The range of the pdf may be changed by wave dynamics such as wave breaking. Extensive application of the theoretical pdf is sometimes risky. The applications of the periodic wave theory for zero-crossing wave properties raise problems. There have been very few studies about the physical properties of zero-crossing waves, of wave period–wavelength/celerity, of wave height/period — water particle velocities, etc. Finally, we have to take into account the statistical properties for 3D irregular waves. Common definitions for wave properties, first of all, become necessary: for wave height, wave period, wavelength, celerity, wave direction, wave crest length, etc.

References 1. J. A. Battjes and G. Ph. van Vledder, Verification of Kimura’s theory for wave group statistics, Proc. 19th ICCE (1984), pp. 642–648. 2. D. E. Cartwright and M. S. Longuet-Higgins, The statistical distribution of the maxima of a random function, Proc. R. Soc. Lond. A 237, 212–232 (1956). 3. G. Z. Forristall, On the statistical distribution of wave heights in a storm, J. Geophys. Res. 83(C5), 2353–2358 (1978). 4. Y. Goda, Numerical experiments on wave statistics with spectral simulation, Report PHRI 9(3), 3–57 (1970).

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5. Y. Goda, On the joint distribution of measured wave heights and periods, Tec. Note of PHRI, No. 272, (1977) 19pp. (in Japanese). 6. Y. Goda, Statistics of wave crest lengths based on directional wave simulations, J. Offshore Mech. and Arctic Eng., Trans. ASME 116, 239–245 (1994). 7. Y. Goda, Numerical simulation of ocean waves for statistical analysis, J. Marine Res. 33, 5–14 (1999). 8. Y. Goda, Random Sea and Design of Maritime Structures, 2nd Edn. (World Scientific, 2000), p. 443. 9. A. Kimura, Statistical properties of random wave groups, Proc. 17th ICCE (1980), pp. 2955–2973. 10. A. Kimura and T. Ohta, Statistical property of the irregular wave heights defined by the zero-crossing method, Proc. Coastal Eng. 40, 146–150 (1993) (in Japanese). 11. A. Kimura and T. Ohta, Probability distribution of the maximum wave height along a sea wall with finite width, Proc. 25th ICCE (1996), pp. 2272–2283. 12. A. Kimura, Probability distribution of 3-D irregular wave directions, Proc. Waves 97, 295–306 (1997). 13. B. M. Lake and H. C. Yuen, A new model for nonlinear wind waves, 1, Physical model and experimental evidence, J. Fluid Mech. 88, 33–62 (1978). 14. M. S. Longuet-Higgins, On the statistical distribution of the heights of sea waves, J. Marine Res. IX(3), 245–266 (1952). 15. M. S. Longuet-Higgins, The statistical analysis of a random, moving surface, Phil. Trans. Roy. Soc. Ser. A 249(966), 321–387 (1957). 16. M. S. Longuet-Higgins, On the intervals between successive zeros of a random function, Proc. Roy. Soc. Ser. A 246, 99–118 (1958). 17. M. S. Longuet-Higgins, On the distribution of the heights of sea waves: Some effects of nonlinearity and finite band width, J. Geophys. Res. 85(C3), 1519–1523 (1980). 18. M. S. Longuet-Higgins, On the joint distribution of wave periods and amplitudes in a random wave field, Proc. Roy. Soc. Ser. A 389, 241–258 (1983). 19. M. S. Longuet-Higgins, Statistical properties of wave groups in a random sea state, Phil. Trans. Roy. Soc. Ser. A 312, 219–250 (1984). 20. S. O. Rice, The mathematical analysis of random noise, Bell System Tech. J. 23, 282–332, 24, 46–156 (1944). 21. S. O. Rice, Selected papers on Noise and Stochastic Processes, ed. N. Wax (Dover, 1945), pp. 133–294. 22. H. Rye, Wave group formation among storm waves, Proc. 14th ICCE (1974), pp. 164–183. 23. M. D. Sawhney, A study of ocean wave amplitudes in terms of the theory of runs and a Markov chain process, Tech. Rep. of New York University (1962), p. 29. 24. R. J. Sobey, Correlation between individual waves in a real sea states, Coastal Eng. 27, 223–242 (1996). 25. M. A. Tayfun, Effects of spectrum band width on the distribution of wave heights and periods, Ocean Eng. 10(2), 107–118 (1983).

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Chapter 8

Generation and Prediction of Seiches in Rotterdam Harbor Basins M. P. C. de Jong Formerly at Environmental Fluid Mechanics Section Delft University of Technology; Presently at Deltares |Delft Hydraulics P. O. Box 177, 2600 MH, Delft, The Netherlands J. A. Battjes Environmental Fluid Mechanics Section Delft University of Technology, P. O. Box 5048, 2600 GA Delft, The Netherlands A resume is presented of recent work that has identified the mechanism through which the seiches are generated that occasionally occur in Rotterdam Harbor basins, and that has led to the development and implementation of an operational warning system for the occurrence of significant seiche events. The generation is due to moving systems of atmospheric convection cells that can arise over the relatively warm water of the North Sea, behind a cold front. These cause fluctuations in wind speed and atmospheric pressure that can generate long waves at sea, as these cells move toward the coast. The average variance spectrum of these long-wave surface elevations is found to have an approximately f −1.5 tail. As these long waves at sea approach the harbor mouth, they can be resonantly amplified inside certain semi-closed basins. The ratio of seiche amplitudes in two different basins, derived through numerical simulations based on a 2DH mildslope model and an incident f −1.5 long-wave elevation spectrum, is in excellent agreement with the mean ratio derived from observations for the same locations. Finally, the principle of a possible system of prediction of the occurrence of seiche events is described. Its predictive potential is proven through a series of hindcasts. Based on these promising results, an operational forecasting system has been developed and implemented by the authorities involved.

8.1. Introduction Resonant harbor oscillations (seiches) occur in a number of basins of the Port of Rotterdam, The Netherlands, during rough weather episodes. Within the Rotterdam 179

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Fig. 8.1.

The Port of Rotterdam.

Harbor system, the highest seiches occur in the Caland Canal, with measured crest heights over 1 m at the closed end near Rozenburgse Sluis, ROZ (see Fig. 8.1). This semi-closed basin has a length of approximately 20 km and a depth of approximately 20 m. For almost all of the observed seiche events at ROZ, the dominant frequency (determined through wavelet analysis) equals the lowest eigenfrequency (quarter-wavelength mode), equivalent to a period of approximately 90 min. Smaller-amplitude seiches, though still quite significant, also occur in other harbor basins. The seiche oscillations in Rotterdam Harbor give rise to some practical concerns. First, water levels used for the design of dikes (in particular their crest height) are required by law to contain an allowance for seiches, superposed on the astronomical high water level (spring tide) and wind setup. The appropriate measure of this allowance is not well known. Second, seiches affect the design and management of a movable storm surge barrier (MSSB) in Rotterdam Waterway (Fig. 8.1), which protects the low areas behind it against flooding. The MSSB must be closed during extreme storm conditions. In that case a semi-closed basin is created on its seaside. Seiches are expected to occur in this temporary basin, with a dominant eigenperiod of approximately 30 min. These contribute to the failure probability of the barrier, but their expected heights were not known. Third, the Rotterdam Port Management needs to guarantee a minimum depth (“nautical depth”) in the approach channel and throughout the harbor for deepdraught vessels. To this end, water levels are predicted operationally including tides and wind-induced setup. Until recently, no method was available for inclusion of seiche events in this short-range prediction, despite their practical importance: the trough of a seiche could cause ships to run aground in case of a narrow predicted tidal window. The aforementioned practical applications in which seiches play a role have led to a recent study on the origin of the seiche events in Rotterdam. The present chapter summarizes the findings. Results are also presented of an analysis of the response of Rotterdam Harbor basins to incoming wave energy in the seiche frequency band using measured and theoretical ratios of seiche amplitudes at the closed ends of two basins. Finally, a new system of routine prediction of seiche events is described,

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based on the newly identified generation mechanism, as well as an indication of its forecasting skill. The above-mentioned research and results have been more fully described by de Jong and Battjes.1,2

8.2. Measurements 8.2.1. Data acquisition The sea surface elevation measurements used in this study were obtained at three offshore platforms located approximately 20–40 km from the harbor mouth: Europlatform (EUR), Lichteiland Goeree (GOE), and Meetpost Noordwijk (MPN). Inside the harbor, surface elevation measurements are available from multiple locations (more than 10 key-locations are monitored). These have been taken continuously since 1995 with floaters, sampled at a 60 s interval. In addition, sea surface temperature measurements have been obtained from 1995 to 2001 at platforms located several hundreds of kilometers to the north and north-west of the harbor mouth in the North Sea (along the path of the relevant storm systems approaching the Port of Rotterdam). These were available as 24-h averaged daily values. Vertical profiles of air temperature and other meteorological parameters, obtained from balloon measurements, were available at the North Sea at Ekofisk platform every 12 or 24 h. 8.2.2. Data analysis To identify seiche events, records of the surface elevations at ROZ were Fourier band-passed filtered, passing the seiche frequency band (0.1–2.0 mHz). A wavelet analysis based on the Morlet wavelet3−6 has been applied to the filtered surface elevation data and meteorological data. This technique is suited for the detection of fluctuations that come in bursts. It has been used to identify the seiche events inside the harbor and the corresponding periods of increased levels of low-frequency energy at sea. Following first identification from the wavelet spectra, the time intervals with seiche events were reviewed in more detail.

8.3. Origin of the Seiches in Rotterdam Harbor 8.3.1. Generation of seiche events by atmospheric convection cells In this section, a short description is given of the mechanism generating the long waves at sea that cause seiche episodes inside the harbor (as mentioned above, a more detailed description can be found in the work of de Jong and Battjes1,2 ). Meteorological measurements and visual inspection of weather charts show that all seiche episodes in Rotterdam Harbor (arbitrarily defined as a seiche-event in which the crest height exceeded 0.25 m at ROZ, a criterion that was applied in several previous studies of seiches at this location) coincided with low-pressure

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systems with cold fronts crossing the southern North Sea toward the Dutch coast. However, the converse does not hold: not all such cold front passages give rise to seiche events. Analysis of measurements and numerical simulations showed that the seiches in Rotterdam Harbor result from resonant amplification of oscillations at sea, which in turn are generated by fluctuations in atmospheric pressure and wind speed following a moving cold front. For the large majority of the seiche events, these fluctuations are caused by atmospheric convection cells that can arise in the area behind a cold front. These cells are a mesoscale atmospheric phenomenon, with typical horizontal spatial scales of 30–100 km.7 Conditions conducive to the formation of convection cells occur when cold (polar) air moves over relatively warm sea water, which is likely to occur in fall and winter (also the seasons of significant seiche episodes). Especially in the area behind the cold front, the relatively warm sea water causes heating of the air in the lower layers, which results in an unstable lower atmosphere that causes air to rise to higher altitudes in narrow plumes, (self-)organized into a more or less regular pattern of convection cells. If sufficient moisture is present in the air, condensation of water vapor can locally cause a release of latent heat, further driving the circulation of air. Narrow bands of clouds can then be formed surrounding clear skies, which can be seen on satellite images (an example is shown in Fig. 8.2) and, in case of precipitation,

Fig. 8.2. Satellite image of infrared frequency band (10:12 GMT, 11 January 1995), taken during a seiche event in the Port of Rotterdam. The image shows a pattern of convection cells that passes the southern North Sea from the northwest. The location of the Port of Rotterdam is indicated by the black and white dot. Source image: Dundee Satellite Receiving Station, Dundee University, Scotland.

