Giorgio Ferrarese ( E d.)
Wave Propagation Lectures given at a Summer School of the Centro Internazionale Matematico Estivo (C.I.M.E.), held in Bressanone (Bolzano), Italy, June 8-17, 1980
C.I.M.E. Foundation c/o Dipartimento di Matematica “U. Dini” Viale Morgagni n. 67/a 50134 Firenze Italy
[email protected]
ISBN 978-3-642-11064-1 e-ISBN: 978-3-642-11066-5 DOI:10.1007/978-3-642-11066-5 Springer Heidelberg Dordrecht London New York
©Springer-Verlag Berlin Heidelberg 2010 st Reprint of the 1 Ed. C.I.M.E., Ed. Liguori, Napoli & Birkhäuser 1982 With kind permission of C.I.M.E.
Printed on acid-free paper
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CON TEN T S
C
0
u r s e s
7
A. JEFFREY
Lectures on nonlinear wave propagation
Page
Y. CHOQUET-BRUHAT
Ondes asymptotiques ••••••••••••••••••
"
99
G. BOILLAT
Urti •••••••••••••••••••••••••••••••••
"
167
S e min a r s D. GRAFFI
Sulla teoria dell'ottica non-lineare
"
195
G. GRIOLI
Sulla propagazione del calore nei mezzi continui •••••••••••••••••••••••••••••
"
215
T. MANACORDA
Onde nei solidi con vincoli interni
"
231
T. RUGGERI
"Entropy principle" and main field for a non linear covariant system ••••••••
"
B. STRAUGHAN
Singular surfaces in dipolar materials and possible consequences for continuum mechanics ••••••••••••••••••••••••••••
"
275
CENTRO INTERNAZIONALE MATEMATICO ESTlVO (C.I.M.E.)
LECTURES ON NONLINEAR WAVE PROPAGATION A. JEFFREY
CIME Session on Wave Propagation Bressanone, June 1980
Department of Engineering Mathematics, The University Newcastle upon Tyne, NEl 7RU, England
9 CONTENTS Lecture 1.
Lecture 2.
Lecture 3.
Lecture 4.
Fundamental Ideas Concerning Wave Equations
1-1
1.
General Ideas
1-1
2.
The Linear Wave Equation
1-2
3.
The Cauchy Problem - Characteristic Surfaces
1-5
4.
Domain of Dependence - Energy Integral
1-9
5.
General Effect of Nonlinearity
1-13
References
1-15
Quasilinear Hyperbolic Systems, Characteristics and Riemann Invariants
2-1
1.
Characteristics
2.
Wavefronts Bounding a Constant State
2-1 2-6
3.
Riemann invariants
2-8
References
2-12
Simple Waves and the Exceptional Condition 1.
Simple Waves
3-1
2.
Generalised Simple Waves and Riemann Invariants
3-2
3.
Exceptional Condition and Genuine Nonlinearity
3-6
References
3-9
The Development of Jump Discontinuities in Nonlinear Hyperbolic Systems of Equations
4-1
1.
4-1
General Considerations
2.
The Initial Value Problem
4-2
3.
Time and Place of Breakdown of Solution
4-2 4-9
References Lecture 5.
Lecture 6.
3-1
The Gradient Catastrophe and the Breaking of Water Waves in a Channel of Arbitrarily Varying Depth and Width
5-1
1.
Basic Equations
5-1
2.
The Bernoulli Equation for the Acceleration Wave Amplitude
5-2
3.
The Amplitude a(x) and its Implications
5-3
References
5-5
Shocks and Weak Solutions
6-1
1.
Conservation Systems and Conditions Across a Shock
6-1
2.
Weak Solutions and Non-Uniquenes&
6-4
10
3. 4. Lecture 7.
Lecture 8.
Conservation Equations with a Convex Extension
6-11
Interaction of Weak Discontinuities
6-13
References
6-14
The Riemann Problem, Glimm's Scheme and Unboundedness of Solutions
7-1
1-
The Riemann Problem for a Scalar Equation
7-1
2. 3.
Riemann Problem for a System
7-3
Glimm's Method
7-5
4.
Non-Global Existence of Solutions
7-8
References
7-10
Far Fields, Solitons and Inverse Scattering
8-1
1-
Far Fields
8-1
2. 3.
Reductive Perturbation Method
8-3
Travelling Waves and Solitons
8-6
4.
Inverse Scattering
8-9
References
8-13
11
Lecture 1 . 1.
Fundamental Ideas Concerning Wave Equations
General Ideas The physical concept of a wave is a very general one.
It includes the
cases of a clearly identifiable disturbance, that may either be localised or non-localised, and which propagates in space with increasing time, a timedependent disturbance throughout space that mayor may not be repetitive in nature and which frequently has no persistent geometrical feature
that can
be said to propagate, and even periodic behaviour in space that is independent of the time.
The most important single feature that characterises a wave
when time is involved, and which separates wave-like behaviour from the mere dependence of a solution on time, is that some attribute of it can be shown to propagate in space at a finite speed . In time dependent
situat~ons,
the partial differential equations most
closely associated with wave propagation are of hyperbolic type, and they may be either linear or nonlinear.
However, when parabolic equations are
considered whicp have nonlinear terms , then they also can often be regarded as describing wave propagation in the above-mentioned general sense.
Their
role in the study of nonlinear wave propagation is becoming increasingly important , and knowledge of the properties of their solutions , both qualitative and quantitative, is of considerable value when applications to physical problems are to be made.
These equations frequently arise as a result of the
determination of the asymptotic behaviour of a complicated system. Nonlinearity in waves manifests itself in a variety of ways, and in the case of waves governed by hyperbolic equations, perhaps the most striking is the evolution of discontinuous solutions from arbitrarily well behaved initial data .
In the case of parabolic equations the effect of nonlinearity
is tempered by the effects of dissipation and dispersion that might also be present.
Roughly speaking, when the dispersion effect is weak, long wave
behaviour is possible, whereas when it is strong a highly oscillatory behaviour occurs, though the envelope of the oscillations then exhibits some of the characteristics of long waves.
12
Waves governed by a linear wave equation arise in many familiar physical situations, like electromagnetic theory, vibrations in linear elastic solids, acoustics and in irrotational inviscid l i qui ds .
However,
these linear equations often arise as a consequence of an approximation involving small amplitude waves, so that when this assumption is violated the equations governing the motion become nonlinear. Not only does this convert the problem to one involving nonlinear partial differential equations, but it also usually leads to the study of a system of first order equations, rather than to a nonlinear form of the familiar second order wave equation.
This happens because the wave equation
usually arises as the result of the elimination of certain dependent variables from f irst order equations (like! or
~
in electromagnetic theory),
and this is often impossible when nonlinearity arises .
Our concern hereafter
will thus be mainly with quasilinear first order systems of equations - that is to say with systems that are linear in their first order derivatives, and for the most part we will confine attention to one space dimension and time. 2.
The Linear Wave Equation Because of the importance of the linear wave equation (1)
let us begin by reviewing some of the basic ideas that are involved, though in the more general context of the variable coefficient equation 3
r i,j-O with a
i j,
(2)
f
b
i,
c , f functions of the four dimensional vector
~
012 (x , x ,x
Not all linear second-order equations of this form describe wave motion , and on account of this it is necessary to produce a method of classification which readily allows the identification of wave type equations from amongst the other types that are possible (i.e. elliptic and parabolic). The form of class ification to be adopted utilizes the coefficients of the highest-order n?-rlva. !:iv ee and ha s an algeb r a i c ba s i s but, as will be s een
3 x).
13 in a subsequent section, this classification in fact effectively distinguishes between equations of wave type and those of other types.
Let us start by
attempting some simplification of the form of equation (2) by changing the independent variables through the linear transformation i
0,1,2,3
(3)
where the cl(ij are constants. A transformation of this form gives an affine mapping of the (xO, xl, x
2,
x 3)-space which is one-one provided
detl~jl; 0.
Employing
the chain rule for differentiation we find that equation (2) may be re-written 3
°.
}:.
i,j,k,R.=O Hence the coefficients a
(4)
of the derivatives u i j' which are functions of x x position, transform to the new coefficients ij
of the derivatives u k t' which are also functions of position.
If, now,
; ;
we confine attention to the set of coefficients a specific point
~
""
~
appropriate to some ij 012 3 in (x , x , x , x )-space, we see that this is exactly
the transformation rule which would apply to the coefficients a
ij
of the
quadratic form 3
L
i,j=O
aijTliTlj ,
(5)
when the Tl i are transformed to 8 by the variable change k 3 Tli a ki8k• k-O
r
Now it is a standard algebraic result that by a suitable transformation a quadratic form with constant coefficients may always be reduced to a sum of squares, though not all of the squared terms need be of the same sign. Furthermore, Sylvester's law of inertia asserts that however this reduction is accomplished , the number of positive terms m and the number of negative
14 terms n will always be the same.
To apply these results to the differential
equation (2) itself with the variable coefficients a attention to a fixed point
~
let us again confine
i j,
o 1 2 3 • !o in (x , x , x , x )-space and attribute to
the a i j the specific values a i j - aij(!o). This then implies that some choice of the numbers a
ij
•
ai j
exists for
which
where m + n < 4.
The number pair (m,n) is called the Signature of the
quadratic form (5) and, being an algebraic invariant, is used to classify the quadratic form.
We shall use it to classify the variable coefficient
partial differential equation (2) at each point
~
= !o'
The effect on equation (2) of using these numbers
ai j
in the transformation
(3) is to yield at ~ - !o a differential equation of the form
m-l
I i=O
u
~i~i
-
m+n-l
3
I
u i i +
i"'1ll
~ ~
I
i:oO
biu i + f t
0
(6)
Equation (6) or, equivalently, (2) is called hyPerbolic at ~ = !o in the o ~ -direction when the signature is (1,3), elliptic when the signature is (4,0) and parabolic when m + n < 4. direction at each point of a region the
o ~ -direction
throughout
If an equation is hyperbolic in the ~O_
n, then it is said to be hyperbolic in
n.
Obviously, if an equation has constant coefficients, then one suitable transformation (3) will reduce it to the form of equation (6) throughout all space.
For example, aside from the trivial transformation to remove the
constant factor I/c 2, the wave equation (1) is already seen to have the signature (1,3).
Thus if a transformation is made at one point of space to
convert the factor llc
2
to unity, then it does so for all points in the space.
The usual effect of variable coefficients and first-order terms in hyperbolic equations of the form (2) is to introduce distortion as the wave profile propagates.
This produces various complications, not die least of
which is the fact that the wave velocity becomes ambiguous and requires
15 careful definition.
Only when there is a clearly identifiable feature of
the wave which is preserved throughout propagation is it possible to define the propagation speed of this feature unambiguously.
Such is the case with
a wave front separating, say, a disturbed and an undisturbed region and across which a derivative of the solution is discontinuous. 3.
The Cauchy Problem - Characteristic Surfaces Fundamental , to the study of hyperbolic equations is the Cauchy problem,
and the associated notion of a characteristic surface.
In brief, when
working with four independent variables the Cauchy problem amounts to the ~etermination
of a unique solution to an initial value problem in which a
hypersurface F is given, and on it the function u is specified together with the derivative of u along some vector directed out of F. directional derivative is
call~d
Such a
an exterior derivative of u with respect
to F, in order to distinguish it from a directional derivative in F which is known as an interior derivative.
In the Cauchy problem it must be
emphasized that the function u and its exterior derivative over the initial hypersurface F are independent, and can be specified arbitrarily. A hypersurface F for which the Cauchy problem is not meaningful because u and its exterior derivative cannot be specified independently is called a characteristic hypersurface.
Let us now see how characteristic hypersurfaces
may be determined. It is convenient to utilize curvi-linear coordinates
~
012 , ~ , ~ ,
~
3
and
to let the hypersurface F on which the initial data is to be specified have the equation
~
o = O.
In terms of the new variables, a derivative with respect
to ~O is a directional derivative normal to F so that it is an exterior derivative, whilst derivatives with respect to
123 , ~ , ~ are interior
~
derivatives. We now utilize this by rewriting equation (2) in a form in which the derivative u
~o~O
is separated from the other second-order derivatives
16
3
+ Here
,
L
L
(7)
f •
i,kwO
signifies that the terms corresponding to k
= i
=0
are omitted from
the summation. Now if we specify u and u 0 independently on F, as is required in the E;
Cauchy problem, the substitution of their functional forms into equation (7) enables the determination of u 0 0' provided only that the coefficient E;
of this derivative does not vanish.
E;
Thereafter, the solution may be obtained
in the form of a Taylor series by determining the coefficients of the series by successive differentiation of the initial data and of equation (7) itself. This is, of course, the idea underlying the classical Cauchy-Kowalewski theorem.
It is, however, very restrictive as an existence theorem since
..
it demands that all functions involved are C • In the event that the coefficient (8)
of u 0 0 vanishes, neither this nor higher-order derivatives of u with E; E; 0 respect to E; can be found . Furthermore, the derivative u 0 0 may then be E; E;
specified arbitrarily on F, and even differently on opposite sides of F. This is not remarkable, because when the coefficient of u 0 0 vanishes, u and u 0 cannot be specified independently over F.
E;
E;
This follows because
E;
they must satisfy the equation which results when the first term is deleted from equation (7), and so we then have insufficient initial data.
As already mentioned, the hypersurface F with the equation E;0
=0
for
which the coefficient (8) vanishes is called a characteristic hypersurface of the differential equation (2). begin by setting Pi H
=
=
3E;
3
r
i,j=O
ai,Pi Pj ' J
olaxi
To examine such hypersurfaces further, we and writing (9)
17 Then the quadratic form H is the coefficient of the derivative u 0 0 in ~ ~
equation (7), and the characteristic hypersurface F will be given by the condition H
= O.
To interpret the condition H
= 0,
differentiable scalar function, then
~ = const.
we first recall that if
grad~
Consequently, by analogy , Pi
~
is a
is a vector normal to the surface
= a~Olaxi
is the ith component of the
four-dimensional gradient of ~O and so is the ith component of a four-dimensional vector
~
normal to the hypersurface F.
Hence the equation H • 0 is a
condition on the orientation of the normal vector
~
to F, and as the a
ij
are
usually functions of position, it follows that this condition will differ from point to point. The quadratic form (9) is, of course, just the same quadratic form we encountered in (5), so that its signature will depend on the type of the equation (7) or, equivalently, (2). ~ =
!o
If the equation is hyperbolic at
the signature will be (1,3), and it follows that at the point the
condition H
=0
determining the characteristic hypersurface can be reduced to (10)
It is obvious that no real characteristic hypersurface exists for elliptic equations, since their signature is (4,0) and the components of the vector
~
need to be complex if they are to satisfy the condition
222 2 H • Po + PI + P2 + P3
0 •
To proceed with the hyperbolic case we now simplify matters by setting O x • t and writing ~O
t - ~(xl, x 2 , x 3)
so that Po • 1 and Pi becomes ~2 + ~2 2 x1 x
+ ~2 3 x
= -~
x
1
i for i
(11)
= 1 ,2,3.
Then the quadratic form (10)
(12)
18
which is a differential equation for the function
~
locally at
~ = ~.
This is, of course, the familiar Eikonal equation from mathematical physics . At any time t
1
~(x ,
=
2
to a real three-dimensional surface S is defined by
3
(13)
x , x )
and this is called a characteristic surface . If equation (7) is a constant coefficient equation it can be reduced to the form of equation (6) with m
= 1,
n
=
3 throughout all space, so that
equation (12) then describes the characteristic surface
= const
~
for all
points in space . In summary, we have established that real characteristic surfaces
oc~ur
in connection with hyperbolic equations, and that across such surfaces a discontinuity may occur in the second normal derivative of the solution.
This
discontinuity in a derivative of a solution is usually identifiable with an interesting physical attribute of the solution, since it represents a wavefront bounding two regions. The discontinuity surface, or wavefront, advances with time, as is shown by the following simple argument. Taking the total differential of ~o
123
dt - dx • 1 - dx • 2 - dx • 3 x x x
=0
and using equation (11) we find
0
or , equivalently dt
= .d£ . gra d.
,
where d£ is the vector differential with components (dx
l,
2 3 dx , dx ) .
Hence 1
(14)
v.n
Igrad" where
Y..
=
dr dt
grad.
Igrad' I
The vector n is the unit normal to the surface •
s
the displacement of a position vector with time, y..
const, and as d£ represents
= d~/dt
is the velocity of
19 displacement of a specific point on the surface as the characteristic surface moves from its position at time t to its position at t + dt .
The
scalar v.n is the normal velocity of propagation of the characteristic surface or wavefront and, in general, is a function of position. By re-writing equation (7) and differencing it across the characteristic surface , we shall see that there may also be a discontinuity in the first normal derivative of the solution and this, like the discontinuity in the second-order derivative, is propagated with the characteristic surface. The equation governing the development of the discontinuities in first- and second-order derivatives is an ordinary differential equation defined along a curve in space and is called the transport equation. 4.
Domain of Dependence - Energy I~tegral The dependence of a wave solution on initial data is most easily
illustrated in terms of the one-dimensional wave equation (15)
with the initial conditions u(x,O)
hex)
au at
and
k(x) •
(x,O)
(16)
The explicit d'Alembert solution h(x-ct)+h(x+ct) 2
u(x,t)
shows how the solution at (xo,t xo - cto
~
x
~
xo + ct
+ J:.... 2c O)
x+c t k(s) ds x-ct
f
(17)
depends only on data in the interval
o
This is called the domain of dependence of the solution at (xo,t
O)'
This
same idea generalises to quasilinear hyperbolic systems and we shall employ it later. In conclusion, to illustrate the important notion of an energy integral that arises when working with equations derived from the conservation of physical quantities, let us prove the uniquenes s of the solution to the
20
?Co
Domain of dependence Cauchy problem for slightly generalised two dimensional wave equation au q(x,y)u - rat u
l
(18)
(x,y) ,
and where we shall assume P, k, r to be positive constants and q(x,y)
(19) >
O.
It will be convenient to consider that (18) governs the motion of a membrane with density P, tension k per unit length, distributed springing under the membrane with spring constant q(x,y) per unit area and fyictional coefficient
r. Then the potential energy within a fixed region R with boundary B of the (x,y)-plane comprises the energy stored in the springing qu
2
dxdy
and the energy stored in the membrane
- .! 2
+.! 2
II I
2u 2 Uk[a 2 + a R ax
1B
alu)
uk au
ax
dxdy
ds
with n the outward dfuwn unit normal to Band ds a length element of B.
The
first integral in ~(t) is the negative of the work done by the tension against the interior of R and the second integral the negative of the work done against the boundary. Green's theorem shows that
so that the total potential energy
21
';
i fiR
H[:iJ'
+
[:;J'] qn') dx +
(20)
dy
The kinetic energy is (21)
so the total energy is
or (22)
It then follows after use of Green's theorem that •
f atau B
-
au an
k -
(23)
ds -
which is the outward flux of energy across the boundary and the loss due to
7
friction.
0
x
Now let R vary in such a way that at t = 0 i t is the smaller domain
~.
and at t = t
l
it is
The surface between R and R we write in the form O I,
t = T(x,y). Integration of the identity
o •
Ro
i,'t H:~J' + k[
f:i)' f:;J'] q.') +
+
-kfa: f~~ ;~) +. a~: (;~ ;;)) + (;~)2 r
22
followed by use of the divergence theorem and some manipulation finally gives the result
dt dx dy +
If
t [k [;: + ;; ~~r
(x,y) in R t=T(x,y)
[;~ + ~~ ~~)
+ k where c
2
= kip
2 + P(l
c
2[
[~;r + [;~r))[~~r + qU2J dx dy,
(24)
and V is the volume of the region concerned.
Now impose on R(t) the condition 1
<
c
(25)
2
Then all terms on the right-hand side of (24) are non-negative, so if u = u
t
=0
at t
=0
in R the right-hand side of (24) must vanish, since O
with zero initial conditions the left-hand side vanishes.
In particular this
means that u t must vanish identically on the top and sides of V. the top R corresponds to any t so that u l l t follows that u
=0
=0
in V.
Since u
=0
However, in R it O
in V.
This proves uniqueness, for if two different solutions v and w exist corresponding to the same Cauchy data (19), u data u
= ut = 0
at t
O.
= v-w
will satisfy the initial
We have seen this implies u
=0
so that v
= w,
and
the solution is unique at all points that cannot be reached by a disturbance starting in R and travelling with a speed.:: c. O
The region R now plays the O
part of the domain of dependence, and the volume V becomes the domain of determinacy. The limiting case 1
= 2" c
may be interpreted in a useful physical manner if we let n be the normal to
23 the ruled surface t dn dt
= T(x,y).
We have
[r~~r
[:~J'r
1
=
TVTf
-
c ,
=
+
so that d~
dt
showing that c is tilt: speed of contraction of the region R.
The volume V
in which the solution is determined by the Cauchy data on R is thus an O inverted cone with base R
O'
5.
General Effect of Nonlinearity It is now necessary to make clear that the effect of nonlinearity in a
wave equation involves more than the loss of superposibility, for it can also change the entire nature of the solution.
This is best shown by a simple
non-physical example. Consider the single first order partial differential equation au + feu) au at ax
=
(26)
0
for the scalar u(x,t) that is subject to the initial condition u(x,O)
(27)
g tx) •
Now the total differential du is given by au dt + au dx at ax
du
so that if x and t are constrained to lie on a curve C, then at any point P on C we have au at
=
r
au + dX) ~ at [dt ax
where now dx/dt is the
gradien~
(28)
of curve C at point P.
Comparison of (26) and (27) now shows that we may interpret (26) as the ordinary differential equation du dt
o
(29)
along any member of th e fami ly of cur ve s C whi.:h ar e the solution curve s of
24 dx dt
(30)
feu) •
These curves C are called the characteristic curves of equation (26).
The
solution of the partial differential equation (26) has thus been reduced to the solution of the pair of simultaneous ordinary differential equations (29) and (30). Equation (29) shows that u • const along each of the characteristic curves C.
The constant value actually associated with any characteristic curve being
equal to the value of u determined by the initial data (27) at the point at which the characteristic curve intersects the initial line t u
= const
Setting
in (30) then shows that the characteristic curves C of (26) form a
family of straight lines. point
o.
So, if we consider the characteristic through the
on the initial line, we find after integrating (30) and using
(~,O)
(27) that the family of characteristic curves C have the equation
=
x where
~
~
+
(31)
tf(g(~»
now plays the role of a parameter.
Expressed slightly differently, we have shown that in terms of the parameter
~,u
•
g(~)
at every point of the line (3l)in the (x,t) plane.
In
physical problems t usually denotes time, so that it is then necessary to confine attention to the upper half plane in which t
~
o.
The solution to (26) and (27) may be found in implicit form if eliminated between u •
g(~),
=
g(x - tf(u»
is
which is true along a characteristic, and the
equation (31) of the characteristic itself. u
~
We find the general result (32)
•
Result (31) shows that if the functions f and g are such that two characteris tics intersect for t > 0, then since each one will have associated with it a different constant value of u, it must follow that at such a point the solution will not be unique.
This can obviously happen however smooth the
two functions may be, since intersection of two characteristics depends merely on the value of characteristics.
f(g(~»
that is associated with each of the straight line
This is to say on the two points
(~1'0)
and
(~2'0)
of the
25 initial line through which they pass.
We conclude from this that such
behaviour of solutions is not attributable to any irregularity in the coefficient f(u), or in the initial data u(x,O)
E
g(x).
Differentiating (32) partially with respect to x gives g'(x-tf(u» l+tg'(x-tf(u»f'(u)
(33)
showing au/ax becomes infinite whenever 1 + tg'(x-tf(u»f'(u) what is often called the gradient catastrophe .
E
o.
This is
In order to extend the
solution beyond this point we will need to introduce the concept of a discontinuous solution called a shock.
This will be done later.
General References [1]
Courant, R., Hilbert, D. Methods of Mathematical Physics, Vol. II, Wiley-Interscience, 1962.
[2]
Garabedian, P. R.
[3]
Hellwig. G.
[4]
Roubine. E. (Editor).
[5]
Coulson. C. A., Jeffrey, A.
Partial Differential Equations, Wiley, 1964.
Partial Differential Equations. Blaisdell, 1964 . Mathematics Applied to Physics, Springer. 1970. Waves, 2nd Ed. Longman, 1977.
26
Lecture 2.
Quasilinear Hyperbolic Systems, Characteristics and Riemann
Invariants. 1.
Characteristics The notion of a characteristic curve needs to be introduced in the
context of the quasilinear system +
-
B
0
(1)
in which U and Bare n element column vectors with elements u , b
un and b l, b Z' elements a ..• 1.J
n,
u
l'
z, ... ,
respectively, and A is an n x n matrix with
The system (1) will be quasilinear if, in general, the
elements a.. of A depend nonlinearly on u l' u 2' ••• , un' 1.J
When B .; 0 the
u ••• , un' elements b i of B may, or may not, depend linearly on u l' 2' will be assumed throughout this section that the elements b
i
and a
It are
ij
continuous functions of their arguments.
Although x, t are the natural variables to use when deriving systems of equations describing motion in space and time, they are not necessarily the most appropriate ones from the mathematical point of view.
So, as we are interested in the way a solution evolves with time,
let us leave the time variable unchanged in system (1), but replace x by some arbitrary curvilinear coordinate
~
and then try to choose
manner which is convenient for our mathematical arguments.
~
in a
Accordingly,
our starting point will be to change from (x, t) to the arbitrary semicurvilinear coordinates ~(x,
~
t)
t'
,
t
.
(Z)
If the Jacobian of the transformation (2) is non-vanishing we may
thus transform (1) by the rule
-
~
L
ax -
~
L
L
dt
L
at
ax
aE; a~
a
+
~
at
at' -
+
l!.~ ax
at' -
a
~
L
~
L
at
ax
a~
a~
+
a
at'
27 where, of course,
a~/at
and
are scalar quantities.
a~/ax
This leads
directly to the transformed equation
o the terms of which may be grouped to yield au
atT
+ 2.§.A)
ax
+
au
at
+
o,
B
(3)
where I is the n , x n unit matrix. Equation (3) may now be considered to be an algebraic relationship connecting the matrix vector derivatives au/at' and
au/a~.
It is then
at once apparent that this equation may only be used to determine au/a~
if the inverse of the coefficient matrix of
au/a~
to say , if the determinant of the coefficient matrix of
exists. au/a~
That is
is non-
vanishing.
This condition obviously depends on the nature of the
~urvilinear
coordinate lines
chosen arbitrarily.
~(x,
t)= const., which so far have been
Suppose now that for the particular choice
~
=$
the determinant does vanish, giving the condition .
I~I at
+
~A
o
ax
(4)
Then because of this the derivative au/a$ will be indeterminate on the family of lines $ • const .
Consequently, across such lines $(x, t)=
const., au/a$ may actually be discontinuous.
This means that each of the
n elements aui/a$ of au/a$ may be discontinuous across any of the lines $
const.
To find how, when they occur, these discontinuities in aui/a$
are related one to the other across a curvilinear coordinate line
~
= const.,
it is necessary to reconsider equation (3). We shall now confine attention to solutions U which are everywhere continuous but for which the derivative au/a$ is discontinuous across
. the particular 11ne
~
= k (say).*
Because of the continuity of U, and the
continuity of the elements a . . of A and b. of B, the matrices A and B will 1J
experience no discontinuity across $ - k.
* We
call this a weak
discontinui~y.
1
So, in the neighbourhood of a
28
typical point P of this line, A and B may be given their actual values at
P.
In equation (3) there is no indeterminacy of au/at' across the lines
~ =
const., and as a/at' denotes differentiation along these lines it
must follow that au/at' is everywhere continuous and, in particular, that it is continuous across the line
~
• k at P.
Taking these facts into account the differencing equation (3) across the line
~
=~
~ A) P [~~] P
+
where [Q]
k at P gives
e
aX
= Q_
across the line
a",
•
(5)
0,
- Q+ signifies the discontinuous jump in the quantity Q ~
z
k, with
Q_
denoting the value to the immediate left
of the line and Q+ the value to the immediate right at P.
As the point P
was any point on this line the suffix P may now be omitted. is differentiation normal to the curves
a/a~
~
= const.,
The operator
so that equations
(5) express compatibility conditions to be satisfied by the component of the
derivative of U on either side of and normal to these curves in the (x, t)plane. This is a homogeneous system of equations for the n jump quantities [
= (au./a~) -
au./a~] ~
-
~
(au./a~) ~
+
and there will only be a non-trivial
solution if the determinant of the coefficients vanishes.
The condition
for this is
I~ at
I
+
~A ax
I
o
However, along the lines
~
(6)
• const. we have, by differentiation,
o so that these lines have the gradient dx dt
~ /a~ ax = - ).
_ at
•
(say).
(7)
Combining (6) and (7) we deduce that ). must be- StICh:- thar:.
IA
-
).1
I
o•
(8)
29 Consequently the A in (7) can only be one of the eigenvalues of A, and since (5) can be re-written
o • the column vector
[oU/o~]must
eigenvector of A.
be proportional to the corresponding right
This. then. 'det ermi nes the ratios between the n elements
[ ou/o4> ] of the vector [ As A is an n
(9)
x
ou/a~]
that we were seeking.
n matrix it will have n eigenvalues.
If these are
real and distinct, integration of equations (7) will give rise to n distinct families of real curves c(l). C(2), •••• C(n) in the (x. t)-plane: C( i ) .• ~ dt
=
A(i)
If x denotes a distance and dimensions of a speed.
(10)
i-I, 2, , .. , n. t
a time. the eigenvalues will have the
Anyone of these families of curves C(i) may be
't aken for our curvilinear coordinate lines 4>
= const.
The A(i) associated
with each family will then be the speed of propagation of the matrix column vector [ au/a~] along the curves C(i) belonging to that family. When the eigenvalues A(i) of A are all real and distinct, so that the propagation speeds are also all real and distinct. and there are n distinct linearly independent right eigenvectors r(i) of A satisfying the defining relation r
(i)
•
for i
= 1.
2••••• n,
the system of equations (1) will be said to be totally hYperbolic.
(11)
We
may. if we desire, replace the words right eigenvector by left eigenvector in this definition. where the left eigenvectors I of A satisfy the defining relation for i
= 1.
2••••• n.
(12)
The families of Curves C(i) defined by integration of equations (10) are called the families of characteristic curves of system (1).
30
The relationship between characteristic curves and the solution vector U to system (1) is illustrated in the Figure in the case of a typical element u i of U.
Here it has been assumed that initial conditions
have been specified for system (1) in the form U(x, 0)
=
'!' (x) ,
where the ith element u
i
of U has for its initial condition ui(x, 0)
I.L~
x, Since it was not necessary that the characteristics
~
aU/a~
should be discontinuous across
• const., it must follow that continuous and
diffe rentiable elements of the initial data ui(x, 0) • propagate along characteristics.
~i(x)
will also
In the case of the element of initial
data at A, this will propagate along the characteristic
~
= kl
(say) starting
from the point (Xl' 0) which is the projection of A onto the initial line. The characteristic
~
= kl
is then the projection onto the (x, t)-plane
of the path AB followed by the element of the solution surface S that started at A.
Characteristics corresponding to k • k 2, k k etc., 3, 4,
may be interpreted i n similar fashion. To distinguish between initial and boundary value problems °i t is necessary to classify arcs r in the (x, t)-plane as being either timelike or spacelike.
This is done by assigning to each characteristic arc
an arrow showing the direction corresponding to increasing t, and then
31 testing to see whether at a point in question all characteristics radiate
r
out to one side of the arc
corresponding to a timelike arc
r.
lie to one side and some the other
or some
This is illustrated in the Figure.
cJ."-')
d-
t4 )
\In-s)
r
-----
timelike arc r at P Supp 0 se th a t th e
~ec
t or r ( i )
r.
arc
spacelike arc r at P Wl' t h
e 1ement s r(i) 1 • r(i) 2 ••• •• r(i) n
the i ,th right 'ei genvect or of A corresponding to the eigenvalue }.
l'S
~ }. (i) •
Then it followes from (9) that across a wavefront belonging to the C(i) family we may write. [aul/a~
]]
(13)
(i) r l
where the elements of r(i)
D
r(i)(u) have values determined by U on the
wavefront. 2.
Wavefronts bounding a constant state In physical situations the solution vector U describes the "state"
of the system described by equations (1).
It is thus convenient to refer
to a region in which U is non-constant as a disturbed state. and a region in which U is constant as a constant state. irrespective of whether or not the system involved described a physical situation.
Our purpose here
will be to examine the simplification that results in equation (13) when a wavefront bounds a constant state.
32 First, as the elements a.o of A are continuous functions of their 1J
arguments, if follows directly that the eigenvalues A(i) of A are continuous functions of a i j, and hence of the elements u l' u 2' • •• , un of U.
Since U is itself continuous across a wavefront we conclude that
A(i)
= A~i) = const.,
where A~i)
on a wavefront bounding the constant state U • U O'
= A(i)(Uo)'
From equations (10) we thus see that if a charac-
teristic curve from the ith family C(i) bounds a constant state, then it must be a straight line. If such a straight line characteristic c~i) belonging to the ith family C(i) bounds a constant state U then because (au/a~)+
=
auo/a~
= Uo
that lies to its right (say),
= 0,
[~ ]
for j
= 1,
2, ••• , n,
(14) Now au/at' is continuous across
c~i) while auO/at' = O. Thus in
the disturbed region immediately adjacent to c~i) the total differential duo reduces to J
duo J
au.
(af i. d~
forj-l,2, ••• ,n.
(15)
By virtue of (13) and (14) this is equivalent to du o J
= Kr~i) d~ J
,
(16)
where K is some constant of proportionality.
It proves convenient to
choose K so that the first element Kr~i) of Kr(i) becomes unity. j - 1 in (16) then gives dU l
d~,
Setting
so that all the other differentials
du 2, dU , •.• , dUn become expressible in terms of du because (16) 3 l, becomes duo J
r
(i) dU j l
fo~
j • 1, 2, •••• n
or
dU
-
r(i) dU
l
. (17)
33 A simple rule that is sometimes useful for deriving results of this form follows by combining the matrix vector form.of (17) and the defining relationship
for the right eigenvector corresponding to the eigenvalue adjacent to the constant state '(A - AOI) dU •
U • Uo this gives the result (18)
where A • A(U ) and A • A(U ) ' O
Immediately
0,
O
O
A.
O
Comparison of this result with system
O
(1) from which it was derived now yields the following rule.
Rule for compatibility conditons for elements of dUo
To find the
relationships that exist between the elements dU
du ••••• dUn of dU 2 l, in the disturbed region immediately adjacent to a wavefront that bounds
a constant
state U • U ' the vector B in (1) should be neglected. the O undifferentiated variables should be replaced by their constant state values. and in the differentiated terms the following replacements should be made
.L at 3.
-+-
-Ad(.)
and
aax
-+-
(19)
d(.).
Riemann invariants This method applies to any totally hyperbolic system of two
homogeneous first order equations involving two dependent variables u
l'
u of the general form 2 aU I at- +
aU
2
at
aU
+
a 2l
axl
aU
+
a 22
ax2
. o,
(20)
which is subject to the initial data ul(x. 0)
.
and
iiI (x)
The coefficients a ••• a .• (u
u
2(x.
0)
.
ii 2(x)
.