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also on radar precipitation images. These satellite and precipitation images showed that the episodes of convection cells passing the southern North Sea more or less coincided with the seiche episodes. The convection cells cause more or less spatiallyperiodic fluctuations in atmospheric pressure and wind speed, which in a fixed point will manifest themselves as temporal fluctuations as the cells are advected with the weather system. Because the system of convection cells moves, these fluctuations can generate low-frequency waves at sea, which in turn can cause seiches in the harbor basins. A sketch of the sea surface elevation response to the wind speed changes caused by the convection cells is depicted in Fig. 8.3. A 1D model based on the linearized shallow water equations was applied for a transect across the North Sea along the main path of the convection cells, forced by sinusoidal variations in atmospheric pressure and wind speed translating at a constant speed, with values that are typical for those observed. Negligibly low responses were found in the more remote, deeper parts of the North Sea. However, the model showed that the fluctuations in wind speed and atmospheric pressure caused by the passing convection cells are particularly efficient at generating long waves with significant amplitudes in the relatively shallow part of the North Sea, the southernmost 400 km near the Dutch coast. This is because in this region, with typical depths of approximately 30 m, the phase speed of shallow-water waves is approximately 17 m/s, i.e., near the advection velocity of the system of cold front and convection cells (typically 15–20 m/s). The result is a near-resonant response of the long waves at sea to the moving pattern of atmospheric convection cells, with amplitudes up to 1–2 dm near the Dutch coast. These waves can in turn be amplified in semi-closed harbor basins through (a quarter-wave) resonance, occasionally forming seiches with observed amplitudes (crest heights) up to a meter. 8.3.2. Energy density spectrum of surface elevation at sea Most previous studies of the seiches in the Port of Rotterdam focused on the relative responses in different basins. To this end, and for lack of better information, those studies assumed a uniform incident energy density spectrum (a “white noise” spectrum). Veraart8 first studied the spectra of measured surface elevations at the

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southern North Sea in the seiche frequency band, obtained during seiche events at ROZ. Based on a relatively small number of seiche episodes he found that, on average, the variance density spectrum in the seiche frequency band approximately followed a f −1.2 trend. From June 2000, numerous processed and validated surface elevation data have become available for a number of storm events. These have now been used to determine the spectrum in the seiche frequency band at sea in more detail (results described here are based on measurements from the platform EUR). The variance density spectra have been estimated from filtered records (0.1– 2.0 mHz) of 4096 min (212 values) approximately centered around the seiche-event, covering the complete storm duration of each event. Spectra of a number of events normalized by the energy content of the seiche frequency band are shown in Fig. 8.4. A least-squares fit of a power law to the averaged normalized spectral densities results in a f −1.5 trend.

8.4. Seiche Amplitudes at Different Locations 8.4.1. Seiche events in different basins of the Port of Rotterdam A total of 49 seiche events were registered at ROZ in the studied time interval (1995–2001). For this location, each such event is defined as a period of enhanced

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seiche activity with amplitudes (crest heights) exceeding 0.25 m. During most of these events, the response at the lowest eigenperiod of the Caland Canal (90 min) was dominant. Surface elevation measurements at Europe Harbor (EH, see Fig. 8.1) have also been studied for the same time interval as the observations at ROZ (1995–2001). These time series were filtered for the frequency band 0.3–2.0 mHz in order to isolate the main eigenfrequencies of this (shorter) harbor basin. Based on a wavelet analysis, and using 0.1 m as an amplitude threshold, 46 seiche events at EH were identified, all at the dominant eigenperiod of 50 min. Measurement data at ROZ were available during 43 of the 46 events that were identified at EH. Of these 43 events at EH, 86% coincided with a seiche-event at ROZ. This percentage depends to some extent on the thresholds for the identification of a seiche-event. These have been chosen subjectively: in case a threshold of 0.15 m is chosen for the identification of seiche events at EH, 100% of the events at EH coincide with a seiche-event at ROZ (with amplitude exceeding 0.25 m). Conversely, of the 49 events at ROZ, 82% coincided with a seiche-event at EH. In this case too, this result depends on the threshold of the definition of a seicheevent. In case a threshold of 0.40 m is applied at ROZ, 100% of the events at ROZ coincided with an event at EH (with a threshold of 0.10 m). The fact that large-amplitude seiche events simultaneously occur in basins of different dimensions indicates that the same mechanism generates seiches in basins of different natural frequencies. 8.4.2. Empirical amplitude ratio Surface elevation signals from ROZ and EH have been filtered for the complete seiche frequency band (0.1–2.0 mHz) in order to compare the crest heights during coinciding seiche events in the Caland Canal and the Europe Harbor. For this purpose, a typical local seiche amplitude could be based on, for example, the standard deviation of the filtered surface elevation signals. However, the time span for which this value needs to be determined (for each event separately) is not obvious. This is problematic in view of the transient nature of the seiche events. Therefore, the maximum crest height of an event at each location has been chosen. Moreover, in order to get a more robust estimate, the average of the five largest crest heights and the five largest trough depths of an event has also been used. The latter parameter is taken as a typical average seiche amplitude of an event. Figure 8.5 shows these average seiche amplitudes for the 30 coinciding events in 1995–2001 for ROZ and EH for which the average amplitude could be determined from at least five crests and five troughs. These data have a correlation coefficient of 0.92. A linear trend line was least-squares fitted to the data, indicating that on average the seiche amplitudes at EH are a factor of approximately 0.56 smaller than those at ROZ. A scatter plot based on the single maximum crest height for all coinciding events (not shown) gave a similar result, though with some more scatter (correlation coefficient 0.7).

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Fig. 8.5. Scatter plot of average seiche amplitudes (the average of the five highest crests and five deepest troughs of an event) of coinciding seiche events at ROZ and Europe Harbor. Trend line based on least-squares estimate.

8.4.3. Theoretical amplitude ratio The calculation of theoretical seiche amplitude ratios (r) at two locations a and b is based on a combination of the energy supply at the harbor mouth and its amplification according to
 S(f )Ra2 (f )df , (8.1) rab =  S(f )Rb2 (f )df where S(f ) is the (normalized) energy spectrum incident from the sea at the harbor mouth and Ra (f ) and Rb (f ) are the amplitude amplification spectra for the considered locations a and b. These had been determined previously using a 2DH mildslope numerical wave model (PHAROS).9 Effects of bottom stress, inflow losses, and seaward radiation were included. Applying these response functions and the empirical seiche energy spectrum with a f −1.5 tail, the theoretical amplitude ratio between EH and ROZ is found to be 0.57, nearly equal to the value of 0.56 that was estimated from the measurements. This result, based on 30 seiche events, supports the validity of the theoretical approach, and it confirms that the energy density at sea in the seiche frequency band typically follows a f −1.5 trend, not just in the three events from which it was estimated.

8.5. Prediction of Occurrence of Seiche Events 8.5.1. Principle of prediction and first verification As noted in the Introduction, practical needs related to shipping or to the movable storm surge barrier require real-time knowledge of expected occurrences of seiches

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in Rotterdam Harbor. However, operational prediction of these cannot be based on measurements at available platforms at the North Sea, because the response at the remote stations is too low, whereas the nearby platforms are too close to allow a useful prediction window. Since moving convection cells are at the basis of the generating mechanism of seiche events in Rotterdam Harbor, circumstances needed for these cells to arise can, in principle, be used to predict the occurrence, and also the absence, of seiche events. The feasibility of this approach was shown by de Jong and Battjes2 using historical data. Based on this promising academic result, a pilot study has been set up using real-time data and weather predictions, as a precursor of a fully operational system. This confirmed that the method was indeed feasible and that it had sufficient forecasting skill to warrant implementation of an operational version. Such a system, in which predictions with windows of 12 h and 24 h are provided to relevant parties throughout the harbor, has now been in operation since 2005 and so far has functioned to satisfaction. The basis for the prediction system is as follows. Convection cells of sufficient strength arise only if the lower atmosphere is (sufficiently) unstable. The criterion for this is that the adiabatic lapse rate (rate of decrease in temperature from the sea surface upward) exceeds some threshold, the value of which depends on the moisture content because of its consequence for possible cloud formation. In case the lapse rate is smaller than 5.5 to 7.5◦ C/km, the lower atmosphere is stable; here 6.5 C/km is used as a representative critical value. Between approximately 10◦ C/km and (say) 6.5◦ C/km, it is conditionally unstable (depending on the moisture content), and a temperature lapse rate larger than 10◦ C/km results in an unstable atmosphere.10 Prediction of the lapse rate and moisture content using multi-layered atmospheric models is part of routine weather forecast systems. This, therefore, also allows operational prediction of the occurrence of significant seiche events on a routine basis, just like common weather prediction. To test the feasibility of this approach, several months of atmospheric temperature and pressure data as well as sea surface water temperature data have been used for “predictions” in a hindcast mode, comparing the outcomes to observations of occurrences (or not) of seiche events. Results of a three-month period are shown here as a typical example of the outcome. The lapse rate can in practice be estimated using the air temperature at sea level (provisionally taken equal to the sea surface water temperature) and at some higher elevation. In meteorology, standard reference levels in the boundary layer are used corresponding to an atmospheric pressure of 925 hPa (average altitude of about 700 m) and 850 hPa (average altitude of about 1500 m). At the latter altitude, cloud formation can occur, which would result in the release of latent heat, implying influence of moisture content on the possible generation of convection cells. Indeed, preliminary tests showed that correct predictions as well as false predictions are found if the average altitude of the reference pressure of 850 hPa is used and moisture is not taken into account. However, generally speaking, the air will not become saturated below the level of the 925 hPa pressure. Therefore, cloud formation will not occur below this level. In this case, no information about moisture content of the air is required because

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the uncertain range of conditional stability is excluded. A lower limit of the lapse rate for unconditional instability of 10◦ C/km then applies. The following “predictions” use this criterion and the estimated temperature gradient between sea level and the observed level of 925 hPa pressure. For comparison, results based on the average elevation of this pressure level (approximately 700 m) are shown as well. The latter would be simpler operationally, albeit at the cost of enhanced errors in the verification of the temperature difference criterion. The results of the aforementioned approach are illustrated by presenting the data of January till March 1999. This time interval was selected because the quality of the data was relatively high during these months (only a small percentage of missing data due to, e.g., atmospheric soundings that were not available on certain days). Weather maps have been used to identify the days during which cold fronts have passed the North Sea that resulted in a flow toward the Dutch coast (including possible successive days during which this flow direction continued). This is obviously needed for seiche events to be generated in Rotterdam Harbor. Figure 8.6 shows the results. The bars in the top panel show the observed difference between the air temperature at the (observed) altitude of the reference pressure 925 hPa (at platform EKO) and the sea surface water temperature (at platform AUK) for each day (not restricted to days of a front passage). Gaps are present for those days for which no data were available. The lower panel shows the

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same data, now restricted to days during which cold fronts have passed the North Sea toward the Dutch coast. The straight dashed lines in both panels indicate the critical temperature difference for an unstable atmosphere based on the average altitude of the pressure level of 925 hPa (∆T = 7◦ C). The wiggly curves in both panels indicate the value of the critical temperature difference for an unstable atmosphere from day to day, based on the temperature at the measured altitude of the reference pressure of 925 hPa for each day. A seiche-event is predicted for those days in the lower panel for which the observed temperature difference (the bar) exceeds the critical one (indicated by the wiggly curve). The asterisks indicate the days for which seiche events have been identified at ROZ. (Consecutive days with seiche events indicated represent a single event that started during the first day and ended on the last day indicated.) The bottom panel shows that an unstable atmosphere is found during a number of potentially seiche-prone cold-front situations, correctly predicting a seicheepisode. All the seiche events that occurred in this time interval are correctly predicted (no misses). A stable atmosphere is found during most other days indicated in the bottom panel and no convection cells are expected during these days (regardless of the availability of moisture in the air), which agrees with the fact that no seiche events occurred. Nevertheless, a limited number of false predictions is also found (17 January and 28 February). On these days, the temperature difference just slightly exceeded the critical value. The resulting false predictions could therefore easily be due to the criterion not being strict enough, or to the spatial variability of the temperature of the sea surface water and the air at the reference levels: although the nominal criterion is met locally, this does not necessarily have to be representative for the complete southern North Sea region since only observations from sparse locations are used. This will not be the case during the correctly predicted seiche events in view of the fact that the prediction on these days is based on relatively large temperature differences, which far exceed the critical values. 8.5.2. Operational seiche prediction The evidence presented above indicates that conditions for the existence of convection cells, together with the information regarding the passage and direction of cold fronts, can be used to identify and predict seiche-prone situations. In an operational setting, the predicted weather chart for the next 24 h is available with relatively high reliability, and the predicted air temperature can be obtained from numerical weather prediction models. The moisture content of the air could also be taken into account. Moreover, a more thorough analysis of the stability of the lower atmosphere is possible, since information of the complete vertical and for the complete southern North Sea area is available (possibly avoiding false alarms due to criteria that are only met locally). This makes the use of specific reference levels unnecessary. Therefore, all relevant parameters are available for the prediction of seiche-prone situations with a 24-h prediction window, sufficient for the port

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authorities and for the closure management of the movable storm surge barrier of Rotterdam (which typically require prediction windows of, say, 6 h). Based on the aforementioned findings, a pilot project was implemented, set up by the Ministry of Public Works and Waterways in cooperation with the Royal Dutch Meteorological Institute (KNMI) for operational prediction of the occurrence of seiche events in the Port of Rotterdam, using parameter values derived from operational numerical weather prediction models. This pilot project has shown that the system has sufficient predictive skill for operational use. Such a system has since been implemented on an operational basis. Experience shows that all major seiche events (above the seiche-amplitude threshold) have been successfully predicted. However, in some cases a false prediction was made in the sense that a warning was issued while no significant seiches occurred. This is a drawback that needs repair, but even as it is, the system has merit because it does not miss significant events. The false alarms mentioned above are related to an important limitation of the above-described method, viz, the fact that so far it only predicts seiche-prone situations but not the expected energy level of the seiche-event. In case additional parameters are taken into account, such as the (range in) sizes of the atmospheric convection cells, the advection velocity, the excess of the temperature difference above the critical value (not just the question whether it exceeds a critical value or not), and the direction of movement of the weather systems, prediction of the seiche energy level might become possible, for example, through the use of a neural network that can be trained with a set of observed values of these quantifiable parameters. This may open the possibility of operational forecasting of seiche-prone conditions together with expected seiche crest heights. In that case, a seiche prediction (warning) would only be issued when seiche-conducive atmospheric conditions are expected that can lead to a seiche-event with amplitudes which justify a warning.

8.6. Conclusions Fluctuations in atmospheric pressure and wind speed, induced by mesoscale atmospheric convection cells that can arise following a cold front passing southward over the North Sea, generate low-frequency gravity waves in the North Sea. These low-frequency waves can subsequently induce a resonant response in coastal harbor basins, thus causing the so-called seiche events. During episodes of enhanced seiche activity in Rotterdam Harbor, the surface elevation energy spectrum at sea in the frequency band corresponding to the typical eigenfrequencies in Rotterdam Harbor basins, say, 0.1–2.0 mHz, varies with frequency (f ) approximately as f −1.5 . An episode of enhanced wave energy at sea in the seiche frequency band causes simultaneous resonant seiche responses in basins with different natural frequencies. The ratio between seiche amplitudes in two different basins can be accurately predicted using an f −1.5 excitation spectrum and numerically calculated spectral transfer functions. Operational prediction of the occurrence of a seiche-episode cannot be based on measurements at available platforms at sea, because at the more remote stations

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the signal is too weak, and for the nearby stations the available prediction window is too short to be of practical use. A hindcast study has shown that the occurrence of significant seiche episodes in Rotterdam Harbor can be accurately predicted on the basis of the prediction of the occurrence of meteorological conditions conducive to the formation of mesoscale atmospheric convection cells behind a cold front. This is presently realized using output of air temperatures and moisture content from numerical atmospheric models that are used in operational weather forecasting. The outcomes of this prediction method are used by the port authorities for ship traffic control and, after further development and testing, can eventually be incorporated into the closuremanagement system of the storm surge barrier in Rotterdam Waterway.

Acknowledgments The first author acknowledges the financial support from the Dr Ir Cornelis Lely Foundation. A part of this study was financed by the Directorate-General for Public Works and Water Management, South-Holland Directorate (Contract number: ZHA 23735). Furthermore, the authors thank the Directorate-General for Public Works and Water Management and the Municipal Harbor Organization, Rotterdam for providing the surface elevation measurements at sea and in the harbor, respectively. The authors thank Dr A. J. van Delden of Utrecht University, The Netherlands, for his extensive information on convection cells. Atmospheric sounding data have been obtained from the Department of Atmospheric Science, University of Wyoming, USA. Wavelet software was provided by Torrence and Compo,6 through URL: http://paos.colorado.edu/research/wavelets/. The figures in this chapter are reprinted from de Jong and Battjes,2 with permission from Elsevier.

References 1. M. P. C. de Jong and J. A. Battjes, Low-frequency sea waves generated by atmospheric convection cells, J. Geophys. Res. 109(C1), C01011 (2004), doi: 10.1029/2003JC001931. 2. M. P. C. de Jong and J. A. Battjes, Seiche characteristics of Rotterdam Harbour, Coastal Eng. 51(5–6), 373–386 (2004). 3. J. Morlet, G. Arens, E. Furgeau and D. Giard, Wave propagation and sampling theory — Part 2: Sampling theory and complex waves, Geophysics 47(2), 222–236 (1982). 4. M. Farge, Wavelet transforms and their applications to turbulence, Ann. Rev. Fluid Mech. 24, 395–457 (1992). 5. S. Mallat, A Wavelet Tour of Signal Processing (Academic Press, San Diego, 1998). 6. C. Torrence and G. P. Compo, A practical guide to wavelet analysis, Bull. Am. Meteorol. Soc. 79, 61–78 (1998). 7. B. W. Atkinson and J. W. Zhang, Mesoscale shallow convection in the atmosphere, Rev. Geophys. 34(4), 403–431 (1996).

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8. C. Veraart, Events of Long Waves in the North Sea, Technical Report RIKZ-94.034, Rijkswaterstaat/RIKZ and Delft University of Technology, Delft, The Netherlands (1994). 9. P. Van den Bosch and J. J. Veldman, Europoortkering-Beerdam, Technical Report H2438, WL | Delft Hydraulics, Delft, The Netherlands (1995). 10. J. R. Holton, An Introduction to Dynamic Meteorology, 3rd edn. (Academic Press, London, 1992).

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Chapter 9

Seiches and Harbor Oscillations Alexander B. Rabinovich P.P. Shirshov Institute of Oceanology, Russian Academy of Sciences 36 Nakhimovsky Prosp., Moscow, 117997 Russia Department of Fisheries and Oceans, Institute of Ocean Sciences 9860 West Saanich Road, Sidney, BC, Canada V8L 4B2 [email protected] [email protected] This chapter presents an overview of seiches and harbor oscillations. Seiches are long-period standing oscillations in an enclosed basin or in a locally isolated part of a basin. They have physical characteristics similar to the vibrations of a guitar string or an elastic membrane. The resonant (eigen) periods of seiches are determined by basin geometry and depth and in natural basins may range from tens of seconds to several hours. The set of seiche eigen frequencies (periods) and associated modal structures are a fundamental property of a particular basin and are independent of the external forcing mechanism. Harbor oscillations (coastal seiches) are a specific type of seiche motion that occur in partially enclosed basins (bays, fjords, inlets, and harbors) that are connected through one or more openings to the sea. In contrast to seiches, which are generated by direct external forcing (e.g., atmospheric pressure, wind, and seismic activity), harbor oscillations are mainly generated by long waves entering through the open boundary (harbor entrance) from the open sea. Energy losses of seiches in enclosed basins are mostly due to dissipative processes, while the decay of harbor oscillations is primarily due to radiation through the mouth of the harbor. An important property of harbor oscillations is the Helmholtz mode (pumping mode), similar to the fundamental tone of an acoustic resonator. This mode is absent in a closed basin. Harbor oscillations can produce damaging surging (or range action) in some ports and harbors yawing and swaying of ships at berth in a harbor. A property of oscillations in harbors is that even relatively small vertical motions (sea level oscillations) can be accompanied by large horizontal motions (harbor currents), resulting in increased risk of damage of moored ships, breaking mooring lines as well as affecting various harbor procedures. Tsunamis constitute another important problem: catastrophic destruction may occur when the frequencies of arriving tsunami waves match the resonant frequencies of the harbor or bay. Seiches, as natural resonant oscillations, are generated by a wide variety of mechanisms, including tsunamis, seismic ground waves, internal ocean waves, and 193

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jet-like currents. However, the two most common factors initiating seiches are atmospheric processes and nonlinear interaction of wind waves or swell. At certain places in the World Ocean, waves due to atmospheric forcing (atmospheric gravity waves, pressure jumps, frontal passages, squalls) can be responsible for significant, even devastating harbor oscillations, known as meteorological tsunamis. They have the same temporal and spatial scales as typical tsunami waves and can affect coasts in a similar damaging way.

9.1. Introduction Seiches are long-period standing oscillations in an enclosed basin or in a locally isolated part of a basin (in the Japanese literature they are commonly known as “secondary oscillations (undulations) of tides”).30,56,59 The term “seiches” apparently originated from the Latin word siccus which means dry or exposed (from the exposure of the littoral zone at the down-swing).31,95 Free-surface oscillations, known as seiches or seiching in lakes and harbors or as sloshing in coffee cups, bathtubs, and storage tanks, have been observed since very early times; a vivid description of seiching in Lake Constance, Switzerland, was given in 1549, and the first instrumental record of seiches obtained in 1730 in Lake Geneva.46,95 Korgen34 describes seiches as “the rhythmic, rocking motions that water bodies undergo after they have been disturbed and then sway back-and-forth as gravity and friction gradually restore them to their original, undisturbed conditions.” These oscillations occur at the natural resonant periods of the basin (so-called “eigen periods”) and physically are similar to vibrations of a guitar string and an elastic membrane. The resonant (eigen) periods of seiches are determined by the basin geometry and depth94,95 and in natural basins may be from a few tens of seconds to several hours. The oscillations are known as natural (or eigen) modes. The mode with the lowest frequency (and thus, the longest period) is referred to as the fundamental mode.41 The set of seiche eigen frequencies (periods) and associated modal structures are a fundamental property of a particular basin and are independent of the external mechanism forcing the oscillations. In contrast, the amplitudes of the generated seiches strongly depend on the energy source that generates them, and can therefore have pronounced variability.31 Resonance occurs when the dominant frequencies of the external forcing match the eigen frequencies of the basin. Harbor oscillations (coastal seiches according to Ref. 19 are a specific type of seiche motion that occur in partially enclosed basins (gulfs, bays, fjords, inlets, ports, and harbors) that are connected through one or more openings to the sea.41,94 Harbor oscillations differ from seiches in closed water bodies (for example, in lakes) in three principal ways68 : (1) In contrast to seiches generated by direct external forcing (e.g., atmospheric pressure, wind, and seismic activity), harbor oscillations are mainly generated by long waves entering through the open boundary (harbor entrance) from the open sea.