(21)
u will. in general, be assumed to l• 2) be functions of the two dependent variables u and u 2• but not to have 1 1J
1J
34 any explicit dependence on the independent variables x and t . (20) will be quasilinear when a •. 1.J
The system
u ) and it will be linear in l• 2 --the special case when the coefficients a . . are all constants. e
a • . (u 1.J
1.J
Defining A and U to be
U
-
[:~J
enables equations (20) to be written (22) when we know that the system will be totally hyperbolic provided the two eigenvalues A(i). i - I . 2 of
IA-AII-O
(23)
are real and distinct and A has two linearly independent eigenvectors. In place of the right eigenvectors r that were useful in the previous section. let us now make use of the corresponding left eigenvectors t defined t(i) A _
A(i) t(i) •
for
(24)
i - I . 2.
If. now. we pre-multiply (22) by t(i) and use (24) we obtain the result l(i)
(au
at
+
A(i)
au) _
ax
0 •
(25)
for i - I . 2.
In this the bracketed expression will be recognised as the directional derivative of U with respect to time along the family of characteristics e(i).
Denoting differentiation with respect to t ime along members of
the eel) family of characteristics by dIdo and differentiation with respect to time along members of the e(2) family of characteristics by d/d6 enables us to replace (25) by the following pair of
ordi~ary
differential equations which are defined along the ell) characteristics by 1 (1) dU
cia
·0.
(26)
35 and along the e(2) characteristics by II.
(2) dU
de
•
0
(27)
Hence B • canst., along C(l) characteristics and a • canst., along C(2) characteristics as indicated in the Figure.
Setting the left eigenvector JI.(i) • (JI.~i), JI.~i», for i
= I, 2 then
enables (26), (27) to be re-expressed as
o
along the e(l) characteristics
(28)
along the e(2) characteristics.
(29)
and
Since, by supposition, A depends only on u
so also will the 2' coefficients lI.~i) of the left eigenvectors 11.(1), 11.(2). Consequently, l
and u
J
both (28) anda9) will always be integrable along their respective characteristics, though they may first require multiplication by a suitable integrating factor
~.
Integrating (28) with respect to a along the eel) characteristics, and (29) with respect to B along the C(2) characteristics gives: along ell) characteristics ~Jl.l(1)
J
dUl
+
J""2(1) dU .." 2
r(B)
(30)
36 and along C(2) characteristics
s(a) ,
csn
where r, s are arbitrary functions of their respective arguments Band a. The two families of characteristics are themselves given by integration of the equations (32)
for i • 1, 2.
The functions r(S) and s(a) are called Riemann invariants and, by virtue of their manner of derivation, rand s are constant along their respective families of characteristics.
To be more precise, r(s) is
constant along any C(l) characteristic, though as it is a function of S, which in turn identifies the characteristics, it will, in general, be different for different characteristics.
Correspondingly, s(a) is
constant along any c(2) characteristic, though here again the constant will be different for different characteristics depending on the value of a associated with each characteristic. Equations (30) and (31) enable u
and u to be expressed in terms 2 l of rand s, the values of which are determined at points of the initial line t • 0 by the initial data (21).
Suppose r(B) in (30) is denoted by
R(u
u and s(a) in (31) is denoted by S(u u Then along the C(i) l, 2). l, 2) characteristic issuing out from the point (x ' 0) of the initial line in O the sense of increasing time we have from (21) and the property of r(S) that (33)
Similarly, along the C(2) characteristic issuing out from the point (Xl' 0) of the initial line in the sense of increasing time we have from (21) and the property of s(a) that
(34)
37 Solving these two implicit equations for u
l
and u
2
then determines
the solution at the point P in the Figure which is the point of intersection of the C(l) and C(2) characteristics along which the respective constant values of Rand S are transported.
In principle the initial value
problem is now solved, since as the points (x O' 0) and (xl' 0) of the initial line were arbitrary, so also is the point P which may be anywhere in tpe upper half plane.
However, in any particular case, the task of
solving the two implicit relationships and of finding the characteristic curves in order to determine their point of intersection P is usually difficult.
Nevertheless, this method of solution can often be used to
solve.problems and it is, in any case, of considerable theoretical importance. References
[1] t2]
[3] [4]
Courant, R., Friedrichs, K. O. Supersonic Flow and Shock Waves, Interscience 1948. Jeffrey, A. Quasi1inear Hyperbolic Systems and ~aves, Research Note in Mathematics, Pitman Publishing, London, 1976. Coulson, C. A., Jeffrey, A. Waves, 2nd Ed ., Longman, 1977. Garabedian, P. R. Partial Differential Equations, Wiley, 1964.
38 Lecture 3.
Simple Waves and the Exceptional Condition
Simple Waves
1.
When one of the Riemann invariants r or s is identically constant, the corresponding solutions of equations (20) of Lecture 2 are known as simple wave solutions.
That is, simple wave solutions occur either when rea)
const., or when sea)
= So
~
=ro ~
const., and we now deduce the basic properties
of this fundamental class of solutions directly from this simple definition. Suppose, for example, that sea)
= sO'
then equations (30) and (31) of
Lecture 2 may be written rCa)
along C(l) characteristics
(1)
and (2)
So
along C
characteristics,
(2)
where f
ij
(u ) j
(i)
J ~R.j
(3)
duj •
This shows that everywhere along a C(l) characteristic specified by S
= 80 = const.,
say, u
l
and u
2
must also be constant, for they are the
sol ut i on of the nonlienar system of simultaneous equations
and
The actual constant values associated with u characteristic are
~
- ul(t), u
2
l
and u2 along this
- u 2(t) determinded by the values of the
initial data (21) of Lecture 2 at the point which this eel) characteristic passes.
(~,o)
of the initial line through
Any function of u
l
and u
2
will also
be constant along this characteristic as, in particular, will be
A(l)(ul(~)' u2(~» - A(l)(t), say.
Consequently, as the eel) characteristic
is found from (32) of Lecture 2 by integrating
39 :~ •
C(1) :
II (1)
m ,
it D1Ust be the straight line
x
t + til (1) m
•
As eo and hence
allowing
~
(4) were arbitrary, this result implies that by
~,
to move along its permitted interval on the initial line, so
(4) will generate a straight line family of C(l) characteristics.
Conversely,
had we set r(e) _ r ' it would then have followed that the C(2) family of O characteristics was a family of straight lines along each of which u u were constant. 2
l
and
Thus simple waves occur adjacent to constant state
regions and one of their main uses is to piece together solutions between different constant states. By analogy with the
sit~ation
in gas dynamics, when a straight line
family of characteristics converges, the associated simple wave is often called a compression wave, whereas when it diverges, the associated simple wave is called -an expansion wave.
Compression waves generate shocks which
are to be discussed later. The property that in a simple wave u
l
and u
2
are constant along the
straight line characteristics means that simple wave solutions are the simplest type of npn-constant solution for the system (1), (2) .
2.
Generalised Simple Waves and Riemann Invariants It is reasonable to enquire whether the notion of a simple wave can
be generalised and extended to systems with more than two dependent variables.
Specifically we shall consider homogeneous systems in one space
dimension and time of the type all (U) a
a
21
(U) a
l2 22
(U)
alo(U)
u
l
(U)
a
(U)
u
2
20
o,
+
u
o
u t
n
x
(5)
~o
which will be said to be reducible in the generalised sense. Since our concern will be with generalised. simple waves we seek an extension of a Riemann invariant from amongst the properties of ordinary simple waves.
The property we choose to generalise is that in an ordinary
simple wave there is a functional dependence between u
and u of the form 2 l Accordingly. we propose to take a generalised simple wave region
z f(u ) . 2 l to be one in which the solution vector U is a function of only one of its
u
n-elements, say of u ' so that U • 1
If in (5) we set U - U(u
l).
U(~).
so that u
i
• ui(u for i • 2,3, . • •• n. an l)
elementary calculation establishes that
• o.
(6)
This system can only have a non-trivial solution if
IA-
o •
\III
where \I • -(aul!at)!(aul!ax).
The n solutions \I(i) to the algebraic equation
(7) are just the eigenvalues ~(i) of A. so that when \I • \I(i) the vector dU!du
l must be proportional to the right eigenvector rei) of A corresponding
to A( i ).
As system (5) is assumed to be hyperollc, there will be n distinct
eigenvectors r(i).
The fact that \I(i) • ~(i) then implies that along the
family of characteristic curves C(i) dx dt
•
[::1) / (::)
~ (i)
(8)
or, equivalently. that
aUl
aUl
--ax dx +--at dt
•
( )
0 along each member of the C i
family.
thereby showing that canst. along each member of the C(i) family. A
(9)
corresponding result applies along each of the n different families of
characteristics C(i). Consider now the k-th such family and let us determine the nature of the generalised simple wave that is associated with it.
The fact that the proposed
41 generalisation of a simple wave region allows a corresponding generalisation of the notion of Riemann invariants will emerge from the fact that we will find that we are able to determine the form of these generalised invariants. Setting i
a
k in (9) shows that u
fact, taken together with U • U(u k-th family of characteristics.
l),
l
a
const along the C(k) family.
This
then shows U a const along members of the
As A • A(U) and U • const along any C(k)
characteristic we conclude that A(k) • const, thereby proving that the C(k) family comprises a family of straight line characteristics.
Now provided
attention is confined to continuous and differentiable solutions, system (6) may be written in differential form by replacing dU/du
l
by dU, when the fact
that in the C(k) family of characteristics dU/du eigenvector r(k) establishes that dU
«
is proportional to the right l r(k) along each member of the family
of straight line characteristics C(k) . This result gives rise to the set of n differential equations dU 2
(10)
"7(k) r
2
in which r (k), r (k), ••• , r (k) are the elements of the eigenvector r(k). l 2 n These n first order ordinary differential equations determine the behaviour of the solution U across what will be called a generalised A(k) - simple wave.
When integrated, (11) will give rise to n-l linearly independent
relations between the n elements of U, though multiplication of (10) by an integrating factor m(u u ••• , un) might be necessary because of the fact l, 2' that r(k) is only determined up to an arbitrary multiplicative factor. These n-l invariant relations along the k-th family of characteristics will be denoted by J (k)(U) - const.for i . 1,2, ••• ,n-l.
(11)
i
They will be called generalised A(k)-Riemann invariants to make clear that (k)
they are associated with the k-th family of characteristics C (k)
relations hold throughout the generalised A
•
These
-simple wave region where they
determine the behaviour of a continuous and differentiable solution.
42 On occasions, when deriving the generalised
~(k)-Riemann invariants
from (10), it is useful to express them in terms of a parameter
~
by writing
(10) in the form du
n - ---o
dt •
(12)
n
The u
may then be determined in terms of t by integrating the system
i
for j • l,2, ••• ,n •
(13)
Elimination of t between these n equations gives rise to 'the n-l generalised
~(k)-Riemann invariants (11). Definition (Reducible System in Generalised Sense) The system U t
+
AU
X
0
in which U is an n x 1 column vector and A is an n x n matrix will be said to be reducible in the generalised sense if the elements of A depend explicitly only on U. Definition (Generalised Simple Wave Region) Let the system
in which U is an n x 1 column vector and A is an n x n matrix be reducible in the generalised sense.
Then any region S in the (x , t)-plane in which the
solution vector U is of the form U - U(u ) , with u j
j
one particular element of
U, will be called a generalised simple wave region . The following theorem is easily established from the previous results. Theorem 1 (Generalised Simple Wave Regions) Let the system U
t
+
AU
X
0
with U an n x 1 column vector and A an n x n matrix be reducible in the generalised sense.
Then if S is a generalised simple wave region:
43 (a)
there is a family of straight line characteristics c(k) traversing S ;
(b)
the solution vector U is constant along members of the C(k) family;
(c)
in S there will be n-l generalised A(k)-Riemann invariants J (k)(U) ~ i const for i = 1,2, •.• ,n-l which will be determined by integrating equations (10).
3.
Exceptional Condition and Genuine Nonlinearity It is now appropriate to introduce two related concepts in connection
with first order quasilinear hyperbolic systems.
These are the notions of
a solution which is exceptional with respect to a particular characteristic -f i el d , and of a system which exhibits genuine nonlinearity with respect to a characteristic field.
Although these ideas may be introduced without
reference to generalised simple waves it will be convenient to use this approach here and to remove this restriction later. For our starting point we take a generalised A(k)-Simple wave region and the associated generalised A(k)-Riemann invariants J (k)(U) • const i
for i= 1,2, . •. ,n-l.
Each of these invariants defines a manifold in the
(u l• u 2 •••.• un)-space, on the i-th of which J (k) must obey the i constraint condition dJ (k) • 0, or i
o• (k)
Now in a generalised A r
j
(14)
-simple wave region we have from (13) that dU • j
(k)d~, so that (14) is equivalent to the condition (VuJ (k»r(k) i
with i
= 1.2 •.•.• n-l.
0 ,
(15)
These orthogonality conditions for the (VuJ (k» i
with
respect to the right eigenvector r(k) associated with eigenvalue A(k) of A were the ones used by Lax to define generalised A(k)-Riemann invariants.
He
then used this definition to establish the properties of solutions in a generalised simple wave region that are given in our Theorem 1. Before proceeding to our main objective let us first use condition (15). together with an argument due to
Friedri~hs.
to prove that the solution
44 adjacent to a region of constant state must be a generalised simple wave region.
This result which might have been conjectured from Theorem 1 will then
complement the results of that theorem. First we notice that from Theorem 1 it follows that if a region A of constant state exists in the (x,t)-plane, then it will be bounded by a characteristic, say by a member C of the C(k)-family.
Any region adjacent to it will also be
bounded by this same line C. Now pre-multiplication of system (5) by the left eigenvector ~(j) of A gives the system along the C(j) family , for j
= 1,2, . •. ,n.
(16)
As the left and right eigenvectors of A are biorthogonal~
so that
o
for j ; k
it follows directly fr~ (15) that the vector ~(j) must be expressible as a linear combination of the vectors (VuJ (k». i n-l
b
L
sal
Accordingly, we set
(V J (k»for j ; k • j sus
(17)
Equations (16) then become n-l
-
dU 0 for j L b js (VuJ s (k» dj s"l which by the chain rule reduces to n-l
L
sal
dJ (k) b j S dj s
-
0
for j
;
k ,
(18)
(19)
; k •
This is now a linear hyperbolic system involving (n-l) equations for the (k)
(n-l) generalised A
-Riemann invariants J l
(k)
specified solution vector U the coefficients b
' J
js
(k)
2
' ••• , I
(k)
n_l
will be known.
For any
The condition
j ; k ensures that the line C common to both the region of constant state and the generalised simple wave region will not be a characteristic of the new system.
Consequently there exists a unique smooth solution that can be
continued across the line C.
Since the solution on one side of C was the
45 constant state solution, the solution that is continued across it will be one for which all the generalised
~(k)-Riemann invariants are constant . Hence
from the nature of generalised Riemann invariants it may be seen that the solution adjacent to a region of constant state must be a generalised simple wave region.
This result also merits a formal statement.
Theorem 2 (Constant State and Generalised Simple Wave) Let the system U
t
+AU
x
-
0
with U an n x 1 column vector and A an n x n matrix be reducible in the Then if A is a region of constant state in the (x,t)-plane,
generalised sense.
the region S adjacent to it is a generalised simple wave region. Let us now examine further the implications of equation (15). the special case in which the eigenvalue J (k),
l
~2 (k) , ••• , I _ (k) n- l
I
'"
n-l il~ (k)
{m-l
~(k) is expressible as a function of
Then we have
n l
(.V), (k) ) u
Consider
~,(k) ilJ (k) m ar (k) ~
_01\__
m
n-l
I
_il~
(k)
m"'l ilJ (k) m
ilJ (k)
m_
,
ilu2
•••
t
aJm(k) ) au
l-
_1 aJ (k) m
n
or, equivalently, n-l
ax (k)
111"'1
ilJ (k)
l
showing that (V
u
generalised
~(k»
m
V J (k)
(20)
um
is a linear combination of the gradients of the
~(k)-Riemann invariants.
After post-multiplication of (20) by
r(k) it then follows directly from (15) that (21) In general, when a quasilinear hyperbolic system exists for which property (k)
(21) is true with respect to the k-th characteristic field C with),
= ~(k),
associated
the system will be said to be exceptional with respect to the
k-th characteristic field.
This will be true irrespective of whether or not the
46
system permits generalised simple wave solutions.
When (21) is not true, the
system of equations will be said to be genuinely nonlinear with respect to the k-th characteristic field C(k).
Expressed differently, condition (21)
asserts that when a system is exceptional with respect to the k-th characteristic field, the directional derivative of A(k) in the direction of the eigenvector r(k) is zero.
We now formulate these ideas generally, without reference to
generalised simple waves or to Riemann invariants. Definition (Exceptional Condition and Genuine Nonlinearity) Consider the quasilinear hyperbolic system
where U is an n x 1 column vector, A · A(U,x,t) is an n x n matrix and B
= B(U,x,t)
is an n x 1 column vector.
Then the system will be said to be:
(a)
exceptional with respect to the k-th characteristic field if
(b)
compl e t el y exceptional if it is exceptional with respect to each of the n characteristic fields corresponding to h(l), A(2), .•• , A(n);
(c)
genuinely nonlinear with respect to the k-th characteristic field if
References [1] [2J [3] [4]
Jeffrey, A. Quasilinear Hyperbolic Systems and Waves, Research Note i n Mathematics 5, Pitman Publishing, London, 1976. Coulson, C. A., Jeffrey, A. Waves, 2nd Ed., Longman, 1977. Friedrichs, K. O. Nichtlineare Differenzialgleichungen. Notes of lectures delivered at GBttingen, 1955. Lax, P. D. Hyperbolic Systems of Conservation Laws II, Comm. Pure App1. Math. 10 (1957), 537-566.
47 Lecture 4. The Development of Jump Discontinuities in Nonlinear Hyperbolic Systems of Equations
1.
General Considerations We shall consider initial value problems leading to the propagation of a
wavefront in quasi-linear systems of equations of the form (1)
where U is a column vector with the n components u l• u 2 , ..• , un' A is an n x n matrix and B is an n element column vector; A and B are assumed to depend on x, t and U.
The system (1) will be considered to be hyperbolic
and so all the eigenvalues of A are real and A possesses a full set of linearly independent eigenvectors. The left eigenvectors of A, 1(i,k) with k • 1,2 •••. ,s corresponding to the eigenvalue A(i) with multiplicity s satisfy the equations 1(i.k) A •
A(i)1{i,k)
•
k •
1.2, •••• 8
(2)
•
They may be used to display the equations (1) in characteristic form and to introduce the n characteristic curves e(i) as follows.
Pre-multiply
equation (1) by 1(i) and, assuming for the moment that the n eigenvalues of
A are distinct, we obtain n equations written in characteristic form which, by virtue of (2). become
+
1 (i) (U
U ) + b(i)
A (i)
x
t
where b(i) • 1(i)B.
The operator
o a~ +
A(i)
a:
(i
1,2, ••• ,n)
(3)
in the ith equation represents
differentiation along the ith characteristic curve e(i) determined by dx • C(i ) •• dt
,(i) 1\
•
(4)
We shall be concerned later with the propagation of a disturbance or wave into a state which is either known (and non-constant) or is constant. when the line bordering these two states, the wavefront, is determined by a relation of the form
Hx.t) The wavefront
O.
• ~
(5)
• 0 is assumed here to be a line acrosa which the solution
48
U is continuous but across which the normal derivative of U is discontinuous. The class of solutions U considered is thus Lipschitz continuous with exponent unity. 2.
The Initial Value Problem Consider the system U
t
+AU
x
+B
(6)
°
subject to the initial condition U(x,O)
t(x) ,
<
x <
(7)
CD
where t(x) is Lipschitz continuous. Using Haar's a priori estimate and a special iteration scheme it may be shown that while the solution on the wavefront remains Lipschitz continuous the solution of (6) and (7) on the wavefront depends boundedly on t he initial values and on the inhomogeneous term.
Accordingly, when the advancing wave-
front ceases to be Lipschitz continuous U will attain a bounded value, say
uc ' behind the wavefront while ahead of the wavefront U will have a value appropriate to the state into which the wave is advancing.
Thus at some
critical time t c and at some critical distance x the solution ceases to he o Lipschitz continuous on the wavefront and a finite jump or shock like discontinuity appears in U with magnitude U - U. c
We shall now obtain exact
analytical expressions determining the initial time t c and the critical distance xc. 3.
Time and Place of Breakdown of Solution We start with a general quasi-linear hyperbolic system (8)
°
and assume that the vector B and the eigenvalues and eigenvectors of A are continuously differentiable with respect to their arguments.
We also suppose
that A and B do not depend explicitly on x and t and so there exists a constant solution U satisfying the equstion
o
°.
(9)
lhia constaut state ~ill be denoted by the subscript 0 and we shall consider
49 a wave advancing into this constant state .
The solution is Lipschitz
continuous normal to the wavefront and initial conditions may be prescribed such that at t continuous at x
= O.
= 0,
U
Denote by t
= Uo for c
x
>
0 and such that U is
and Xc the critical time and critical
distance, respectively, at which the solution ceases to be Lipschitz continuous on the wavefront. Let us now assume that there exists at least one positive eigenvalue of A so that the wave proceeds in the direction of the positive x-axis. We identify the velocity of the wavefront with one of the positive eigenvalues, say
~~~), and introduce the curvilinear coordinates
constant ,
t'
constant
through the equations (lOa)
t
t'
and . (lOb) Ftom equation (lOb) we see that (11)
but along
~ =
constant we may write
o and so from (11) and (12) we see that the
(12) ~
= constant
lines are characteristics
and so dx dt
•
~(~) along ~ = constant.
Thus, since equation (13) is only valid along
~ =
constant and from (lOa) we
have t' • t, equation (13) is identical with
ax
(13' )
at' which is a result that will be required later. As ~ is a solution of (lOb) we must specify it by giving initial
conditions .
These should reflect the fact that it is a coordinate variable
50
and so should be assigned monotonically.
We choose
~(x,t)
by imposing the
initial condition
Hx,O)
x
when the wavefront is given by Hx,t)
0
and, in the region of constant state ahead of the wavefront,
Hx,t) > 0 • The transformation introduced through equations (10) is non-singular provided the Jacobian •
1 ~x
(14)
is non-zero and finite.
The initial condition on
~
ensures that
x~
is initially equal to unity and
so we may assume the non-vanishing of the Jacobian for at least a finite time
= O.
after t
Let us denote by L the open region lying to the left of the
advancing wavefront (x,t) • 0 and bounded on the left by the characteristic (x,t)
=
0 also issuing out of the origin and chosen so that no other
characteristics enter L.
Then, since no characteristics enter L, U will
remain smooth in L for at least a finite time. operations on the side
~
< 0
All subsequent limiting
of the wavefront will be assumed to be performed
in L. Let l(j) be the left eigenvector of A corresponding to the eigenvalue A(j) then, from equation (3)
o . Employing the identities
...!. =!t ...!.. + l!'...!.. at - at
a~
at at'
and
...!. =!t...!.. + l!•...!.. ax
- ax
we see that
a~
ax at'
o
$1 Thus, using these results and equation (lOb) together with condition (14) to ensure the non-vanishing of the Jacobian, we obtain R. (j)
(x41 ...!. at'
+ (>. (j) - >. w)
...!.) a41
U
+
k
-
bo(j)
o.
x 41
(15)
In particular, if >.(j) - >.(41), we have ,(4I,k) + b(4I,k) Ut' ..
o ,
1,2, ••• ,r
(16)
41
is the multiplicity of the eigenvalue >.(41). 41 By virtue of our choice of coordinates the wavefront 41 - 0 is a
where r
characteristic, and since the solution is Lipschitz wavefront, jump
con~nuous
across the
discontinuities in derivatives with respect to 41 may take
place across 41 - O.
Accordingly we define the jumps across 41 - 0 as follows:
41-0- 41-0+
U is continuous :
[U ]
Ut' is continuous
[Ut']4I-O+
41=0-
-
0
or
U(O,t') - Uo
(constant)
0
U is discontinuous 4I and X
41
is discontinuous
We note that since both IT and X depend on 41 they are not independent and we shall later determine their precise relationship [see equation (21)]. From the definition of X we see that X + (x
- (x4')4I_0_,whi1e 4l)4I=0+ Hence, in a neighbourhood of the wavefront
(x4')4I-O+ - x 0 (say) is finite. 41 condition (14) is seen to be equivalent to the condition
x+
x 0 is finite and non-zero. 41 The significance of the non-vanishing of the Jacobian may easily be
(14')
seen by noting that in L and along 41 - 0 we have U
x
whence
So, if x
41
vanishes while U4' remains finite, U ceases to be Lipschitz continuous
and we have the gradient catastrophe.
52 In the simple case that U
o
m
constant the jump conditions on n and X
reduce to n(t' )
and X(t') •
(17)
So, since t(j) is assumed to be continuous across the wavefront and Ut' is continuous across the wavefront with U , t
=0
in the constant region we
have, by using (9) and by considering equation (15) at a point P in Land letting the point tend to a point on the wavefront, that j
(18)
r.p+l' ••• , n
where aga in the subscript 0 signifies the constant state appropriate to (9). We now differentiate equation (16) with respect to .p at point P in L to obtain
or,
where T denotes the. transpose operation and where V is the gradient operator u with respect to (u
l'
u
2,
•••• uu)-space.
Again, letting P tend to a point
on the wavefront and using the fact that U is continuous across the wavefront t, with (Ut').p_O+= 0 we obtain the equation k
m
1,2, •• • ,r.p
(19)
If, now, we differentiate equation (13') with respect to .p at a point P in L we obtain 0
~
(~~, )
(V A(,» u
U,
and so, 0
at' (x,)
(V A(.p»U u ,
(20)
53 Thus, again letting P lend to a point on the wavefront, and using (17) we find that (21)
i~j) are linearly independent vectors we may use equations
Since the (18) to express TI.
(n-r~)
components of TI in terms of a certain
components of
r~
Introducing these expressions into equations (19) leads to
r~
first order
ordinary differential equations with constant coefficients for the say TIl' TI
. •. , rrr~ .
2,
r~
unknowns,
Introducing TI thus determined into equation (21) and
integrating the result we may obtain an expression for X.
However, before
doing this it is necessary to define certain limiting operations that will be necessary in the integration process. ~
If Q is a quantity defjned only in L we define the operation Q to be the limit
Q_ lim
t' ->()
(Q)~~o-
adjacent sides of ~ taken along
=0
For a jump quantity P depending on the state On we define the operation
P to
be the limit
o.
'~ ~
P = lim
t' ->()
P
Thus, integrating equation (21) with respect to t' between 0 and T and noting that X is a jump quantity defined across operation
x ..
X just
~
.. 0 we may use the limiting
defined to obtain the result
x+ ITo (V A(~) u
TIdt' •
(22)
0
This equation describes the variation of X along the wavefront
~
.. 0 with
advancing time and in writing equation (22) we have tacitly assumed that multiplicity
r~ of A(~) remains unchanged .
and at t' .. Tl(T
l
the
Should this assumption not be true
< T) the multiplicity changes, then TI must be re-determined
for the interval t' > T l•
We remark here that although an initial condition
on Ux may be prescribed arbitrarily by specifying lim (Ux)t-O' this limiting x->() operation is not in L and so in general
Ux is
not equal to this limit and
although on the initial line may not be prescribed arbitrarily. Equation (22) may be displayed in a slightly different form from which the critical time t
c
may be determined as follows.
By definition
54 it = or
when equation (22) becomes
However, from (14') we see that the left-hand side of -t hi s expression is simply the Jacobian of the transformation and so is required to be finite and non-zero in order that the transformation is unique. critical time Jacobian x$
T
= tc
=X+
x 0 $
So, if there is a
at which condition (14') ceases to be valid and the -
0, this is given by
(23) Geometrically the vanishing of the Jacobian is equivalent to the point at which the $ - constant lines first intersect the wavefront $
= O.
(i.e ., the
family of characteristics $ - constant intersect at a cusp) . To determine t
c
in terms of U we divide equation (23) by x$ to obtain x
o. The vector
n must
be determined from equations (18), (19).
(24)
In the particularly
simple case that B • 0 it follows directly from (18) and (19), provided the multiplicity of A($) is constant, that n is a constant equal to its initial value
.
n,
and so
n • and thus
ux Using the definitions of X and n we see that
55
when x
4
..
but
and so U is given in L by the expression x
UIII + X
U
X
(V
u
~(4»
0
Ut'}. x
(25)
Thus Ux becomes unbounded if the denominator vanishes for some t c > O.
If
(Vu~(4»o = 0 it follows directly from equation (21) that X is a constant and so t
c
is infinite.
The discontinuity in this case is propagated but
remains finite for all time.
Systems for which this property is true are
a special case of those which are exceptional with respect to the
~(4)
characteristic field. The general case when A, B depend on U and also explicitly on x, t has been discussed in detail in [1].
A different approach to the problem
that involves three space dimensions and time has been described by Boillat [2] and Chen [3). References [1] [2] [3]
A. Jeffrey, Quasilinear Hyperbolic Systems and Waves. Research Note in Mathematics No .5, Pitman Publishing, London, 1976. G. Boillat, La Propagation des Ondes. Gauthier-Villars, Paris,1965. P. J. Chen, Selected Topics in Wave Prop~gation. Noordhoff, Leyden, 1974.
50
Lecture 5. The Gradient Catastrophe and the Breaking of Water Waves in a Channel of Arbitrarily Varying Depth and Width
1.
Basic Equations To illustrate the gradient catastrophe in a physical context, let us
show how to obtain an explicit form for the amplitude of an acceleration wave that propagates into water at rest which is contained in a vertical walled channel with slowly varying width W(x) and an arbitrarily varying depth hex) below the equilibrium water level.
The method we describe is
taken from the joint paper submitted for publication to ZAMP by the author and J. Mvungi [1]. As usual, let the x-axis lie in the equilibrium surface of the water
in the direction of propagation, with the y-axis pointing vertically
= o.
upwards, and write the equation of the bottom of the channel as y + hex)
Then, if the elevation of the water above the equilibrium level is n(x,t), g is the acceleration due to gravity and the x-component of the water velocity is u(x,t), the equation of motion in the x-direction is as derived by Stoker
[2], namely (1)
However, the equation corresponding to the conservation of mass will now be different on account of the width variation of the channeL
To derive
it, all that is necessary is to observe that the cross-sectional area S(x,t) of the water at any given place and time (x,t) is S(x,t) and that the flow through this area is S(x,t)u(x,t).
= W(x)(n(x,t)
+
hex»~,
Thus, equating the
time rate of change of S to the negative flux through it, we find
-(Su)
x
(2)
,
from which it follows that
o
(3)
The governing equations for flow in a variable width channel of arbitrary depth are thus equations (1) and (3).
The assumption of a slow
variation in the width is necessary because the transverse movement of the
57 water has been neglected in these one-dimensional long wave equations, . and this will cease to be a good approximation if the width changes too rapidly. 2.
The Bernoulli Equation For The Acceleration Wave Amplitude Suppose the wave moves in the direction of increasing x, starting
from x • 0 at t • 0, and that it moves into water at rest.
Then, across
the wavefront: (i)
u and n are continuous, with u(x,t) • n(x,t) • 0 ahead of the advancing wave,
(ii)
the first and second derivatives of u and n suffer at most a jump discontinuity, so that the wavefront being propagated on the surface is an acceleration wave.
Using a superscript minua sign to denote the value of a function immediately behind the advancing wavefront (i.e . at the edge of the disturbp.d region) we conclude from (i) that u
n
o
(4)
Taking the total differential of equations (4) gives, just behind the wavefront, n-dx + n-dt x t
~+ u~dt and 0
-0
or, equivalently, u
t
-cux
and
nt
-c n
(5)
x
where c • dx/dt is the speed of propagation of the wavefront which is, of course, a characteristic curve for the system (1), (3) . Immediately behind the wavefront (1) and (3) become
o
and
n~ + hu; -
(6)
0,
where it is understood that h • hex) is the depth at the wavefront. n; ; 0 equations (5) and (6) imply the standard result c
2
If
• gh.
Now define the amplitude of the acceleration wave to be
a
a(x)
• nx '
(7)
58 when (5) and (6) become
"e- ,. -ga
and
ux •
(8)
ga/c.
Now notice that the operation of differentiation with respect to x along the characteristic followed by the wavefront, behind which u~ and u~ are defined, takes the form (9)
It then follows immediately from this that •
c2
U
xx
-
U
(10)
tt
To obtain the differential equation governing the behaviour of the amplitude a of the acceleration wave we first differentiate (1) partially with respect to t and (3) partially with respect to x . n
xt
Then eliminating
' and using (7) and (8), we find gcW_ 3 2 c 2u- - u- + (2lh __ x + _ x ) a +.=.lL xx tt c W c
a2
,.
o.
(11)
Combining (10) and (11) and using (8) brings us to the required Bernoulli type equation for the amplitude a(x), W) 3a2 da + 3h (4h x + 2; a + 2h dx
,. o ,
(12)
in which use has been made of the fact that, as c 2
gh, we have (dc/dx)
=
gh/2c. 3.
The Amplitude a(x) And Its Implications The standard substitution a • b
-1
reduces the Bernoulli equation (12)
to a linear first order equation, and a simple calculation then shows that (13)
in which a
I(x)
O
o•
• a(O), W
3h 3/4w 1/2 o 0 2
W(O) and
(14)
59 A wave of elevation corresponds to a to
"o
O
< 0 , and a wave of depression
The wave will be said to break if for some x
> O.
g
x
c
the water
surface behind the wavefront becomes vertical, so that the amplitude a(x (a)
g
c)
Since lex)
m.
~
0, we conclude from the form of (13) that:
A wave of elevation (a . < 0) in a variable width channel always
o
breaks in water of finite depth provided lex) is such that 1 + aOI(x 0, and Xc > 0 is finite.
at x
~
c)
g
If the depth of the water shelves to zero
t, say, so that h(t) • 0, a wave of elevation propagating
towards the shore will break before reaching the shore line if laol > l/l(t), and at the shore line if laol ~ let). (b)
A wave of depression (a
O
> 0) in a variable width channel can only
break if the depth of the water shelves to zero, and then only at the shore line provided let) < When we set W(x)
= WO'
m.
these general conclusions agree with the
special case of waves climbing a beach that was studied by Greenspan [3]. This is because the one-dimensivnal long wave equations do not distinguish between flow in a parallel channel and unrestricted one-dimensional flow. Result (14) shows that the integrand of importance in this case combines the depth function hex) and the width function W(x) in the form (h(x»-7/4(W(x»-1/2.
Thus any modification of the depth and width that
leaves this combination invariant will lead to the same conditions for breaking provided a O' hO and Wo are unchanged. As special cases of results (13) and (14) we observe first that in
a parallel channel of constant depth h we obtain Stoker's result [2], that breaking occurs when x
c
2h - 3a O
at a time t
c
2 - 3a
O
(E.g) 1/2
(
Secondly, when hex) • h - mx so that the bottom has a constant slope, we obtain Jeffrey's result [4] that breaking occurs when
(15)
60
x
c
(16) Result (16) also shows that when the water deepens at a constant rate (m
<
0), then although a wave of elevation will normally break, this will
not occur in the special case that 2a
O
s
m.
This was the result found in
[4J which used the transport equation approach that has been presented in a general form in [5J.
The equivalence of the method used here and of the
seemingly different one used in [4J and generalised in [5J has been established by Boillat and Ruggeri [6J.