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(2) Energy losses of seiches in closed basins are mostly associated with dissipation, while the decay of harbor oscillations is mainly due to radiation through the mouth of the harbor. (3) Harbor oscillations have a specific fundamental mode, the Helmholtz mode, similar to the fundamental tone of an acoustic resonator.54 This modes is absent in closed basins. Because harbor oscillations can produce damaging surging (or range action) — yaw and swaying of ships at berth in a harbor — this problem has been extensively examined in the scientific and engineering literature.4,11,12,36,41,46,47,62,67,68,70,78,79, 94,95 One of the essential properties of oscillations in harbors is that even relatively small vertical motions (sea level oscillations) can be accompanied by large horizontal water motions (harbor currents); when the period of these motions coincides with the natural period of sway, or yaw of a moored ship, further resonance occurs, which can result in considerable motion and possible damage of a moored ship.82,94 Harbor oscillations can also break mooring lines, cause costly delays in loading and unloading operations at port facilities, and seriously affect various harbor procedures.79,80 Tsunamis constitute another important problem that have greatly stimulated investigations of harbor oscillations. Professor Omori (Japan) was likely the first to notice in 1902 that the dominant periods of observed tsunami waves are normally identical to those caused by ordinary long waves in the same coastal basin (see Ref. 30). His explanation was that the bay or portion of the sea oscillates like a fluid pendulum with its own period, i.e., the arriving tsunami waves generate similar seiches as those generated by atmospheric processes and other types of external forcing (see also Ref. 30). Numerous papers on the spectral analysis of tsunami records for various regions of the world ocean have confirmed this conclusion.13,48,69,74,75,84,90 Catastrophic destruction may occur when the frequencies of arriving tsunami waves match the resonant frequencies of the harbor or bay. One of the best examples of strong tsunami amplification due is the resonant response of Port Alberni (located at the head of long Alberni Inlet on the Pacific coast of Vancouver Island, Canada) to the 1964 Alaska tsunami.28,54 9.2. Hydrodynamic Theory The basic theory of seiche oscillations is similar to the theory of free and forced oscillations of mechanical, electrical, and acoustical systems. The systems respond to an external forcing by developing a restoring force that re-establishes equilibrium in the system. A pendulum is a typical example of such a system. Free oscillations occur at the natural frequency of the system if the system disturbed beyond its equilibrium. Without additional forcing, these free oscillations retain the same frequencies but their amplitudes decay exponentially due to friction, until the system eventually comes to rest. In the case of a periodic continuous forcing, forced oscillations are produced with amplitudes depending on friction and the proximity of the forcing frequency to the natural frequency of the system.84

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9.2.1. Long and narrow channel Standing wave heights in a closed, long and narrow nonrotating rectangular basin of length, L, and uniform depth, H, have a simple trigonometric form35,95 : ζ(x; t) = A cos kx cos ωt,

(9.1)

where ζ is the sea level elevation, A is the wave amplitude, x is the along-basin coordinate, t is time, k = 2π/λ is the wave number, λ is the wavelength, ω = 2π/T is the angular wave frequency, and T is the wave period. The angular frequency and wave number (or the period and wavelength) are linked through the following well-known relationships: ω = kc, λ T = , c

(9.2a) (9.2b)

√ where c = gH is the long-wave phase speed and g is the gravitational acceleration. The condition of no-flow through the basin boundaries (x = 0; x = L) yields the wave numbers: k=

π , L

2π , L

3π nπ ,..., , L L

(9.3)

which are related to the specific oscillation modes [Fig. 9.1(a)], i.e., to the various eigen modes of the water basin. The fundamental (n = 1) mode has a wavelength equal to twice the length of the basin; a basin oscillating in this manner is known as a half-wave oscillator.34 Other modes (overtones of the main or fundamental

Closed basin

Open-ended basin

n=1

n=0

n=2

n=1

n=3

n=2

n=4

n=3

0.00

1.00 0.00

L

1.00

L

Fig. 9.1. Surface profiles for the first four seiche modes in closed and open-ended rectangular basins of uniform depth.

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√ Table 9.1. Normalized periods, Tn∗ = Tn gH/(2L), for a closed and open-mouth rectangular basin of uniform depth. Mode Basin

n=0

n=1

n=2

n=3

n=4

— 2

1 2/3

1/2 2/5

1/3 2/7

1/4 2/9

Closed Open-mouthed

“tone”) have wavelengths equal to one-half, one-third, one-fourth, and so on, of the wavelength of the fundamental mode (Fig. 9.1(a), Table 9.1). The fundamental mode is antisymmetric: when one side of the water surface is going up, the opposite side is going down. Maximum sea level oscillations are observed near the basin borders (x = 0; x = L), while maximum currents occur at the nodal lines, i.e., the lines where ζ = 0 for all time. Positions of the nodal lines are determined by xm n =

(2m − 1)L , 2n

m, n = 1, 2, . . . ; m ≤ n.

(9.4)

Thus, for n = 1, there is one nodal line: x11 = L/2 located in the middle of the basin; for n = 2, there are two lines: x12 = L/4 and x22 = 3L/4; for n = 3: x13 = L/6, x23 = 3L/6 = L/2 and x33 = 5L/6 and so on. The number of nodal lines equals the mode number n [Fig. 9.1(a)], which is why the first mode is called the uninodal mode, the second mode is called binodal mode, the third mode the trinodal mode, etc.31,95 The antinode positions are those for which ζ attains maximum values, and are specified as xjn =

jL , n

j = 0, 1, 2, . . . , n.

(9.5)

For example, for n = 2 there are three antinodal lines: x02 = 0, x12 = L/2, and = L. Maximum currents occur at the nodal lines, while minimum currents occur at the antinodes. Water motions at the seiche nodes are entirely horizontal, while at the antinodes they are entirely vertical. The relationships (9.2) and (9.3) yield the well-known Merian’s formula for the periods of (natural) in a rectangular basin of uniform depth68,78 : x22

2L , Tn = √ n gH

(9.6)

where n = 1, 2, 3, . . .. Merian’s formula (9.6) shows that the longer the basin length (L) or the shallower the basin depth (H), the longer the seiche period. The fundamental (n = 1) mode has the maximum period; other modes — the overtones of the main fundamental — “tone” — have periods equal to one-half, one-third, one-fourth, and so on, of the fundamental period (Fig. 9.1(a), Table 9.1). The fundamental mode and all other odd modes are antisymmetric, while even modes are symmetric; an antinode line is located in the middle of the basin.

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The structures and parameters of open-mouth basins are quite different from those of closed basins. Standing oscillations in a rectangular bay (harbor) with uniform depth and open entrance also have the form (9.1) but with a nodal line located near the entrance (bay mouth). In general, the approximate positions of nodal lines are determined by the following expressions (Fig. 9.1(b), Table 9.1): xm n =

(2m + 1)L , 2n + 1

m, n = 0, 1, 2, . . . ; m ≤ n,

(9.7)

j, n = 0, 1, 2, . . . ; j ≤ n.

(9.8)

while antinodes are located at xjn =

2jL , 2n + 1

In particular, for n = 1 there are two nodal lines: x01 = L/3 and x11 = L and two antinodal lines: x01 = 0 and x11 = 2L/3; for n = 2 there are three nodal lines: x02 = L/5, x12 = 3L/5 and x22 = L, and three antinodal: x02 = 0, x12 = 2L/5, and x22 = 4L/5. The most interesting and important mode is the lowest mode, for which n = 0. This mode, known as the Helmholtz mode, has a single nodal line at the mouth of the bay (x = L) and a single antinode on the opposite shore (x = L). The wavelength of this mode is equal to four times the length of the bay; a basin oscillating in this manner is known as a quarter-wave oscillator.34 The Helmholtz mode, which is also called the zeroth mode,a the gravest mode and the pumping mode (because it is related to periodic mass transport — pumping — through the open mouth,36,41 is of particular importance for any given harbor. For narrow-mouthed bays and harbors, as well as for narrow elongated inlets and fjords, this mode normally dominates. The periods of the Helmholtz and other harbor modes can be approximately estimated as84,95 Tn =

4L √ , (2n + 1) gH

for mode n = 0, 1, 2, 3, . . . .

(9.9)

Using (9.9) and (9.6), the fundamental (Helmholtz) mode in a rectangular open√ mouth basin of uniform depth H is found to have a period, T 0 = 4L/ gH, which √ is double the period of the gravest mode in a similar but closed basin, T 1 = 2L/ gH. Normalized periods of various modes (for n ≤ 4) are shown in Table 9.1. Expressions (9.4)–(9.9), Table 9.1, and Fig. 9.1 are all related to the idealized case of a simple rectangular basin of uniform depth. This model is useful for some preliminary estimates of seiche parameters in closed and semi-closed natural and artificial basins. Analytical solutions can be found for several other basins of simple geometric form and nonuniform depth. Wilson95 summarizes results that involve common basin shapes (Tables 9.2 and 9.3), which in many cases are quite a In many papers and text books,8,94,95 this mode is considered the “first mode”. However, it is more common to count nodal lines only inside the basin (not at the entrance) and to consider the fundamental harbor mode as the “zeroth mode”.41,68,78,80,84 This approach is physically more sound because this mode is quite specific and markedly different from the first mode in a closed basin.

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good approximations to rather irregular shapes of natural lakes, bays, inlets, and harbors. The main concern for port operations and ships and boats in harbors is not from the sea level seiche variations but from the strong currents associated with the seiche. As noted above, maximum horizontal current velocities occur at the nodal lines. Therefore, its locations in the vicinity of the nodes that are potentially most risky and unsafe. Maximum velocities, Vmax , can be roughly estimated as84 :
g , (9.10) Vmax = An H where An is the amplitude of the sea level oscillation for the mode. For example, if An = 0.5 m and H = 6 m, Vmax ≈ 0.64 m/s. 9.2.2. Rectangular and circular basins If a basin is not long and narrow, the 1D approach used above is not appropriate. For such basins, 2D effects may begin to play an important role, producing compound or coupled seiches.95 Two elementary examples, which can be used to illustrate the 2D structure of seiche motions, are provided by rectangular and circular basins of uniform depth (H). Consider a rectangular basin with length L (x = 0, L) and width l (y = 0, l). Standing oscillations in the basin have the form35,41 ζ(x, y; t) = Amn cos

nπy mπx cos cos ωt, L l

where m, n = 0, 1, 2, 3, . . .. The eigen wave numbers (kmn ) are  1/2 mπ 2  nπ 2 kmn = + , L l and the corresponding eigen periods are78 −1/2   m 2  n 2 2 Tmn = √ + . L l gh

(9.11)

(9.12)

(9.13)

For n = 0 expression (9.13) becomes equivalent to the Merian’s formula (9.6); the longest period corresponds to the fundamental mode (m = 1, n = 0) which has one nodal line in the middle of the basin. In general, the numbers m and n denote the number of nodal lines across and along the basin, respectively. The normalized eigen ∗ periods Tmn = Tmn /T10 and spatial structure for the different modes are shown in Table 9.4. For oscillations in a circular basin of radius r = a, it is convenient to use a polar coordinate system (r, θ) with the origin in the center: x = r cos θ,

y = r sin θ,

where θ is the polar angle. Standing oscillations in such basins have the form ζ(x, y; t) = Js (kr)(As cos sθ + Bs sin sθ) cos ωt,

(9.14)

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Table 9.2.

Modes of free oscillations in closed basins of simple geometric shape and constant width (after Ref. 95). Periods of free oscillation Mode ratios Tn /T1 values for n

Basin type Description

Dimensions

2

3

4

2L/(gh0 )1/2

1.000

0.500

0.333

0.250

h(x) = h0 (1 − 2x/L)

1.305[2L/(gh0 )1/2 ]

1.000

0.628

0.436

0.343

h(x) = h0 (1 − 4x2 /L2 )

1.110[2L/(gh0 )1/2 ]

1.000

0.577

0.408

0.316

h(x) = h0 (1 − 4x2 /L2 )2

1.242[2L/(gh0 )1/2 ]

1.000

0.686

0.500

0.388

L 0

Triangular (isosceles)

h0

h0

L0 h(x)

h0

Quartic

h0

x

∠ 2

L Parabolic

x

h(x)

0 h(x)

x

∠ 2

L

0 h(x)

∠ 2

x

h(x) = h0

(Continued)

A. B. Rabinovich

Rectangular

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Fundamental T1

Profile equation

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Table 9.2.

(Continued ) Periods of free oscillation Mode ratios Tn /T1 values for n

Basin type Dimensions

Profile equation

2

3

4

1.640[2L/(gh1 )1/2 ]

1.000

0.546

0.377

0.288

1.000

0.546

0.377

0.288

L1 /L2 = 1/2 4L2 /(gh2 )1/2

1.000

0.500

0.250

0.125

L1 /L2 = 1/3 3.13L2 /(gh2 )1/2 L1 /L2 = 1/4 2.73L2 /(gh2 )1/2 L1 /L2 = 1/8 2.31L2 /(gh2 )1/2

1.000 1.000 1.000

0.559 0.579 0.525

0.344 0.367 0.371

0.217 0.252 0.279

L 0

Triangular (right-angled)

h1

Trapezoidal

h0

h(x)

h1

h(x) = h1 x/L

L

0

h(x) = h0 + mx m = (h1 − h0 )/L

x h(x) h1

L1 Coupled, rectangular

x

L2 0

x h(x) h 2

h(x) = h1 (x < 0) h(x) = h2 (x < 0) h1 /h2 = 1/4

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Description

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Modes of free oscillations in semi-closed basins of simple geometric shape (modified after Ref. 95).