-References - - - [1) [2J
[3)
(4)
(5) (6)
A. Jeffrey and J. Mvungi, On the breaking of water waves in a channel of arb itrarily varying depth and width . ZAMP (submitted for publication . J . J . Stoker, Water Waves. Wiley-Interscience, New York, 1957 . H. P. Greenspan, On the breaking of water waves of finite, amplitude on a sloping beach. J. Fluid Mech. 4 (1958), 330-334. A. Jeffrey, On a class of non-breaking finite amplitude water waves. Z. angew. Math . u. Phys. (ZAMP) , 18 (1967), 57-65. See also Addendum, A. Jeffrey, Z. angew . Math. u. Phys. (ZAMP), 18 (1967), 918. A. Jeffrey, Quasilinear Hyperbolic Systems and Waves, Research Note in Mathematics 5, Pitman Publishing, London, 1976. G. Boillat and T. Ruggeri, On the evolution law of weak discontinuities for hyperbolic quasilinear systems. Wave Motion 1 (1979), 149-151.
Lecture 1.
6.
Shocks And Weak Solutions
Conservation Systems and Conditions Across a Shock In what follows it will be assumed that the system of equations
involved is hyperbolic and capable of expression in the generalised conservation form.
That is, when the system involves n dependent
and is formulated in
]R3
variabJ ~s
x t, we assume it can be written in the divergence
form 3F
111: + div G with U
c
F
U(~,t),
vectors and G -
H,
(1)
= F(U,~,t)
G(U,~,t)
and H
= H(U,~,t)
an n x 3 matrix .
all n element column matrix
The matrix G in (1) is in effect
to be regarded as a tensor so that div G has the meaning div G -
r ~s s-l 3
3 (s)
where 8(s) is the s-th column of G. Systems of this type are of considerable importance because of their frequent occurrence in phys{cal problems where they arise from integral formulations of quantities that are conserved.
Indeed, since an integral
formulation is more fundamental than the related differential equation and it permits the integrand to be discontinuous, we shall make use of it to discuss discontinuous solutions for system (1). Discontinuous solutions have considerable physical significance, since they may be interpreted in terms of physical phenomena such as a shock wave in a gas.
If a discontinuous solution exists across a surface, the first
problem to be resolved is how the solutions on adjacent sides of the surface are to be related one to the other and to the speed of propagation of the surface .
In the case of a shock wave in a gas this involves determining the
relationship connecting gas pressures and densities on opposite sides of the shock with the speed of propagation of the shock. Theorem 1 (Integral Rate of Change Theorem) Let F be an n x 1 column matrix with elements· which are continuous scalar
62
functions of position and time defined throughout the volume V(t), which is itself bounded by a surface S(t) moving with
velocity~.
Then the rate of
change of the volume integral of F is given by
I
d
dt
V(t)
FdV-
JV(t) atilF
dV
+
I
S(t)
F
~.d.§.
where d.§. is the vector element of surface area. Let us now identify the column matrix F in Theorem 1 with the n column matrix F in system (1) and assume that a surface
a(~,t)
x
1
= const
exists across which the matrix vector U, and hence F, G and H are discontinuous. Next we choose the volume V(t) bounded by surface S(t) moving with velocity so that an arbitrary part SO(t) of the discontinuity surface divides it into the two sub-volumes V+(t) and V_(t).
a(~,t)
~
= const
Denote by S+(t) and S (t)
those parts of S(t) that bound V+(t) and V_(t), respectively, excluding the dividing surface SO(t) which, we assume, also has velocity
~.
Integrating (1) over V(t) • V+(t) u V_(t) gives
f
V+UV_
~:
dV +
JV+UV_ div G dV
JV+uV_H dV
or, from the matrix form of the Gaussian divergence theorem applied separately to V+ and V in which F, G are continuous and differentiable, (3)
where G.d.§. denotes the scalar product of G now regarded as a tensor and vector dS.
Combining (3) with the result of Theorem 1 applied separately to V+ and
V then gives the next result in which, it must be remembered, the dividing surface So(t) that is part of ddt
a(~,t)
= const
also moves with velocity
~
J
FdV V+uV_
If, now, we subtract from (4) the corresponding expressions integrated over the separate .vo l ume s V+(t) and V_(t), and bounded, respectively, by S+(t)USO(t) and S_(t)USo(t) we arrive at the result
(4)
63 (5)
where
d~
and
d~
_ are the outward directed surface elements with respect to
the volumes V+(t) and V_(t).
This situation is illustrated diagramatically
in the figure which shows an arbitrarily thin volume element taken across a(~,t)
= const.
The effect of differencing to obtain (5) is to make the
volume contribution and the contribution due to the surface element dS' directed along
~'
parallel to a(x,t) • const vanish in the limit as the
cylinder collapses onto the area element dS O•
-
r1_
Vofume element divided by discontinuity surface Since
d~
a(~,t)
• const.
are both normal to the same discontinuity surface a(x,t) = const,
but are oppositely directed so that n = -n
,we have
d~ =-d~_
= ~dSo showing
that (5) may be re-written as (6)
The fact that dS is arbitrary then gives an algebraic jump condition across O a(.!.,t) • const of the form (7)
It is useful to re-express this result by observing that the scalar quantities d~
~.~
and dS
and y_
.~
are the normal speeds of propagation of the elements
on opposite sides of, and moving with,
must be continuous across SO(t).
a(~,t)
= const,
and as such
So writing ~ = ~.E. = y_.~ enables the
jump condition (7) to be expressed in an alternative form using the speed ~ normal to S
64 (8)
which is sometimes written
~ ~ F ]] •
I[G]~.!!.,
(9)
with [Q]] denoting the jump in Q across discontinuity surface SO(t).
The
arbitrary nature of dSO also implies that U+ varies continuously over S. Because of the similarity of (8) to a corresponding condition in gas dynamics this result will be called the generalised Rankine-Hugoniot condition for system (1).
In general, ~ is ~ equal to a characteristic speed ~.
Theorem 2 (Generalised Rankine-Hugoniot Condition) Consider the conservation system
~~
+ div G •
with F •
F(U,~,t),
H, G•
G(U,~,t)
and H •
H(U,~,t).
Then, if this has a
discontinuous solution across a surface S, on the adjacent sides
± of
S the
solution varies continuously and is related by the jump condition
in which n is the normal to S and A is the normal speed of propagation of S, with G+ regarded as a tensor and G ..!!. denoting the scale of prodvct of G and the t unit vector.!!. normal to S. Definition (Shock Solution) A discontinuous solution to a system of equations expressed in conservation form which satisfies the generalised Rankine-Hugoniot condition will be called a shock. 2.
Weak Solutions and Non-Uniqueness In the development of the concept of a solution to a quasilinear hyperbolic
system, care has been taken to distinguish between classical once differentiable so called
cl
solutions, and piecewise differentiable C1 solutions separated
by shocks across which both U and its derivatives are discontinuous.
It
would be desirable, if possible, to unify these two types of solution by generalising the whole concept of a "solution" to system (1) in such a way that strict differentiability and continuity are no longer required.
This is
65 precisely the motivation underlying the notion of a weak solution.
For
simplicity, the argument that follows will be confined to a scalar equation, but the extension to a system may be made without requiring any essentially new ideas. For our starting point we take the equation
au + feu) au • at ax
0
(10)
'
suoject to the initial condition u(x.O)
g(x)
(11)
and assume that feu) is a continuous differentiable function of u.
Then the
first point to notice is that (10) can be expressed in conservation form by defining F(u)
If(U)dU •
(12)
to obtain
au +.!!. • o . at ax
(13)
Let us consider the half-plane t > 0 and recall that in general a unique solution to (10) and (11) will only exist for a finite time.
As we
have seen in Section 1, a conservation equation possesses discontinuous solutions or shocks, corresponding to a non-unique solution along an arc.
Accordingly,
and with reference now only to a general function f and initial condition g. l let us consider some strip 0 < t < T in Which the classical unique C solution exists everywhere except on certain shock lines across which the solution is bounded.
Adapting the notation of Section 1 we denote by u_ and u+ the
limiting values of u to the left and right of the shock under consideration, which from Theorem 2 are seen to vary continuously along the shock. Then the bounded function u defined in the half plane t > 0 will be called a weak solution of (10) if in this half plane it satisfies the condition (l4)
66
for every twice continuously differentiable function w(x,t) that vanishes outside some finite region in the half plane t >
o.
Such functions ware
called test functions and the closure of the region in which they are nonzero is then known as the support of the test functions.
As a general
l classical C solution to (10) subject to (11) has been found, we already 1
know that if a weak solution satisfying (14) is also piecewise C , then it must be a classical solution wherever it is solution coincides with a piecewise
cl.
cl classical
1
Thus a piecewise C weak solution, as would be
expected of any reasonable extension of the concept of a solution . L~t
us now show that there is a further common property shared between
l weak and piecewise C classical solutions.
This is that a piecewise Cl weak
solution satisfies the generalised Rankine-Hugoniot condition across a shock. Consider the region R bounded by the closed arc aR and traversed by the line L across which a shock occurs.
Denote the two sub-regions so defined
by R_ and R+ and their boundaries by aR_ and aR+, and let the directed arcs along adjacent s ides of L be aL_ and aL+, as in the Figure.
t L
Shock line L dividing R Then R • R_ UR+ and aR • aR_UaR+.
The test functions w in (14) will be
assumed to have their support in R so that the test functions w will vanish on
oR.
Thus (14) may be written
ffJ:~ u +
:: F(U») dxdt
o
(15)
67 Now multiply (13) by wand integrate over R_ to obtain
IIR (w ;~ + w ;~) dxdt
-
0 ,
which may also be written in the form
II
R
(a~;U)
+
a~:F»)
dxdt -
II
R
(;~
u +
~:
F) dxdt
•
O .
(16)
Applying Green's theorem to the first terms in this result then transforms (16) to
1 -wFdt + wudx - Ir(~w jaR uaL
JJgt
u + aw F) dxdt ax
•
(17)
O.
However as the support of the functions w lie in R, w will be zero on aR so that (17) reduces to
1 -wF(u_)dt
raL
+ w u_dx -
Ir(aw u + aw F) dxdt JJat ax
-
o.
(18)
A similar result applies with respect to R+ where we find
f
-w F(U+) dt + w u+dx aL+
Ir(:~ ~+
u + :: F) dxdt
-
(19)
0,
the integration along aL and aL+ being oppositely directed, as indicated in the Figure. If (18) and (19) are now added, the sign of the line integral 'in (18) is reversed with a corresponding replacement of aL_ by aL+ and result (15) is used we find (20)
where as the point (x,t) is now constrained to lie on aL+ the term (dx/dt) represents the speed of propagation
~
of the shock along L.
As w is arbitrary,
(20) can only be true if (21)
which is the one dimensional form of the generalised Rsnkine-Hugoniot condition.
This holds degenerately when u is continuous across L.
If, now, the support of w is allowed to be arbitrary, the same form of
68 argument proves that piecewise Cl solutions of (13) satisfying (21) across a shock will also be a weak solution of (13).
We thus arrive at the following
definition and theorem. DefInition (Weak Solution) The function u will be called a weak solution of ~
at
+
aF(u)
ax
_
o
if for all twice continuously differentiable test functions w with support in t > 0 the function u is such that
lat-fJ(raw
u
aw F(u) ) dxdt + -ax
o,
the integration being extended over the upper half plane t > O. Theorem 3 (Properties of Weak Solutions) Let u be a weak solution of
o. The following results are then true: (a)
If u is piecewise Cl in addition to being a weak solution it is also l a piecewise C classical solution.
(b)
a piecewise Cl weak solution satisfies the generalised Rankine-Hugoniot condition
across a discontinuity moving with speed Aj (c)
a necessary and sufficient condition for a piecewise
cl classical
solution to be a weak solution is that across a discontinuity moving with speed A it satisfies the generalised Rankine-Hugoniot condition . The general objective when introducing a weak solution was to lift the requirements of strict continuity and differentiability that need to be imposed on classical solutions.
In this respect the notion of a weak
solution is successful and, furthermore, because of its method of definition l the class of weak solutions is even wider than the class of piecewise C
69 functions so that considerable
generality has been achieved.
However,
this generality has been obtained at the cost of the uniqueness of a weak solution.
More precisely, unlike a strict classical Cl solution, a weak
solution is not determined uniquely by the initial data.
This is most easily
demonstrated by means of a simple example. Consider a Riemann problem for an equation of the form ilu
ilt
+.1.. (1 3} ilx [3 u J
-
0.
with u(x,O)
-
for x <
O l' {
for x >
° °
so that in (13) we have F(u) • u 3/ 3. Then, &s the equation
~s
homogeneous, when it is differentiable a
non-constant solution u will be a function of x/t, and it is easily verified that the function
o u(x, t)
{
(x/t);
I
is a
cl solution
° for ° ~ x/t ~ I
for x/t <
for x/t
>
1
subject to the initial condition.
This solution is continuous
everywhere for t > 0. and it is differentiable everywhere except along each of the lines x •
° and
x - t on which, due to the continuity of u, the
generalised Rankine-Hugoniot condition holds in a degenerate form.
It is a
simple matter to verify directly that this piecewise Cl classical solution is also a weak solution.
The form of this solution 1s shown in the Figure.
This
is simply a centred rarefaction wave of the type mentioned in Lecture 3. ~
The continuous piecewise C1 solution resolving a discontinuous initial condition.
70
Another weak solution follows by observing that a discontinuous function that is a Cl solution away from the lines of discontinuity will be a weak solution provided the discontinuity condition.
L~t
us seek an even simpler
{:
u(x.t)
~atisfies
fer x/t
< k
for x/t
>
w~ak
the generalised Rankine-Hugoniot
solution of the form
k •
by choosing k to satisfy the generalised Rankine-Hugoniot condition. into (2l) coupled with the fact that the speed of shock
SubB~itution
propagation h .. k then g1ves k(l - 0)
(~-
0)
or k
..
1/3 .
The second weak solution is thus for x/t < 1/3 u(x. e) for x/t > 1/3 • and this weak solution is piecewise constant. but is discontinous across the line 3x .. t. In physical problems only one solution is permissible. so that if the
class of weak solutions is considered some selection principle must be devised to choose a unique weak solution with the appropriate physical properties.
This is usually achieved on the basis of the stability of the
solution and leads to selection methods known as entropy conditions.
This
name derives from the gas dynamic case in which both compression and rar~faction
shocks are
shock is physically ~es
~thematically
r~alisable
since it is only in that case that the entropy
not decrease ,across the shock.
~ed
possible. though only the compression
In the example just examined the
rarefaction wave is the physical solution since the shock wave is
nat. .sub.1e.
71 Some account of entropy conditions and of the associated literature is to be found in the work of Lax [1], Jeffrey [2] and in the paper by Dafermos [3].
3.
Conservation Equations with A Convex Extension When the conservation system involved is symmetric hyperbolic, the
ideas of Section 2 may be pursued in some detail without giving rise to undue difficulty.
This we do now. basing our approach on the paper by
Friedrichs and Lax [4]. Consider a system of conservation equations
3U + 3G • 3t
3x
0
(22)
•
with U and G • G(U) each n x 1 vectors and integrate it over an arbitrarily large interval [-a,a] of the x-axis .
Integrating the second term by parts
then gives rise to the equation
r -a
au at
dx
+ GI
- GI
a-a
• o.
Now for the class of solution vectors U that vanish sufficiently rapidly for large Ixl. so that G(± a,t)
+
0 as a + m, we see from the above result
and the degenerate form of Theorem I that
showing that the integral
is a conserved quantity because it is independent of t. The problem we now consider is, when is a new conservation system
av + aK • 0
at
ax
(23)
•
with V, K functions of U, a direct consequence of the original law (22). To resolve this we need
to
make a direct comparison between (22) and (23)
lib that fittlt we perform the indicated differentiations, wben these equations becOllle, rlaSp,ect:1ve1y.;
72
au + at
(V G) U
au .. ax
0
(24)
and (V V)
U
au +
at
(V K)
U
~ ax
..
0
(25)
0
Employing the summation convent ion, the j-th component of ( 24) may be written
~ +~
aUt aUt ax
at
..
o ,
(26)
while equation (25) itself becomes
1Y... .-:.J + ~ ~ .. aU
j
at
aUt ax
0
(27) 0
Consequently, comparing (26) and (2 7), we conclude th at (25) will be a consequence of (26) only if (28) Let us now assume that this cond it io n is true , and dif fer ent i ate i t with re s pec t to
~,
~ a aUt [ a~
when we find 2g o
( aU av ) ) + av aUj (a j
~)
The second term on the left hand s i de and the right hand side are both symmetric in t and h, so that the f irst term must al s o be s ymmetr ic.
~e
have thus shown that if (28) is true , then
~ fa~ [~:J) . ~ [a: [~:J)
(29)
t
If, now, we multiply (26) by a
~ ~+--..iL ~ ~Uj ~~ at aUj a~ aUt
2
v/aUja~
aUt ax
..
and sum with respe ct to j we find
o .
This will be equivalent to (22) if the matrix {a
(30) 2
v/ aUj a~}
is non-singular,
and we here take note of the fact that system (30) is symmetric.
Hence,
whenever (22) is hyperbolic, and (28) is true, the equivalent system (30) will be
symmetric hyperbolic.
It can be shown that initial value problems
73 for symmetric hyperbolic equations are unique and will exist in some neighbourhood of the initial data. As the hyperbolicity of (30) implies that the matrix {a
2
v/aUja~}
is
positive definite we may assert that V is a convex function of the elements ~
and so arrive at the following conclusion.
Theorem 4 (Uniqueness Theorem) If the system of conservation equations (22) is such that it implies a new conservation equation (23) with the property that the new conserved quantity V is a convex function of the original elements u l' u
z' ... ,
un
of U, then the initial value problem for (2Z) has a unique solution in the neighbourhood 4.
ot
toe initial time.
Interaction of Weak Di,continuities We conclude this lecture by adding a few remarks about the interaction
ot ~
weak
discont inui~y
propagated along a cha r a c t e r i s t i c
c(~)
and a shock.
Here we use the term weak to refer to a solut ion which is continuous across
C(~) though its normal derivative i s di scontinuous.
This is in contrast to
the strong discontinuity pf the Rankine-Hugoniot type wher e the solution itself is discontinuous.
This situation is illustrated i n the Figure where D is the
shock line corresponding to a conservative system of the form U + A(U)U + B(U) t x /;
~
0
v.. o In general at
~where C(~) meets D, there will lie r characteristics of
the system to the right of D entering the state U+ and s to the left reflected back into
~he
state
:U~
•
74 By writing down the transport equation for the incident weak discontinuity along C(~), as in Lecture 4.it is possible to determine its nature as it approaches P from the left . 5
Then. using the fact that the r transmitted and
reflected weak discontinuities must propagate along characteristics. it
is possible to resolve the jumps across all characteristics at P in terms of the original system of equations. the differentiated Rankine-Hugoniot equation across D at P and the initial discontinuities propagating along the r + s characteristics together with C(,). Provided the set of equations that results at P is properly determined the reflected and transmitted weak discontinuities may be determined. Special cases arise. like the coincidence of D with a characteristic on either side at P, and the fact that the system may be exceptional with respect to one or more characteristic fields. A general accoUDt of these ideas is to be fouod in Jeffrey [1]. while attention was drawn by Bolllat and Ruggeri [5] to thp necessity to perturb the shock speed in cases where the interface D can move. References [1] [2] [3]
[4] [5]
Lax. P. D. Hyperbolic Systems of Conservation Laws and the Mathematical Theory of Shock Waves. SIAM Regional Conference Series in Applied Mathematics. 11. 1973. Jeffrey. A. Quasilinear Hyperbolic Systems and Waves. Research Note in Mathematics. 5. Pitman Publishing. London. 1976. Dafermos. C. H. Characteristics in Hyperbolic Conservation Laws. A Study of the Structure and the Asymptotic Behaviour of Solutions in Nonlinear Analysis and Mechanics: Heriot-Watt Symposium. Vol. 1. Research Note in Mathematics, 17. Pitman Publishing. London. 1977. Friedrichs. K. 0., Lax. P. D. Systems of conservation equations with a convex extension. Proc. Nat. Acad. Sci •• USA 68 (1971). 1686-1688. Boillat. G., Ruggeri. T. Reflection and transmission of discontinuity waves through a shock wave. General theory including also the case of characteristic shocks. Proc. Roy. Soc. Edin. 83A (1979). 17-24.
75 Lecture 7.
The Riemann Problem, Glimm's Scheme and Unboundedness of
Solutions
1.
The
Riema~~
Problem for a Scalar Equation
To illustrate ideas we consider the single equation for the scalar u already encountered in connection with weak solutions in Lecture 6, namely:
o
or, equivalently,
0, (1)
The Riemann problem for this equation is then the resolution of the discontinuous initial data u(x, 0) where
o and
for
x
for
x > 0
< 0
u
are two arLitrary constants. More generally, it may be l extended to include a number of such discontinuities located along the U
initial line. The characteristics of (1) are the curves (2)
along which the equation may be written in tne form du dt
o•
K
Hence for x
(3) <
0 the characteristics are parallel straight lines with
2
o ' whereas for x
slope A • u
> 0 they are parallel straight lines with
slope A • u 2 • l If
U2
o
< u 2 t h ese two f am~' 1'~es
l
0
f ch aracter~st~cs ,. d'~verge , as 1n ,
Figure (a), when the wedge shaped region W is not traversed by any of these characteristics.
However, if u~
>
ui the two families of
characteristics intersect from the start, leading to non-uniqueness and shock formation of the type first indicated at the end of Lecture 1,
76
f:'
Is~k /i.,e.
I
!I %
(b)
(a)
Centred simple wave in W
Shock speed
we thus arrive at the result that the condition for a physically admissible shock solution lor (1) is
(4) Now (1) is invariant under the replacement of x and t by aX and at, 50
that its solution depends only on the ratio t • x/to
all pass through the origin, and along them u • const.
The lines t • const. They thus fill in
the wedge shaped region Win (a), and as they are characteristics the wave solution described by them in W is called a centred simple wave.
In this
case the centre is at 0 which is the location of the discontinuity in the initial data . Taking the particular case
o•
• 1 and setting u(x, t) • u(t) l in (1) leads to the differentiable solution for region W given in U
Lecture 6, and illustrated there by 0
u(x, t)
for
(x / t ) I { l
for
x/t
<
0, u
3
Figure:
0 0 s x/t S 1
for
x/t > 1
Notice that the non-physical shock
(5)
(weak solution) given in Lec!:ure 6.
namely u (x, c)
{:
for x/t
<
1/3
for x/t > 1/3 ,
(6)
77
lies in region Wand so is
~
produced by the intersection of characteristics.
It is for this reason that it is not physically realisable and so must be rej ected. A physical shock occurs in the situation illustrated in Figure (b) however and emanates from the origin. u(x, 0)
for
It < 0
for
x > 0
Using the initial data
as a typical example, we find from the Rankine-Hugoniot condition that
A = 1/3. Thus in this case the resolution of the initial discontinuity merely involves its propagation along the shock line t • 3x. We conclude from this that for a centred simple wave (rarefaction fan) to occur, the characteristics must diverge from a point, leaving a wedge shaped region to be filled by the centred simple wave.
A shock will only
occur when the characteristics converge and intersect. 2.
Riemann Problem for a System Let us now consider the reducible hyperbolic system
subject to the initial data U(x, 0)
{::
for
x
< 0
for
x
>
0 ,
where Uo and Un are constant n element vectors.
(8)
The Riemann problem now
becomes the resolution of the initial vector discontinuity at x • 0, though as with the scalar case it may be extended to include a number of such discontinuities along the initial line. We look for the solution of this problem in terms of generalized simple waves and shocks , which will be the analogue of the situation just discussed for a single equation.
The generalized Rankine-Hugoniot
condition is of the form
A[
U])
[ F]) ,
(9)
78 once (7) has been expressed in the conservation form 3U + 3t
~ (U)
ax
•
O.
(10)
This implies n possible types of shock with speeds ~(I)C; ••• ~A(n) and we shall need a
p~1sical
we did in the simpler case.
admissibility criterion for them, just as The extension of our earlier result (4) that
provides the criterion we need is due to Lax Who requires that for some integer k with I S k
~
n
while
(11)
This condition ensures that k characteristics converge onto the shock line from the left and n - k + 1 from the right.
There is thus a
total of n + 1 conditions provided by characteristics which when taken together with the n - 1 results that follow from (9) after A has been eliminated enable the determination of the 2n values taken by U on the left ( t)
and right (r) of the shock.
The shock that satisfies (11) for some
index k is called a k-shock. Now differentiable solutions to system (7) are also expressible in terms of the ratio t • x/t, so that this system permits a generalisation of the notion of a centred simple wave.
The general solution to the
Riemann problem (7), (8) thus consists of n fans of waves, each consisting of shocks and centred simple waves, arranged in order of increasing k from left to right, and separated by sectors in Which the solution assumes constant values. As already mentioned, this generalisation of the Riemann problem may be extended to the case of an initial vector that is piecewise constant along the line t • O.
It is this very idea that is basic to
Glimm's method for the numerical solution of conservation laws (7) with arbitrary initial da ta in place of (8), and it is this that forms our next topic.
79 3.
Glimm's Method This is a method for the numerical solution of a system of hyperbolic
conservation laws
au at
+
aF (u) ax
o
(12)
with arbitrary initial data U(x, 0)
t(x) •
(13)
It is a method of first order in accuracy and the basic idea is to replace the arbitrary initial data (13) by a piecewise constant approximation in spatial intervals of length h.
Then, until such time as the
centred simple waves and shocks that result from the discontinuities interact, an analytical solution to the piecewise constant approximation to the , initial data is Riemann problem.
provi~ed
by the solution to the appropriate
If this solution is used for a suitably small time step
k, a new Riemann problem may be derived from the analytical solution at time 't .. k, and thereafter the process may be repeated to advance the solution step by step in time . The special feature in Glimm's method lies in the way in which the new Riemann problem is derived from the analytical solution.
Unlike the
averaging process over the spatial interval h used by Godunov who also employed the Riemann problem approximation, Glimm chose the constant value for each interval of length h by random sampling within that interval. Let us elaborate on this process sufficiently for its basic ideas to become clear.
First. however, we need to recall the notion of a
domain of dependence.
In Lecture 1. when discussing a second order wave
equation, the domain of dependence of a point P was defined as the interval produced on the initial line by tracing backwards in time the two characteristics passing through P until such time as they intersected the initial line. P.
Only data on this interval influenced the solution at
Now. in the case of systems (12). the analogue is to trace backwards
80 in time from a point P in the (x, t)-plane the n characteristics that are associated with system (12).
The interval on the initial line contained
between the extreme characteristics is then the domain of dependence of
P, and the Figure shows a typical example of such a situation.
o Only initial data lying within this interval can influence the solution at P . • 1
Suppose we now think of a domain of dependence of fixed length h, then for the ith interval there will be a time t is determined by the data on this interval.
i
up
to which the solution
If our initial data (13) is
approximated in a piecewise fashion at intervals of length h, then provided our time step k
t •• ;, t there will be no n} l, 2, interactions between the centred simple waves and sho cks that will result <
inf {t
from the Riemann approximation to our problem.
Expressed differently, if
A* • sup {A(i)} where A(i) are the eigenvalues of V F (A* corresponds to U~
u
the fastest propagation speed), then we need to take k
<
h/2A* •
(14)
This is, of course, the familiar Courant-Friedrichs-Lewy stability condition for the numerical solution of hyperbolic equations. Having thus generated an approximate solution at time t • k, Glimm's method then attributes to each interval of length h at this time a value of the analytical solution at a randomly chosen point in
tl~t
interval.
A new Riemann problem is thus gener.ated and the process is repeated to advance the solution a further time step.
81 Symbolically, if the subintervals are I
(sh, (s + 1) h) ,
s
(15)
then we set (16)
for x ( Is, where un (x, t) is the exact solution in the nth strip t -< t n+l' and
{~
n
~
a uniform n } is a sequence of random numbers havino -0
distribution in the interval (0, 1). Let us use an example due to Lax to illustrate how the method liOrks in the case of two constant states "t, and
~
(';.
>
~)
separated by a
shock moving with speed ~, which bas the solution u(x, t)
•
[~
At
for
x <
for
x> At
(17)
For ease of illustration, let us take all time steps equal and denote them by k.
The first step of Glimm's scheme gives
[~
for
[:
1£
(18)
for
where now J
1
•
(19)
1£
The result of n such steps with the scheme is to give
un(x,nk) •
[~
for
x < J h n
for
(20)
where we have set In
•
number of
Q
j
<
Ak 11 '
(21)
The law of large numbers tells us, with probability 1,
I
n • ni(*) + ndn
(22)
t
82 where d
n
O(l/Iii") •
(23)
The consequence is that (20) differs from the exact solution (17) by the error d in the location of the shock, though the discontinuity itself is n represented perfectly sharply as a true shock. By employing stratified sampling , Chorin has obtained more accurate results t han by the simple random sequences proposed by Glimm. 4.
Non-Global Existence of Solutions We conclude this lecture by presenting two simple examples that show
how even when a hyperbolic system is in the form of a set of conservation laws, and it has so weak a nonlinearity that it is completely exceptional (also called linearly degenerate in some papers), the solution itself may still become unbounded within a finite time.
When this happens no extension
of the solution is possible, so that a global solution no longer exists . Consider the system proposed by Jeffrey and
o
av
at
+ .&i!!L f(v)
au _
ax
0,
subject to the initial data u(x,O)
and
v(x,O)
This is easily seen to be hyperbolic, and as the eigenvalues are ±l it follows that it must also be completely exceptional.
The characteristic (+)
curves belong to the two families of parallel straight lines C -
given by
solving dx
±l.
dt Defining
u = fg(u)du
and
v•
ff(v)dv the system reduces to the linear
hyperbolic system in conservation form
au + av _ at ax
0
and
The general solution is simply
83 u(x,t)
•
F(x + t) + G(x - t) ,
v(x,t)
•
-F(x + t) + G(x - t) ,
with F,G arbitrary differentiable functions. We now take
two
special cases to illustrate the unboundedness
(blow-up) of the solution. Example 1
2 Take uO(x) • x , vO(x) •
2 f • ltv , g - 1.
~l,
u(x, e)
-
Then it follows that
1 --#
2xt-l
There is thus an eecape time t. > 0 for the solution v(x,t) when t. •
l/2x
(x > 0) •
This is not due in any way to the intersection of characteristics within a family, for they are parallel straight lines.
However, in this case the
initial data becomes unbounded for large x so that it might be considered this is the cause of the unboundedness of the solution.
To show this is
not the case consider this next example. Example 2 2 Take uO(x) • a tanh x, vO(x) - 1, f • l/v and g - 1, when we find
f~)
u(x,t)
[tanh(x + e) + tanh(x - t)] 2
v(x,t)
2+a[tanh(x+t)-tanh(x-t)]
Here u(x,t) remains finite for all x,t but v(x,t) becomes unbounded at an excape time t. given by t
...
tanh
-1
In this case, by making
a
suitably small, the deviation of the
initial data from constant values may be made as small as desired, but the finite escape time still persists.
84
References [1] [2] [3] [4] [5] [6] [7]
Jeffrey, A.
Quasilinear Hyperbolic Systems and Waves, Research Note in Mathematics, 5, Pitman Publishing, London, 1976. Lax, P. D. Hyperbolic Systems of Conservation Laws and the }~thematical Theory of Shock Waves, SlAM Regional Conference Series in Applied Mathematics, II, 1973. Glimm, J. Solutions in the large for nonlinear hyperbolic systems of equations, Comm. Pure Appl. Math. 13 (1965), 697-715. Godunov, S. An interesting class of qua s ilinear systems, Dokl . Akad. Nauk SSSR 139 (1961), 521-523. Godunov, S. Bounds on the discrepancy of approximate solutions constructed for the equations of gas dynamics, J. Comput. Math. and Math. Phys . 1 (1961), 623-637. Chorin, A. Random choice solution of hyperbolic systems, J . Compo Phys. 22 (1976), 517-533. Jeffrey, A. The exceptional condition and unboundedness of solutions of hyperbolic systems of conservation type, Proc. Roy. Soc. Edinb. 77A (1977), 1-8.
85 Lecture 8.
Far Fields, Solitons and Inverse Scattering
Far Fields
1.
There are many different types of higher order equations and systems of equations that characterise nonlinear wave propagation in
~x
t, either
A simplification frequently takes place in
with or without dispersion.
the representation of the solutions to initial value problems to such equations after a suitable lapse of time or, equivalently, suitably far from the origin, particularly when the initial data is localised and so has compact support.
These s implified forms of solution are often asymptotic
solutions, and are appropriately called far fields. The simplest examples of these are the types of far field behaviour exhibited by the ordinary linear wave equation and by a homogeneous quasilinear hyperbolic system with n dependent variables.
Due to its
linearity, the wave equation const.)
(c
(1)
may be written either in the form (au _ (2.at + 2.) ax at c
cau) -
ax
o ,
(2)
0
(3)
or as (au + c au) (2at. _ 2.) ax at ax c
Then, if u(+) is the solution of au(±) ax
c-
o.
(4)
it follows that u(t) is a degenerate solution of (2) and u(-) is a degenerate solution of (3).
The general solution of (4) is then
f(±) (x
+ ct)
,
(5)
with f( ±) arbitrary Cl functions. Thes e travelling wave solutions are such that u(+) propagates to the right and u(-) to the left with speed c.
We t hus have the s ituation that u(±)
86 are special simple types of solution to the wave equation (1), in the sense that they only satisfy a first order partial differential equation. whereas the wave equation itself is of second order.
Such special solutions
become of considerable interest when the initial data f(±) is differentiable
o
with compact support. initial data lies in
that f
50
Ixl
(+)
O-
(x)
!
1
CO'
Then. if the support of the
d. after an elapsed time dlc the interaction
<
between waves moving to the left and right ceases and only the solutions u(-) and u(+) are observed to the left and right of the origin, respect ively. These are the far fields of the wave equation (1).
Since u(+) is transported
along the C(+) characteristics x-ct· ( and u(-) along the C(-) characteristics X+ct = n. and neither family of characteristics intersects itself. the far fields of the wave equation will propagate indefinitely after the interaction has finished. The situation is different in the case of the homogeneous quasilinear hyperbolic system of hyperbolic type
l£ + A(U) au • at ax
0
(6)
•
in which U is an n x 1 vector with elements [aij(u
l•
~.
u ••••• un and A(U) • 2
u2 • •••• un)] is an n x n matrix with elements. depending on the
elements of U. If. now. we seek a special solution of (6) in which n-l elements of U are functions of only the one remaining element. say u Direct substitution into
au au _) 1 I + 1 A(U) (___ at ax
(6) j'"
~U
~
l•
we may set U· U(u
l).
shows that
•
o•
in which I is the unit matrix. A non-trivial solution of this form only exists when
.0, showing that if A is an eigenvalue of A,
(8)
87
(9)
Since the system (6) is hyperbolic there are n real eigenvalues A(l), A(2),
....
A(n) of A, from which it follows that when (6) is totally hyperbolic there are n different solutions
a
:~
a
(i)
ul i )
(i)
+ A(i)(U) :;
for i - 1,2, ••• ,n.
satisfying
-
(10)
0
The solutions (10) are, of course, simple wave solutions,
and for initial data having compact support they represent the far field bolutions after the interaction has finished.
The characteristic curves
e(i) in this case are given by solving (11)
for i · a 1,2, ••• ,n. The characteristics comprising each family e(i) are again straight lines, but now they are no longer parallel within the family as the gradient of a characteristic depends on the value of the solution that is transported along it.
This leads to a breakdown of differentiability when members of a family
of characteristics e(i) intersect, and to the formation of a discontinuous solution at some finite elapsed time corresponding to the solutions ui
i)
t~i). Thus the simple waves U = U(ui i » of (10) can only form far fields in the
time interval between the end of the interaction period for initial data with compact support and the breakdown time for the system t t
(2)
c
2.
' ••• ,
t
(n)
c
c
-
min{t~l),
}.
Reductive Perturbation Method The far field equations discussed so far are very special, since the
equations that gave rise to them involved neither dissipation nor dispersion, and one was, in fact, linear.
In more general situations both dissipation and
dispersion may be present, and typical of the far field equations that then result are the following nonlinear evolution equations:
88 Burgers' Equation (dissipative) av + v av at ax
2v a \/--2 • ax
-
(v
(12 )
> 0)
KdV Equation (weakly dispersive)
3 av + v av + 1.1 a v at ax ax3 Nonlinear
Schr~dinger
2v
av 1 a +- + at 2 ax2
i -
_ O.
(1.1 > 0)
(13)
Equation (strongly dispersive)
alvl 2v - o.
(14)
An important scalar equation that has either (13) or (14) as a far field equation. depending on the circumstances. is the Boussinesq equation "(15)
in which u(x,t) is a one-dimensional field, c is the phase velocity in the long wave limit and
1.1
is the dispersion parameter.
This occurs in the study of
water waves . A very general quasilinear system that contains as special cases many of the systems that are of physical interest has the form
au +
at
A(U)
au + ax
B+
[~Il-l a-I~ [HaIl 2... + K!' 2...).) at a ax
U
=0
•
(p ~ 2)
(16)
Here U is an n x 1 vector with elements u u • • • ' un' the matrices A, l' 2' H:, K: are all n x.n matrices depending on U and B is an n x 1 vector depending on U.
When wave propagation is involved, it is weakly dispersive when B - 0,
and strongly dispersive when B
O.
~
We now outline the so-called reductive perturbation method due to Taniuti and Wei, referring for all the details involved to that paper or to the review by Jeffrey and Kakutani . Considering the weakly dispersive case (B - 0) we apply the GardinerMorikawa transformation T
c
a+1
t,
a
l!(p-l) for p
to system (16) where ). is taken to be a real eigenvalue of A.
~
2
It is not
(17)
89 necessary that all of the eigenvalues of A are real, but when they are, and the corresponding eigenvectors span the space En associated with A, the first order system comprising the first three terms of (16) will be hyperbolic. Set
(18) where Uo is a constant solution of the homogeneous form of (16) (i.e. B - 0). Then, rewriting the system in terms of derivatives with respect to t and T. and equating like powers of E, we obtain the results
(19)
an,
(-AI
+ Here ·AO' H: O'
au
au
+ Ao) ~: + -!. +' {U • (II A) }-!. o~ ilT 1 u 0 ilt
r ~ [- x HaOa + KaOa ) ilE;P ilPU l a-I a-I
K:O and
_
(20)
o.
(VuA)O indicate quantities appropriate to the solution
u-
UO' while Vu denotes the gradient operator with respect to the elements of U. Then if t and r denote the left and right eigenvectors of A corresponding O to the eigenvalue >., so that
o
and
o,
(A - >'1) r O
(21)
equation (19) may be solved in the form
(22) with
~
one of the elements of U and VI an arbitrary vector function of l
The compatibility condition for (20) when solving for ilU t ilUl + t[V (V A) ] ilUl + ilT l' u 0 ~
I.
i
s-i
[_m eoa + Keoa )
~
a-I
Then taking the boundary condition U ~ U as x
o
VI : 0, we find that
~
~~.
ilt·: ilE;P
2!ilE;
is
_ 0 •
so that we may set
satisfies the nonlinear evolution equation
T.
(23)
9° c
ilP~ __ 1
•
2 ilf;P
o,
(24)
where
and
When p • 2 equation (24) becomes Burgers' equation, and when P • 3 the KdV equation.
The scalar equation (24) thus governs the far field behaviour
of the homogeneous form of system (16) that is associated with the eigenvalue A.
There will be such a far field for each real eigenvalue A of A. In concluding this section we remark that although in what follows we
shall be referring to properties of exact solutions of some far field equations, it should be remembered that these far field equations are in the main only asymptotic approximations to the solution that is of interest .
There are,
in fact , a variety of different methods by which the nonlinear evo lution equations characterising far fields may be derived, and for an account of three o ther methods we refer to the paper by Jeffrey and Kawahara , to the Scheveningen Conference paper by Jeffrey and to the AMS paper by Whitham. Before moving on to discuss soliton solutions to equations l i ke (2 4), we remark that it is a simple matter to show that the coefficient c i n fact proportional to (VuA)'r.
in (24 ) is
This means that when the characteristic
field associated with A is exceptional c vanishes.
l
l
=0
and the nonlinear term in (24)
Further analysis is required to derive the nonlinear evolution
equation that then governs the far field, and it has been shown by Jeffrey and Kakutani that a modified KdV type equation then results. 3.
Travelling Waves and Solitons We have seen that in nonlinear hyperbolic equations, waves propagate
with a shape change so that no travelling waves can exist for such
t Lons ,
That is to say, there is no reference frame moving with a constant spee1 s
91 in which the wave appears staionary.
However, when dispersion or dissipation
are present in a nonlinear evolution equation, in the sense that the linearised equation exhibits these effects as described in Lecture 1, travelling waves become possible due to the competition between the effects of non linearity and dispersion or dissipation. In
~
x t, travelling wave solutions have the form
vet) ,
v(x,t)
x - st ,
s
const. ,
and, in addition to the nonlinear evolution equation, they must satisfy some appropriate boundary conditions at infinity. determine the permissible range of values of s.
In general, these will In the case of Burgers'
equation and the KdV equation which are, respectively, examples of purely dissipative and purely dispersive nonlinear evolution equations, we find when seeking solutions for which all the derivatives tend to zero as Ixl 4~, the well known solutions : Burgers' Shock Wave (Purely dissipative)
vet)
1 + 1 + + I(V... + v...) - I(v... - v...) tanh [(v... - v...)1;/4v] •
•
(25)
satisfying lim
viti
Itl .....
•
v±
...
...
with v
>
+
v~
and a
-- - - - - - - -- ----- Uoo
\f+ ----«I:J
o KdV Solitary Wave (Purely dispersive) (26)
92 satisfying lim V(I;)
v..
11;1--
with v..
~
0
and s
z
v..
+ a/3.
o The Burgers' shock wave, as the solution (25) is called, is seen to propagate with a speed s • (v..-
+ v..+ )/2 that is uniquely determined by the
boundary conditions, but is invariant with respect to an arbitrary fixed spatial translation.
In general, all solutions v(x,t) of Burgers' equation
are invariant with respect to a Galilean transformation.
This solution links
two different constant states at plus and minus infinity. The KdV solitary wave, as the solution (26) is called, is different and is a pulse shaped wave that, relative to the same constant value v.. at plus and minus infinity, tends to zero together with all its derivatives as
11;1--.
Its speed of propagation relative to v.. is proportional to the
amplitude a, and its width is inversely proportional to the square root of the amplitude. by
In this travelling wave solution the speed is not determined
the boundary conditions, but by the amplitude a
>
O.
Like Burgers' equation,
the KdV solitary wave is also invariant with respect to a Galilean transformation. Z3busky and Kruskal found numerically that a KdV solitary like a particle.
wav~
behaves
Specifically they found that when two different amplitude
waves of this type are such that the one with the greater amplitude starts to the left of the one with theless e r amplitude, then the larger one overtakes the smaller one and , after interacting with it, the waves have merely inter-
93 changed positions.
This is a nonlinear interaction yet the pulse shapes
are preserved exactly after the interaction , though the phases of the pulses (the location of their peaks) is affected by this process.
On
account of this Zabusky and Kruska1 invented the word "soliton" for a wave that preserves its identity exactly in this sense after a nonlinear interaction.
Thus KdV solitary waves are solitons.
The recent interest in solitons derives from the fact that the KdV equation is often found to arise as a far field equation and, furthermore, arbitrary initial data for the KdV equation evolves into a train of solitons together with, possibly , an oscillatory tail.
This means that solitons are,
in a sense, fundamental solutions of the KdV equation.
An
extensive
literature now exists on this topic, and we refer to the articles and to the references contained therein, in Jeffrey and Kakutani and in the various articles by Kruskal, Lax, Ablowitz, Newell and Segur in the AMS publication Nonlinear Wave Motion listed at the end of this lecture.
Many different
types of nonlinear evolution equation have been found to possess soliton solutions and for more information on this topic we refer to the review paper by Scott, Chu _ and McLaughlin for both a good account of some of them and also for the basic references, and also to the edited collection of papers by Bul10ugh and Caudrey. 4.
Inverse Scattering The behaviour of soliton solutions to the KdV equation is suggestive
of linear behaviour and this motivated Gardner, Greene , Kruskal and Miura to try to find a linearising transformation of the type used by Hopf and Cole to transform Burgers' equation to the heat equation (see Scott, Chu, Mclaughlin).
No such transformation was found, but during their search
they discovered an important connection between the KdV equation and an eigenvalue problem for the Schr6dinger equation in terms of the inverse scattering method used in
qu~ntum
mechanics .
It is this result that has
94 come to be known as the inverse scattering method in the context of solitons. We can do no more here than outline the ideas that are involved. The basic problem to be considered is how a general solution of the KdV equation
o
(27)
subject to arbitrary initial data u(x,O) • uO(x) may be obtained.
The factor
-6 is included here for convenience, but it may easily be removed by a trivial transformation if required .
In essence, the approach to this
question by Gardner et al . proceeded as follows .
When v satisfies the
modified KdV equation v
t
- 6v
2v
x
+ vxxx •
0,
(28)
they noticed that the quantity u which is given by u
v
2
(29)
+ vx ,
satisfies the KdV equation u - 6uu + u t x xxx
•
(30)
O.
Equation (29) is a Riccati equation for v if we consider u to be given. Therefore, we can use the well known transformation which linearizes the Riccati equation, (31)
This gives
o ,
(32)
where u is a solution of the KdV equation (30).
This ia a natural extension
of the Hopf-Cole transformation since the KdV equation has a third order space derivative and it is one order higher than that of the Burgers' equation. However, if we merely use (32) in the KdV equation we obtain a complicated result that is not useful .
Now the KdV equation is Galilean invariant, and
so allows the replacemen ts u ... u - )., x ... x + 6),t, so tha t (32) can be generalised to
95 (33) This is simply the eigenvalue problem for the Schr6dinger equation for ljJ with the "potential" u, where u is the solution we are seeking . Equation (33) differs essentially from the eigenvalue problem of the Scbr6dinger equation in quantum mechanics because, as u must be a solution to the KdV equation, it is time-dependent. considered as a parameter in (33).
That is, the time should be
In other words, it is required that
(33) must hold at every instant with u(x,t) at that same instant.
Thus,
generally speaking, the eigenvalues A would be expected to be time-dependent. Rather surprisingly, after some calculation,it can be shown that they are time independent (and constant), provided u decreases sufficiently rapidly at infinity . Let us deduce the relationship between the KdV equation (30) and the Schrodinger equation (33) . O~
If we let u
gives the dispersion relation w +
becomes _k
2•
~
0 at infinity,
the KdV equation
k 3 • 0, and the phase velocity A p
For .s uf f i c i ent l y small lui, therefore, we have a plane wave
propagating in the negative direction.
For large values of lui, the
nonlinearity dominates to give rise to solitons.
The linear approximation
is also valid at infinity, since lui becomes 0 at infinity. In the case of solitons, the wave decreases exponentially at infinity and k becomes purely imaginary with k • i K ' A. KP2 > 0 • p p propagates in the positive direction.
On the other hand, in the case of
the Schr6dinger equation (33), it follows that -ljJ xx small
Iu, I
and we obtain
ljJ ~
e
±1kx
Thus a soliton
2
, A • k.
~
AljJ for sufficiently
This approximation is still
valid for an arbitrary value of u, provided k is thought to be sufficiently large. and
ljJ
For a bound state, the eigenvalue becomes A • -Kp decreases exponentially at infinity.
one bound state corresponds to one soliton. soliton solution
2
<
0, i.e. k • iKp'
Therefore we can deduce that In fact, if we substitute the
96
u
(34)
•
at t • 0 into (33) and solve the eigenvalue problem we can get only the bound state ~ •
-K
2/4•
P
Thus we can see that the eigenvalue soliton
K
p
2.
~
corresponds to the speed of a
If the "potential" u<x,O) (our initial data) provides N-bound
states, the solution u(x,t) as t
+ m
is given by N-solitons propagating with
speeds four times as large as the value of each eigenvalue and by wavetrains decreasing algebraically with respect to time .
The wavetrains can be
determined in relation to the Bcattered 8tate for the potential u(x,O).
This
is one of the most important re.ult. an4, in ,eneral, the far field of purely dispersive systems for which the I4V aquacion i. cha far f1eld equation may as t
+ m
be approximated by
M-~~11ton.,
For a given arbitrary valul of u(x,O), u(x,c) can be obcained exactly by the following procedure: (1)
Direct problem Find the discrete eigenValue
~
for
I
,ivan pocenCial u(x.O) of the
Schr6dinger equation (33) and &1.0 find the .eatt.rin, date at Ixl • for the wave function
~
m
(i.e., the refl.ction or tranam1s.1on coefficients
for u(x.O» . (2)
Time evolution of scattering data Find the time evolution of the scattering data and the asymptotic form of ~ at Ixl •
(3)
m.
Inverse problem Find the potential u(x,t) in terms of the scattering data at time t. This potential is then the exact solution to the KdV equation (30) subject to the arbitrary initial data u(x.O) • uO(x).
97 Inverse Scattering Method t x xxx I u (x, 0) 1---------------------------u -6uu + u
'" 0
~
u (x, t)
(1) direct problem
eigenvalue A scattering data at t .. 0, Ixl .. ell
(2) time evolution of scattering data
scattering data at t .. t,
Ixl
-+
ell
The properties of the evolution of initial data comprising only solitons follows directly from this approach and gives rise to nonlinear superposition laws for solitons .
There are, however, easier ways of obtaining these than
by means of the inverse scatter ing method which takes account of arbitrary initial data, and not just data comprising a train of solitons.
For the
details of the various steps involved we refer, for example, to the paper by Scott, Chu, Mclaughlin.
The basic paper by Ablowitz, Kamp, Newell and Segur
presents an alternative treatment of this same problem. References [1) [2) [3) [4) (5) [6) (7) [8) [9) (10)
Taniuti, T., Wei, C. C. Reductive perturbation method in nonlinear wave propagation I, J. Phys. Soc. Japan 24 (1968), 941-946 . Jeffrey, A., Kakutani, T. Weak nonlinear dispersive waves: a discussion centred on the Korteweg-de Vries equation, SIAM Appl. Math. Rev. 14 (1972), 582-643. Jeffrey , A. , Kawahara, T. Multiple scale Fourier transform: an application to nonlinear dispersive waves, Wave Motion, 1 (1979), 249-258. Jeffrey, A. Far fields, Nonlinear evolution equations, the B~cklund transformation and inverse scattering, Scheveningen Differential Equations Conference, August 1979, Springer Lecture Notes (in press). Whitham, G. B. Two-timing, variational principles and waves , in Nonlinear Wave Motion, Ed. A. C. Newell, Lectures in Applied Maths, Vol. 15, Am. Math. Soc. (1974) , 97-123. Zabusky, N. J., Kruskal, M. D. Interaction of solitons in a collisionless plasma and recurrence of initial states, Phys. Rev. Lett. 15 (1965), 240-243. Scott, A. C., Chu, F. Y. F., McLaughlin, D. W. The soliton. A new concept in applied science, Proc. IEEE, 61 (1973), 1443-1483. Bu11ough, R. K., Caudrey, P. J. (Editors). Solitons, Lecture Notes in Physics, Springer 1979 . Gardner, C. S., Greene, J . M., Kruskal, M. D., Miura, R. M. Method for solving the KdV equation, Phys. Rev. Lett. 19 (1976), 1095-1097. Ab1owitz. M. J., Kamp, D. J ., Newell, A. C., Segur, H. The inverse scattering transform - Fourier analysis for nonlinear problems, Studies in App1 . Math. 53 (1974), 249-315.
CENTRO INTERNAZIONALE MATEMATICO ESTIVO (C.I.M.E.)
ONDES ASYMPTOTIQUES
YVONNE CHOQUET-BRUHAT
101
ONDES ASYHPTOTIQUES.
Yvonne Choquet-Bruhat. I.M.T.A. Universite Paris VI
Dedie
a
la memoire de Carlo Cattaneo.
Introduction.
Nous allons, dans ces
le~ons,
exposer les grandes lignes de la methode ge-
nerale de construction des ondes asymptotiques et approchees pour les systemes d'equations aux derivees partielles. Le procede utilise a son origine dans la methode W.K.B (Wentzel-Kramers-Br illouin) consi stant
a
chercher pour
une equation differentielle une solution de la forme Aei ¢ ou A est une amplitude lentement variable et ¢ une phase rapidement oscillatoire. Dans la premiere
le~on
nous exposerons la construction de sol ut i ons des
equations de Maxwell qui donn ent, en premiere approximation, les lois de l'optique geometriques. Dans les
le ~ons
II et III nous exposer ons , dans le
cas d'un systeme du le~ ordre, la theorie genera1e de J. Leray des deve loppements
as)~ptotiques
pour les equations lineaires, dans la
1e~on
IV nous mon-
trerons comment les resultats sont modifies quand les equations ne sont pas 1ineaires, et nous donnerons dans 1a
l e ~ on
V une application aux ondes dans
les f1uides parfaits re1ativistes. Dans la
le~on
VI nous formulerons en termes de geometrie symplectique les
constructions effectuees; dans 1es
1e~ons
VII et VIII nous etudierons les pa-
rametrisations des varietes 1agrangiennes et nous construirons un developpement asymptotique au voisinage d'une caustique.
103
I LES EQUATIONS DE MAXWELL ET L'OPTIQUE GEOHETRlQUE.
Les solutions asymptotiques dont nous allons parler dans ces conferences ont leur origine dans les travaux de Debye, Sommerfeld et Runge pour retrouver les lois de l'optique geometrique
a
partir de la theorie electromagneti-
que. On a montre depuis que cette methode de solution s'appliquait
a beaucoup
d'autres prob lemes et permettait d'obtenir des renseignements, tant qualitatifs que quantitatifs sur des phenomenes physiques varies rapidement oscillants. Pour justifier l'introduction des developpements asymptotiques generaux que nous etudierons par la suite nous allons considerer d'abord Ie cas original des equations de Maxwell.
1 - Milieux isotropes non conducteurs.
Les equations de Maxwell verifiees par Ie champ electrique E et Ie champ magnetique H dans un dielectrique parfait homogene s'ecrivent :
aE
.
j
aH
.
0
(1-1)
rot H-
€
~
(1-2)
rot E +
~
~
104
ou j est Ie vecteur courant e1ectrique;
€
et
~
sont des constantes.
Prenons pour j Ie courant produit par un dipole situe
a
l'origine, de fre-
quence w. Alors j s'ecrit, en representation complexe (Ie courant reel est
.1t e
j)
OU o(x) est 1a mesure de Dirac a l'origine (de l'espace m3 ) et M un vecteur donne de (3(si M est reel, Ie dipole est dit polarise rectilignement). Si on cherche un regime "stationnaire", solution des equations de Maxwell, c'est-a-dire de 1a forme, en representation complexe
H • v(x) e iwt
E • u(x) e itut
3
xE.1R ,
t~\R.
on trouve que les vecteurs complexes u et v doivent verifier 1es equations (1-3)
rot v - i e III u • M 0
(1-4)
rot u + i u III V
•
0
on introduit Ie potentie1 vecteur A en posant
v d I apres (1-4)
»
rot A
, div v • 0), alors u est de 1a forme, d'apres (J-4) u - - i
jJ III A
+ grad V
Vest appe1e potentiel sca1aire. On Ie choisit en lui imposant la relation dite "condition de jauge de Lorentz" V • _._J_ div A
(1-5 )
alors ( 1-5 (1-6)
~W€
) devient
105
avec u • - i(~ w A +
JL WE
grad div A)
2 Une solution elementaire de l'operateur A + W E
~
est la fonction locale-
ment integrable
G(x) • on
lEU Ixl Ixl
exp(- i w
I
4n
Ixl • {
3 E i •
en deduit une solution A de (1-6) A
• - G:t
oM
•
exp(- iw,/£ji Ixl ) M
4nlxl
Pour cette solution A on a v • rot A • exp (- i w ~ avec v
on
o
- m/<. x
•
2
Ixl
,-
v
yEll
1
i
m·
mAX
•
~3
u
.!. rot
i
wH
4n
trouve aussi
ou v
o
., ~/£ X
Ixl
.
I
£
v
0
Iii V1AX
{l!.
Ixl
£
Les champs E et H correspondants. solutions exactes des equations de Maxwell
a
source dipolaire sont : E
•
c: exp r Liw(- veu
H
•
exp
II ' + t)J
(u o
+;;;1
t»)
(v o
+~vl ')
X
[iw(-"£~ Ixl
+
u1
+.!. w2
u2
}
on constate que ces solutions sont produit d'un terme oscillatoire. de
106
frequence w comme 1a source, par une somme de termes qui ne dependent de w -k 0 que par un facteur w • Si west grand les termes en w , c'est ~a-dire U o V o ' sont preponderants . Le terme oscillatoire est de la forme ~(x,t) • -
IEiJ IXI +
exp i
w~x,
e) avec
fonction ~(x,t) satisfait
t , La
a
l'equation dite ei-
kona1e de l'optique geometrique :
IVlfl2 Les surfaces rayon R • t/lEU
\f.
cte sont pour chaque t donne des spheres de centre
0,
de
,e11es sont identiques aux fronts d'onde (normales aux
rayons lumineux) correspondant
a une
source ponctuelle en
0,
pour une vitesse
de propagation de 1a 1umiere II ~. On cons tate de plus que
sont orthogonaux au rayon x (i1 en est o ainsi aussi de H, mais pas de E). done tangents au front d'onde et orthogoV
o
et
U
naux entre eux. On voit aussi qu'ils verifient 1es equations : d (r u o) dr
ou
d dr
.
a xi --.ax1
0
..!. (r vol dr
et
r
2
• r
0
(i)2
i
donc
qui exprime 1a conservation sur Ie front d'onde Ie long du rayon x de la densite associee au scalaire luol2 + Ivol2 • Ces differentes proprietes sont carasterisciques du champ de l'optique geometrique .
107
2 - Milieux non homogenes, et conducteurs.
Les equations de Maxwell sont maintenant (2-1 )
aE rot H - e at
(2-2)
rot E +
e: ,
~
-
aH at
~
J" +
4
IT
aE
0
ainsi que la conductivite electrique a sont des fonctions continues du
point x. On cherche encore une solution de representation complexe
E
(2-3)
correspondant
a
= e i wt
u(x)
H- e
la source dipolaire j - e
iwt
iwt
v(x)
M6. On peut encore introduire Le
potentiel vecteur A, tel que v - rot A et fixer Ie potentiel scalaire V par Ie choix
)-5
• mais 1'equation satisfaite par A ne peut plus etre resolue
de fa90n exacte explicite.
On cherche done directement u et v sous la forme (2-4)
u _ e i w ljI(x) U
(2-5)
v - e
U-
i w ljI(x) V
V-
u _(x) E_n _ n (iw)n v __ (x) E _n n (iw)n
E et H satisfont aux equations de Maxwell si U et V satisfont en dehors de l'origine x - 0 aux equations suivantes : (2-6)
rot V + i w 'iI ljI
V - i w e:
(2-7)
rot U + i wV ljI
U+iw~V
U- 4 -
IT
a U
o
0
ou on a pose 'illj1 - grad ljI • En reportant dans ces equations les series 2-4 et 2-5 et annulant Ie terme
108
en
O W
on trouve
(2-8)
V1/J .A v
(2-9)
V1/J A u
0
0
- e: u
0
- \.I v
0
0
0
les equations ont une solution non nulle (uo' Yo) si et seulement si
c'est-il-dire 9i 'f(x,t) :: 1/J(x) + t
aatisfait l'equation eikonale il coeffi-
cient e:\.I dependant de x 3 1:
•
0
i •
Les vecteurs uo et
o sont encore perpendiculaires en chaque point x il la di-
V
rection (spatiale) du rayon lumineux, V1/J , perpendiculaire au front d'onde 1/J(x) pour
= constante,
Ivwl
2
et perpendiculairea entre ewe : la solution generaIe de 2-8,9
• e:\.I est telle que v
elle est determinee, par exemple, par Ie choix d'un vecteur
U
o
o
• 0
orthogonal il
V1/J • Un moyen remarquable, qui s'averera tout equations de propagation de
o et
U
a
fait general, pour trouver les
o consiste il ecrire que lea equations doi-
V
vent etre verifiees il l'ordre suivant, c'est-a-dire
a annuler
les termes en
W-) dans 2-6, 2-7, on trouve:
V1/J ..A v) - e: u)
- rot
V1/J A u) + \.I v J
- rot u0
V
o
+ 4
1T
a u
0
on sait que si IVl/I!2 ~ e:1l (impose par les equations d'ordre zero) les premiers
109
membres de ces equations ne sont pas 1ineairement independants - i1 en est donc de meme des premiers memhres : un calcul purement algebrique fournit les relations satisfaites par ces seconds membres, qui seront donc des equations aux derivees partiel1es du premier ordre pour
U
o
et v
o'
On trouve que ces e-
quations differentie11es Ie long des rayons, trajectoires du vecteur VW, qui s'ecrivent du I ---do + -2 (6 ~ + 4 ~ 0 ~)u + (u • T u 0 0
avec
6~ 1jI
d
dT -
~
r -1!l!..i i
.
ax
.a, r i ( .!.u ait
a
ai ~) axl.
6 ~ E
-
E
r
a
i -ax
I
~)
e axl.
Les equations differentielles que l'on vient d'obtenir sont appelees "equations de transport" - la methode qui les fournit s'etend aux systemes quelconques, comme nous Ie verrons dans la prochaine
le~on.
References.
M. Kline, 1. W. Kay "Electromagnetic theory and geometrical optics" Interscien-
ce, 1965.
110
II ONDES ASYMPTOTIQUES POUR U:S SYSTEMES D' EQUATIONS AUX DERIVEES PARTIELLES LINEAIRES.
La methode indiquee dans la
le~on
I pour la recherche d'ondes de frequence
elevee solution des equations de Maxwell. developpee par Luneburg Kline
I II
puis par
121 a ete generalisee par P. Lax 131 qui construit. pour un systeme
d'equations aux derivees partielles lineaires du premier ordre
a
caracteris-
tiques simples un developpement asymptotique
u
'"
e
iw
(V
o
+
1w v I
+ ••• )
P. Lax utilise ce developpement pour la resolution approchee de Cauchy
a
donnees initiales oscillatoires, et pour la demonstration de certains theoremes gene raux concernant les equations hyperboliques. D. Ludwig
151 generalise encore la methode en rempla~ant la fonction e i w
par une famille de fonctions f j , et construit des developpements asymptotiques de la forme :
u
'"
E
j • 0
avec
111
pour 1es equations aux derivees partie11es 1ineaires hyperbo1iques dans 1e cas d'un systeme du premier ordre ou d'une equation d'ordre que1conque.
J. Leray
141
puis L. Garding, T. Kotake, J. Leray 16\construisent pour des
systemes d'equations aux derivees partie11es 1ineaires d'ordre que1conque, des deve10ppements asymptotiques tres gene raux, du type
u
'"
(x point d'une variete differentiable, reel) dans 1e cas ou 1a phase
\f
~(x)
fonction sca1aire, w parametre
a une
correspond
caracteristique simple du
systeme . C'est 1a methode de J. Leray qui s'etend 1e mieux aux systemes d'equations aus derivees partie11es non 1ineaires. Nous allons l'exposer, dans un but de simp1icite, dans 1e cas d'un systeme du premier ordre.On sait d'a il1eurs que tout systeme d'equations aux derivees partie11es peut s'ecrire sous forme d'un systeme du premier ordre.
1 - Definitions
Nous designons par X une variete differentiable de c1asse C~, de dimension i, par x un point de X; x O , xl, ••• xi-I sont 1es coordonnees locales de x. Un operateur differentie1 [systeme d'equations aux derivees partie11es] 1ineaire du premier ordre sur X est une application L de l'espace des sections C~ d'un fibre vectorie1 E sur X dans l'espace des sectionsC~d'un autre fibre
F sur X. Nous supposons ici dim E = dim F = i + N, a10rs L s'ecrit en coordonnees locales (1-1 )
(lui (lX>"
+ b.
1
J'
u
i
i,j
1, ... N ; >.. = 0, .. . i-I
112
jA b i j sont des fonctions donnees, c= , des coordonnees locales. ou~ 1es ai' Designons par ui(x,~), q - 0, I, ••• , des fonctions q
a valeurs
plexes, derivables, de x E X et d'un parametre numerique reel rametre reel (nomme frequence), par
~(x)
reelles ou com~
, par w un pa-
une fonction reelle derivable sur X
(nommee phase). On
definit la fonct ion
u~ • w ~ sur X par :
( 1-2)
I4 I
Nous dirons avec J. Leray
que la serie formelle
i '"
(1-3)
est une onde asymptotique
r
q .. 0
pour Ie systeme d'equations aux derivees partielles
si en reportant formellement 1-3
dans
I-I , compte tenu de 1-2
on trouve un
developpement en serie de puissance de w qui s'ecrit, formellement :
=
( 1-4)
E q .. 0
W-qFi.W'f q
dont chaque coefficient Fj est nul quels que soient x et ~ • q
Les developpements de Lax correspondent au cas particul ier :
e i ~ v qi
2 - Determination de la phase
~
(x)
•
Derivons formellement, il vient en posant au (2-1)
i 9
(x , ~)
u
i q
aiq
(x,~)
113
(2-2) Dans Ie cas des developpements de Lax les formules 2-1 et 2-2 se reduisent
a: e i~
aA V i
q
(x )
c'est-ii-dire
On trouve dans Ie cas general en reportant dans I-I
une expression de la
forme 1-4 OU les coefficients pi sont : q
(2-3)
pi
(2-4)
pi
-I
0
uoi
-
a~>'
-
">' i a~ CU I 1
1
d),- 'f a),
'r
+
a),
i) 0
+
b~1. u 0i
(2-5) Les equations 2-3 ont une solution non nulle la matrice a
i
J" ),
uoi
en x € X si et seulement si
d),f est singuliere, c'est-ii-dire si Ie symbole principal de
l'operateur L est non iniectif pour Ie vecteur covariant p = (a>. ~), ce symboIe etant suppose non identiquement nul (systeme regul ier au sens de Cauchy Kovalevski) : les equations 2-3 auront une solution-non nulle sur un ouvert
n de X si et seulement si, dans cet ouvert la phase
~
verifie l'equation aux
derivees partielles du premier ordre de degre N - dite equation eikonale du systeme : (2-6)
A(x ,
'f x)
• 0
qui s'ecrit en coordonnees locales
OU
'f x
= (V 'f)(x)
114
Une phase le~on
~
verifiant l'equation eikonale 2-6 etant supposee choisie (cf.
VI), la resolution du systeme lineaire 2-3 - puis les equations sui-
vantes - depend du rang de ce systeme. Nous etudierons Ie cas Ie plus simple ou ce rang est N-I.
3 - Cas ou
'P
correspond
a une
caracteristique simple.
Designons par A(x,p) Ie polynome caracteristique du systeme I-I , c'est-a-. dire Ie polyname des composantes
PA du vecteur covariant
p defini par :
'A
det ( a~
A(x,p)
i.
I' equation 2-6 exprime que Ie gradient
'f x de
la phase l' est une racine du poly-
nome caracteristique; supposons que cette racine soit simple, { A A (x, p) }
p -
If x
«0
c'est-a~dire
que
pour tout x E.. X
oU on a pose A
A
(x, p)
aA A(x,p)
Le determinant A est alors necessairement de rang N-l pour p - ~x • V'x ~ X i Determination de u o
'r etant choisi
~
solution de 2-6 on deduit de 2-3
()-I )
oU hi(x) est une solution du systeme d'equations lineaires homogenes
115
o
(3-2)
et V0 (x, F;) une fonetion arbitraire. Puisque A(x,tp x) es t de rang N-I, hi. est determine
a un
faeteur pres.
On deduit de 3-1 : (3-3)
(3-4)
i
et wo(x)
des fonetions arbitraires de x.
Les equations les N ineonnues
F~
a
0
sont , si
If
verifie 2- 6 ,N equations lineaires pour
u~ , a determinant nul. Les u~ ne pourront done exister, re-
guliers, que si (3-5)
o
oil on a pose (3-6)
et ou on a designe par h. une solution du systeme transpose de 3-2, e'est-aJ
dire telle que
(3-7) 3-5 s'eerit
ii. J
hi a~'>' d>. U + ~ 0 + b
j
h
ii.
On a, puisque hi (respeetivement
J
ii. a~>' J
~
U d>. hi 0
hh U + ai' >' d>. wi + b~ h. wi 0 0 ~ J 0
ii.) est proportionnel a A~ pour ehaque J
J
j
116 fixe (respectivement chaque i fixe) k
(3 -8)
A~J
(x,
i
ou Aj(x,P) designe Ie mineur de a
\D) Ix
jA PI. dans Ie determinant A(x,p) et k une i
fonction de x qui depend du choix de hi et
a avoir
k
=
h. J
(on peut les choisir de maniere
I). Or, d'apres la loi de derivation d!un determinant
dA(x,p) dPA
(3 -9)
= A~J
a
(x,p)
dPA
(a~lJ p ) lJ
1
D'ou (3-10)
•
x (x, f
k A
x )
On deduit de 3-10 que l'equation 3-5 s'ecrit
•
(3-1 I)
AI. __d_ dX"
0
est la derivation Ie long du rayon correspondant
\f(x) • cte • On a pose
60 est nul si w~ • 0 (3-13)
ne depend que des equations donnees e t du choix de Calcul de
'f .
B
Une expression interessante de Best obtenue aisement Lenme
on a, si k • I B(x) = .!. 2
(3-14)
+
'1.
a ax
1.(
A A x,
1 -2
Ih • J
x
)
., "hi
aJi\
i
_0_
di
+ _
h
I dai
i -
h (---j ~
dx"
h i a.j 1
oh' I :.::r ox
+
a
la surface d'onde
117
Preuve
un simple calcul, puisque alors
Corollaire
si l'operateur differentiel L : L
est autoadjoint, on peut choisir hi de sorte que
B se
reduise A
(3-15)
= 0,
L'equation de propagation 3-1!, avec 00 2
de conservation pour U o
se
~eduit
alors A une equation
Ie long des rayons.
Preuve: Lest autoadjoint si, pour u et v A support compact on a l'egalite . . sca I a1res L2 d es prod U1ts
(v,Lu)
(u,Lv)
'A
c'est-A-dire si les matrices a A I a '2;:; On
a alors h et
jours k
=
(a.1 J ) sont symetriques et si
a.i A _ b.i 1 .
1
=
0
h proportionnels. On peut choisir h tel que h = h (avec tou-
1).
L'equation 3-11 peut alors s'ecrire
o
(3-16)
Remarque
D'apres les regles classiques de derivation d'un determinant on a:
(3-17)
si x
'1':
= x(t,y)
verifie Ie systeme differentiel des rayons associes
a
la phase
118
et on a designe par
d
de·
~
()
A ()x~
la derivation le long de ces rayons.