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202

Table 9.3.

Periods of free oscillation Mode ratios Ts /T1 [n = (s + 1)/2]

Basin type Description

Dimensions 0

0

b(x)

b1

x

h(x)

n=0

1

2

3

h(x) = h1

2.000[2L/(gh1 )1/2 ]

1.000

0.333

0.200

0.143

h(x) = h1 x/L

2.618[2L/(gh1 )1/2 ]

1.000

0.435

0.278

0.203

h(x) = h1 (1 − x2 /L2 )

2.220[2L/(gh1 )1/2 ]

1.000

0.409

0.259

0.189

b(x) = b1 x/L h(x) = h1

1.308[2L/(gh1 )1/2 ]

1.000

0.435

0.278

0.230

x

h1

0

b(x)

b1

x

0

h1

h(x)

L Rectangular

Triangular L

x

b(x)

b1

0

x

h1

h(x)

L Rectangular

Semi-parabolic

b1

L Triangular

x

0

h(x)

h1

x

Rectangular

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L

b(x)

0

0

A. B. Rabinovich

L x

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L Rectangular

L Rectangular

Fundamental T0

Profile equation

(Continued)

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Table 9.3.

(Continued ) Periods of free oscillation Mode ratios Ts /T1 [n = (s + 1)/2]

Basin type Description

Dimensions

1

2

3

b(x) = b1 x/L h(x) = h1 x/L

1.653[2L/(gh1 )1/2 ]

1.000

0.541

0.374

0.283

b1 /L = 2 = 4/3 =1 = 2/3

2.220[2L/(gh1 )1/2 ]

1.000

0.707 0.554 0.447 0.317

0.578 0.493 0.468 0.455

0.378 0.323 0.264 0.185

h(x) = h1 (1 − r 2 /L2 )

2.220[2L/(gh1 )1/2 ]

1.000

0.707

0.578

0.500

L

b(x)

0

b1

x

0

h1

h(x)

L Triangular

x

Triangular L

0

b1

x

0

L Semi-elliptic

h1

x

Semi-paraboloidal

L

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n=0

Seiches and Harbor Oscillations

Fundamental T0

Profile equation

L 0 b 1

0

r h(x)

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h1 r

Semi-circular

Semi-paraboloidal

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Mode parameters for free oscillations in uniform depth basins of rectangular and circular geometric shape.

Rectangular basin (l = 0.5L)

1

2

1

2

1

2

0.500

2

0

0.500

0

1

0.447

1

1

0.354

2

1

+

+

+ + + +

+

+

0

2

0.243

1

2

ωa/c

Tsm /T10

—

—

1.841

1.000

—

—

3.054

0.603

+

0.628

—

3.832

0.480

0.719

—

5.331

0.345

0.766

—

6.706

0.275

0.343

0.787

7.016

0.262

0.449

0.820

8.536

0.216

+

+

+

+

0.250

r2

+ +

+

+ +

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0

0

Relative period

+

1

1

Normalized frequency

r1

2a

+

1

1.000

l

+

Circular nodal lines

+

0

m

+

0

s

+

2

Tmn /T10

+

0

L

Mode forms

+

1

Relative period

9.75in x 6.5in

n

Mode numbers

Mode forms

A. B. Rabinovich

m

Circular basin

+

Mode numbers

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Table 9.4.

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where Js is the Bessel function of an order s, As and Bs are arbitrary constants, and s = 0, 1, 2, 3, . . .35,41 These oscillations satisfy the boundary condition: Js (kr)|r=a = Js (ka) = 0.

(9.15)

The roots of this equation determine the eigen values ksm (m, s = 0, 1, 2, 3, . . .), with corresponding eigen modes described by Eq. (9.14) for various k = ksm . Table 9.4 presents the modal parameters and the free surface displacements of particular modes. As illustrated by Table 9.4, there are two classes of nodal lines, “rings” and “spokes” (diameters). The corresponding mode numbers m and s give the respective exact number of these lines. Due to mass conservation, the mode (0, 0) does not exist in a completely closed basin.41 For the case s = 0, the modes are symmetrical with respect to the origin and have annular crest and troughs.35 In particular, the first symmetrical mode (s = 0, m = 1) has one nodal ring r = 0.628a (Table 9.4). When the central part of the circular basin (located inside of this ring) is going up, the marginal part (located between this ring and the basin border) is going down, and vice versa. The second symmetrical mode (s = 0, m = 1) has two nodal rings: r = 0.343a and r = 0.787a. For s > 0, there are s equidistant nodal diameters located at an angle ∆θ = π/s from each other; i.e., 180◦ for s = 1, 90◦ for s = 2, 60◦ for s = 3, etc. Positions of these diameters are indeterminate, since the origin of θ is arbitrary. The indetermatability disappears if the boundary deviates even slightly from a circle. Specifically, the first nonsymmetrical mode (s = 1, m = 0) has one nodal diameter (θ = π/2), whose position is undefined; but if the basin is not circular but elliptical, the nodal line would coincide with either the major or minor axis, and the corresponding eigen periods would be unequal.35 The first unsymmetrical mode has the lowest frequency and the largest eigen period (Table 9.4); in this case, the water sways from one side to another relative to the nodal diameter. This mode is often referred to as the “sloshing” mode.78 Most natural lakes or water reservoirs can support rather complex 2D seiches. However, the two elementary examples of rectangular and circular basins help to understand some general properties of the corresponding standing oscillations and to provide rough estimates of the fundamental periods of the basins. 9.2.3. Harbor resonance Let us return to harbor oscillations and consider some important resonant properties of semi-closed basins. First, it is worthy to note that expressions (9.7)–(9.9) and Table 9.3 for open-mouth basins give only approximate values of the eigen periods and other parameters of harbor modes. Solutions of the wave equation for basins of simple geometric forms are based on the boundary condition that a nodal line (zero sea level) is always exactly at the entrance of a semi-closed basin that opens onto a much larger water body. In this case, the free harbor modes are equivalent to odd (antisymmetric) modes in a closed basin, formed by the open-mouth basin and its

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mirror image relative to the mouth.b However, this condition is not strictly correct because it does not take into account wave energy radiation through the mouth into the open sea. The exact solutions may be obtained based on the Sommerfeld radiation condition of free wave radiation through the open boundary.35,41 Following application of the appropriate mouth correction (α), the nodal line is located close to but outside the entrance. In other words, the effect of this correction is to increase the effective length of the basin.95 The mouth correction depends on two parameters: the basin aspect ratio q = l/L, which relates the width of the basin (l) to its length (L); and the aperture ratio ϑ = b/l, in which b is the actual width of the mouth. Mathematical determination of α is rather complicated but, as a rule, it increases with increases of q and ϑ. For example, the fractional correction to Eq. (9.9) for the fundamental mode in a rectangular basin of uniform depth and open mouth (ϑ = 1.0) is determined as30,95     q 3 πq α= − γ − ln , (9.16) π 2 4 where γ = 0.5772 is Euler’s constant. Roughly speaking, radiation into the external basin and the mouth correction are important when the semi-closed basin is broad and has a large open entrance, and negligible when the basin is long and narrow (i.e., when q is small); in the latter case, expressions (9.7)–(9.9), as well as those presented in Table 9.3, are quite accurate. The character of natural oscillations in a bay or harbor is strongly controlled by the aperture ratio ϑ = b/l, which can vary from ϑ = 1.0 to ϑ = 0.0. These two asymptotic cases represent a fully open harbor and a closed basin, respectively. It is evident that the smaller is ϑ (i.e., the smaller the width of the entrance), the slower water from the external basin (open sea) penetrates into the harbor. Thus, as ϑ decreases, the periods of all harbor modes for n ≥ 1 in Table 9.1 increase, tending to the periods of the corresponding eigen modes for a closed basin, while the period of the fundamental (Helmholtz) harbor mode tends to infinity.c This is one of the important properties of harbor oscillations. Another important property is harbor resonance. The amplification factor for long waves impinging on a harbor from the open sea is H 2 (f ) =

(1 − f /f0

)2

1 , + Q−2 (f /f0 )2

(9.17)

where f is the frequency of the long incoming waves, f0 is the resonant frequency of the harbor, and Q is the quality factor (“Q-factor”), which is a measure of energy damping in the system.47,95 Specifically, Q−1 =

b This

c This

dE/dt = 2β, ωE

approach is used for numerical computation of eigen modes in natural 2D basins.70 is the reason for calling this the “zeroth mode.”

(9.18)

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where E = E0 e−2βωt is the energy of the system as it decays from an initial value E0 , β is a dimensionless damping coefficient, and ω = 2πf is the angular frequency. The power amplification factor attains the value Q2 at resonance (f = f0 ), decreases to unity at f = 0 and goes to zero as f goes to infinity. Therefore, Q for harbor oscillations plays a double role: as a measure of the resonant increase of wave heights for waves arriving from the open ocean and as an index of the time decay rate of wave heights inside the harbor. The higher the Q, the stronger will be the amplification of the incoming waves and the slower the energy decay, i.e., the longer the “ringing” of seiche oscillations inside the harbor. In closed basins, like lakes, bottom friction is the main factor controlling energy decay. Normally, it is quite small, so in lakes with fairly regular topographic features (low damping), a high Q-factor may be expected. Consequently, even a small amount of forcing energy at the resonant frequency can produce significant seiche oscillations that persist for several days.31,95 In contrast, the main factor of energy decay in semi-closed water basins, such as gulfs, bays, fjords, inlets and harbors, is wave radiation through the entrance. In their pioneering work, Miles and Munk47 concluded that narrowing the harbor entrance would increase the quality factor Q and, consequently, the amplification of the arriving wave. This means that the construction of dams, dikes, and walls to protect the harbor from wind waves and swell could so constrict the entrance width that it leads to strong amplification of the resonant seiche oscillations inside the harbor. Miles and Munk47 named this harbor paradox. As pointed out by Miles and Munk,47 there are two limitations to the previous conclusions: (1) A time of order Q/π cycles is necessary for the harbor oscillations to adjust to the external forcing. This means that harbors with high Q would not respond to a strong but short-lived incoming disturbance. In most cases, this limitation is not of major concern because atmospheric disturbances (the major source of open-sea long waves inducing harbor oscillations) are likely to last at least for several hours. Even tsunami waves from distant locations “ring” for many hours, resonantly “feeding” harbor seiches and producing maximum oscillations that have long (12–30 h) durations that persist well after the arrival of the first waves.74,75 This contrasts with the case for near-field sites, where tsunamis normally arrive as short-duration impulsive waves. Such tsunamis are much more dangerous at open coastal regions than in bays or harbors, as was observed for the coast of Thailand after the 2004 Sumatra tsunami.86 (2) As the harbor mouth becomes increasingly narrower, the internal harbor dissipation eventually exceeds energy radiation through the mouth. At this stage, further narrowing does not lead to a further increase in the Q-factor. However, normally internal dissipation is small compared to the typical radiative energy losses through the entrance. Originally, Miles and Munk47 believed that their “harbor paradox” concept was valid for every harbor mode provided the corresponding spectral peak was sharp and well defined. Further thorough examination of this effect37,46,78 indicated that the harbor paradox is only of major importance for the Helmholtz mode, while for higher modes frictional and nonlinear factors, not accounted for in the theory,