L'equation 3-16 exprime la conservation le long des rayons de la densite de
dt d
[u2o D(y)] D(x)]
0
119
III ONDES ASYMPTOTIQUES D' ORDRE q, ONDES APPROCHEES. EXEMPLE.
I - Determination des termes successifs u i
q
Uo(X'~) etant determine verifiant 3-5 de la le~on lIon pourra trouver u~ verifiant pj • O. sa forme generale sera o (1-1 )
ou
VI(x.~)
est une fonction arbitraire et
i
VI(x.~)
est une solution des equa-
i et U
des primitives par
tions lineaires (1-2)
On deduit de I-I en designant par rapport
a~
u(x.~)
(x.~)
de V et Vi :
(1-3)
Supposons alors construites des fonctions u i • p < q • verifiant les equations pjp (1- 4)
=
0 • p < q-I et soit uqi de la forme u
i q
p
120
ou U
q
i (x.~) es t arbitraire et U est une solution quelconque, mais fixee . des q
equations lineaires
tiqi
(1-5)
+
gi
q-I
'). i _ a1.,J ~,u
• 0
1\
q-I
. i + b.J u
1.
q-I
i i Pour ces fonctions up , u on a aussi q
o
(1-6)
Les equations Fi • 0 s'ecrivent q (1-7)
.i uq+ 1 +
a . iA d). 1.
gi - a J' A d u i i A q q
gi • 0 q
+
b~1. uqi
on pourra t rouver u. i + 1 S1.· e t seu I ement S1.' q
o
(1-8)
Cette equation est une equation differentielle du premier ordre lineaire pour Uq ' analogue
a
II. 3-5. qui s'ecri t :
k AA d U + B U + cS q q A q
(1 -9)
ou Best la fonction de x
est connu quando
0
II - 3-13 et :
u~_1 e tant determine . U~ a ete choisi .
2 - Ondes approchees .
Nous dirons que Ie developpement fini
121
(2-1)
u
i
r
E p -
o W 0
est onde approchee d'ordre r -I si il existe une constante M telle que pour tout w
(2-2)
ou
j
IlL (u)1
Ix designe
{2-3) On
une norme convenable, par exemple
IILj(u)ll x
Sup xES: X
{I)
ILj (u) I
voit que la condition 2-2 sera realisee si les u
j i verifient F • 0, P < r p p
et sont bomes, ainsi que leurs derivees partielles par rapport aux variables
x. D'apres les equations )-7 les
u~+)
dependent lineairement des
ra trouver une onde approchee d'ordre r-\ si u
a t;
i o
u~
; on pour-
admet r primitives par rapport
uniformement bornees , pour t; Eo. R, ainsi que leurs derivees partielles par
rapport aux variables x . Cette condition pourra toujours etre verifiee, par un choix convenable d'une donnee initiale, au moins dans un domaine de regularite des rayons . Cette construction d'ondes approchees d'ordre quelconque ne sera en general pas poss ible pour les equations non lineaires.
3 - Exemple
ondes dans les flu ides compress ibles )-dimensionnels linearises.
Les equations classiques non lineaires sont
(I) Pour les equations hyperboliques on est amene normes .
a
considerer d'autres
122
ou t est Ie temps, x la variable spatiale. p , n et u respectivement la dens ite, la pression et la vitesse. Les equations lineairsees au voisinage d'un etat de repos sont
(3-1)
ou r et a sont des fonctions donnees de x. Cherchons une solution asymptotique approchee d'ordre I, avec
(3-2) v • v 0 (x, e,
W
'f
'f(x,t) :
I 'f) + w v I (x, t.w If )
on trouve pour l'annulation des termes en w,avec 'fx· 3tr / 3x, 1ft • 3'f/3t F
I -I
F
2 :: r V• ~ t + Po -I o
-
(3-3)
On
pourra verifier
Po \ft + a
I
rvo
\.f x
'fx
0 0
F~I • F: 1 • 0 avec Po et Vo pour tous deux nuls si l'e-
quation eikonale suivante est verifiee :
(3-4) II Y a ici deux familIes de phases possibles qui sont si a est une constante \F(x,t)
= ~(x
- at)
et
f(x, c)
~(x +
at)
Les rayons correspondants etant alors les droites x - at • c~ et x + at • c~~ si a est une fonction donnee de x les deux familIes d'ondes sont obtenues par
123
resolution des equations lineaires du premier ordre
'ft - a 'fx • 0
(3-4 b)
et
(3-4 a)
Les rayons spatio-temporels correspondant
a
la premiere fsmille sont les cour-
bes solutions de l'equation differentielle dx dt
a(x)
soit x • f(t,y)
f(o,y) • Y
Ie rayon issu d'un point (o,y). Supposons l'equation inversible (c'est toujours Ie cas pour t assez petit puisque f(o,y) • y) on en deduit que si (x,t) est un point du rayon issu du point y on a : y
•
g(t.x)
g(o.y)
y
La solution de (3-4 a) telle que (3-5)
If(o,x)
•
t/J(g)
au t/J est une fonction donnee, est (3-6 a)
puisque
'r(t,x)
If est
•
t/J(g(t.x»
constant sur Le r!1yon spatio-temporel (bicaracteristique de
l'eikonale) issu de y. De meme la solution de (3-4 b) telles que (3-5) soit verifiee est
ou g-(t,x) est obtenu par inversion de la solution de dx dt
•
- a (x, e)
La solution generale des equa tions pour un
tp
de la premiere fsmille est
124
(3-7) (on a integre en
et supprime Ie terme independant de
~
O W est equivalente
L'annulation des termes en
~
).
a:
(3-8)
Pour
If2t
- a
2
r 2x ces equations impliquent
(3-9)
rempla~ant V
o et Po par 3-7 on trouve l'equation de propagation pour Uo' de
la premiere fami lle
(If t
= -
a
r x)
(3-10)
equation de conservation Le long du rayon du facteur a r de
rl-o ,
independanment
~ .
On
resoud ensuite les equations
ou PI' VI
F~ - 0, i = 1,2, sous la forme
est une solution particuliere, par exemple, U satisfaisant o
a
1-10
d'ou ou U designe une primitive de 0 Si on calcule Lu
=
Uo(t,x,~)
par rapport
a
~
.
2 (L (p,v), L (p,v)) pour la somme 3-2 ou Po' PI' v o' vI I
sont les fonctions qu'on vient de determiner on trouve :
125
I L (p,v)
1. F 1
-
w
2
1
=
1. (a w t PI
1. F 2 =1. (r at w 1 w
L (p,v) -
+ a
2
r ax vI)
+ ax vI)
donc ILi(P,V)I
~
-1
w
si les derivees partielles en t et x de ceci sera realise si et
~,
Uo(t,x,~)
et ceci pour tout
Uo satisfaisant
a
~
et
M PI(x,~,~)
Uo(t,x,~)
vl(x,t,~)
sont bornees :
sont uniformement bornes en t, x
E R.
l'equation de transport 1-10 (condition de conservation)
sera uniformement borne en
~,
dans tout domaine du plan (x,t) ou les rayons
ne representent pas de caustiques (equations x inversibles) s'il en est ainsi est :
et
a
= f(t,y)
donnant les rayons
l'instant initial taO. Un choix possible
126
IV DEVELOPPEMENTS ASYMPTOTIQUES POUR DES
E~ATIONS
QUASI LINEAIRES A CARACTERISTIQUES SIMPLES.
r Definitions
m
Nous designons par X une variete differentiable de classe C de dimension n ~.
• • de par x un p01nt
X.'
xO.x l
•
~ l I es d e x. •••• x ~-! sont d es coord onnees oca
Nous considerons un systeme de N equations aux derivees partielles du premier ordre sur X. quasi lineaires. aux N inconnues uk(x). fonctions(!) sur X
a
valeurs complexes (1-1 )
en posant u
L u
=
I
u ,
••• , u
N
o
et en des i.gnanz par u ... L u une application (non
lineaire) de l'espace des suites {uk(x)} de N fonctions differentiables contenues dans un polycylindre P :
(I) les inconnues uk(x) peuvent aussi etre des tenseurs sur X. l'application
u » Lu s'ecrit encore !-'3 en coordonnees locales. les u i etant les composantes des tenseurs envisages,
127
uk(x) fonctions donnees ,
(1-2)
o
dans l'espace des suites de N fonctions, ayant dans chaque systeme de coordonnees locales une expression de la forme 'A i _ a~ (x,u) au, + bj(x,u) 1. ax"
(1-3)
tx
s
0
i,j
s
I, •• ,N; A = 0 ••• ,t-I
.
at (x.u) et bJ(x.u) sont des fonctions de classe C~ en x, analytiques en u sur P et egales
a
la somme de leur developpement en serie entiere
(1-4)
bj(x,u) ou on a pose
Designons par u
i q
• q - 0.1 •••• , des fonctions
(x.~)
vables. de x €X et d'un parametre numerique reel (norome frequence), par
~x)
~
a valeurs
complexes. deri-
• par w un parametre reel
une fonction derivable sur X (noromee phase).
Posons (1-5)
u
i
u
q
i q
0
w
i u (x, q
w'f' )
Nous dirons comme dans Ie cas lineaire (cf. II) que la serie formelle (1"-6)
u
i
est une onde asymptotique pour le systeme d'equations aux derivees partielles si en reportant formellement 1-6 dans 1-3, compte-tenu de )-4 on trouve un developpement en serie de puissance de w qui s'ecrit, formellement :
12 8
( 1-7)
w-q
E q
=0
dont chaque coefficient F
j q
F~
• W
If
=
0
est nul que1s que soient x et
2 - Determination de 1a phase
~
~.
•
On pose. comme dans I I .i
( 2-1)
u
q
on a (2-2)
On trouve a10rs en r eportant dans \-3 une expression de 1a forme \-7 au les coefficients F
j
q
sont
(2 -3)
(2-4)
j F 0 F
(2-5)
j I
(ui a 'f - a~A 10 I A
-
a
+ Cl " u i ) +a jA uh u.i Cl 'f + b j o 1 0 A 0
j A •i j" h i. i h .i jA i + j i u i o 2 Cl A'f + a i o Cl" u 1 b i u \ +aih{u l(Cl "Uo+ul Cl,,' f) + u2 uO ClA'f}
129
Nous etudierons Ie cas ou
U
i
o
est une solution donne e, i ndependant e de
S,
des
equations 1-3. (2-6)
Les equations Fj • 0 s'ecrivent alors q
(2-7)
j F-I
(2 -8)
F
(2-9)
F
j 0
j 1
-
0
.i u - a~A 10 CIA 'f 1
-
a~A
ClA~
10
• 0
.i + a jA CIA u i + a jA CIAIf uh u.i ih 1 1 2 1 io
U
'A
i
j)
+ (aih CIA u + b 0 h
(2-10)
F
j q
-
u
h 1
.
0
a~A CIA If' u.1'+ + a jA CIA ui + a jA CIA ih q io 10 q 1 'A
+ {alh (CIA
OU fj ne depend que des u i q
p
,
.i u p
'P
'f
h .i u u 1 q
. i + CI ui) + b j } uh + f j • 0 u1 A 0 h q q
, CIA upi OU p < q.
.i L'equation 2-8 entraine, si l'on veut que u t 0 1
(2-11)
A(x,
'f' x)
o
2-11 est une equa tion aux der i ve es part ielles du premier ordre qui exprime
que la phase
'f
est solut ion de l' equation caracteristi que approchee, obtenue
en donnant aux coefficien ts principaux de 1-3 la valeur qu 'i1s prennent pour la solution donnee ui(x) . o On designe par A(x,p) Ie polynome caracteristique du systeme 1-3 correspondant
a
la solution
u~ , c' est a dire Ie polynome des composantes PA du vecteur
covariant p defini par : A(x,p)
' A p,)
det(a~
10
J\
130
I' equation 2-11 exprime que Le gradient
'f x
de la phase
'f
es t une racine de
ce polynome et on suppose que cette racine est simple, c'est pour tout
a dire
que
x Eo X
ou on a pose
aA A(x,p) Le determinant A est alors necessairement de rang N-I pour p =
l'
'f x'
Vx Eo X.
etant choisi solution de 2-11 on deduit de 2-8
(3-1) ou hi(x) est une solution du systeme d'equations lineaires homogenes
(3-2) et
o
VI(x,~) une fonction arbitraire . Puisque A(x, fx) est de rang N-I. hi est
determine
a un
facteur pres.
On choisit : (3-3)
(3-4) Les equations F
j 1
.i les N inconnue s u 2
=0
sont. si
If
• a determinant
verifie 2- 11, N equations lineaires pour nul. Les u.i ne pourront d onc exister 2
,
131
reguliers, que si (3-5) ou on a pose (3-6)
et OU on a designe par h. une solution du systeme transpose de 3-2, c'est J
a
dire telle que h. J
a~).
a, ID T
designe Ie mineur de
af~
p). dans Ie determinant A(x,p), et k une
(3-7)
1.0
•
0
1\
On a (cf . II)
(3-8) ou
A~(X,P)
fonction de x qui depend du choix de hi et
a
h.J
(on peut les choisir de maniere
avoir k • I) et
(3-9)
On deduit de (3-9) que l'equation 3-5 s'ecrit A aUI (x,!:;) k A A
(3-10)
ax
+ a(x) UI(x,;) UI(x,;) + B(x) UI(x,;)
est la derivation Ie long du rayon correspondant
a
E
0
la surface d'onde
cte
On a pose (3-11) (3-12)
i
a et Bne dependent que des equations donnees et de la solution u o•
132
Le coefficient
ea
une expression analogue
aire, en 11.3.14, une fois la solution u R. 11 .3.14 par b i + aR.i a). Uo j
Le terme aU
j).
I
01
i o
a
ce11e obtenue dans Ie cas line-
rempla~ant b~1 dans
choisie, et en
n'existe pas dans Ie cas lineaire. La non nullite du coef-
ficient a va provoquer un effet, lie
a
la non linearite, de dis torsion des
signaux. Pour ealculer Ie coefficient a on remarque que a
(3-13)
•
or, d'apres 3-8 on a (3-14)
=u • 'f~
en eonsiderant que
on demontre (cf Boillat ~ • d er ive e
On
')'
A(x,u,p)
(3-15)
au aura:
det (ai
IV-~)
(x, u) p).)
que la quantite 3-14 est proportionnelle
de la vitesse de propagation eorrespondant
a
a
a
la
l'onde Y'(x) • ete.
0
si (3-16)
hi
{OA }
auR.
u. u p •
o
'f~
les varietes earaeteristiques possedant la propriete 3-16, rene ontree par P. Lax (meeanique classique) et G. Boillat (meeanique relativiste) dans l'etude des discontinuites ont ete appelees par eux exeeptionnelies .
133
4 - Integration •
L'equation 3-11
est une equation aux derivees partie1les du ler ordre qua-
si lineaire, son integration se ramene done
a
celle d'un systeme differentiel
ordinaire, son systeme bicaracteristique. Celui-ci s'ecrit :
(4-1)
dt
la solution x
(4-2)
= x(t,y)
x(o,y)
=y
des t+ 1 premiers rapports est le rayon correspondant ~x)
= cte
a
la surface d'onde
passant par le point y.
La solution
UI(x,~)
(4-3)
passant par la variete initiale s(y) • 0,
( l: )
{UI(x,~)} x ~
=Y• = n
WI(y,n)
(s et WI fonctions donnees) est engendree par les courbes bicaracteristiques (solutions de 4-1) s'appuyant sur l: , c'est
a dire
est obtenue par elimina-
tion de t, y, n entre les solutions de 4-1 x • x(t,y)
(4-4 a) (4-4 b)
(4-4 c)
~
•
~(t,y,n)
verifiant x(o,y) • y , UI(o,y,n)
= WI(y,n),
~(o,y,n)
• n avec s(y)
= o.
Remarque : Si la sous-variete S de Vn, s(y) = 0, est differentiable et transversale aux rayons (c'est
a
dire que son plan tangent ne contient pas Ie vecteur tangent
134
au rayon en ce point) les equations 4-4 a sont inversibles pour t assez petit; en dehors d'un voisinage de s(y) = 0 cette inversion peut ne pas etre possible (cas ou les rayons admettent des caustiques). Les rayons x ~(t,y,n)
= x(t,y)
etant supposes connus les fonctions U1(t,y,n) et
sont donnees par les quadratures t
(4-5 a)
UI • exp( -
(4-5 b)
~
• n +
f:
fo 8(X(T,y» a(X(T,y»
dT) WI (y ,n) U1(T,y,n) dT
c'est il dire
(4-6 a)
U • WI (y,n) $ (t,y) I
(4-6 b)
~
n + WI (y,n)
~(t,y)
ou
$
= exp
(-
ou
~
=
a
r 0
Remarque
~
dT )
dT
on a ~
(4-7) d'ou
J: a
an
(~(o,y) =
(t,y,n)
aWl an
+-- ~
0) :
l i (o,y,n) • an
(4-8)
~ etant une fonction continue de t, d'apres les hypotheses faites au
§
I on
deduit de 4-8 l'existence pour t petit d'une fonetion differentiable n
PI us
~ . ~ t prec~semen
pour
It I ~
•
n(t,y,~)
on pourra t·~rer n (t; ,y,~t') d e
4-6 b pour
I t I $.,
t0
at' s~. ~
to' On a , pour t petit (done ~ petit)
qui traduit, reporte dans 4-6 b de petites perturbations de la forme du
roJ 0
135 signal, dependant de cette forme.
On remarque d'autre part que, puisque tjl > 0
si a est de signe constant
~
est une fonction monotone de t et en general
(independamment du phenomene des caustiques) valeurs de t (dependant de y) si
aWl
an-
~ s'annulera pour certaines
est de signe oppose a a.
La plus petite valeur de t pour laquelle l'equation 4-6 b cesse d'etre inversible est appelee Ie temps critique .
Le temps critique effectif a ete de-
termine dans un certain nombre de situations physiques realistes de la mecanique des fluides compressibles et de la magnetohydrodynamique par A. Greco , M. Anile et leurs collaborateurs.
5 - Conclusions.
Le premier terme u
u~
0
W~
=
i l
0
wf
de l'onde asymptotique 1-6 est donc
WI (y(x),n(t(x), y(x), W'f(x»)
cI>
(t(x), y(x»
hi(x)
c'est-a-dire que ce terme es t, comme dans Ie cas lineaire, proportionnel au vecteur propre a droite hi de la matrice A(x, facteur de proportionnalite WI d'une fonction
cI>
cI>
~x) (valeur propre zero). Le
est, comme dans Le cas Li.nea i t e , produit
qui ne depend que de l'equation et de la surface d'onde don-
nee (et ici de l a solution u
i)
o
et que l'on calcule par integration Ie long
des rayons correspondants par une fonction WI' dite facteur de forme, qui depend d'un choix initial. Cette fonction est, dans Ie cas lineaire, constante Ie long des rayons; elle ne l'est plus dans Ie cas non lineaire, si a n'est
136
pas nul : nous dirons que la non linearite provoque en general une dis torsion des signaux, et nous enoncerons Ie theoreme : I. Le long de varietes caracteristiques exceptionnelles les signaux se pro-
pagent sans dis torsion. 2. Si Ie facteur de dis torsion a est de signe constant il existe un temps critique au dela duquel la distorsion du signal conduit
a
la disparition de
l'onde asymptotique (formation d'un choc).
References.
[11
Leray J., Particules et singularites des ondes ..• Cahiers de Physique, t.
[2J
IS, 1961,
p
373-381.
Garding L., Kotake T."
Leray J., Uniformisation et solution du probleme
de Cauchy lineaire ••• Bull . Soc. Math. 92, 1964, p 263-361.
[31
Lax P., Asymptotic solutions of oscillatory initial value problem, Duke mat . J., 24, 1957, p. 627-646 .
[4]
Ludwig D., Exact and asymptotic solutions of the Cauchy problem, Comm. on pure and appl. Math. 13, 1960, p 473-508.
[5J
Choquet-Bruhat Y., Ondes asymptotiques pour un systeme d'equations aux derivees partielles non lineaires, J. Maths pures et appliquees, 48, 1969, p. 117-158.
[6]
Anile A.M. and Greco A., Asymptotic waves and critical time in general relativistic magnetohydrodynamics, Ann. I.H.P . vol XXIX,nD3, p. 257,197&
[7J
Boillat G., Ondes asymptotiques non lineaires, Ann. Mat. pura et Appl.
iL,
1976, p. 31-44.
137
V APPLICATION AUX EQUATIONS DES FLUIDES PARFAITS RELATIVISTES.
I - Equations des fluides parfaits relativistes .
En l'absence de courant de chaleur, les equations s'ecrivent (1-\ )
(1-2)
(1-3)
(p + p) a
u
U
a
'V
a
uS _ yas
a
a
a
d p+ (p + p) 'Va u a
ua d S a
p
K
K
.
0
0
(E)
0
p , p et S sont respectivement la densite d'energie , la pression et l'entro-
pie spec i f i que , Ce sont des "grandeurs d'etat" fonctions donnees de deux d'entre elles, par exemp1e p et S pour 1'equation d'etat ( 1-4)
On
p
p(p , S)
sait que si l'on pose p
r(I + e)
ou rest 1a densite materie1le propre et e 1'energie interne specifique,
138
1-3 est, compte-tenu de l'equation thermodynamique de • T d 5 + P r- 2 dr equivalente
a
l'equation de conservation de la matiere introduite par Taub [2]
On a pose
a,S·0 , 1,2,3
(1-5)
ou gaS est une metrique hyperbolique donnee ; V designe la derivee covariante ex correspondante et d ex On peut ecrire les equations 1-2, 3, 4, sous la forme
I,J· 0,1, • •• , 5
(1-6)
en posant u u
I
4
- u
a
pour I • 0, •• • 3
- P
2 - Determination de la phase
If .
Cherchons une solution asymptotique des ces equations, au voisinage d'une solution donnee uo' Po' So' c'est
a dire
perturbe d'un mouvement donne. Soit
un mouvement vibratoire du fluide,
139
P
S +
S
o
L q =
en supposant que l'equation d'etat p • p(p , S)
est une fonct ion analytique de P et S, et on ecrit (2-1)
p(p,S)
Po + P~
(p - po) + Ps
o
(S - So) + ••• 0
en posant
Les termes independants de w donnent
, S
0, • • • ,3
(2-2)
On a pose
Ces equations lineaires en ~~
PI'
81 n 'ont
une solution non nulle que si
Ie determinant A est nul. On trouve pour A, comme prevu, Ie determinant caracteristique du systeme (E)
ou on a pose
140
a
~
a 'P o Ia
u
La condition A ~ 0 exprime que les varietes ~x) • c t e sont des varietes
a dire
caracteristiques, c'est solution
des surfaces d'onde, des equations (E) pour la
u~, ondes acoustiques (a 2 + b p' Po
0) ou ondes materielles (a • 0).
3 - cas ou la phase verifie l'equation des ondes acoustiques .
Supposons que (3-1 )
A
A est, pour A
-
bp' + a Po
aB :: {p' g - u uo(p~ 0 po· 0
2
-I}'fa 'fB
=
0
a u l ' PI' 51) est donne par
I
0, de rang N-I • 5, {u! }
(3-2) ou U1(s,x) est une fonction scalaire et les hl(x) sont une solution de 2-2, c'est a dire proportionnels aux mineurs d'une meme ligne de A. Calculons, par 1
exemple les mineurs de la quatrieme ligne A on trouve : 4,
A4 • (p + p)4
4
0
as
Nous prendrons hA _
aA
I
pp Yo 0
(3 -3)
'fa
h4 • (p + p) 0 a 5 0 h
141
Appliquons la methode generale pour trouver l'equation ditferentiel le verifiee par U en egalant
:
I
a
zero les termes en w-I des equations on trouve
(3-4) o~
(l'indice I indique que l'on prend dans la quantite ecrite Ie coefficient -I
de w )
a s[( Pp') I P• ; + (Ps ') I - Yo
51]
'fa + (p + p)o uao
M v
a
uSI
(3-5)
Les equations 3-4 ont une solution
·1
z si
U
(3-6)
ou les pIe
hJ sont proportionnels aux mineurs d'une mieme colonne de A, par exem-
a~ .
On
a
A4
A
4 A 5 On
prendra
4
(p + p)o a
as - Yo
4
'f a PS'
0
4 A 4
'fA 4
(p + p) 0 'fs a
3
=
4
( p + p)o a
5
142
(3-7)
a
J
Formons hJ gl en
2
u
I
I
trouve que U doit satisfaire 1 l'equation differentielle du premier ordre suivante rempla~ant
1
par U h • 1
On
(3-8) avec
(ACl est
=-
k
(3-9)
;.a
2
2 a (p + p)o
1 aX • '2 dpex
bien Ie rayon acoustique)
et, compte-tenu de 3-1 : 2
ex = ~ {2(1 - p' ) 2 Po
(3-10)
(p + p)o )
alors que -I
u
+ pIt
ex
o
(p+p) 0 -
2
(P+P) _0
2 p' P Po -I (p+p)o -~{PS Po
oil on a designe par
a
I' operateur des ondes acoustiques
ex" + uex }) V :: (p' Yo 0 0 ex Po 23 A r:; rap aPA} ex a" ) ex PA = 'fA
O'f (on a
a
J
=
'2
Si la solution de base es t
a densite
~"
et entropie constantes, et
a
lignes de
143 courant
a divergence
nulle
o
(3-12)
a se
reduit
o
o
a
a •
t
Q
!p a
on remarque que les conditions 3-12 impliquent, si u o' Po' Po' So est une solution de (E) que ua Va u a • 0 o
0
'
d'ou on deduit que
Remarque
a est nul si • 0
et c'est
a dire
viste cf
en particulier si Ie fluide est incompressible (au sens relati-
[IJ),
c'est
a dire
admet l ' e qua ti on d'etat p • p + cte
Dans ce cas les andes acoustiques sont exceptionnelles au sens defini preced eement ,
Pour les fluides reels on a p' ~ P
1
p" p
>, 0
donc a
~
0
Si p'p < 1, ou p"p > 0 on a raidissement des signaux. On peut aussi trouver un
dev~loppement
asymptot ique correspondant aux ondes
materielles; celles ci apparaissent comme multi ples (en accord avec la theor ie genera Ie de Boillat, cf [4J). Le probleme de Cauchy oscillatoire est
144
reso1ub1e, au premier ordre d'approximation, pour 1es equations des fluides parfaits. II n'en est plus de meme en presence de phenomenes dissipatifs, par exemple pour un fluide charge
a
conductivite non nulle (cf [3
J) .
References.
[.]
Lichnerowicz A., Hydrodynamique et magnetohydrodynamique, Benjamin 1967.
[2]
Taub A.H., High frequency gravitational waves and average lagrangian, General Relativity and Gravitation, Einstein Centenary volume, A. Held ed, Plenum.
[3}
Choquet-Bruhat Y., Ondes asymptotiques pour un systeme d'equations aux' derivees partielles non lineaires, J. Maths pures et appliquees, 48, 1969, p . 117-158. Coupling of high frequency gravitational and electromagnetic waves, Actes du Congres Marcel Grossmann, Trieste, Juillet 1975).
r4}
Boillat G., Ondes asymptotiques non lineaires, Ann. Mat. Pura et Appl. ~,
[51
1976, 31-44.
Anile A.M. and Greco A., Asymptotic waves and critical time in general relativistic magnetohydrodynamics, Ann. I.H .P., vol.XXIX, n03, 1978,p .2S7.
[6]
Breuer R.A. and Ehlers J., Propagation of high-frequency electromagnetic waves through a magnetized plasma in curved space-time, to appear in Proc. Roy. Soc. A.
145
VI DETERMINATION DE LA PHASE. BICARACTERISTIQUES. VARIETES LAGRANGIENNES.
I . Definition
des varietes lagrangiennes.
L'equation eikonale 1i laquelle doit satisfaire la phase (I-I)
A(x,
'fx>
If:
= 0
est une equation aux derivees partielles du premier ordre non lineaire, A est un polynome de degre N, homogene, en
~x'
Nous allons rappeler comment on in-
tegre une telle equation, en utilisant Ie langage de la geometrie symplectique. On designe par T*X l'espace fibre cotangent
a
la variete X de dimension ~.
Un point de T*x est un couple (x,p) ou pest une I-forme sur l'espace tangent TxX
a x en
x, c'est
a dire
un vecteur covariant.
Une solution de l'equation )-1 dans au-dessus de
n C.X est une section du fibre T:tX
n: par
telle que
(d
'f) x
• < p, dx >
x .... (x, p , Ip(x» A • 0, • ••
~-I
x lR,
146
et
o
A(x, p)
sur une te11e section on a
Soit IT : T~ + X la projection canonique (x,p)'+ x de T~ sur X. On definit une (-forme sur T~, appelee I-forme fondamentale ou forme de soudure par: B(
x,p ) (u)
c
u €.
p(lI' (u»
T T*x X,p
son expression en coordonnees locales est : B
Ph dx
x
La 2-forme
(j
est fermee et de rang (j
2~,
=
de
une telle 2-forme est dite symplectique. La 2-forme
munit T~ de sa structure symplectique canonique . Une sous-variete de T*x
qui annule
(j
et qui est de 1a dimension maximum possible, c'est
a dire
:
~
,
est dite lagrangienne. Si Vest une variete de dimension ~, immergee dans T*x par une application f, on dit que (V,f) est une sous-variete 1agrangienne [immergee] de T*x si
o
sur V
ou
f
2 Recherche d'une sous-variete lagrangienne (V,f) de T~ telle que A(x,p) = O.
Le prob1eme est celui de la recherche des varietes integra1es (immergees
147 dans T~)de dimension ~ du systeme differentiel exterieur
o
a
(2-1)
La fermeture de ce
s~steme
o
(2-2)
contient, outre 1es equations precedentes, 1'equa-
tion exterieure sur T~ :
(2-3)
o
dA
Le systeme caracteristique de 2-1, 2-2 est Ie systeme associe de 2-1, 2-3. 11 est constitue par les champs de vecteurs v sur T1x tels que : (2-4)
i
v
a
k d A
avec
k€:JR
c'est a dire en coordonnees canoniques (XA,PA) de T*X ou v A - 0, ••• ,
= (vA,VA+~)
~-I
(2-5) Un champ de vecteurs vA possedant la propriete 2-4 est dit hami1tonien pour 1a structure symp1ectique a et 1'hamiltonien A. On remarque que vA est tangent
a
Jl: (sous-varie te
A(x,p)
0 de T1x)
Une trajectoire du champ de vecteurs hamiltonien vA est appelee une (courbe) bicaracteristique de l'equation aux derivees partielles A(x,
~x) =
o.
On suppose que l'hamiltonien A n'a pas de point critique sur ciC(c'est
dire que dA sur
en: , et
~
0 quand A
=
a
0), Ie champ hamiltonien vA n'a alors pas de zero
on d.emontre Le theoreme fondamental suivant (cf par exemple
Ill):
Soit Y une sous-variete compacte de dimension t-I de T*x verifiant a
=0
et A
= 0,
transversale en chaque point au champ hamiltonien vA. Soit f t
Ie flot du champ de vecteurs vA' alors (Y x :JR ,f)
ou
f: Y x :JR ... T~
par (y,t) ..... ft(y)
14 8
est une sous-variete 1agrangienne immergee de T~.
3. Determination de 1a phase.
Etant donnee une sous-variete 1agrangienne (V,f) immergee dans X, il existe dans chaque ouvert U C V simplement connexe une phase constante additive pres, satisfaisant
a
~
, determinee
a
une
l'equation
(3-1 )
puisque l'on a, sur U, fX de = fXo
= O. v
Remarque : 11 existe toujours, globalement, sur Ie recouvrement universel V de V, projete sur V par
v
IT, une fonction
v ~
v d ~
On deduit de IT
0
~
(f 0
telle que v IT) x
, connu dans uc V, une phase
~
e dans n ex si l'application
f : U ~ nest inversible.
L'application f etant une immersion il existe toujours un sous-ouvert, encore note U, de U, tel que f soit un diffeomorphisme de U sur feU). L'application IT
0
fest alors inversible sur U si la projection IT : T*X ~ X restreinte
a feU) est inversible : il en sera ainsi au voisinage de tout point ou feU) n' est pas tangent
a
la fibre de TXX, c'est
a dire
n'a pas un plan tangent
"vertical". Soit x€.n
ex,
tel que IT-I (x) ne soit pas tangent
1es points de feU) tels que IT(Yi) = x
a
feU). Soit Y1, .. • Yk
149 Le poin t x admettra un voisinage dans X, encore note n tel que IT-Icn) soit l'union disjointe d'ouverts de fCU) :
C3-2)
U
i
A chaque U, pour une meme phase
~
= I, •.• ,k
fCUi)
sur V, correspond une phase
~
i sur
n
donnee
par :
If i
= ~ 0 (IT
i
0
f)
-)
,
Dans les applications la donnee physique est souvent la variete lagrangienne V, provenant de la geometrie du probleme et de sa dynamique : les bicaracteristiques, c'est
a dire
les trajectoires du vecteur hamiltonien vA sont les
rayons lumineux Cdans l'espace des phases) dans les problemes d'optique, les trajectoires des particules materielles dans d'autres problemes. Nous allons cons i d e r er Ie cas ou la projection de V, supposee sous-variete de simplifier, sur Q
c: r%x
-I IT cm = V ........ L =
=
U
i = 1, •• • k
oU chaque res tric tion
pour
n'est pas bijective, mais OU il existe un sous-ensem-
ble L de V (son "contour apparent") tel que IT-I cm (3-3)
r*x
1\ de
IT
V " L soit de la forme
U.
L
a Ui IT .
L
U.
L
~
Q
est un diffeomorphisme.
4. Solutions asymptotiques.
II est nature 1 de chercher une solution asymptotique du systeme differentiel
150
d'equation eikonale A(x,
~x)
= 0,
correspondant a une variete lagrangienne du
type 3-3 sous la forme : k
(4-1)
E L •
ufx)
ou les i'i sont des phases, sur
i,
e
n c. X,
w 'f i (x)
vi (x)
correspondan tala varie te
v.
On
sera
aide dans ce calcul par la methode de la phase stationnaire qui montre (cfVII) comment 4-) liee a l'evaluation asymptotique d'une integrale. Des developpements du type 4-1, et les equations de transport correspondantes, sont utilisees pour determiner l'intensite lumineuse en presence de caustiques (enveloppe des rayons lumineux en projection sur l'espace-temps X). Remarque : Chaque phase
~i
n'est connue qu'a une constante additive pres,
puisque la variete lagrangienne V ne determine
Wqu'a
l'addition pres d'une
constante, depourvue de signification physique. La theorie des integrales asymptotiques et de l'incide de Maslov permet de determiner des relations entre ces constantes, puis des conditions sur la variete V, dites
cor~itions
de quan-
tification de Maslov, pour qu'il corresponde a V une solution asymptotique globale, avec une phase
[3l
et
[51
de VIII).
\f
determinee globalement sur n
= n(V)
(cf references
151
VII PHASE STATlONNAlRE. PARAMETRlSATION D'UNE VARlETE LAGRANGlENNE.
I. Methode de la phase statiounaire (une variable).
On se propose d'evaluer, pour w grand, une integrale de la forme
lew) -
a
00
ou a et f sont des fonctions C , et a est J
O
)
On suppose que ~~
support compact.
ne s'annule pas sur Ie support de a; on deduit alors
de
a aa
(e i w f)
que
lew)
I
z
iw
•
iw
l'integrale etant bornee par un nombre M independant de w on a -I
Mw
152
Par iteration du procede on trouve, pour tout n
~ ~
0 (w- n)
I(w)
2°) On suppose que f s'annule en un point et un seul que
a2 f ---2 r ao
0 pour
0
= 0
0,
a dire
c'est
0
0
du support de a et
que f a un point critique, et un seul.
non degenere sur Ie support de a . On montre que f peut alors s'ecrire dans un voisinage de
ment de variable
o~
=
g(o) +
I(w)·
e
iwg(o)
t t2
J::
donc
=
0,
par un change-
t(o) tel que t(oo) - 0, sous la forme
g(t) - f(o(t»
_ ou b I t )
0
e: •
e i we:/2 t
2
b(t) dt
do a(o(t» dt
l'integrale existe, absolument convergente, puisque b a un support compact. On peut en particulier la calculer par passage
I(w)
A
f-A
lim A = +
00
eiwe:/2 t
2
a
la limite
b(t) dt
On pose alors b I t ) = b(o) + t c(t)
On sait que lim A=oo
+A
f-A
b(o) e
iwe:/2 t 2
dt
b (0)
On va. estimer I I (w) ::
lim A=+oo
+A
f
eiwe:/2 t
2
c (c)
t dt
-A
Puisque b a un support compact on a c(A)
c(-A) done
(2II») /2 e iIIe:/4 w
153
I(
(
W
)
_1_
m
iwe:
+A
lim
A =
J-A
<:D
e
2
iwe:/2 t
c'(t) dt
2 l'integrale est bornee parce que c' est une fonction de c1asse c bornee ain2 . d' ord re.;;: 2 ( on montre que JAB e i we:/2 t c'(t) dt tend s r. que ses d-errvees vers zero quand A et B tendent vers l'infini en faisant des integrations par parties). Le raisonnement peut s'appliquer
a nouveau
pour l'estimation de
l'integrale figurant dans IJ(w). On a montre que :
Ona
Ona
sa dt
=
au point t = 0, a
a o on a done
done da dt
I
m
t=O
,-1/2
a =a
o
D'ou finalement l'evaluation asymptotique
I(w) ou 3°) Supposons que f a un ensemble fini a., j E: J, de points critiques non J
degeneres sur Ie support de a. En utilisant une partition de l'unite sur ce
154
support, isolant les points critiques, on demontre que e
I(w)
ou
ilk ./4 J
e
iwf (a.) J
a(a.) J
E. J
2. Methode de la phase stationnaire (plusieurs variables).
Considerons l'integrale I(w)
=
a(y) eiwf(y)
JY
ou a et f sont des fonctions
a valeurs
d~(y) oo reelles, C , sur une variete riemanien-
ne Coo, Y, d'element de volume d~(y), de dimension n .
a
1°) Supposons que a est
support compact et que f n'a pas de point critique
sur Ie support de a. on a a Iors , pour tout 1'1 E. II : I(w) • O( w
-N)
En effet on a : e
d'ou e
iwf
•
iw v
i
e
iwf
en posant (si f n'a pas de point critique on a v
i
iwf
155 c'est
a
dire e
iwf
1
= iw
done I
iw
lew)
l'integrale etant bornee on a
= O(w-I).
N) s'obtient
Le resuitat O(w-
par iteration . 2°) Supposons que f a un point critique et un seuI, Yo' sur Ie support de a, et que ce point critique est non degenere c'est (l2 f
a dire
que Ie hessien
en ce point est une forme quadratique non degeneree, a q carres po-
sitifs et p = n - q car res negatifs. D'apres Ie lemne de Morse il existe des l
coordonnees locales t , ... t f(t)
f(y(t»
a
=
n
au voisinage de y
o
f(o) + Q(t)
telies que: t
avec
Q(t)
q
I
= 2"
E i
. 2 (t~) -
p+q . 2 E (t~) i = q+1
I
2"
=
Dans ces coordonnees :
avec bet) = e
iwf(o)
a(y(t»
-
[d et g .. 1J
1
1/2
ou g •. sont les composantes de la metrique dans les coordonnee s t. La quanti~J
te det Hess f/ det g est un scalaire; on a au point Yo' en co ordonnees t
done, en ce point
15 6
Idet c'est
Idet g . . 1 1J
g.1J. 1
a dire b(o) - e
iwf(y )
1/2 / a(y) Idet g . • 1 1J y=yo o
0
On montre, par un raisonnement analogue
e
ou 0 = s ign Hess f Iyay
o
a
Idet Hess fl
y=yo
celui fait pour une variable que
ill/40 iWf(y 0) a(y ) Ide t gl 1/2 o
e
Idet Hess fl
7'"Y
I 2
0
y·yo
- q- p
3° ) Si f a un nombre fini de points critiques y . non degeneres sur Ie supJ
port de a, I(w) est estime par une somme de termes de la forme de l'expression precedente .
3. Integrale dependant d'un point x
e. X.
Dans Ie cas OU les fonctions f et a sont definies sur Ie produit de deux varietes X x Y et ou on considere l'integrale I(w)·
(3-1)
I
a(x,y) eiwf(x,y)
y
on obtient l'estimation, pour w grand:
(3-2)
I(w)
1: jE.J
e
iIl/40.
J e
iw'f'(x) J
1/ 2 a j (x) Idet g 1y=y .(x) 1/2 J Idet Hess ' fly=y (x) o
ou
OJ
=
sign Hess flYaYj(x) , ou Yj(x), j £ J designe l'ensemble suppose fi-
ni et non vide des points crit iques, supposes non degeneres , de f considfree comme fonction de y pour x fixe et ou on a pose
~j(x)
= f(x'Yj(x»,
157
a.(x)
a(x,y.(x').
=
J
J
Remarque
L'expression 3-2 est
Maslov-Leray . Pour que la phase
a
la base de la methode de quantification de
'f
soit determinee globalement sur n i l faut
que 1I
modulo 2 1I
W'Pj(X) + 4" OJ
ce qu'on ecrira, si y est un I- c yc l e sur V, puisque 'fi(x) = et
deW
0
~-I) =
(W
0 ~
-I
)(Yi(x»
e mod 1
ou [nJ
y
161 de
est la variation sur y de "l'indice de Maslov" de V (cf VIII). Dans les applicat ions
a
131, 141, 15 1,
la mecanique quantique on prend W
=
I/~.
4. Parametrisation d'une variete lagrangienne V.
Soit V une sous-variete lagrangienne de T*x. Si la restriction
a V de
la
projection 1I : T*x ~ X est un d iffeomorphisme V ~ n,des coordonnees locales xA sur X fournissent aussi des coordonnees locales sur V. Une phase fournit une phase
'f
sur
Wsur
V
n et l' application n ~ T*x par x I~ (x, 'f') est un
diffeomorphisme de n sur V, qu'on appelle une parametrisation de V. On
generalise la definition de parametrisation, dans Ie cas ou 1I1v n'est
pas un diffeomorphisme
en plongeant X dans un espace
a
un plus grand nombre
de dimensions. Nous exposerons ici le cas le plus s imple ou l'on introduit un seul parametre supplementaire. Designons par f(x,a) une fonction sur X x (x,a)~
Xx
~
(If
tels que (la (x,a)
=
O.
~
et par C l'ensemble des points f
15 8
Definition
xx
La fonction f
:JR ... :JR est dite une parame t r i.aa t i on de V si
l' app lica tion
est un diffeomorphisme de C sur V. f Si fest une parametrisation de V la fonction W = f
V, c'est dans
a dire
T*x.
verifie sur cette variete dW
0
~-) est une phase sur
i*e ou i est l'inclusion de V
Une condition equivalente, puisque ~ est un diffeomorphisme de C f
sur V, est en effet
,",veT x,a
C
f
Cette relation est verifiee d'apres la definition de e et Ie fait que. sur
Une fonction f parametrisant V fournit done aussi une fonction phase un ouvert
n de
X diffeomorphe par la projection IT
a un
ouvert
U. de V. 1
~i
sur
On
pose : f
c'est
~
0
-)
0
IT
-)
i
ou
a dire
If i (x) ou l 'application ~- I
0
a
f(x, a i (x)
rr~1
: x>+ (x.ai(x»
de l'equation de Cf , ~~ (x, a) localement) si
=
est determine par resolution en a
O. Cette resolution est possible (au moins
d2 f
---2 (x ,u) f O. L'ens emble singulier L de la variete lagrandU
gienne V (cycle de Maslov) est l'image par ~ des points de X x :JR •ou af = 0 da 2 d f
et - -
dcl
= O.
La definition d'une parametrisation d'une variet~ lagrangienne s'etend en rempla~ant
X' x :JR par un espace fibre de dimension £ + m au dessus de X (par
159
exemple T*x dans lequel Vest deja plonge). L'obtention d'une parametrisa tion donne la pos s Lbi l i.t.e de fixer dans l' ouvert n C dant a chaque U diffeomorphe par i
ITlu.
a
n
X une phase
'f'i cor r esporr-
et de remplacer 1a valeur asymp-
1
totique d'une integrale du type 3-1 en uti1isant 3-2 par une somme de termes faisant intervenir ces phases. Exemple : supposons que nous ayons une seule variable d'espace x et une variete lagrangienne dans Ie plan (x,p) qui passe par Ie point (0,0), et est tangente en ce point a l'axe des p. p
v
11 est alors naturel pour "deplier la singula-
rite", c'est a dire i c i pour trouver la var Lete et la fonction f(x, a), de chercher cet en-
C
f
x
semble C dans X x lR sous la forme : f
x ce sera 1 'ensemble
Ct
2
~. 0 si f est de la forme aCt f(x,Ct)
a(x) +
XCt -
3a
3
f sera une parametrisation de V si af ax
a'ex) + a
~
est tel que l'on a
V-
{x, a'ex) + Ct} x
= a2
c'est a dire si Vest compose des deux ouverts u\ =
{x, a'(x)
Ix }
x > 0
U2 =
{x, a'(x) - Ix }
x > 0
+
et
et de l'ensemble singulier : L
(0,0)
160
VIII DEVELOPPEMENT ASYMPTOTIQUE AU VOISINAGE D'UNE CAUSTIQUE .
I. Caustiques du premier type.
La forme d'une parametrisation d'une variete lagrangienne est liee ture des singularites de l'inverse de la projection
n:
a
la na-
T*x + X restreinte
a
V. Les singularites des applications differentiables ont ete classifiees par Thom. On montre que dans Ie cas Ie plus simple ( pliage, auquel correspond l'exemple
a
la fin de VII) on peut parametriser V par une fonction de la
forme : f(x,a)
o(x) + p(x)a -
3
a :r
ou a et p sont des fonctions regulieres du point x E X. L'ensemble C est alors f
~~ _ p(x) - a 2
0
. a2 f L'ensemble singulier E de Vest l'image de l'ensemble ---2 II se projette sur X en p(x)
=0
u e; ±
et on a
aa
=- 2a
=0
de Cf'
161
La, figure represente la projection sur
X : l'ensemble singulier E se projette sur l' ensemble C appe Ie "caustique". L'ensemble p(x) < 0 n'est la projection
p(x) < 0
d'aucun point de V. l'ensemble p(x) > 0 est recouvert deux fois par la projection de V : deux "rayons" passent par chacun de ses points.
2. Integrale asymptotique. fonction d'Airy.
Pour construire des solutions asymptotiques d'un systeme differentiel correspondant
a une
variete lagrangienne du type precedent on va etudier la va-
leur asymptotique de l'integrale u(x) •
f e iwf(x.a) a ( x.a ) da = - e iwo(x)
fe
iw(p(x)a
3
~
ex 31)
a(x.a) da
On sait que (theoreme de preparation de Malgrange) il existe des tonctions
2 + h(x.a) (p(x) - a )
done puisque
p(x) - a
2
•
ilf
ilex feiWf(x.a)da + al(x) +
Le dernier terme peut s'eerire
J
Jeiwf(x.a) a da
e iwf(x.a) h( x.a ) ~ ila d a
162
a
aa h(x,a) da
ioo il est done d'ordre
100 .
En repetant Ie meme raisonnement on trouve un develop-
pement de la forme u(x)
'\,
1:
Jeioof(x.a) da +
oo- q b (x) q
q
r
00-q c
(x)
q
q
_ a(x) + p(x)
- a 3/3
J e ioof(x ,«) ada
Dans Ie cas considere f(x.a)
on introduit la fonction d'Airy 1
.A,(t) :;
2II
+ ...
J- ...
e
itt - t
3
/3 dt
on a alors feioof(X,a) da
:/3
•
eiwa(x)
Jt(00 2 / 3
p(x»
00
et, en derivant sous Ie signe d'integration par rapport
J
e ioof(x,a) a da
1
·213 00
eiwa(x)
.:i' (i/ 3
a
p:
p(x»
D'oii
On
a les estimations suivantes pour la fonction d'Airy
.A: (t)
~
.:/C' (e)
:lL
J
~4 _ t l/4
"n-
2 3/2 cos(} t
114 )
pour t > 0
. 2 3/2 un(} t -
114 )
pour t > 0
grand
.
. grand
(peut etre obtenu par la methode de la phase stationnaire). On
voit que pour p > 0 et
pement :
2 3 / grand on a pour Ie premier terme du develop-
00
163
'V
u ( x) _
L
eiwcr(x) i jj 2/j
vu w
(w
p)
1/4 [b O cos(
2w3P2/3
-1!.4) - Co p I/ 2 Sin(2Wp23/3 - 1!.4
J
cette expression coincide avec celIe obtenue dans les etudes d'optique geometrique en presence de caustique, on cons tate en particulier un changement de phase de n/2 entre les deux termes. Remarque
pour t < 0,
cit (t) it' (t)
'V
'V
Ie]
grand on a -:
un 1
- 2/3 (- t)3/2 1 e(_t)1Y4
un
(_ t) 1/4 a - 2/3 (- t) 3/2
Si l'on admet que l'integrale donne aussi une estimation de u pour p <
o.
On trouve que u devient rapidement petit avec I/lpl , p < O. On trouve aussi
ici une estimation pour p • 0 (c'est
a dire
sur la caustique), en utilisant Ie
fait que pour t voisin de zero on a : ~(t)
3. Onde
a haute
~ 0.355) 0.259 t
frequence, solution approchee d'ordre I d'un systeme differen-
tiel lineaire.
Considerons Ie systeme differentiel lineaire (3-1 )
o
j • I, ••• N
de polynome caracteristique A(x,p). On cherche une solution approchee, pour
w grand, sous la forme :
164
ui
(3-2) ou b
= e iWO"
(it:<w2/ 3
b
i + w- 1/3
dt'(i / 3 p)
c i)
i et c i sont des fonctions de x de 1a fol:'llle
(3-3)
b
tandis que f(x.a)
0
i _ bi + 0
1.w b iI
et p sont lies
= o(x)
c
a la
i • ci + 0
1. bi w I
geometrie des rayons du prob1eme pose
3
+ p(x)a - a / 3 est une parametrisation
de 1a variete lagran-
gienne engendree par les rayons dans l'espace des phases. On voit aisement sur son expression que la fonction d'Airy verifie l'equa-
tion differentielle : J!"(t) + t Je(t)
(3-4)
o
En reportant 3-2 dans 3-1 et annulant Ie terme en w on trouve. compte tenu de 3-3 et 3-4. et du fait que systeme lineaire en b i c i o·
eft et Jr'
sont des fonctions Lndependances , Le
0
o qui est equivalent pour P > 0 nus en faisant
£ •
a
l'ensemble des deux systemes lineaires. obte-
+ I ou £ - - I dans
On en conclut. en accord avec les resultats precedents, que les fonctions
'f £
•
£ -
doivent etre solutions de l'equation eikonale
+ I
ou
165 Un ca1cu1 analogue
a ce1ui
fait en l'absence de caustique (cf II) peut alors
etre fait pour determiner les equations de transport de b o et co'
References.
[IJ
Choquet-Bruhat Y., De Witt-Morette C., "Analysis manifolds and physics" nd 2 edition, North Holland 1980.
[2]
Ludwig D. Uniform asymptotic expansion at a caustic. Comm. pure and app. Maths. Vol XIX p. 215-2 50, 1966.
[3J
Leray J ., Solutions asymptotiques des equations aux derivees partielles (une adaptation du traite de V.P • • Maslov). Convegno International, Metodi valutativa della fisica matematica, Acad. Naz. dei Lincei 1972.
[4J
Leray J ., Seminaires du College de France 1976-1977 et livre
a parattre
aux M.I.T University press.
[5]
Guillemin V. et Sternberg S. "Geometric Asymptotics" A.M.S., Providence
1977 (Math. Surveys nO 14).
CENTRO INTERNAZIONALE MATEMATICO ESTlVO (C.I.M.E.)
U R T I
GUY BOILLAT
Guy BOILLAT
1.Preliminari Introduciamo un vettore che e un insieme di N funzioni delle variabili ina dipendenti x (a .. 1, 2, ••• .n)
a
u(x )
La variabile
O X ..
t rappresenta usualmente il tempo mentre xi (i
1,2, ... ,
n) sono variabili di spazio. Scrlviamo ( 1.01)
dove
u. =
t)u/~x4l(
e le A" Bono matrici NX N generalmente dipendenti dal campo u e delle variabili x~ • La somma
e sottointesa
sugli indici ripetut~. • Un tale sistema si
chiama quasi lineare. Se 1e matrici Bono indipendenti da u si ha un sistema semi lineare; se poi anche la funzione sorgente f non dipende da u il sistema si dice lineare. Dimentichiamo ora la dipendenza esplicita da tema
pu~
essere riscritto coal
x«
.11 5is-
170
o
i
A (U)U + A (U)U t i
= feu)
(1.02)
Definizione di iperbolicita. Gli autovalori di A
n
O
spazio) rispetto a A sono tutti reali per ogni
1
i
...
A n . (n versore della ~
ed esiste una base di autoO
vettori delle spazio di u. Questo implica la regolarita di A e pertanto il sistema (1.02) si puo mettere sotto la forma
u All'autovalore
t
\(i)
~
i + A (U)U i
= feu) .
di
moltepl~cita
(1.03) (1)
m
(i) devono corrispondere m autovetto-
ri (destri e sinistri) linearmente indipenaenti cosl definiti
= 0,
I
= 1,2,-••• ,m
(i) (1.04)
(j) J :: 1,2, ... ,m
che denoteremo
anche
se~plicemente
lJ ' d
I• In particolare se tutte Ie matrici di (1.01) sono simmetriche, cioe
O
ed inoltre A
e
definita positiva, (1.01) viene chiamato sistema di Friedrichs .
E chiarD che tali sistemi sono iperbolici. In generale un sistema qualunque non si puo mettere nella forma simmetrica; pero i ~atica
sis~emi
della Fisica mate-
8i possono ricondurre, come vedremo , ad una tale forma .
2. Sistemiconservativi Questi hanno una
~orma
speciale nel senso che si scrivono come divergenza
nello spazio-tempo di certi vetteri f~(U)
"
?.. f'
(u)
= f'(u)
(2.01)
oppure, 'con l'introduzione del gradiente rispette al campo u, (2.02) chs corrisponde a (1.01) con f4
A
= Vr' .
171 O
Poiche A non e s\ngolare 5i puo quindi scegliere
"e
i
rO
come ca mpo u ed allora
r(u)
+ ';)i t (u)
(2 .04)
Per un fluido, ad e5empio. k (>u
=
U
i k \'ik eu u + p
r
~
i
tu
(e+ p)u
£ dove
~
=
eu212
i
(2.05) i
e e l ' energia totale. e l' energia interna. i
+
e +
pIt
l 'entalpia libera legata all'entropia da di
= TdS
+
dr/t
Come conseguenza segue la legge di conservazione del1'entropia (2.07) Abbiamo qui un sistema con un equazione i n piu ; l'ultima e pero una conseguenza delle altre . In generale dato un sistema conservativo (2 .08)
e un'equazione scalare conseguenza (2.09) se si fa la derivata rispetto a t e si sostituisce (2 .08) si ha 1'identita i i Vh(r - Au) - Vh u
i
i
che deve essere vera per ogni u
i
=g
da cui i
1.2 , •••• n
(2.10)
Friedrichs e Lax hanno fatto vedere [1] chef definita 1a matrice hessiana H
la matrice HA
i
=
VVh.
e simmetrica. Basta allora
ottenere un sistema del tipo
m01t~plicare 1a
(2 .08) per H per
172
Hf
(2.11)
che e un sistema simmetrico ne1 senso di Friedrichs purche 1a funzione h(u) sia una funzione convessa di u , cioe H Quando si un cambio
mo1t~lica
definita positiva.
(2 .08) per H si perde 1a forma conservativa, perc con
e: di variabi1i s1
puc ritrovare e si perviene ad un sistema conserva-
tivo e simmetrico. S1 introduce i l campo nuovo dato da
r2J
~
(2.12)
u' =Vh e qua ttro funz10ni scalari
(2 .13) In partico1are per ~
=0
= u' .u
h'
- h,
(2.13' )
h'o;& h',
e una trasformata di Legendre. Deriviamo la (2.13) r ispetto a u'
e da (2.10) r1su1ta
1-" = V'h,l(
(2.14)
= V'h'
(2.15)
In partico1are
~ Ne segue
A,at(u,)U' at
= s,
(2.16 )
cioe 1e nuove matrici non solo sono simmetriche rna anche hessiane. La forma (2.16) e stata introdotta da Godunov 131 con tre esempi. Notare 1a differenza
1n (2.11) e (2.16).
Ie matrici simmetriche sono
rispettivam~nte
173 3. Equazionl dl Eulero Applicando un principio variazionale alla lagrangiana L
= L(q:
• qS)
(3.01)
dove le qS(x~) sono funzioni dello spazio-tempo e ~
~~qS 3i arriva al1e
equazioni di Eulero
tdo(OL/~) _';)L/~qS
=0
(3.02)
che si possono mettere nella forma conservativa (2.04) con ~Lj)qS
u
s
=
r.i
qi S
q
~L~s
':>L/)q;
0
-qs ~ j o i
• f
=
0
0
(3.03)
s qo
Se L non dipende esplicitamente da qS basta eliminare la terza riga. D'altra parte si puo definire una quantlta con due indici (3.04) tale che
CJT~-O fa f( -
(3 .05)
se le equazioni di campo (3.02) sono soddisfatte. Le (3.05) rappresentano quattro equazioni supplementari. La conservazione dell'energia corrisponde a ~
=0
che sceglieremo
come equazione (2.09) con (3.06)
Si deve ricavare u'. il che significa che dobbiamo valutare le derivate parziali di h rispetto alle componenti della u. Indichiamo le componenti di u nella maniera seguente u
u '"
u
o S
i
s
174 s i ricava u' i n
questo modo
";)h/au o =~qsl) uo 'h/~qS r 0 r 0 _ ()L/dqS )qS/hO o r 0
+ qS ':> (~L/dqs)/) uo
o
_ ~L/~ q~ ~q~/)u° 1. r
0
r
_ ~L/)qS !)qS,1u o r
r qo e similmente per le altre componenti. Infine s i ha u' u'
u'
S 0
i
S qo - ")Ll)q~
5
(3.07)
1.
_h/~s
u' S
S
(5 i tolga la terza r iga se L
L(~).)
Oi conseguenza - u,j s
- u' s
_u,s ~j o i
ri
f =
u'
0
cioe
~,
f
(3.08)
0
s 0
sono funzion i lineari di u' ; le equazioni di Eulero prendono la
forma (4) . (53 . (3 .09) L'uni ca nonl inearita e dovuta al coefficiente della derivata tempor ale. Le matric i A,i, B' sono co stanti. le prime s i mmetriche. l' ultima emisimmetrica V i _ A,i A' ,
oJ
B'
=-
B'
5e L non dipende esplicitamente da qS, f e i l secondo membro della (3.09) sono nUlli; un caso gUI con siderato da Godunov
C3:! .
5tudiamo adesso la conves s ita di h . ~u.lu' = ~uHSu .,
o,
'f
~u -F O.
Con (3 .03 ,07 ) s i ottiene tor nando a Ile va ri abili inizia li
175
1u.}u'
(3.10)
somma di tre forme quadratiche ciascuna delle quali deve essere definita positiva. 5e cerchiamo Ie velocita caratteristiehe basta far eorrispondere
e si ottiene -
Quando la veloeita
~~u
e
+'
l'n
=0
•
nulla segue subito da (3.08)
= O.
0,
5e inveee supponiamo ~ ~ 0
(3.11)
da (3.02) si ha +
,,2 eJ
"r
LI"q .
~
s
:"1:2 s
o.
~ q . n , n . )" q J ~ J
Deve risultare nullo il determinante delle quantita tra parentesi. Le sue radiei
~(t) usualmente sono diversi da zero .
Ad
ogni autovalore
~I
0
e
asso-
eiato un eerto autovettore. 5upponiamo ehe per un eerto valore di~ la veloeita diventa nulla, signifiea ehe la molteplieita. dell'autovalore ~= 0 aumenta, ma il numero di autovettori assoeiati
e
sempre 10 stesso, dato da (11).
5i perde eosl l'iperbolieita. 5i deve dunque evitare ehe
J (~) = 0;
la matri-
ee
deve essere regolare. Una eondizione ovviamente soddisfatte se vale la eonvessita (vedi (3.10).
176
4 . Urti Supponiamo che attraverso una superficie ~(x~)
o
che si muove col tempo,
di normale,
e velocita
s:_f(/"'f\ t il campo u
e
discontinuo
(u) :
U -
U
o
f. o.
(4.01)
Al sistema (2 .01) si appliea il teorema di flusso-divergenza e si arriva a11e equaz ioni di Rankine-Hugoniot ehe si serivono
[rlseuJ: n I ,e ,
t
0,
[6)
i
n
r (u)n .
1.
(4.02)
t
r (u) -
B U
n
I1 problema
= r (u ) - s n
0
U
0
e
trovare il campo dopo l'urto in termini del eampo U prima o l'urto e di s (velocitA dell'urto) u
= Uo
h(u ,s,~).
+
(4.03)
0
Deriviamo (4.02) rispetto a s e
(A - 81)& - h n
U
o
=0
(4.04) (4.05)
Supponiamo ehe l'urto sia debole. Faceiamo tendere b a zero; si ha
(A - 81)& on 0 II che signifiea ehe s 8
=
e
O.
autovalore di A on
~(u ,'it) , Ii =d..d(u~) . o
o
0
177
Da (4 .03) segue •
h(u • ~ ', n ) ~ 0 0
(4.06)
0
e. facendo la derivata rispetto a
U
o
V0 h + il V ~ = O. 00 0 Prendendo il dterminante della
(4.05) si ottiene
= det
det (I + V h) o
(4.07)
(A - sI)1 det(A - sI). on n
L'equazione precedente d~ subito il valore del primo membro
=1
det(I + V h) o
-
V ~ .il 0
0
0
=1
-
~.
0
II secondo membro si presenta jn una forma indeterminata
11 cui limite - 1/(~' - 1)
o
deriva facilmente della regola di L'Hospita1. Pertanto 1- ~
o
o
= II (1
significa che
v0~.d 00
-
0
oe
V}. .il o 0 0
=
~ (~
/I:0 i.
0
-
2)
= O.
(4.08)
quindi (4.09)
O.
Torneremo su questa condizione di "eccezionalita ". cosl chiamata da Lax e supporremo per il momento
i. o s
= 2,
che non e verificata (urto normale). Allora ~
= ~0
= (~+ ~ o )/2
+
\' (s - ~ ) + ••• ,
~o
+ O(s _
0
~
)2
(4.10)
0
cioe la velocita di un urto debole e il valore medio della ve10cita caratteristica.
17 8
Questo mostra che s u
o
e
compresa fra
..
10 s tate tale che
~(u ,n) o
~
o
~. 5i PUQ qu indi sempre ch iamare
e
...
< s < C'\ (u.ri )
(4 .11)
esc l udendo i l caso Keccez ionale" (4.09 ). Unat a 1e disuguaglianza insieme aIle equazioni (4 ,5) conduce aIle condizioni di Lax [71 : a non raggiunge mai una veloci ta cratteristica ma
e
compreso
fra due autovalori conaegutlvl sla per 11 campo u che per il campo u
o
(4.12)
(4.13) per ogn i ~, in modo tale che (11) viene verificata quale che sia k. Cosl gl i urtl ven gan o classificati secondo il valore di k.
5. Entropia. Funzione generatrlce 5upponi amo adesso l'es istenza dl una legge di conservazione supplementare
(2.09) e, in analogia con Ie equazioni di Rankine-Hugoniot cons lderiamo la fun zione de l l ' ur t o
(5.01) In condizione di dlfferenziabilita (2.09) ebbe pensare che
~
e
e
conseguenza di (2 .08) e s i potr-
nulla se valgono (4.02) . 1nvece, di sol ito, non
e ve-
. ' . 5e si i ns er i s ce (4.03) i n (1) sl ottlene
(5.02) Deriv lamo rispetto a s
i
sVh}1i -
=
(hI = ~'(A • n
~'h - [h]
- a1)it - [h]=
W
(5.03)
(5.04)
t ene ndo conto di (2 . 10 ) e (4. 04). Essendo h una fu nz ione convessa di u ,
179
hj u ) - h l u) - 'i1h(u - u ) =:!(~ - ;-;) H(u ) (~ -~) '? 0 , o 0 2 0 c 0 u
u
c
o
+ 2;"( u - u ), 0
e pe rtanto w
"70,
"f
(5.05)
h " O.
Da (3) e dalle condizioni di Lax segue '( = I s
wds
~o
purche l'urto sia nullo quando s = ~ • Di conseguenza o
[81 (5.07)
che traduce come vedremo la crescenza dell'entropia . Facciamo la derivata di ~ lispetto a u
o v
>J
V"1=u'(A -sl)(I+Vh)-u'(A -sl) o (n 0 0 on
= ('i;' -
~')( A o
on
- sl)
in virtu di (4.05). Se conosciamo la runzione
'7
~
(uo ,s,n) possiamo ricavare
1'urto in termini delle variabi1i u' V (
u 'J =
Vo /VII (A on
- s L)
-1
(5.08)
L'entropia genera l 'urto. Accanto a questa proprieta la formula mette in risalta i l r uolo importante di u' ch iamato da T. Ruggeri pale
[91
,If
campo princi-
If
6. Esempi Per il f1uido (§ 2) la legge di conservazione supplementare (2 .09) che sceglieremo e quella dell'entropia (2.07) e vedremo che
e funzione co nvessa di u
(2 .05) . Inratti (10)
.....,
2
Tdh = u.d(eu ) + (G - u !2)d{' - de. dove G
- TS e l'entalpia l ibera e
180
~
u
(6 .03)
u' - 1
che coincidono con Ie variabili introdotte direttamente da Godunov [3J . Per la convessi ta basta considerare [10"1 w
Ma
e pertanto deve essere
(~~T)
LG) -
o
(~Gj)p) [p) <0 .
[TJ -
0
Ne risulta che - G deve essere una funzione convessa di peT, una condizione verificata per I' equilibrio termodinamico [111. D'altra parte
(6.04)
i rna dall'equazione di conservazione della massa segue la continuita di
e dunque
=
i\f ,
I)
\ 0
(5 - u
on
)rsJ .
(6.05)
Da (5.07) deriva a110ra 1a crescenza de1l'entropia
Infine 1a funzione generatrice
'( (U
o
• s,
2¥
exp ("1/e c ~ M ) = M /(1 o " v 0 0 0 dove c
e 1a
velocita del Buono, M o
r
2
=
«(-
l)/()'+ 1),
(u
on
1) si
-I"2
esprime ne1 modo seguente 2 2 "tL
2
+ r M ) 1(1 +("')14
- s)/c
)' = ep /c v
o
0
2. 0
-I"
i1 numero di Mach,
2
~
181
C
p
e C sono i calori specifici costanti per un fluido politropico. v II grafico della ~ si compone di due rami corrispondenti all'urto lento e
all'urto veloce, con due punti di flesso per s = u
~
Co ' due
'1 = 0
V
~ Co «(- 1)/2Y e un punto isolato on dente all'urto caratteristico. s = u
on
asintot~
per
per s= u corrisponon
Per i materiali iperelastici della meccanica dei continui de = Td5 - T dF ij ij
~6.06)
dove T = (T 0 0) e i l tensore di Piola-Kirchhoff, F = (F .. = ')u .I~X.) ~J
~J
J
~
tensore
gradiente di spostamento. 5e poi v , =I)u.l~ e la velocita di spostamento, 0 _ ~
~
~
la densita costante nella stato di referimento, b
o
~
Ie forze esterne, il siste-
rna (2.04) e definito cosl (su una base ortonormale di vettori ~)
....e
~
e,t
T ji j
u=
F
, f'
i
... e... - v ~
=
e
f =
i
V/j 1
0
...... e*b.v
con la conservazione dell'entropia
~S/~ = Da h
o .
- 5 e (6.06) si deduce
...v
u' =
T- 1
(6.07)
- T - 1
5i dimostra come nel caso del fluido che -5 e una funzione convessa di u purche questo sia vero
p~e(5,
I
F, 0)' Ne segue la crescenza dell'entropia quando ~J
la veloci ta dell' urto e positiva
£12
J.
II quadrivettore (2.13) ha Ie componenti h'
2 = (Ov /2 \~
- TooF , - G)/T, ~J i J
T u" J
.IT
J
182
7. Urti caratteristici Finora si
e
considerata una soluzione (4.03) delle equazioni di Rankine-
Hugoniot (4.02) dipendente da un solo parametro s. Adesso ci chiediamo se
e
possibile di trovare una soluzione
= uo
u
+ h(u
0
parametri u
• u
poi rispetto a u
I'
~
• n},
(7.01)
=0
(7.02)
I
(A - sI)~ h I
n
I
~IS h
e eliminiamo h
Se p)l esiste almena uno dei vettori che non sariamente un autovalore
e
nullo . Ne segue che s
e
neces-
di A
n
Studiamo, in generale, questa possibilita. Supponiamo per primo che s
....