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dampen this effect.95 However, the Helmholtz mode is the most important mode in natural basins and is normally observed in bays, inlets, and harbors with narrow entrance, i.e., in semi-closed basins with high Q-factor. Significant problems with the mooring and docking of ships (and the loading and unloading of their cargo) in ports and harbors are often associated with this fundamental mode and most typically occur in ports with high Q.4,6,41,62,66–68,78–80 Rabinovich67 suggested reducing these negative effects in ports by artificially increasing the internal dissipation. The idea is the same as that widely used in rocket technology to damp eigen oscillations in fuel tanks.44,45 Radial piers in ports and harbors play the same role as internal rings and ribs in rocket tanks, efficiently transforming wave energy into vortical motions which reduce the wave energy and therefore the intensity of the seiches and their associated horizontal currents. As shown by Rabinovich,67 the logarithmic attenuation factor, δ0 = π/Q, for the Helmholtz mode associated with the jth pier, is given by  ˆ   ∆E0j 1 j bj ω0 ζ0 j = Cx , (9.19) δ0 = 2E0 6 r0 h0 σ0 where E0 is the energy of the mode inside the harbor, ∆E0j is the energy dissipated at the pier over the mode period (T0 = 2π/ω0 ), bj is the length of the pier, r0 and h0 are the mean radius and depth of the harbor, ζˆ0 is the mean amplitude of the Helmholtz mode in the harbor, Cxj is a dimensionless resistance coefficient, and σ0 = (gh0 )1/2 /(πr0 ). Thus, the rate of damping of oscillations in a harbor depends on the number of piers (N ) and a number of dimensionless parameters: specifically, the relative amplitudes of the oscillations, ξ0 = ζˆ0 /h0 ; the normalized harbor frequency, Ω0 = ω0 /σ0 ; the relative lengths of the piers, Bj = bj /r0 ; and the coefficient Cxj . The parameter ξ0 depends on the intensity of the external forcing while the two other parameters Ω0 and Bj do not depend on forcing but only on the characteristics of the harbor. The coefficient Cxj strongly depends on the Keulegan– Carpenter (KC) number which relates hydraulic resistance in oscillating flows to those for stationary currents.32 For typical values Bj = 0.3, ξ0 = 0.1, Ω0 = 1.0, N = 8, and Cxj = 10, we find δ0 ≈ 0.4 and Q ≈ 8. Another important aspect of the harbor oscillation problem is that changes in port geometry, and the construction of additional piers and dams can significantly change the natural (eigen) periods of the port, thereby modifying considerably the resonant characteristics of the basin.6 Helmholtz resonators in acoustics are used to attenuate sound disturbances of long wavelengths, which are difficult to reduce using ordinary methods of acoustical energy dissipation. Similarly, side channel resonators are suggested as a method for attenuating incident wave energy in harbors.6,66,78 In general, estimation of the Q-factor is a crucial consideration for ports, harbors, bay, and inlets. For a rectangular basin of uniform depth and entirely open mouth (ϑ = b/l = 1.0), this factor is easily estimated as: Q=

L , l

(9.20)

which is inversely proportional to the aspect ratio q = l/L. This means that high Q-factors can be expected for long and narrow inlets, fjords, and waterways.

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Honda et al.30 and Nakano and Unoki59 examined coastal seiches at more than 110 sites on the coast of Japan and found that strong and highly regular seiche oscillations are most often observed in such elongated basins and that the periods of these oscillations are in good agreement with the approximate period (9.9) for the Helmholtz mode (n = 0): 4L T0 = √ . gH

(9.21)

If the aperture ratio ϑ < 1.0, corresponding to a partly closed entrance, it is more difficult to estimate the Q value and the resonant mode periods analytically. In practice, special diagrams for a rectangular basin with various q and ϑ are used for these purposes.80,84 For natural basins, these parameters can be estimated numerically or from direct observations. If the respective spectral peak in observational data is isolated, sharp and pronounced enough, then we can assume that Q  1. In this case, it follows from (9.17) that the half-power frequency points (f1/2 ) are given by the following expression47:   1 ± f1/2 = f0 1 ± , (9.22) 2Q and the relative frequency bandwidth is simply ∆f = Q−1 , f0

(9.23)

+ − where ∆f = f1/2 − f1/2 and f0 = 1/T0 is the resonant frequency. This is a useful practical method for estimating the Q-factor and amplification for coastal basins based on results of spectral analysis of observational data. However, the spatial structure of different modes, the distribution of currents, and sea levels inside a natural basin, influence harbor reconstruction based on changes in these characteristics, and many other aspects of harbor hydrodynamics, are difficult to estimate without numerical computations. Numerical modeling has become a common approach that is now widely used to examine harbor oscillations.4,14,70,92

9.2.4. Harbor oscillations in a natural basin Some typical features of harbor oscillations are made more understandable using a concrete example. Figures 9.2 and 9.3 illustrate properties of typical harbor oscillations and results of their analysis and numerical modeling. Several temporary cable bottom pressure stations (BPS) were deployed in bays on the northern coast of Shikotan Island, Kuril Islands in 1986–1992.13,14,68,70 All BPSs were digital instruments that recorded long waves with 1-min sampling. One of these stations (BPS-1) was situated at the entrance of False Bay, a small bay with a broad open mouth [Fig. 9.2(a)]. The oscillations recorded at this site were weak and irregular; the respective spectrum [Fig. 9.2(b)] was “smooth” and did have any noticeable peaks, probably because of the closeness of the instrument position to the position of the entrance nodal line. Two more gauges (BPS-2 and BPS-3) were located inside Malokurilsk Bay, a “bottle-like” bay with a maximum width of about

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Sea of Okhotsk

45°

K

Hokkaido I.

sla il I ur

s nd

Iturup I.

4

Kunashir I. Shikotan I. 5

1 2

False Bay

3

Malokurilsk Bay

Pacific Ocean

Shikotan I. 0.5 km

(a) 40° 140°

145°

150° 10 5

Spectra (cm 2 cpm -1)

10 4

18.6

10 4 1

10 4 2 10 3

10 2

95%

10 3

18.6 3

10 2

4

10 0

10 -2 10 -3

95%

95%

10 2

(b)

10 1 10 -2

10 -1

10 -3

10 1

(c)

10 0 10 -2

10 -1

(d) 10 -2

10 -1

Frequency (cpm)

Fig. 9.2. (a) Location of cable bottom pressure stations near the northern coast of Shikotan Island (Kuril Islands) and sea level spectra at (b) BPS-1, (c) BPS-2 (both in autumn 1986) and (d) BPS-3 and BPS-4 (October–November 1990).

1300 m and a narrow neck of 350 m [Fig. 9.2(a)]. The oscillations recorded by these instruments were significant, highly regular and almost monochromatic; the corresponding spectra [Figs. 9.2(c) and 9.2(d)] have a prominent peak at a period of 18.6 min. An analogue tide gauge [#5 in Fig. 2(a)] situated on the coast of this bay permanently measure oscillations with exactly the same period.70 It is clear that this period is related to the fundamental mode of the bay. The Q-factor of the bay, as estimated by expression (9.23) based on spectral analysis of the tide gauge data for sites BPS-2 and BPS-3, was 12–14 and 9–10, respectively. The high Qfactors are likely the main reason for the resonant amplification of tsunami waves that arrive from the open ocean. Such tsunami oscillations are regularly observed in this bay.13,69 In particular, the two recent Kuril Islands tsunamis of 15 November 2006 and 13 January 2007 generated significant resonant oscillations in Malokurilsk Bay of 155 cm and 72 cm, respectively, at the same strongly dominant period of 18.6 min.76

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n=0 18.9 min

n=1 6.5 min

n=2 3.8 min

n=3 3.4 min

n=4 2.6 min

n=5 2.3 min

Fig. 9.3. Computed eigen modes and periods of the first six modes in Malokurilsk Bay (Shikotan Island). Black triangles indicate positions of the BPS-2 and BPS-3 gauges (from Ref. 70).

Figure 9.3 shows the first six eigen modes for Malokurilsk Bay.70 The computations were based on numerical conformal mapping of the initial mirror reflected domain on a circular annulus (for details, see Ref. 77) and the following application of Ritz’s variational method to solve the eigenvalue problem. The computed period of the fundamental (Helmholtz) mode (18.9 min) was close to the observed period of 18.6 min. The spectra at BPS-2 and BPS-2 indicate weak spectral peaks (three orders of magnitude less than the main peak) with periods 4.1, 3.3, and 2.9 min (the latter only at BPS-3), thought to be related to modes n = 2, 3, and 4. The first mode (n = 1), with period of 6.5 min, was not observed at these sites apparently because the nodal line for this mode passes through the positions of BPS-2 and BPS-3. Thus, the computed periods of the bay eigen modes are in good agreement with observation; plots in Fig. 9.3 give the spatial structure of the corresponding modes. However, this approach does not permit direct estimation of the bay response to the external forcing and the corresponding amplification of waves arriving from the open ocean. In actuality, the main purpose of the simultaneous deployments at sites BPS-3 and BPS-4 [Fig. 9.2(a)] in the fall of 1990 was to obtain observed response parameters that could be compared with numerically evaluated values.14 The spectrum at BPS-4, the station located on the outer shelf of Shikotan Island near the entrance to Malokurilsk Bay [Fig. 9.2(d)], contains a noticeable peak with period of 18.6 min associated with energy radiation from the bay. This peak is about 1.5 orders of magnitude lower than a similar peak at BPS-3 inside the bay. The amplification factor for the 18.6 min period oscillation at BPS-4 relative to that at BPS-3 was found to be about 4.0. Numerical computations of the response characteristics for Malokurilsk Bay using the HN-method14 gave resonant periods which

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were in close agreement with the empirical results of Rabinovich and Levyant70 (indicated in Fig. 9.3). Resonant amplification of tsunami waves impinging on the bay was found to be 8–10. 9.2.5. Seiches in coupled bays A well-known physical phenomenon are the oscillations of two simple coupled pendulums connected by a spring with a small spring constant (weak coupling). For such systems, the oscillation energy of the combined system systematically moves from one part of the system to the other. Every time the first pendulum swings, it pulls on the connecting string and gives the second pendulum a small tug, so the second pendulum begins to swing. As soon as the second pendulum starts to swing, it begins pulling back on the first pendulum. Eventually, the first pendulum is brought to rest after it has transferred all of its energy to the second pendulum. But now the original situation is exactly reversed, and the first pendulum is in a position to begin “stealing” energy back from the second. Over time, the energy repeatedly switches back and forth until friction and air resistance eventually remove all of the energy out of the pendulum system. A similar effect is observed in two adjacent bays that constitute a coupled system. Nakano56 was probably the first to investigate this phenomenon based on observations for Koaziro and Moroiso bays located in the Miura Peninsula in the vicinity of Tokyo. The two bays have similar shapes and nearly equal eigen periods. As was pointed out by Nakano, seiches in both bays are very regular, but the variations of their amplitudes are such that, while the oscillations in one bay become high, the oscillations in the other become low, and vice versa. Nakano56 explained the effect theoretically as a coupling between the two bays through water flowing across the mouths of each bay. More than half a century later Nakano returned to this problem60 and, based on additional theoretical studies and hydraulic model experiments, demonstrated that two possible regimes can exist in the bays: (1) co-phase oscillations when seiches in the two bays have the same initial phase; and (2) contraphase when they have the opposite phase. The superposition of these two types of oscillations create beat phenomenon of time-modulated seiches, with the opposite phase modulation, such that “while one bay oscillates vigorously, the other rests”. Nakano and Fujimoto suggested the term “liquid pendulums” for the coupled interaction of two adjacent bays. A more complicated situation occurs when the two adjacent bays have significantly different eigen periods. For example, Ciutadella and Platja Gran are two elongated inlets located on the west coast of Menorca Island, one of the Balearic Islands in the Western Mediterranean [the inlets are shown in the inset of Fig. 9.5(a)]. Their fundamental periods (n = 0) are 10.5 min and 5.5 min, respectively.50,71,73 As a result of the interaction between these two inlets, their spectra and admittance functions have, in addition to their “own” strong resonant peaks, secondary “alien” peaks originating from the other inlet.38 This means that the mode from Ciutadella “spills over” into Platja Gran and vice versa. The two inlets are regularly observed to experience destructive seiches, locally known as “rissaga”.18,22,49–51,85 Specific aspects of rissaga waves will be discussed later (in

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Sec. 9.4); however, it is worth noting here that the coupling between the two inlets can apparently amplify the destructive effects associated with each of the inlets individually.38

9.3. Generation Because they are natural resonant oscillations, seiches are generated by a wide variety of mechanisms (Fig. 9.4), including tsunamis,14,28,54,69 seismic ground waves,2,15,34,40 internal ocean waves,8,19,21 and jet-like currents.30,54,57 However, the two most common factors initiating these oscillations in bays and harbors are atmospheric processes and nonlinear interaction of wind waves or swell (Fig. 9.4).62,68,95 Seiches in lakes and other enclosed water bodies are normally generated by direct external forcing on the sea surface, primarily by atmospheric pressure variations and wind.31,95 In contrast, the generation of harbor oscillations is a two-step process involving the generation of long waves in the open ocean followed by forcing of the

Period (sec) 10 5

10 4

10 3

10 2

101 sec

AT M O S P H E R E

Storm surges

Scattering nonstationarity

Nonlinearity

Long waves

Meteorological waves

Swell Wind waves

Infragravity waves

Tsunami waves

OCEAN Seismicity

BOTTOM 10 6

10 5

10 4

10 3

10 2

10 1

10 0

Wavelength (m) Fig. 9.4.