= s(uo , n ) ,
autovalore di A on
Sostituendo nella (4.02) e facendo la derivata rispetto a
e
u
I
viene
pertanto s deve essere anche autovalore di A . Scriviamo dunque Ie equazioni n
(4 .02) con s
=~(u,
ri)
(7.03)
pe r ottenere I, 2, ...• p
e d'altra parte, facendo la derivata rispetto s u (A
-~I-hV~(I+Vh) =A
n O M
(7 .04)
o -
~I
(7.05)
queste equazioni per un autovettore corrispondente all'sutova(i) lore ~ di molteplici t8 m
Mol~lichiamo
183
(7.06) - (1II·h)V~(I + Voh) I'=1,2, ... ,m Se II I · h
1
I'
(A
on
- ~I) ,
(7.07)
(i)
~ 0, dalla prima segue ';)1~
O. Se invece vale l'uguaglianza,
la seconda equazione d~ 1
I'
(A
on
-~I)=O,
cioe ~ e autovalore anche di A , dipende soltanto da u e dunque abbiamo' on 0 ancora
o e da
(7.08)
(7.04)
o
(7.09)
i. e. h puo dipendere da tanti parametri quanti possono essere i vettori (i) ~Ih indipendenti, vale a dire m I=1,2, ... ,m
(i)
(7.10)
La (8) si scrive
oppure tenendo conto di (9) (7.11)
Questa uguaglianza, che deve va1ere qual1{he sia u e ~ e la condizione di ecceziona1ita di Lax. La (8) fa vedere che ~ e indipendente di u zione
n~11a
u
= Uo
I
e siccome amme~amo la solu-
(corrispondente per esempio alIa nullita di tutti i para-
metri u ), segue
.... = ~(u ,n). .... o
~(u,n)
(7.12)
La co ndizione (11) si incontra spesso in Fisica ma t ema t i ca ; basta citare
184
Ie onde di materia, di Alfven, gravitationali, di Born-Infeld, della corda relativistica, ecc. [13)
. E'sempre verificata per Ie onde moltiple di un sis-
tema iperbolico conservativo (14) • Oeriviamo (A
n
-~r)d
r =0
nella direzione dell'autovettore d
I,
Scriviamo la stessa cosa cambiando gli indici e facciamo la differenza. Perchll 11 sistemail conservativo An .. Vt'n
e
VVfndI,d
I
=VVt'ndldI' . Res t a
Ne r i s ul t a necessariamente (11) . Nel caso di moteplicita esistera dunque sempre un urto che si propaga con una velocita caratteristica e che chiameremo urto caratterlstico. Nel caso di un autovalore semplice la condizione (11) di ortogonalita del gradiente di ~ e dell'autovalore corrispondente pu~ anche essere soddi 5f at t a , p. e. per gli urti di Alfven. Sia ~ (x~) .. 0 l'equazione del fronte d'urto che veri fica l'equazione caratter-t s t I c a
(7.13)
(7.14) S1 ha (7.15)
e derivando rispetto a (7.16)
1 At4. I
Moltiplichiamo Ie equazioni del campo per II
IrA Ma lIA~ ~~
e un
It
(7.17)
uee " lIt
operatore di derivata tangenziale alIa superficie d'urto come
5i vede da (13,15). 01 conseguenza s1
pu~
50stituire nella (17) il valore
185
(1) di u sulla superficie 0(1'\
lIA ~(u ..
0
+
It
h(u • u , n»
°
= 1 .r ; I
I. I'
= 1,2,
(1)
••.• m
E~un sistema di equazioni differenziali ordinarie per gli u I• Infatti, essendO~Ih un autovettore destro di An
paiono
(vedi 9) Ie derivate dei coefficienti ap-
tramite termini del tipo
" d ~t( u I' lI A I,
(7.18)
che tenendo conto di (16) si mettono sotto la forma (7.19) dove
! la derivata lunge i raggi dell'urto (7.20) Po~ch! il sistema! iperbolico la matrice di coefficienti lI.d!.
e
invertibile
e (14) puc essere risolto rispetto aIle derivate. Le quantita mente
e una
~i~ sono Ie componenti della velocita radiale. La ~ chiara-
funzione
omogen£~del
primo grade rispetto aIle
f• .
Pertanto
dal teorema di Eulero
oppure
E'interessante di notare che mentre la velocita normale ~ questo non
e vero,
e
continua (12),
in generale,per la velocita radiale [141. Derivando
(13) rispetto a 'f.tenendo u
o
costante si ha
';)~ t = ~04cr +V'r~~h
I ~i crJ = -lVflV c\ .")ioh e il secondo membro non
e,
di solito, nullo
Per esempio, in un fluido esis-
te un urto, cosl chiamato di contatto, che si propaga con la velocita carat-
186
teristica
= ~.n =-;ro .n • Invece e bene conosciuto radiale ~ e discontinua t ~:1 f, o.
continua s
urto, la veloci ta
che , per questo
8. Soluzione esplicita
e debole,
Abbiamo gia visto che, quando l'urto
e piu deboIe si o ancora dare una forma esplicita del saIto, non di u rna di u', grazie
10 sottospazio degli autovettori di
puo
il saIto di u appartiene al-
1.
alIa funzione generatrice
I)"'=~'(A I (
n
~ . Quando l'urto non
Derivando la (5.01) con s
-~I)?h
=~ ,
=0,
I
in virtu di (7.09). Sieeome ~ e nulla per l'urto nullo ne risulta
!11 I
,~ ,h)!f
(u
0
0
(8.01)
0
Pertanto. la sua derivata rispetto a
U
o
e anehe nulla e tenendo conto dalla
(5.03) viene
ehe inserita nella (5.08) fornisee
h' (A on
~ 0 I) = -
-
wV
~
h' = u ' - u'
o
0 0
(8.02)
Adesso introduciamo un vettore g(u,~) cos1 definito g(A n
-h) = -v~,
I
= 1,2, ... ,m (i)
(8.03)
La soluzione esiste proprio perche vale la condizione di eccezionalita (7 .11),
e
unica perche e ortogonale agli autovettori d
puo mettersi sotto la forma ..,
h'
=u
I
[151
• 11 saIto del campo princi~~
I
-t
~
(8.04)
1 (u , n) + w g(u , n). I
0
0
Nel caso lineare g sarebbe nullo. II secondo termine rappresenta dunque la parte non lineare dell'urto caratteristico. I due vettori dipendono sol tanto I
dello stato prima dell'urto e sono conosciuti. Resta da determinare w(u ,u ). Deriviamo (5.04) rispetto a u
o
I
..,
=
'i> I h'.h
187
cioe (8.05) purche (8.06) Per la derivata seconda I) "}
II'
11'
w
='a
w g
o'
h'
II" h
h
+~h'HI(U')~ I
I'
h'
v
+') h'H'~ h'
I
I'
e
(8 .07)
9. StabilitB dell'urto caratteristico Sia n,
1
= cost.,
(9.01)
una soluzione ovvia di (7 .13). Scegliamo come parametri u sono del tutto arbitrari ) delle funzioni lineari di
Se supponiamo che
tf.
I
(che, a priori,
Segue da (8.05, 07)
qua~he sia l'urto ,
(9.02) come 10 e quando l'urto e debole, allora
la derivata prima cresce e siccome e nulla quando l'urto e nUllo, e positiva. Ne risulta che w tende all'infinito con '( • Ma ai vede faci1mente che hvh ' ~ w
188
e dunque anche l'urto non
e
limitato app ena si sposta la superficie d'urto
rappresentata ad esempio da un'equazione del tipo (1)
e e -If = x i n Questo
e
- 1
~-0 t
'f~o.
il caso dell'urto di contatto gia citato. In un urto stabile la quan-
tita (8.06) cambia di segno [16] • Vedremo come si traduce questa condizione per Ie equazioni di Eulero.
10. Evoluzione del1'urto caratteristico di Eulero DaIle equazioni di Eulero scritte nella forma u
(vedi §3)
(10.1)
t
derivano Ie condizioni di Rankine-Hugoniot A'h' -~h n
= O.
(10.2)
Da cui segue, tenendo conto del 1egame (§ 2)
e di (8.04), h = H'h' -
wV')./).t
(10.3)
000
Definiamo gli autovettori 1'(A' -~H')= 0, I n
(A' -c\H')d' n I
0,
(:L0.4)
tale che,
l' I
= II
~
d' I
H'd' I
~II'
•
Allora da (3) viene semplicemente, 1 .h 10
I' =S = uI _II' u
g .h
= 0(0 w
mentre o
dove c(
o
=0(
u , 0
1) e
,
(10.5)
una funz ione conosciuta del campo prima dell' urto
189
( 10 . 6 )
La (8.05) diventa (10.7)
che si integra subito J
(1 -~ w)
2
o
=-
lui 2
~
0
+a(u,1:),lul o
="-'S(
II' !I'u u i.
La costante di integrazione si determina sapendo che l'urto e nullo I, que anche gli u vedi 8.04) quando w e nullo (5.05) e r17j J
(1 - .... w)
2
o
e(
= 1 -
0
lui 2 .
(e dun(10.8)
Vediamo da (5) che l-gh=l-o(W
o
0
L' instabilita corrisponde a a( " O. Invece se 0( '> 0, Iu I , w e di conseo 0 guenza l'urto e limitato; la quantita 1 - goh cambia allora di segno • Non
e difficile
di vedere che la velocita radiale
(A' n
H' )V'd
I'
I
Moltiplichiamo a sinistra per l' e a destra per I
vata del autovettore
Derivando (4)
= o.
- H'd' V'). -~V'H'd' I
e continua.
~id'I che rappresenta la deri-
rispetto a ~1
'V.~ ~id' = -~l'V'H'd·)id' I
I
I
I
(10.9)
Se adesso s1 moltip11ca a destra per di &ottiene
Der1vando rispetto a ~ e ricordando che H' =V'V'h' s1 trova che la quantita al secondo membra d1 (9)
e
nulla. A causa della condizione di ecceziona11ta
(7.11) r1sulta che
che equivale ancora a f} ~ilj, ["iu.-J I ~ = 0 e pertanto U T . = O.
190
Una conseguenza immediata e che, per quan to abbiamo detto alIa fine del §7, non esiste per il fluido un principio variaz ionale con una densita h (3.06) convessa . Dopo pass~gi lunghi assai [17) la legge di evoluzione (7.17) dell'urto si " esprlme ln un mo d0 mo It 0 semp I l' c e ; un s i sterna d'1 m(1) equazlonl d 1i f'f erenzla I'1 '
per quantita VI legate in modo non lineare agli u
I
0
0
• Quando ~ e negativo 0
l'urto puo effettivamente tendere all'infinito dovuto in particolare alIa formazione della caustica.
11. Limitatezza della velocita dell'urto
Per un urto normale, cioe non caratteristico, l'evoluzione non si fa lunge Ie cara ttertstiche. Da ciascun lato della superficie i campi u e u
o
soddis-
fane i1 sls tema a derivate parziale. Le due soluzioni si devono poi raccordare sulla superficie tramite Ie equazione di Rankine-Hugoniot. II saIto del campo dipende come abbiamo visto della velocita dell'urto che, a priori, se non si assumono Ie condizioni di Lax, puo anche andare all'infinito .Invece con una densita di energia convessa la velocita e limitata (18j Abbiamo i l seguente teorema (19:l (dove fu pubblicato per la prima volta ?). Sia f una applicazione continuamente differenziabile di un aperto convesso di R in R , Se la parte simmetrica della matrice jacobiana
N
cazione
N
e
e
definita, l'appli-
globalmente invertibile.
La matrice jacobiana di (4.02) rispetto aIle variabili u' e simmetrica A' - s H' • n
Basta allora che s ') sup
max ~(i)
u'6D'
i
inf u'fiD'
min
oppure s
<.
~(i)
i
(0' aperto c onves s o ) purche esiste una soluzione unica che ovviamente
e
191
u' = u· o
i. e. assenza d'urto. Di
con~nza
i valori estremi delle ve locita caratte-
ristiche costituiscono limiti per la velocita dell'urto in accordo con le condizioni di Lax. In particolare in una teoria relativistica tutte le ve l oc i ta caratteristiche sono minori,in val ore assolutto,della velocita della luce. Ne segue, a causa de lla convessita, anche la limitatezza della veloc ita dell 'urto
lsi' c.
Ringrazio la Dott.ssa Franca Franchi per gli appunti presi da lei durante 11 corso e che sono serviti come base per la stesura del testo.
192
Bibliografia
[11
K. O. FRIEDRICHS & P. D. LAX, Proc. Nat1. Acad. Sci. U. S. A., 68 (1971) 1686.
[21
G. BOILLAT, Comptes rendus, 278 A (1974) 909.
[3J
S. K. GODUNOV, SOy. Math. , 2 (1961) 947.
[4J
G. BOILLAT, Ann. Mat. pura ed app1., 111 (1976) 31.
[5}
10., Comptes rendus, 283 A (1976) 539.
[6]
A. JEFFREY, Corso C.I. M. E. , in questo libro .
[7]
P. O. LAX , Comm . Pure App1. Math., 10 (1957) 537.
[8J
G. BOILLAT, Comptes rendus, 283 A (1976) 409 .
[91
T. RUGGERI, Corso C. I. M. E. , inquesto 1ibro; T. RUGGERI & A. STRUMIA, Ann. Inst. H. Poincare, in corso di stampa.
(10)
O. FUSCO, Rend. Sem. Mat. di Modena, in corso di stampa.
[ l lJ
o.
ter HAAR & H. WERGELANO, Elements of Thermodynamics, Addison-Wesley
Publ. Co., Reading, Mass. , 1966. [12)
G. BOILLAT & T. RUGGERI, Acta Mach., 35 (1980) 271.
r13J
10., Boll. Un. Mat. Ital., 15 A (1978) 197.
[14J
G. BOILLAT, Comptes rendus, 274 A (1972) 1018.
1;5]
10., Comptes rendus, 280 A (1975) 1325.
I1~1
10., Ibid., 284 A (1977) 1481.
[171
10., J. Math. pures et appl., 56 (1977) 137 .
[18J
G.BOILLAT & T. RUGGERI, Comptes rendus, 289 A (1979) 257.
[l ~J
M. BERGER & M. BERGER. P~rspectives in nonlinearity, W.A. Benjamin, Inc . New York (1968), pag.137.
CENTRO INTERNAZlONALE MATEMATlCO ESTlVO
(C.l.M.E.)
SULLA TEORIA DELL'OTTlCA NON-LlNEARE
DARIO GRAFFl
SUlla teoria dell'ottica non-lineare Dario GraEfi U~versitA
1.
di Bologaa
Come i! aot o, un campo elettromagnetico in ua dominic ep del-
--- --
10 spazio i! rappresentato da cinque vettori (che chiameremo vet-
---
tori elettromagnetici) ........ E, H, -.. D, B, J detti, rispettivamente,campo elettrico, campo magnetico, vettore spostamento, vettore induzione, dens itA di corrente. Questi vettori sono, in generale, Punzioni del punto)(e P
e del tempo t quindi, a rigore, si do-
vrebbe scrivere, in luogo di
E: E(X, t)
altri vettori .elettromagnetici.
Per~,
e analogamente per gli per semplicitA, useremo
quest'ultima scrittura solo nei casi in cui i! necessaria per Come
e
nato, i vettori elettromagnetici sono legati Era lora
dalle equazioni di Maxwell: , (1.2) che esprimono leggi £isiche £ondamentali e
perci~
sono val ide
sempre e ovunque. Ma Ie (1.1) e (1.2), anche corredate da opportune condizioni inizia1i e alIa £rontiera, non sono su££icenti per determinare un campo e1ettromagnetico e perci6 biso-
19 8
gna aggiungere opportune equazioni costitutive che nell'elettromagnetismo ordinario (supponendo, come £aremo sempre i. segIli to, esterne a 5) le sorgenti del campo e1ettrOlllagnetico e,t=»e.r' oW , isotropo i1 mezzo in "P ) sono: (1.3)
--t
-.
D::. eo E.
, (1.5)
-J.: rE
dove E. '.Y" y" sono rispettivamente la costante die1ettrica, la permeabilitA magnetica, la conduttivitA del mezzo nel punto
X in cui
si considerano i vettori che compaiono rispet-
tivamente in (1.3),(1.4),(1.5). Be poi i1 mezzo 1e E ,
r-
Poiche Eo ,
e
anisotropo
' f vanno sostituite con tensori -d opp d , V ~ dipendono solo da1 mezzo e non da1 campo e-
lettromagnetico, le (1.1),(1.2),(1.3),(1.4) e (1.5) costituiscono un sistema 1iaeare, percio 1'elettromagnetismo
ordina~•.
rio si puo chiamare anche e1ettromagnetismo 1ineare. Per6 in quei die1ettrici (ai quali ci ri£eriremo sempre in segIlito) dove si manifestano i £enomeni de11'ottica nOft lineareJ mentre restano valide 1e (1.4) e (1.5), la (1.3) va sostituita con una relazione non lineare ira DeE che scriveremo: (1.3') sic r.he i mezzi in cui vale (1.3') si possono chiamare dielettrici non 1ineari. In questa 1ezione cerchero di .stabilire alcune proprietA
de1~e
onde elettromagnetiche che si propa-
"',,"Oot'"
gano nel dielettricorcosl da interpretare qualche £enomeno de11'ottica non lineare.
2.
Ri£eriamo i punti dello spazio a un sistema di coordinate
cartesiane ortogona1i (O,x,y,z) e supponiarno i1 dominio coincidente con una lamina di spessore s riempita da un
~
199
dielettrico non 1ineare omogeneo. Porremo l'origine 0 e l'asse z del sistema di assi in modo che 1e facce della lamina abbiano equazione z=O, z=s. A11'esterno della lamina supporremo i1 vueto che, da1 punto di vista elettromagnetico, si puo identificare con l' aria. Indicheremo con £. 1a costante dielettrica del vuoto, mentre ammetteremo la r
che compare nella (1.4) ugua-
le a quella del vuoto (ipotesi non restrittiva dal punta di vi5ta fisico) cioe
ammetteremo~ identica
i. tutto 10 spazio.
Nel semispazio z < 0 sia posta una sorgente che generi un'onda elettromagnetica piana con direzione di propagazione parallela all'asse z. Supponiamo la lamina tagliata e disposta in modo che il campo elettromagnetico dipenda solo da z e t; anzi, con un'opportuna disposizione degli assi x e y si possa scrivere, per ogni punta delle spazio, :
-
-
1noltre supporremo D para11elo ad E cioe;avremo: (2.3)
Al10ra la (1.3') diventa (sottintendendo Ie variabi1i z e t) · l'equazione sca1are: (2.4)
e la
£unzi~ne
D{E) verra supposta di classe
c~
in qualunque
intervallo limitato del1'asse reale. Le equazioni di Maxwell nella lamina si riducono a:
2 00
L~
(2.5) e (2.6) valgono anche all'esterno della lamina
si ponga
r
Ammetteremo inoltre, con£orme
ci~ D
~~
.0, .5., in luogo di
Punzione crescente di
purch~
•
l'esperienz~:
E
e D{O).O.
Stabiliamo ora alcune condizioni sui pian! che limitano la lamina,piu precisamente sui piani z=+O, z=s-O ; si
~
scritto +0 e
5-0 per identificare 1e facce dei piani rivolte verso l'inter-
nO della lamina
o/pi~
brevemente,facce interne.
Ora, nel semispazio z <
° si
avranno due onde, una che diremo
diretta, emessa dalla sorgente e che si propaga nel verso positivo de11'asse z, l'a1tra ri£lessa da11a lamina e che si propaga nel verso negativo del1'asse z. Detti E~(z,t), H~{z,t)
E~{z,t),
HdJz,t),
rispettivamente il campo dell'onda diretta e
i1 campo de11'onda rifles sa, si ha:
) Ora, co:ne
e
noto, su un piano che separa due mezzi diversi so-
no continue le
co~ponenti
tangenziali a1 piano del campo e1et-
tromagnetico (ovviamente z=-O. z=s+O sono 1e Eacce della lamina rivolte verso l'esterno
( ~' .:» (2.9)
0
facce esterne). Si ha cosi:
£.ol(- 0, t) + E"" (- 0, t;) : E
(+ 0, t:: ) H
Ora, per note proprietA delle onde e1ettromagnetiche piane si ha: (2.10) H.,l(-O'~)"'~ E~(-Or~) ,./
Sostituendo (2.10) e (2.11) in (2.9) e sommando con (2.8) moltiplicata per ~
r
si eliminano EIt- e HI(. • A11ora, riservando
201
il simbolo E(z,t), H(z,t) al campo entro la lamina ed ometteado, per semplicita di scrittura e perehe ora non vi e luogo ad equivoc0, i segni + e - davanti allo 0, si ha:
Nel semispazio z » s si ha solo un' onda che diremo trasmessa e che si propaga nel verso positive dell'asse z (non si avere riflessioni perche per z campi indicheremo con
~s
POSSOftO
il mezzo e omogeneo) i cui
E~(z,t), H~(z,t).
Per la continuita del-
le componenti del campo elettromagnetico sul piano z=s (ora si possono evitare i simbe1i +0 e -0) si ha:
e poiche "~vale
J~ ~t si ha subito:
Le (2,12), (2.14) in cui
E~(O,t)
si suppone assegnato, costi-
tuiscono condizioni a11a frontiera per (2.§) e (2.6). Ad esse si possono eventualmente associare opportune condizioni inizia1i, sicche il campo entro 1a lamina resta determinato. Le (2;12) e (2.14) si devono al Frof.Cesari (1) [2] [3] [4] il qua1e ha dimostrato import anti teoremi di esistenza, di unicita, di dipendenza continua dai dati per Ie soluzioni delle equazioni sia noto
(2.~)
e (2.6) corredate da (2.13) e (2.14), qualora
Eo~(O,t)
per ogni t (positivo
0
negativo). Nel caso,
importantissimo per 1e app1icazioni, in cui E~(O,t) co rispetto al tempo e con periodo T,
~nche
e
periodi-
i1 campo entro la
lamina risulta periodico con 10 stesso periodo. Noto il campo entro la lamina, mediante (2.8), (2.9) e (2.13)
e
colare il campo riflesso e trasmesso da11a lamina.
facile cal-
202
I teoremi di Cesari sono stati dimostrati per valori della spessore s della lamina non troppo elevati. Torner6 in seguito su questi risultati, per ora noter6 che il Prof. Bassanini
[5]
ha dimostrato che i valori di s per cui sana validi i teoremi ora citati risultaao superiori allo spessore delle lamine usate in pr-at Lca ,
3.
Passiamo ora a ricereare una soluzione di notevole interes-
se di (2.5) e (2.6) supponendo (come £aremo sempre in seguito)
r
aO.
A questa scopo poniamo,
ricord~do
(2.4'),
(3.1) ~
Nel caso lineare (si ricordi (1.3»
=c
(costante die-
lettrica) ed esiste una soluzione delle (2.5) e (2.6) per cui il campo elettrico ha l'espressione: (3.2)
dove G(u)
E(z,t)
e
= G(u)
una funzione di classe C4 della u per u variabile
in qualunque intervallo limitato dell'asse reale; G(u) se u=t vale il campo elettrico suI piano z=O e all'istante t, sicche Ie proprietA della fupzione di t,E(O,t), sono Ie stesse di G(t) o G(u).
Ora, nel caso line are peE): Je~'
;
viene pereio naturale con-
getturare valide Ie (3.2) anche nel caso generale sostituendo pera nell'espressione di u a V~
,
peE) come de£inita da (3.1)
e con segno positivo. 5i ha cosi: (3.3)
E
= G(u)
u = t - p(E)z.
203
Ora. portando G(u) al
~rimo
membro di (3.3) si ha l'equazione
che definisce implicitamente E in Punzione di t e z :
(3.4)
E - G( t - p(E)z )
Questa equazione
e
risolubile per ZFO
=0
ovviamente risolubile per z=O. Affinche sia
e
suf£icente, per il teorema delle Punzioni
implicite. che sia : )
condizione certamente soddisfatta per z=O. Ora. riservandoci Cal n.5) di discutere meglio la (3.5). ammettiamo, come del resto e intuitive, che esista un numero h> 0 tale che per ogni Z
e (O.h). t
€
(-'Ji,T) (T, e T positivi e del resto arbitrari)
sia valida (3.5). Fino ad avvertenza in contrario, ammetteremo z e t nei limiti ora indicati. Cia
pr~messo,
vediamo di determinare il valore del campo magne-
tico H che, associato al campo elettrico espresso da (3.3), sia tale da soddisfare le equazioni di Maxwell (2.5) e (2.6). A questo scopo ricordiamo che il Prof. Jeffrey ha dimostrato. nella prima delle Sue lezioni. che E, come espresso da (3.3). soddisfa all'equazione a derivate parziali: (3.6) che ora verificheremo direttamente. A questa scopo osserviamo che, derivando (3.3) prima rispetto a z e poi rispetto a t, si ha: (3.7)
';) E
~l
(-1..+ <;.'(l«.) '!J~i~.11) BE
+
G.'(~) r (E) ::.
0
2 04
Sommando (3.7) con (3.8) mo1t iplicata per peE), tenendo presente (3.5), segue subito (3.6). In base aIle (3.1), (3.6) Ie (2.5) e (2.6) si possono scrivere:
Poniamo ora:
f
fi
(3.10)
B. == i.
7
In base alIa (3.10) H
pCB) dE
0
e
una funzione di E e tramite E di t e z.
Dalla (3.10), derivando rispetto a z e a t si ha:
0 1-4
(3.11)
d Eo "
~)~E D!-
..r
9H .. .J..h(e)f,>£ ~~
./1.1-
r
~ t:
che coincidono con Ie (3.9), quindi (3.3) e (3.10) rappresentano una coppia di va10ri di E e H che soddisfano aIle equazioni di Maxvell.
4.
Passiamo ora allo studio delle soluzioni delle equazioni
di Maxve11 trovate nel numero precedente. Supponiarno anzitutto la lamina di spessore infinito, ossia s=
"0
,
sd.cche 1a lwdna occup a i1 semispazio z .s14
~o
•
Supponi arno che per t=t~, z=O'fEocG(to) e proponiamoci di determinar e i v alori di z e t per cui E rimane uguale a Eo, valori che rappresenteranno una caratteristica dell'equazione (3.6), come ha osservato i1 Prof. Jeffrey. Si ha percio l'equazione:
certamente soddis f atta se:
205
Di££ereDZiando s1 ha subitOI (4.3)
Ora dz
~
10 spostamento del campo elettrico di valore
tempo dt. quiJldi
Re-.,)
E~
nel
~ la veloci ta con cui si propaga il
campo elettrico che all'istante to aveva il valore
E••
Notiamo che la (4.2) si puo ricavare in altro modo d1 va11dita
pi~
amp1a. Si osservi in£atti che se G(t - p(E.,)z)
e
costan-
te e uguale a Eo. per la (3.6) s1 hal
da cui integrando e teJlendo conto che per t=t o
,
z.o , s1 r1-
trova (4.2). E' bene nctare che, essendo ~1/~r pos1tiva, 11 campo s1 propaga Bel verso positivo dell'asse z, qu1ndi la soluz1one del numero precedente rappresenta un'onda che s1 propaga nel verso positivo dell'asse z. In particolare, se il campo
~
nullo per z=o
in un certo istan-
te to , esso s1 propaga con veloc1ta l/p(O). Hel caso per noi
pi~
interessante in cui G(t)=O per t ... 0, si
c ompz'ende che per t> 0 si avra un £ronte d' onda, cioe ua piano/ che si sposta col Z
temp~ di
ascissa z 0
..
Zo
(t ) tale che per
:;>zo, E(z,t)::o, per z
k ;>0 e del resto qualsiasi. Poiche E
~
COD.
uguale a zero per ogni
t suI £ronte d'onda, la sua velocita sara la velocitA del campo nullo,
ci~
il £ronte d'onda si sposta con velocitA l/p(O).
5. Passiamo ora a discutere la (3.5). Anzitutto se Gt (u) e !dppr; hanna (se diversi da zero) per ogai u e per ogn1 E 10 stesso segno (per esempio G(u) e peE) sono
206
funzioni crescenti, la prima rispetto a u, l'altra rispetto a E), la (3.5)
e
sempre soddisfatta e h =00
per ogni t.
In questa caso, se Ie condiziOfti iniziali sono nulle per z?O e suI piano z=O
e
assegnatc per ogni t positive il campo elet-
trico, per un teorema di unicitA del campo elettromagnetico, (3.3) e (3.10) (pureM si assuaa G(u)=o per U -< 0) rappresentaBO
i l campo elettromagnetico cOlllpatibile con Ie condizioni ini-
ziali e alIa frontiera e che si propaga nel verso positive dell'asse z. Tornando al case generale, cerchiamo di dimostrare l'esisten... za del numero h> 0 di cui si
e
accennato al n.3.
A questo sc opo, aggiungeremo un'ipotesi
pi~
che plausibile dal
punta di vista fisico. Cioe la funzione G(t) (0 che so G(u»
e
10 stes-
che rappresenta il campo E(O,t) sia limitata assieme
alla sua derivata G' (u) per t
E (--
,T); i. altre parole esi-
stano due numeri positivi M e M' tali ehe per ogni
U 4:- ( -
00
,T)
sia:
1G.1(1.L)\~M'. Inoltre per Ie nostre ipotesi
I g ~(E~E \
Ci~
limitata da
UJl.
.fi"D
.9
e-h,e..
\ E
\-s: M
sarA
numero N.
premesso, fissato un istante t, esisterA un numero positi-
vo h(t) tale ehe per z € [O,h(~)') , (3.5)
e
verificata e quiJl-
di (3.3) risolubile. Allora per questi valori di z, t, eM
l ~ 1< N,
inoltre
(E(z,t)I::IG(u)\~M, sic-
IG' (u) l~ MI.
Dimostriamo ora che esiste un nwnero h o tale ehe h(t») h o ' t E. (- coo ,T). Infatti sostituendo h(t) in (3.5) e- tenendo conto ehe
e
soddisfatta se
Gt(u)?JI~)
si ha:
207
Ouiadi:
(5.3)
h(t) >
~M'
=
~ ..
come si era arPermato. Assumeremo h T
pu~
Per~
~
he> l'estremo inPeriore delJli h(t) per t
essere aache inPinito nel caso G(t)=O per
soddisPatte) e t non
purch~
.T).
sia soddis£atta (5.1).
t~O (sicch~
e molto
t (- 0 0
(5.1) sono certamente
elevato, segue h-
00.
InPatti
sia p~> 0 il minimo valore di peE) per IE\~ M, allora se valgono le relazioai:
(5.4)
t
~
T
t
il valore di u che
(5.5)
u
=t
< P.
Iho..
compar~
- p(E)h(t)
E;
h
0
= r......, NM'
nella (5.2)
~:
t - 'D ...... h < P.-.. h - P-. h
Ma allora il G'(u) della (5.2)
~
= O.
nullo e questa equaziORe nOB
puo essere soddisfatta per h(t) finito. Deve essere h= 00 10. che
~
10 stesso, la soluzione (3.3)
e
valida,per valori di t
soddisfacenti (5.4), per ogni z, ed essa rappresenta il campo elettromagnetico in tutto il semispazio. Si noti chef come vedr~~o ~el
numero seguente, N ~ molto piccOlo; l'intervallo di
tempo in cui la (3.3)
e
valida PUQ essere sufficentemente gran-
de per Ie applicazioni pratiche. Hel caso in cui non siano soddisfatte Ie ipotesi ora esposte. fissato t PUQ esistere un valore:z di z per cui la (3.5)
e
nul-
la, e se G(t - p(E)z) risulta diverso da zero, da (3.6) e (3.7)
I
fl. ' 9E . ". ' l;lE segue che \ ~ ~J: =-t- " 0 , I i-:i I.)f; I = ... c>o , C10e Ie derivate di E per z _z tendono a diventare infinite. Si ha cioe,
conforme a una locuzione del Prof.JefPrey, una catastroPe. Si PUQ cosi interpret are l'accennato risultato di Cesari per cui i Suoi teoremi sono validi solo mina
e
suPPicentemente piccOlo.
~e
10 spessore della la-
208
In seguito cornunque ammetteremo che (3.3) e (3.10) rappresentino il campo elettromagnetico, almeno per valori di t e z sU££icentemente grandi per Ie questioni pratiche.
6.
=
Hel case s
notiamo che, mentre (2.12) rimane valida,
00
(2.14) non ha pi~ signiPicato e si pu~ sostituirla can la con-
dizione che il campo sia nullo all'infinito,
0
meglio che il
campo rappresenti un'onda che si propaga nel verso positivo dell'asse z, condizione questa, come si
~
osservato al •• 4, sod-
dis£atta dalle (3.3) e (3.10). Supponiamo ora l ' onda E
t:L"
senl.Dt
(A 0 e
w costanti)
•
Converra introdurre la Punzione di Heaviside let), (l(t)=l per tpO, l(t) =0 per t<:O) vita di scrittura,
sicch~
E~(t)
a
lo:i-l".IL)
si avr! (sostituendo.' per bre-
E~(O,t),
e analoga semplilicazione-
.faremo in seguito per i termini che compaiono in questa equazione) : (6.1) Ci~
premesso, per ottenere formule semplici, supporremo, come
avviene spesso in pratica, debole la non linearit! cioe che sia lecito scrivere: (6.2)
D(E)=
dove F(E) guit~,
Eo E +
1.. e.
peE)
una £Unzione di E che specificheremo meglio in se-
~
~ UK
parametro adimensionale molto piccOlo in modo da
poter traseurare in seguito termini in il cbe non (3.1) :
~
,1. . SUpporremo inoltre,
af.fatto res trittivo, F(O)=O. Quindi si avra, da
209
e da (3.10) : (6.4)
Allora sostituendo in (2.12) si ha l'equazione per E(t). campo elettrico sul piano Z=O e sUlla faccia rivolta verso l'interno della lamina:
Non sarebbe difficile dimostrare che la soluzione E(t) della (6.5)
~ limitata~
ometteremo per brevita questa dimostrazione o
Per risolvere esplicitamente (6.5) useremo un ben nota procedimento di approssimazione ponendo: (6.6)
E ( t ) = Eo ( t )
+
1. E.f (t) •
Ora. come e noto. si puo scrivere. applicando poi il teorema del valor medio e indicando con e ,1 :
l
(6 • 7H f (
~ (t) + '1. fAit))" '1 F(~(~))40
9-
nn numero compreso fra 0
,.t F(~Lt)
=1 F(Eo(t-)) "Hl'l. F'( E,,{t-h~,E.(t-))E.clt) tenendo conto che l'ultimo termine •
•
~
+
1.EAt-V - F
;&
(E"ol~»)
=
"'l. F(toU"))
trascurabile perche del-
'1.
I 'ordJ.ne dJ. "l • Con questo ragionamento si giunge alIa formula piu generale (A(t) e B(t) due funzioni limitate di t ):
Si ha cosl. sostituendo (6.6) in (6.5)
210
Quindi: ( 6 . 9 ) Eo/l;;):: 1~ a MM~t --let): _~_ l ~+J£ 0 ..- 4+_
dove a=
V:
o
aoMMwt- -itt)
e 1'indice di ri£razione del dielettrico in assen-
za di non linearita. Si ha poi:
Specializziamo ora la F(E) considerando la relazione non --lineare pill semplice ira DeE cioe (c{ costante) :
(6.12)
'L
= l(t) E~(t). _ ~ c( _u_
Quindi essendo (l(t»
t(4+t\-)
[..(ot '")L
a: ~wt
Posta: (6.13)
si ha quindi:
e il valore di
E~(t),
cioe
de~
campo elettrico dell'onda ri-
flessa suI piano z=O, ricordando (2.8) e (6.12) :
Per avere il campo ri£lesso nel punto di coordinate z, basta porre nella (6.15) al posto di t, t + ~ z, perche ora l'on./
da si propaga nel verso ~egativo dell'asse z, con velocita ~ • JE;t. ./
211
Nell'onda riPlessa vi sono tre termini, uno di Prequenza W che si ha nel caso lineare, mentre la non linearita porta per t
- ~~
z> 0 a un termine costante (rispetto al tempo) di scar-
so interesse pratico e a un termine sinusoidale di Prequenza
20 , cioe nell'onda riPlessa si ha un'onda di Prequenza doppia dell'onda incidente. In altre parole, dalla riPlessione su un dielettrico non lineare si PUQ avere il cosiddetto Penomeno della
duplicazione di Prequenza. Per esempio, inviando la lu-
ce emessa da un laser, di lunghezza d'onda
6940 AO (cioe luce
rossa), si PUQ avere, hella luce ri£lessa, anche luce di lunghezza d'onda 3470 AO, cioe luce violetta.