Sketch of the main forcing mechanisms generating long ocean waves.

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harbor oscillations as the long waves arrive at the harbor entrance where they lead to resonant amplification in the basin. Seiche oscillations produced by external periodic forcing can be both free and forced. The free oscillations are true seiches (i.e., eigen oscillations of the corresponding basin). However, if the external frequency (σ) differs from the eigen frequencies of the basin (σ = ω), the oscillations can be considered forced seiches.95 Open-ocean waves arriving at the entrance of a specific open-mouth water body (such as a bay, gulf, inlet, fjord, or harbor) normally consist of a broad frequency spectrum that spans the response characteristics of the water body from resonantly generated eigenfree modes to nonresonantly forced oscillations at other frequencies. Following cessation of the external forcing, forced seiches normally decay rapidly, while free modes can persist for a considerable time. Munk53 jokingly remarked that “the most conspicuous thing about long waves in the open ocean is their absence.” This is partly true: the long-wave frequency band, which is situated between the highly energetic tidal frequencies and swell/wind wave frequencies, is relatively empty (Fig. 9.5). For both swell/wind waves and tides, the energy is of order 104 cm2 , while the energy contained throughout the entire intermediary range of frequencies is of order 1–10 cm2 . However, this particular frequency range is of primary scientific interest and applied importance (Walter Munk himself spent approximately 30 years of his life working on these “absent” waves!). Long waves are responsible for formation and modification of the coastal zone and shore morphology;5,68 they also can strongly affect docking and loading/unloading of ships and construction in harbors, causing considerable damage.41,78,79,96 Finally,

100

24 12 5

hr

Period

2 1 30 10 5

Tides

min

2 1 30 10 5

sec

2 1

Swell, wind waves

10 3

10 1 -2

10 0 Storm surges

Meteorological waves IG- waves

10 -1

Variance (cm 2 )

10 2 -5

10 -2 Tsunami 10 -4

10 -3

10 -2

10 -1

10 0

10 1

10 2

10 -3

Frequency (cpm) Fig. 9.5. Spectrum of surface gravity waves in the ocean (modified from Ref. 68). Periods (upper scale) are in hours (h), minutes (min), and seconds (s).

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and probably the most important, are tsunamis and other marine hazardous long waves, which are related to this specific frequency band. The recent 2004 Sumatra tsunami in the Indian Ocean killed more than 226,000 people, triggering the largest international relief effort in history and inducing unprecedented scientific and public interest in this phenomenon and in long waves in general.86 Because of their resonant properties, significant harbor seiches can be produced by even relatively weak open ocean waves. In harbors and bays with high Qfactors, seiches are observed almost continuously. However, the most destructive events occur when the incoming waves have considerable energy at the resonant frequencies, especially at the frequency of the fundamental mode. Such a situation took place in Port Alberni located in the head of long Alberni Inlet on Vancouver Island (Canada) during the 1964 Alaska tsunami, when resonantly generated seiche oscillations in the inlet had trough-to-crest wave heights of up to 8 m, creating total economic losses of about $10 million (1964 dollars).28,54 9.3.1. Meteorological waves Long waves in the ocean are the primary factor determining the intensity of harbor oscillations. If we ignore tsunamis and internal waves, the main source of background long waves in the ocean are atmospheric processes (Fig. 9.4).10,53 There are three major mechanisms to transfer the energy of atmospheric processes into long waves in the ocean68 : (1) Direct generation of long waves by atmospheric forcing (pressure and wind) on the sea surface. (2) Generation of low-frequency motions (for example, storm surges) and subsequent transfer of energy into higher frequencies due to nonlinearity, topographic scattering and nonstationarity of the resulting motions. (3) Generation of high-frequency gravity waves (wind waves and swell) and subsequent transfer of energy into larger scale, lower frequency motions due to nonlinearity. Long waves generated by the first two mechanisms are known as atmospherically induced or meteorological waves.d Typical periods of these waves are from a few minutes to several hours, typical scales are from one to a few hundreds of kilometers. The first mechanism is the most important because it is this mechanism that is responsible for the generation of destructive seiche oscillations (meteorological tsunamis) in particular bays and inlets of the World Ocean (Sec. 9.4). “Meteorological waves” can be produced by the passages of typhoons, hurricanes or strong cyclones. They also have been linked to frontal zones, atmospheric pressure jumps, squalls, gales, wind gusts and trains of atmospheric buoyancy waves.10,59,68,71,87,95 The most frequent sources of seiches in lakes are barometric fluctuations. However they can also be produced by heavy rain, snow, d The Russian name for these waves is “anemobaric”68 because they are induced by atmospheric pressure (“baric”) and wind (“anemos”) stress forcing on the ocean surface.

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or hail over a portion of the lake, or flood discharge from rivers at one end of the lake.27,31,95 9.3.2. Infragravity waves Long waves generated through the nonlinear interaction of wind waves or swell are called infragravity waves.5,63 These waves have typical periods of 30 s to 300–600 s and length scales from 100 m to 10 km. The occurrence of relatively high-frequency long waves, highly correlated with the modulation of groups of wind or swell waves, was originally reported by Munk52 and Tucker.89 Because the waves were observed as sea level changes in the nearshore surf zone, they became known as surf beats. Later, it was found that these waves occur anywhere there are strong nonlinear interacting wind waves. As a result, the more general term infragravity waves (proposed by Kinsman33 ) became accepted for these waves. Recent field measurements have established that infragravity waves (IG waves) dominate the velocity field close to the shore and consist of superposition of free edge waves propagating along the shore, free leaky waves propagating in the offshore direction, and forced bound waves locked to the groups of wind waves or swell propagating mainly onshore.3,5,68 Bound IG waves form the set-down that accompanies groups of incident waves, having troughs that are beneath the high short waves of the group and crests inbetween the wave groups.39 They have the same periodicity and the same lengths as the wave groups and travel with the group velocity of wind waves, which is significantly smaller than the phase speed of free long waves with the same frequencies. Free edge IG waves arise from the trapping of swell/wind wave generated oscillations over sloping coastal topography, while free leaky waves are mainly caused by the reflection of bound waves into deeper water.5,63 The general mechanisms of the formation of IG waves are shown in Fig. 9.6.e IG waves are found to be responsible for many phenomena in the coastal zone, including formation of rip currents, wave setup, crescentic bars, beach cusps and other regular forms of coastal topographies, as well as transport of sediment materials. Being of high-frequency relative to meteorological waves, IG waves can induce seiches in comparatively small-scale semi-closed basins, such as ports and harbors, which have natural periods of a few minutes and which may pose a serious threat for large amplitude wave responses. Certain harbors and ports are known to have frequent strong periodic horizontal water motions. These include Cape Town (South Africa), Los Angeles (USA), Dakar (Senegal), Toulon and Marseilles (France), Alger (Algeria), Tuapse and Sochi (Russia), Batumi (Georgia) and Esperance (Australia). Seiche motions in these basins create unacceptable vessel movement which can, in turn, lead to the breaking of mooring lines, fenders and piles, and to the onset of large amplitude ship oscillations and damage.62,67,68,82,94–96 Known as surging or range action,78,79 this phenomenon has well-established correlations with (a) harbor oscillations, (b) natural oscillations of the ship itself, and (c) intensive swell or wind waves outside the e Figure 9.6 does not include all possible types of IG waves and mechanisms of their generation; a more detailed description is presented by Bowen and Huntley5 and Battjes.3

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Fig. 9.6.

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Generation mechanisms for infragravity waves in the coastal zone.

harbor. Typical eigen periods of a harbor or a moored ship are the order of minutes. Therefore, they cannot be excited directly by wind waves or swell, having typical periods on the order of seconds.96 However, these periods exactly coincide with the periods of wave groups and IG waves. So, it is conventional wisdom that surging in harbors is the result of a triple resonance of external oscillations outside the harbor, natural oscillations within the harbor, and natural oscillations of a ship. The probability of such triple resonance is not very high, thus surging occurs only in a limited number of ports. Ports and harbors having large dimensions and long eigen periods (>10 min) are not affected by surging because these periods are much higher than the predominant periods of the IG waves and the surging periods of the vessels. On the other hand, relatively small vessels are not affected because their natural (eigen) periods are too short.82 The reconstruction of harbors and the creation of new harbor elements, can significantly change the harbor resonant periods, either enhancing or, conversely, reducing the surging.f Another important aspect of the problem is that ship and mooring lines create an entirely separate oscillation system.79 Changing the material and the length of the lines and their position, f A famous example of this kind is the French port Le Havre. Before World War II it was known for very common and strong surging motions that created severe problems for ships. During the war a German submarine torpedoed by mistake a rip-rap breakwater, creating a second harbor opening of 20–25 m width. After this, the surging in the port disappeared.68

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changes the resonant properties of the system (analogous to changing the material and the length of a pendulum). It is important to keep in mind that each oscillation mode has a specific spatial distribution of sea level variability and associated current (as emphasized in Sec. 9.2.1, maximum currents are observed near the nodal lines). The intensity of the currents varies significantly from place to place. Moreover, topographic irregularities within the harbor and the presence of structure elements (dams, dykes, piers and breakwaters) can create intense local vortexes that may significantly affect the ships.67 So, the effect of surging on a ship strongly depends on the exact location of the ship, and even on its orientation, in the harbor. In summary, harbor oscillations arise through co-oscillation of sea surface elevations and currents in the harbor with those at the entrance to the harbor. Seichegenerating motions outside the harbor typically have periods of several minutes and most commonly arise from bound and free long waves that are incident on the harbor entrance. 9.3.3. Tsunami Tsunami waves are the main factor creating destructive seiche oscillations in bays, inlets and harbors.30,41,53,54,95 Tsunamis can produce “energies” of 103 –105 cm2 , although such events are relatively rare (depending on the region, from once every 1–2 years to once every 100–200 years). The main generation mechanisms for tsunamis are major underwater earthquakes, submarine landslides and volcanic explosions. Great catastrophic trans-oceanic tsunamis were generated by the 1946 Aleutian (magnitude Mw = 7.8), 1952 Kamchatka (Mw = 9.0), 1960 Chile (Mw = 9.5), and 1964 Alaska (Mw = 9.2) earthquakes. The events induced strong seiche oscillations in bays, inlets, and harbors throughout the Pacific Ocean.90 The magnitude Mw = 9.3 earthquake that occurred offshore of Sumatra in the Indian Ocean on 26 December 2004 generated the most destructive tsunami in recorded history. Waves from this event were recorded by tide gauges around the world, including near-source areas of the Indian Ocean (Fig. 9.7), and remote regions of the North Pacific and North Atlantic, revealing the unmatched global reach of the 2004 tsunami.42,74,86,88 In general, the duration of tsunami “ringing” increased with increasing off-source distance and lasted from 1.5 to 4 days.74,75 The recorded oscillations were clearly polychromatic, with different periods for different sites, but with clear dominance of 40–50 min waves at most sites. The analysis of various geophysical data from this event indicates that the initial tsunami source had a broad frequency spectrum, but with most of the energy within the 40–50 min band. Therefore, although tsunami waves at different sites induced local eigen modes with a variety of periods, the most intense oscillations were observed at sites having fundamental periods close to 40–50 min. Differences in spectral peaks among the various tide gauge records are indicative of the influence of local topography. For example, for the Pacific coast of Vancouver Island (British Columbia), the most prominent peaks in the tsunami spectra were observed for Winter Harbor (period ∼30–46 min) and Tofino (∼50 min). In fact, the frequencies of most peaks in the tsunami spectra invariably coincide with

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West and Central Indian Ocean 200 cm

E

Colombo (SL)

Hanimaadhoo (Md)

Sea level

Male (Md)

Salalah (Om)

Pointe La Rue (Sl)

Port Louis (Mt)

26

27 December 2004 (UTC)

28

Fig. 9.7. Tsunami records in the Indian Ocean for the 2004 Sumatra tsunami for six selected sites: Colombo (Sri Lanka); Male and Gan (both Maldives); Salalah (Oman); Pointe La Rue (Seychelles); and Port Louis (Mauritius). Solid vertical line labeled “E” denotes the time of the main earthquake shock (from Ref. 74).

corresponding peak frequencies in the background spectra. This result is in good agreement with the well-known fact that periods of observed tsunami waves are mainly related to the resonant properties of the local/regional topography rather than to the characteristics of the source, and are almost the same as those of ordinary (background) long waves for the same sites. For this reason, the spectra of tsunamis from different earthquakes are usually similar at the same location.30,48,69,g It is therefore difficult to reconstruct the source region spectral characteristics based on data from coastal stations. Rabinovich69 suggested a method for separating the effects of the local topography and the source on the resulting tsunami wave spectrum. This method can be used to reconstruct the open-ocean spectral characteristics of tsunami waves. The approach is based on the assumption that the spectrum S(ω) of both the tsunami and background sea level oscillations near the coast can be represented as S(ω) = W (ω)E(ω),

(9.24)

g The resonant characteristics of each location are always the same; however, different sources induce different resonant mode, specifically, large seismic sources generate low-frequency modes and small seismic sources generate high-frequency modes.

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where W (ω) = H 2 (ω), H(ω) is the frequency admittance function describing the linear topographic transformation of long waves approaching the coast, and E(ω) is the source spectrum. It is assumed that the site-specific properties of the observed spectrum Sj (ω) at the jth site are related to the topographic function Hj (ω) for that site, while all mutual properties of the spectra at all sites are associated with the source (assuming that the source is the same for all stations). For typical background oscillations the source spectrum has the form, E(ω) = S0 (ω), where S0 (ω) = Aω −2 , and A = 10−3 −10−4 cm2 .68,69 During tsunami events, sea level oscillations observed near the coast can be represented as ζobs (t) = ζt (t) + ζb (t),

(9.25)

where ζt are the tsunami waves generated by an underwater seismic source and ζb are the background surface oscillations. If the spectra of both tsunami, St (ω), and background oscillations, Sˆb (ω) and Sb (ω) (during and before the tsunami event, respectively) have the form (9.24), and the admittance function, W (ω), is the same for the observed tsunami and the background long waves, then the spectral ratio R(ω), is estimated as 

ˆ0 (ω) E(ω) + S ˆ St (ω) + Sb (ω) (9.26) R(ω) = = = A−1 ω 2 E(ω) + 1.0. Sb (ω) S0 (ω) The function R(ω), which is independent of local topographic influence, is determined solely by the external forcing (i.e., by tsunami waves in the open ocean near the source area) and gives the amplification of the longwave spectrum during the tsunami event relative to the background conditions. The close similarity of Rj (ω) for various sites confirms the validity of this approach.69 The topographic admittance function Hj (ω), which is characteristic of the resonant properties of individual sites, can be estimated as

1/2

1/2  1/2 j Sb (ω) Sbj (ω) Sj (ω) = =ω . (9.27) Hj (ω) = E(ω) S0 (ω) A The same characteristic can be also estimated numerically. 9.3.4. Seismic waves There is evidence that seismic surface ground waves can generate seiches in both closed and semi-closed basins. In particular, the Great 1755 Lisbon earthquake triggered remarkable seiches in a number of Scottish lochs, and in rivers and ponds throughout England, western Europe and Scandinavia.95 Similarly, the Alaska earthquake of 27 March 1964 (Mw = 9.2) induced seismic surface waves that took only 14 min to travel from Prince Williams Sound, Alaska, to the Gulf Coast region of Louisiana and Texas where they triggered innumerable seiches in lakes, rivers, bays, harbors, and bayous.15,34 Recently, the 3 November 2002 Denali earthquake (Mw = 7.9) in Alaska generated pronounced seiches in British Columbia and Washington State.2 Sloshing oscillations were also observed in swimming pools during

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these events.2,15,40 The mechanism for seiche generation by seismic waves from distant earthquakes is not clear, especially considering that seismic waves normally have much higher frequencies than seiches in natural basins. McGarr40 concludes that there are two major factors promoting efficient conversion of the energy from distant large-magnitude earthquakes into seiches: (1) A very thick layer of soft sediments that amplify the horizontal seismic ground motions. (2) Deeper depths of natural basins, increasing the frequencies of eigen periods for the respective water oscillations. It should be noted, however, that seismic origins for seiches must be considered as very rare in comparison, for example, with seiches generated by meteorological disturbances.95 9.3.5. Internal ocean waves In some regions of the World Ocean, definitive correlation has been found between tidal periodicity and the strong seiches observed in these regions. For example, at Palawan Island in the Philippines, periods of maximum seiche activity are associated with periods of high tides.21 Bursts of 75-min seiches in the harbor of Puerto Princesa (Palawan Island) are assumed to be excited by the arrival at the harbor entrance of internal wave trains produced by strong tidal current flow across a shallow sill located about 450 km from the harbor.8 Internal waves can have quite large amplitudes; furthermore, they can travel over long distances without noticeable loss of energy. Internal waves require 2.5 days to travel from their source area in the Sulu Sea to the harbor of Puerto Princessa, resulting in a modulation of the seiche oscillations that are similar to those of the original tidal oscillations. Similarly, large amplitude seiches on the Caribbean coast of Puerto Rico are also related to tidal activity and are usually observed approximately seven days after a new or full moon (syzygy). Highest seiches in this region occur in late summer and early fall, when thermal stratification of the water column is at its annual maximum. The seven-day interval between syzygy and maximum seiche activity could be accounted for in terms of internal tidal soliton formation near the southwestern margin of the Caribbean Sea.34 A theoretical model of seiche generation by internal waves, devised by David Chapman (Woods Hole Oceanographic Institution), demonstrated that both periodic and solitary internal waves can generate coastal seiches.8,20 Thus, this mechanism can be responsible for the formation of seiches in highly stratified regions. 9.3.6. Jet-like currents Harbor oscillations (coastal seiches) can also be produced by strong barotropic tidal and other currents. Such oscillations are observed in Naruto Strait, a narrow channel between the Shikoku and Awaji islands (Japan), connecting the Pacific Ocean and

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the Inland Sea. Here, the semi-diurnal tidal currents move large volumes of water back and forth between the Pacific and the Inland Sea twice per day with typical speed of 13–15 km/h. This region is one of the greatest attractions in Japan because of the famous “Naruto whirlpool”, occurring twice a month during spring tides, when the speed of tidal flow reaches 20 km/h. Honda et al.30 noticed that flood tidal currents generate near both coasts significant seiche oscillations with a period of 2.5 min, which begin soon after low tide and cease near high tide; the entire picture repeats with a new tidal cycle. No seiches are observed during ebb tidal currents (i.e., between high and low water) when the water is moving in the opposite direction. Nakano57 explained this phenomenon by assuming that a strong current passing the mouth of a bay could be the source of bay seiches, similar to the way that a jet of air passing the mouth piece of an organ pipe produces a standing oscillation within the air column in the pipe. Special laboratory experiments by Nakano and Abe58 demonstrated that jet-like flow with a speed exceeding a specific critical number generates a chain of antisymmetric, counter-rotating von Karman vortexes on both sides of the channel. The checker-board pattern of vortexes induce standing oscillations in nearby bays and harbors if their fundamental periods match the typical vortex periods, Tvor =

l , u

(9.28)

where l is the distance between vortexes, and u is the speed of the vortexes (u = 0.4V to 0.6V , where V is the speed of the tidal currents). For the parameters of the Naruto tidal currents, the laboratory study revealed that values of Tvor agreed with the observed seiche period of 2.5 min. Apparently, the same mechanism of seiche generation can also work in other regions of strong jet currents. 9.3.7. Ice cover and seiches It seems clear that ice cannot generate seiches (except for the case of calving icebergs or avalanches that generate tsunami-like waves). However, an ice cover can significantly impact seiche motions, suppressing them and impeding their generation. At the same time, strong seiches can effectively break the ice cover and promote polynya creation. Little is known on the specific aspects of ice cover interaction with seiche modes. Hamblin26 suggested that the ice cover in Lake Winnipeg influences the character of seiche activity. Schwab and Rao83 assumed that absence of certain peaks in the sea level spectra for Saginaw Bay (Lake Huron) in winter may have been due to the presence of ice cover. Murty55 examined the possible effect of ice cover on seiche oscillations in Kugmallit Bay and Tuktoyaktuk Harbor (Beaufort Sea) and found that the ice cover reduces the effective water depth in the bay and harbor and in this way diminishes the frequency of the fundamental mode: in Kugmallit Bay from 0.12 cph (ice-free period) to 0.087 cph (ice-covered); and in Tuktoyaktuk Harbor from 1.0 cph to 0.9 cph.

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9.4. Meteorological Tsunamis As discussed in Sec. 9.3.3, tsunamis are the main source of destructive seiches observed in various regions of the World Ocean. However, atmospheric disturbances (atmospheric gravity waves, pressure jumps, frontal passages, squalls) can also be responsible for significant, even devastating, long waves, which have the same temporal and spatial scales as typical tsunami waves. These waves are similar to ordinary tsunami waves and can affect coasts in a similar damaging way, although the catastrophic effects are normally observed only in specific bays and inlets. Nomitsu,61 Defant,10 and Rabinovich and Monserrat 71,72 suggested to use the term “meteorological tsunamis” (“meteotsunami”) for this type of waves. At certain places in the World Ocean, these hazardous atmospherically induced waves occur regularly and have specific local names: “rissaga” in the Balearic Islands, “ˇs´ciga” on the Croatian coast of the Adriatic Sea, “marubbio” (“marrobio”) in Sicily, “milghuba” in Malta, “abiki” and “yota” in Japan, “Seeb¨ ar” in the Baltic Sea, “death waves” in Western Ireland, “inchas” and “lavadiads” in the Azores and Madeira islands. These waves are also documented in the Yellow and Aegean seas; the Great Lakes; northwestern Atlantic; Argentina and New Zealand coastal areas; and Port Rotterdam.7,9–12,16,27,29,30,34,43,51,64,68,71,72,91,92 Table 9.5 gives a list of destructive harbor oscillations, which apparently have the same atmospheric origin and similar resonances due to similarities in the characteristics of the atmospheric disturbances and local geometry and topography of the corresponding basins. Because of the strong likeness between “meteotsunamis” and seismically generated tsunamis,51,88 it is quite difficult sometimes to recognize one from another. Catalogues of tsunamis normally contain references to numerous “tsunami-like” events of “unknown origin” that are, in fact, atmospherically generated ocean waves. “Rissaga” (a local Catalan word that means “drying”, similar to a Spanish word “resaca”) is probably the best known example of meteorological tsunamis.h These significant short-period sea level oscillations regularly occur in many bays and harbors of the