7. Passiamo ora allo studio della propagazione dell'onda trasmessa dal dielettrico. Per la (3.3), poiche G(u) e espressa da (6.14), si ha: (7.1)
E (t,1) ~ ll)~A.t.o (t -t'LEl~, t))~) .{ (t-- to (c(~/t:l) x.)
-
- 't [ b- ~ Co,2.w (b- r(et!',t))!)1 {[ ~ - r( E(l!-, ~))'l- ) Ora, per (6.3),(6.11), trascurando sempre i termini in ha:
(7.2)
rlEl
l , \;\) ,.
~ (-!.i"
"t c(
nendo presente anche (6.7'), si ha:
rCE (~,! J)"~ [i .. ('1.
0( Cl
si
€(tl~))
Sostituendo (7.1) in (7.2) e trascurando i termini in
(7.3)
\2 ,
'l.
t '
te-
~ IV l~- r(f(~,Hh) /{ U·-r(E(J/t-lj~)J
Ora, ponendo in (7.3) (7.2) e sempre trascurando i termini in 1-
'\ si ha: (7.4)
)
r(6(t,'f))"~ (i+ 1d-a.-~~~t-J!j~) -i(t--~ l);
2 12
Ora poniamo per brevita:
~
(7.5)
..
k-~E)
50stituendo (7.4) e (7.5) in (7.1)
si trova, con Ie semp1ifi-
cazioni suggerite dq (6.7') (7 .6)
€(~,l-). a.Jf- (~_, ~ O'tC><'a.,~&' ,( (t- ~.)) -l (t- ~ l) - "L (b - 6Co,:>l~) -{ (~- vy.. l') .
Ora, con uno sVl1uPPo di Maclauri"relativo alla variabile accorciato fino al termine in
t'2..,
't
e
si ha, per i1 primo termine
a secondo membro di (7.6) (a meno s'intende del termine int'L):
Q.~(+-"t'J~U>'E-~a.HM"8'-{(I--~1))i (~-~ 'I) ~ "" (a.r \9"_ a. ~ Q- ~ ~ W"" ol a ~ e- ) -l (~- rY" l) • e,
(ar e- -
to( a~~
t.O"t- MNt. 'l
ty) -{ (l-- ~ ,.)
In conclusione si ha:
Quindi in un punto zein tutti gli istanti t > J5-.. ~ si ha che il campo dell'onda trasmessa
e
la somma di tre campi, uno
di frequenza eo , cioe il campo qualora si trascura la non linearit3., uno costante e infine uno di frequenza 2 w
,cioe, co-
me nell'onda rifles sa, si ha una duplicazione di frequenza. 5i noti che per z
_00, i1 campo di frequenza doppia tendereb-
be all'infinito, pero per z grande non valgono Ie precedenti approssimazioni, anzi non sarebbe neppure valida 'la (3.5) e quindi la soluzione (3.3) e (3.10) delle equazioni di Maxwell. S' bene anzi not are
~he
nelle ricercpe sperimentali dell'otti-
ca non lineare si considerano ovviamente lamine di spessore finito s. 5e s
e
inferiore al valore di h calcolato al n.5, i1
213
che avviene in pratica, Ie (7.3) e (3.10) sana soluzioni (senza catastroEi) delle equazioni di Maxwell. Esse pero rappresentano l' onda entro la lamina e quella che ne esce attraversata solo se si trascura :l'inEluenza
do~averla
>r.\ • dell~facce
~!e....
in-
terne della lamina stessa. Comunque non mi sembra inutile la seguente osservazione. Eseguiamo i l rapporto r Pr-a i due termini di Erequenza 2"" , dhe compaiono in (7.7). Ricordando il valore (6.12') di b e che
w~. lUJh,.~ •
J,lrhV /
~
dove'). e la lunghezza d'onda,
nel vuoto, corrispondente alla Erequenza
~
, si ha:
Ora s vale 10·1 mm . , poiche n+l/2 e dell'ordine di unitA, r all'uscita dalla lamina (cioe per z=s)
e
io. Pereio nella (7.7), nei termini in
dell'ordine del miglia 2~
, prevale il primo,
e
l'ampiezza dell'on-
almeno all'uscita dalla lamina. Poiche b da riElessa, si puo aEEermare che
e
molta piu facile osserva-
re la duplicazione di Erequenza nell'onda trasmessa che nella riElessa. Notiamo che la velocita di propagazione delle onde che compaiono in (7.7) vale l/{fj-
, cioe
tale velocitA non e altera-
ta dalla non linearitA. II risultato non esempio se F(E)
e
e
pero generale, ad
proporzionale al cubo di E si ha alterazio-
ne di velocitA, ma su cio non insisto.
214
Bibliogra£ia 1
L.Cesari-Rend.Sem.Mat.Fis.Univ.Milano 45, 139 (1974) •
2
L.Cesari-Ann.Scuola Normale Sup.Pisa 4, (311) (1974).
3
L.Cesari-Riv.Mat.Univ.Parma 13, 107 (1974).
4
L.Cesari-Rend.Accad.Naz.Lincei 56-1 (1974); 57,303 (1974).
5
P.Bassanini-ZAMP 27, 409 (1976).
6
D.Graffi-nNon linear partial differential equations in physical prob1ems M Pitman London (1980).
CENTRO INTERNAZIONALE MATEMATICO ESTlVO (C.I.M.E.)
SULLA PROPAGAZIONE DEL CALORE NEI MEZZI CONTINUI
GIUSEPPE GRIOLI
SULLA PROPAGAZIONE DEL CALORE NEI l-1EZZI CONTINUI
Giuseppe Griol1 Universi~a d! Pad ova Introduzi one II problema della propagazione del calore nei mezz! con~1nui ha richiamato in questi ultimi tempi l'attenzione di molti studiosi i quali hanno prospettato teorie di vario tipo con 10 scopo di arrece~e un contributo alIa formulazione delle equazioni costitu~ive dei continui e nel contempo superare il cosidetto paradosso della propagazione del calore con velocita infinita di cui e affetta la teoria classica. Le varie teorie proposte poggiano 0 su generalizzazioni di talune funzioni termodinamiche di stato (a parer mio, discutibili dal punta di vista !isico matematico) 0 au una qualche modifiea de lla classica legge di Fourier che lega il vettore flus so termico al gradiente della temperatura 0 su ambedue Ie cose. Ben nota e vastissima e la tett~ratura relativa alla termodinamica dei continui e alle equazioni costitutive.Mi limitero a citare taluni lavori attinenti aIle questioni considerate. Una soddisfacente formulazione della teoria della propagazione del calore per conduzione non puc prescindere dall'influenza dei fenomeni meccanici concomitanti ad eccezione,se mai, dei casi ideali di continui a temeratura ignorabile e de! corpi rigidie Gie il tener conto della completa interazione tra fenomen! termici e fenomeni ceccanici da luogo a equazioni non pin affette dall'accennato paradosso ma rende la velocita di propagazione del calore uguale a quella del suono. Tale risultato puc essere accettabile ma non e certo un privilegio della teoria classica il fatto ehe la velocita di propagazione delle onde meecaniche non sia sostanzialmente in!luenzata dalla propagazione
218
termica anche in casi non isotermi ne adiabatici. Oio e dovum al iatto che nella trattazione abituale le derivate prime della temperatura sono ritenute continue anche attraverso il ironte d'onda. Recentemente e stata proposta una teoria in cui s1 continua ad ammettere la validita della legge di Fourier ma si abbandona l'ipotesi di der~ate prime della temperatura continue attraverso il ironte d'onda.Ne nasce un problema analitico del tutto nuovo il cui studio righiede l'uso delle discontinuita iterate secondo Thomas[12l. Un'ipotesi costitutiva ben diversa dalla legge di Fourier e stata gia. considerata da Haxwell(l]. In essa si ammette un legame lineare tra vettore ilusso di calore, la sua derivata temporale e il gradiente della temperatura.L'ipotesi,abbandonata dallo stes s o riaX\·rell che ha ritenuto di dovere sopprimere il termine con la derivata prima temporale ricadendo nella legge di Fouri er , e s tata ripresa da vari Autori «(2], ••• 6 J ) e anche generalizzata(lO]. La relazione,associata alIa nota uguaglianza dell'entropia (che in generale e in realta una disuguaglianza) ds. luogo a un problema non pin parabolico e implica velocita di propagazione finita. Trr~tavia, il necessario procedimento di eli minaz~pne, ove si tenga conto, com'e desiderabile~dell'inte r azione Qecc ani c a , porta la presenza di derivate terze delle componenti di s rostamento che insieme alle seconde sono discontinue attraverso il ironte d'otda, modificando totalmente il classico problema delle onde di discontinuita.L'inconveniente non si presenta soltanto nel caso dei corpi rigidi. Una teoria della propagazione termica fondata su un'ipotesi costitutiva anaLoga a que.l.la pr-opos t a da ;':axwell che sembra molto interessante rna che tenga conto in modo completo dell'interaziozi one tra renomeni termici e renomeni meccanici si ottiene dando alla relazione di l·jaxwell un significato un po pin generale, adatta aIle esigenze della meccanica dei continui, come mi propongo di mostrare. Le ipotesi amcease aono le seguenti: continuita attraverso il rronte d'onda della temperatura, del vettore flusso di calore, dello s posta~ento e delle sue derivate prime; possibilita di discontinuit a di prima specie per le derivate prime della temperatura,del vettore flus so di calore e delle derivate
.
,e
219
seconde delle spostamento.A titolo applicativo e di indagine ho considerato il caso dei tluidi non viscosi comprimibili e incomprimibili e quello dei corpi elastici isotropi poco deformabili. In ognuno dei casi considerati, si giunge, per la determinazione delle possibili velocita di propagazione, a un'equazione risolvente di quarto grado. In as senza di vincoli interni, tale equazione ammette in generale due radici reali positive, una maGGiore e una minore di quella ben nota che de la velocita di propagazione delle onde acustiche longitudinali. Mentre nel caso elastico Ie onde possibili sono ne trasversali ne longitudinali, nel caso di un fluido non viscoso comprimibile esse sono,invece, solo longitudinali ma, come nel caso elastico, sono possibili due distinte velocita di propagazione, una maggiore e una minore di quella solitamente ammessa alla quale esse si riducono se si fa tendere a zero il eoefficiente di rilassamento. Si presenta,cioe, un secondo suono la cmi possibilita e gia stata segnalata (vedi, ad es.,[n1). Differentemente vanno Ie cose nel caso dei fluidi non viscosi incomprimibili in quanto, pur dipendendo ancora il problema da un'equazione di quarto grado" puo accadere che sia possibile una sola velocita di propagazione, come capita certamente nel caso che la perturbazione si propaghi in un mezzo in quiete a tem~eratura uniforme. E' interessante notare che ove l'incomprimibilita sia totale (ci06, la densita non 4ipende neppure dalla temperatura) il comportamento del fluido per quanto concerne la propagazione di onde termomeccaniche e analogo a quello dei corpi rigidi. Nel seguito riteriro in modo esplicito solo suI caso dei fluidi non viscosi incomprimibili, rinviando per altri casi a una nota lincea in corso di stampa[141. 1.- Quel che osservazione sull'eguazione di Fourier. Siano CeO' due configurazioni del continuo in evoluzione teroomeccanica, delle quali la prima,fissa, e la configurazione di riferimento, l'altra quella attuale (all'istante t). Denotero con x r' Yr Ie coordinate rispetto a una medesima tema di riferimento trirettangola levogira di punti corrispondenti P, P' di
220
oe
OI.La corrispondenza tra C e CI g1ianze
e
caratteriz zata dalle ugua-
continue,invertibili e a jacobiano,D, positivo. Denotero,inoltre, con T e S la temperatura assoluta e il vettore che caratterizza i1 flusso di calore in pt. Denotando con E llentropia per unita di massa, dalla disuguaglianza di C1ausius-Duhem,in assenza di un'eventuale sorgente di calore,inesr,enziale per quanto seguira, segue (1. 2)
•
E +
1
T
divplS
~ 0,
ove i1 punt o denota derivazione materiale rispetto al tempo. Tradizionalmente, alIa (1.2) si associa la re1azione cositutiva
ove L rappresenta un operatore lazione
~atriciale
soddisfacente alIa re-
(1.4)
Suppor r o che llentropia, ~ lienercia interna, il cui val ore per unita di massa indichero con J, dipendano dalla te~peratura e dal complesso di altre variabili al,a~, ••• ,an caratterizzanti c: la defor~azione e il ~oto del continuo, mentre eventuali vincoIi interni siano esprimibi1i nella forma (1.5)
r
z
1,2, •••
Sussiste di conse3uenza la nota relazione dell i e n t r o p i a
221
(1.6) ove i coefficienti Pi rappresentano delle incognite reazioni vincolari da considerare tutti nulli in assenza di vincoli interni. Da (1.2),(1.6) segue (1.7)
•
cT - T
)2 J 7.J:?a s
B:s- ~ (P1ti+ Piti) + divpdl
.
~ 0,
ove (1.8)
e
-T
)2 J
rw-
denota 11 calore specifico sotto configurazione costante. E' ben noto che nel caso di sistemi a trasformazioni reversibi11, come accade,ad es., nel caso di corpi iperelastici e di fluidi non viscosi, nella (1.7J vale il segno di uguaglianza, rna anche in caso di irreversibilita si suole ritenere che la (1.7~ considerata come uguaglianza e associata alIa (1.3),rappresenti l'equazione che regola la pr opag a zi one del calore. Spesso si ritiene trascurabile 1'interazione tra fenomeno termico e fenomeno meccanicG,sopprimendo nella (1.7) il secondo termine e provocando cosi 1'insorgere del paradosso della velocita infihita di propagazione del calore. Si supponga, per semplicita, che non vi siano vincoli interni e si interpreti l'uguag1ianza associata alIa (1.7), associata alla (1.3), come equazione del calore. Si denoti con la sbarretta la derivazione rispetto a.lLe xr e con Cw] la discontinuit a (di prima specie) attraverso il fronte d'onda di una qualunque funzione w. Supponendo che L co me J possa dipendere dalle as e da T e, c.om'e abitua1e,continua la te l1lperatura attraverso 11 fronte d'onda, da (1.3),(1.7) segue
= o.
222
L'ipotesi di continuita delle derivate price della temperatura fa si cbe i1 pr obl ema della determinazione della ve10cita delle onde acusticbe non risentevdella propa6azione termica.La (1.9) serve successivamente,per la determinazione delle discontinuita delle derivate seconde di T le quali si propagano, pertanto, con 1a medesima velocita delle onde puramente acustiche. E' pos5ibile riconoscere che nel caso dei corpi elastici poco deformabili solo le on4e longitudinali trasportano una discontinuita termica. A parte ogni considerazione di tipo fisico, una trattazione di quella appena richiamate presenta i1 grosse inconveniente di non consentire di asseEnare a piacere i valori iniziali delle di5contint,ita termiche e cio fa ritenere che il problema sia mal posto. Osservando che le (1.2),(1.6),(1.7),(1.9) discendono dai principi generali della meccanica e della termodinalica, 5i conclude che unico modo possibile di modificare 1e cose e quello di riconeiderare l' equazione costitutiva (1.3) e,al fine di avere una effettiva interazione tra problema meccanico e problema termico, di ritenere che le derivate prime della te mperatura possano non ee sere continue attraverso il fronte d'onda rna possano ivi presentare delle discontinuita di prima specie. 2 . -~
nossibile teoria della
nropa~azione
termomeccanica
La s ener al i zza zi one della (1.3) a cui si e gia accennato e accolta da vari Aut or i e espressa da1l'equazione ( 2.1)
o ,
ove beL sono dei coefficienti non ne Dativi. ; :~1e1l gi a considero un'ipotesi del cener e rna poi ritenne di so pprimere il termine in ricadendo nella (1.3). Facendo sistema tra la (1.7) jriva del termine nel1e as e la (2.1) 8i ottiene tm sist ema che ~ediant e l'elininaz ione dei vettori ~ e II porta a un ' oqua zi one pili ;ene ral e di que l l a di Fourier e non pili parabolica e ch e d~ luozo a veloc ite di prop a~azione
g
223
finita[2] • La relazione (2.1) e stata giustificata con considerazioni di meccanica statistica(2) e anche gener al i z zat a [l O] • L4eliminazione dei vettori g e ~ tra le (1.7),(2.1) non e agevole (e spesso impossibile) nel caso di deformazioni finite, facile nel caso di corpi rigidi. Comunque, essa da luogo a un grosso inconveniente qualora nella (1.7) 8i tengano 1 termini nelle ~s' com'e corretto fare, per il fatto che compaiono le derivate delle as e cio crea delle compltcazioni non lievi e in un certo senso stravolge il classico problema delle onde di accelerazione.Ad es., nel caso dei corpi elastici il sistema differenziale risolvente dipende dalle derivate terze delle componenti di sposta8ento, dando luo~o a un problema del tutto nuovo e alla necessita dell'applicazione della teoria delle discontinuit a iterate. I~oltre, l'ipotesi di co~tinuita delle derivate prime della temperatura toglie l'influenza della propagazione termica su quella meccanica. ~ostrero come sia possibile formulare una teoria priva dei vari inconvenienti segnalati semplicecente usando una relazione differenziale piu generale della (2.1) nella Quale si tenga conto in modo completo della necessaria interazione tra fenomeni termici e fenomeni meccanici.A tal fine e fondamentale l'osservazione che la (2.1) si puo derivare dalla relazione eostitutiva
., s = -hL Se-hSg(t_s)ds
,
e
ove g denota 11 gradiente spaziale di T, nell'ipotesi di h e L indipendenti sia dall~ deformaz ione sia dalla temperatura e da t. Volendo tener conto della naturale interazione tra fenomeni termici e fenomeni meccanici, si ammetta,invece, ancora va lida la relazione costitutiva (2. 2) senza escludere pero che L possa d~ pendere, in un modo che per ora non occorre precisare, dalla temperatura e dalla deformazione del continuo.In tale ipotesi e facile verificare che g soddisfa non alla (2.1) rna all'equazione differenziale
2 24
• -1) g + Lgradp,T = zg• + ( 1 - zLL
0,
ove z e un coefficiente costante di rilassamento (~=l/h) mentre L e un operatore mat r i c ial e dipendente da T e dalle as' Si den ot i con V la velocita di pr opaga zi one , con a e £ 10 scal ar e e i l vettore caratteristici delle discontinuit a delle derivat e prime di T e g, con B il versore della normale al fronte d'on~a supposto dotato di normale orientata (nel verso dell' avanz amento). Suppost o che, co me generalmen t e accade, 1 coefI f 1c ien~ trs, t r chec compaiono nel le equazioni dei vi~coli siano funzioni note delle as e di ~ , l'applicazione de lle formule di HUG on i ot- Hadamard alle (1. 7) (pensata come uguaglianza) e alla ( 2.5), de luogo alle relazioni
'}ti 1'i! -
(If-Pi
( 2.4-)
1
-zV£
+
a( Ls
cV)a
+
zV
~* L-l~) - z
ment r e dalle (1. 5) si deduce o. Da ( 2. 4, 2) 51 trae ( 2. 6 ) La
o. ( ~. 6),
confrdntata con la ( 2. 4,1), de
225
La (2.7) va assoCiata alle equazioni sulle discontinuita che provengono dalle equazioni dinamiche, all'equazione di continuita e alle (~.5). Osservazione I~ Il caso Q.g = 0 va considerato a parte; e da presurnersi che soltanto in casi eccezionali esso non dia luogo a incompatibilita. Osservazione II~ Se il continuo e assimilabile a un corpo rigido le equazioni della dinamica non hanno influenza sulla velocita di propagazione di onde termiche.ln tal caso 11 problema e descritto dalla sola equazione (2.7), ridotta al solo termine in a e supposto che L sia un coefficiente di proporzionalita dipendente al piu dalla temperatura.Si ha, pertanto,
(2.8)
2 ,L zcV - z ~ g.g V - L
0,
la quale ammette due radici reali di cui una sola positiva, per ogoi valore positivo di z. Osservazione IlIa. Il va10re z=o,posto nelle relazioni precedenti, porta a incompatibilita, avendo supposto g continuo anche attraverso il fronte d'onda e,invece, discontinue le derivate prime di T.eio non puo sorprendere: infatti, per z = 0 la ( 2.3) si riduce alla le ~[e di Fourier, incompatibile con tale sup posizione. r;eglio si puo dire che llipotesi z = 0 ha come c onseguenza la continuita delle derivate prime di T e riporta alla teoria tra-dizionale con gli inconvenienti in essa contenuti. D'altronde, se si riflette che g in realta esprime una velocita di flusso di calore se@bra naturale attribuire ad esso una -densita e a z il significato di coefficiente d1inerzia termica.ln tal mpdo il calore viene assimilato a un fluido dotato di massa e l'annullarsi di z non a9pare plausibile.
3.- Fluidi n£g viscosi
inco~2rimibili
Supporro che il continuo sia un fluido non viscoso sos setto al vinco10 interno di incompri~libilita.lntendero tal e vincolo in senso generalizzato rispetto alIa sua cor.suetn accezione; riterro,cioe, che oeni sua porzione possa variare di vomume se e sol-
226
tanto se varia la te nperatura. Basta richiamare l'equazione di continuita per riconoscere che un tale tipo di vincolo e esprimibiIe nella forma §=F(T)T,
ove ~ denota la densit a all'istante t e F(T) e una funzione della temperatura. La (~.l) Bostituisce il gruppo di equazioni (1.5). Denotando con p la pressione e con y la, velocit a in pI, il gruppo delle equazioni dinamiche e di continuita e
0.2)
P /r
.
= - ) vr '
Detta p' l'incognita pressione di reazione vincolare e osservato che l'enersia libera de ve ritenersi funzione di T e ~, Ie equazioni costitutive, tenuto conto del vincolo di 1ncomprimib1lita, si scrivono
.2L ')'r
+ E - p'F
=
0,
+ pI
= o,
Tenuto conto di (3.3), si ha
&vend o i nd i c at o con llapice l a der i vaz i one rispet to a T. 81 denoti con ~ il vet tore che c arat t eriz za La di s co ntinuita del le de r i vate pr ime de l la v eloc it e. Data l a continuit a attraverso 11 fronte d'onda di T,g,y e p, l' applica zione delle not e formule di HU50niot - Hadamard po r ta a I l e r elazioni
[T 1
227
In base alle equazioni (3.1),(3.2), 8i ha
In definitiva, tenuto conto di (3.5),(3.6),da (3.4) si deduce - (F
V3 -~- + mV)a,
2
pur d'intendere 0.8)
=
m
Tenuto conto di (3.5),(3.7) e ritenendo L funzione di T e di ~, il sisteQa delle equazioni (1.2),(2.3) da luogo al1e secuenti re1azioni Bulle discontinuita 0,
l&-
zbV - [(
-
1~
F +
JL
)z
J"T
+
q + LqJ a
.lJ
= o.
Esc1udendo l'ipotesi Q'll = 0 che in generale porta a discontinuita terQica nulla (cioe,ad a = 0 ), da (3.9) segue l'equazione risolvente 0.10)
A
Z ~.I! 2_V_4 ~
? + "zV-( £II
f"\r /I
J
1<'
'Jf~.
)r.LJ\
+ )~ZSl'll
V
LT -
L
T
o.
Risulta, com'e facile riconoscere, 0.11)
A(o)
-L
<
0,
Cio assicura che la (5.10)
aw~ette
in ogni casm due radici
22 8
re ali, una pos i t iva l' altra ne sativa. Se am> 0 la ( 3.10) non ha altre r adici r ealijinfatti,in t al caso, l a fun zione A(V) ha de r i vat a seconda se mpre positiva e i l corris ponde nt e di agr amma la concavita rivolta sempre verso Ie A pos i t ive . Se si riflette che V i ndi c a una ve locita di pr opasazi one rispetto al me zzo continuo ed a, pert ant 0 , espressa da V = V'-y.g, se con V' si denot a la velocita di avanzament o nello s pazioR della superficie d'onda e si suppone che 10 stato di riferimento c oincida con quello attuale, si comprende co me possano es sere ac cettabili anche valori neg~tivi di V. La scelta del segno di V presuppone una discussione che ninuncio a fare. Se la perturbazione si propaea in un mezzo iniz i almente in quiete e a temperatura uniforme, nella regione i mperturbata e per continuit 3 suI fronte d' onda risulta ~ = cost., ~ = c os t , , ! = g = 0 e la ( 5.10), tenuto conto di (3. 8), diviene (5.12)
+
mzV
2
L -:c-
0,
l a quale ammette due radi ci re ali in V2 di c ui ~a sola pos i t i va . Per t ant 0 , in un mezzo non perturbato a possibile solo una velocita di propa~azione. Osserva zi one . Se i l vincolo di ihcomprimibilit a a a ssoluto, c Loe se ne l l a ( ,,; 1 ) si suppone F = 0 e, pert an to, risulta ~ = cost., l'eq~ione (3.10) s i identifica con la ( 2. 8), valida nel caso dei corpi rigidi.Pertanto, sotto questo aspetto il fl uido non viscoso assolutamente incmprimibil e si co mporta come un corpo rigido. BIBLIOGRAFIA
[11 [2]
[3] (4)
[5] [6J r7J
~~r.#e l l , J . C . Phi l .Tran s . rt oy.3 0c . 1 57 A 49 (1867) Cavt aneo,C. Atti del S e~inario oat ecat i c o e fisico dell'universit o. di ~;odena .5 . ( 1948 ) Ver not t e ,P .Compt . Rend .Ac ad. Sc i . 21~6 ( 1958) Cattane o , C. Compt. Rend. Acad. 3c i. 247 (1958 Re ttl et on , R. F .Phys . Flui d s 3 (lS60) Chester, M Phy s .Re v . 131 (196 3) Gurt i n ,N and Pipkin, A. Arch . Ra t. Nech. a nd AnaL31 (1 968)
229
j';eixner _~_rch.Rat.Eech.andAna1.39 (1970) Carrassi, 1'1 e Hor r o , A. Nuovo Cimento 9B (1972) Carrassi,M. Nuovo Cimento 46 B (1~7a) LindsaY,K.A. and Straughan,B. Arch.Eat.Mech. and Anal. Ge (1978) [12J Bressan,A. In corso di stampa nelle Eemprie dell' Accademia Nazionale dei Lincei Grioli,G. Nota I a, in corso di stampa nei Rend. dell'Accademia Nazionale dei Lincei Grioli,G. ibidem
La] (9] lio) [11]
CENTRO INTERNAZIONALE MATEMATICO ESTIVO (C.I.M.E.)
ONDE DEI SOLIDI CON VINCOLI INTERNI TRISTANO MANACORDA
ONDE NEI SOLIDI Calf. vmoorr IlflrERNI
Tristano Manacorda Universitk di Pisa ~-Vincoli
interni
La nozione di vincolo di incomprimibilitk in un fluido ideale ~
ben nota fin dai primordi;la stesaa nozione
ta nella teoria dei solidi,ben piu
~
stata introdot=
recentement~.Per quanto
J
6 a
Mia conoscenza la si trova in Poincar6 [1I~] f19 e soprattutto nel fondamentale articolo di Hellinger [i2 .Solo assai piu re= centemente,sono state sviluppate considerazioni generali sul
v~
colo di incomprimibilita. t ugger i t e inizialmente dallo studio del comportamento della gomma 1a qua1e
a dilatazione cubica nulla
~
in ognl sua trasformazione 1soterma [5]
colo cinematico studiato di .recente
~
quel10 della inestendibi1!
ta in una direzione introdotto da Riv1in te studiato da Adkins
[2J
[24J .A1tro tlpo diva:
[20]
e poi ampiamen=
ad altri,anch'esso suggerito dal
comportamento della gamma rinforzata da una fitta trama di fill dl ny10n.Vinco1i piu comp1essi Bono stati considerati da Wozniak [26] .Una teoria generale dei vinvoli cinematlci non pub non far riferimento all'articolo d1 ~ruesde11 e Noll
[25]
dello
Handbuch der Physik. E' quasi apontaneo, a questo punt 0, la introduzlone di vincoli in= terni non puramente cinematici ma dipendenti anehe dalla tempe= ratura assoluta (11 (1) Ne11a ,teoria termodinamica di MU11er,sviluppata ampiamente da Alta 3] per solidi vincolati,l' esistenza della temperatu= ra aasoluta 6 provata inveee ehe ammeasa.Qui per semplicltk s1 aceettera la temperatura assoluta come nozione primitlva.
r
Una prima estens10ne s1 ha quando si ammetta che la d1latazio=
234
ne cubica sia una funzione della temperatura assoluta la quale ai riduce a zero nella trasformazioni isoterme ( solidi incomp primibili secondo Signorini
[11] ,cfr.
(151.
anche Manacorda
Piu in generale si pub ammettere come vincolo interno una lazione finita
fra deformazione e temperatura,cfr.Amendola [4]
[15]
e Manacorda
r~
.Green,Naghdi,Trapp
(91
hanno introdotto
vincoli eepressi da forme differenziali non com)letamente
int~
grabili nelle.quali perb non compare la derivata temporale de! la temperatura.La teoria piu generale di vincoli termomeccanici ~ dovuta a Gurtin e Podio-Gu1dugli
tll] .Essi hanno provato la
impossibilita che il vincolo riguardi tutta Ia storia della de= formazione e della variazione di temperatura se si vuole
soddi~
fatta la disuguaglianza di Clausius-Duhem da ogni processo pos= sibile.I lore risult nti
50no stati ritrovati in modo piu ele=
[15] .
mentare,ma con condizioni piu restrittive da Manacorda Restrizioni suI gradiente di temperatura sono state te da Trapp s en er al e
t 23]
conside~
• Per il seguito, ~ bene tratteggiare La teoria
di un solido vincolato nell'ambito della moderna mec=
caniea dei continui. Di un continuo
~
,formato da elementi
canto ad una configurazione di riferimento ne istantanea
B.
X ,si considera ac= Bo,la configurazio=
X indica la terna di coordinate di
rispetto ad un sistema cartesiano fiseo, mentre corrispondente in Ii
B
B.
la
terna
Data la corrispondenza biunivoca
traB o
~
1. 1 )
.!
=
X in Bo
=
X (~
J
.t)
mentre il gradiente di deformazione
, !=
J-J ('!:
...,F= Grad X .....
J
~
-t)
(2)ha Ie comi!
(2) te lettere maiu8cole indicano che Ie derivate 80no fatte r! spetto alle .!. ponenti
cartesiane
235
t,
1.2 ) Naturalmente 1.3 )
J = det
e s1 ammette
o ,
J)
r..
= x l 'R
PH
=~
oX
Ipi l;l
0
durante tutto il moto del corpo.
Altri tensori di interesse cinematico nella meccanica
dei
continui Bono : 11 tens ore di Cauchy-Green
=
( 1;4 )
( .!,,"eil
trasposto di
! )
( 1.5 )
1e
,j"". l E ,
P
=1 ( C - 1 ),
2' ...
11 tensore unitario.
solido
-
e il tens ore di deformazione
E
Lo stress d1 Cauchy
T F
-)
rappresenta gli sforzi interni nel e ad esso corrispondono, il tens ore degli T
TR
sforzi di Piola-Kirchhoff ( 1.6 )
!R =
_I ! (
!T )-1
e il tensore lagrangiano degli sforzi Piola-Kirchoff ( 1.7 )
T
"V O
Men~re
~R
non
= p-l
T
--R
=J
(
0
secondo tensore di
p-l T ( FT )-1
,...,
e simmetrico,l'equazione
-
,..,
di bilancio
del mo=
mento della quantita di moto e l'assenza di coppie interne 1m=
T
plica la simmetria di
!o'
e di
Le equazioni fondamentali di b1lancio per un continuo sono: a)
l' e quaz i one di conservazione della maaaa, Se
ta materiale di lP.>
in
per ogni sottodominio ( 1.8 dove
bo
e l'insieme
B, e
b
di
di
Eo
~
e la
So e la densitll. di d3
dens;!.
in Bo
Hie
Sl.~ dv ~ }J'odV
corr1spondente a
b • Sotto ov=
vie di regolarita ( 1.8 ) equivale a ( 1n forma rispettivamen= te lagrangiana ed euleriana ( 1.8' )
~ J = ~o
,
236
b)
l'equazione di bilancio della quantita di moto. Si scrive ( 1. 9 ) s, J5 dv : k 1" (!'" M dt b b ~b
f ~
J\'£
Sotto opportune condizioni di regolarita , ( 1.9 ) equLvale a • ( 1.10 ) ~ ! = 3! 1" d1v! in questa f e la dens ita di forze di massa e ,..V la de:
.
rivata molecolare della velocita, Alla
v = X( X , t ), .;.
.....,
'"
""
.v
..
=X = ;"... ; -..;
1.10 ) a1 puc dare forma lagrang1ana. 3i ottiene
•=
~D!. + D1v 1 R c) b11anc±o del momento della guantita di moto.In assenza
( 1.10'
~
.!
coppie distribuite, si limita ad imporre la aimmetria
,!
,
Zo
TT
(loll) e in conseguenza
,...
-
T
: T
!!o
(dr. ( 1.6 »
=
T ..,T "'R ,...
=
di
di
~o
la condizione
.., TT - -R
d) bilancio della anereia. In aSsenza di sorgenti interne,si scrive per ogni ( 1.12 ) ove eaterna a
b
2..
e la b
dt
f :B ,
(~f.
= ( q.n dEi + (btr(:£ grad! )dv ~~~ ~ ), bdensita di flusao termico, n la normale
J.
dv
u b.
nei punti di
Queata equivale ,nelle conauete ipotesi di regolarita a 1.13 ) .$ f." = div q + tIz.( ~grady ) = div.s + tr ( ! Q ) tenuto conto della simme7ria
di
.!
,con
Al1e equazioni di bilancio va aggiunta
E: 1. ( la 2
grad v +(,t4tll1r) disuguaglian=
za dell'entropia.Qui ,per semplicita,in assenza di Borgenti,e assunta nella forma (1.14)
J.. [.""'iCM1' J ..
_
,O[t
1's'e b
~ J.~ ~ 0
equivalente,nel1e consuete ipotesi di regolqrita,a ( 1.15 )
~61-.div4
+
.~.,--l9 »o
237
e
In questa, div
e 1a
temperatura asso1uta.L' e1iminazione di
.3 tna. ( 1.13 · )
(1.15
e
(1.15
) consente di scrivere 1a
) nella forma della disuguag1ianza di C1ausius-Duhem.
Le ( 1.13)
e
(1.15) si possono scrivere in forma
la=
grangiana,rispettivamente, ' .
So £ = Dl.v ~R $0 e'1 - Di v Sa +
1.13:
T
tr (~R
+
j
•
1.15 ' ) 'sR' Grad e)j 0 L'introduzione dell' energia 1ibera l' f. ~ di scrivere
(1.15)
~(f-"le) -
.'J )0 (r"~ 8 -
( 1.15' )
=
~R = J F
!),
,,8
-1
S
,consente
e ( 1.15' ) nella forma
(;g.D) - 1 q. grade~ 0 T' ~~ tr ( lR X) - 1 .9-R. Gr a d e ~ 0 tr
f
Le equazioni fondamentali { 1.10
e ( 1.13 ) vanno comple=
tate da equazioni costitutive. Un solido e detto termoelastico se
r
=
If ( z-
e, §
( 1.16 ) ove g = grade
....
)
s.e, s ), 1 = j { !' e, s ) ~ = s ( !' e, § ) ~ =
1 (
e sottintesa
ed
l'eventuale dipendenza da
x,
La incondizionata va1idita di ( 1.15 ) per oeni processo am= missibile, implica che
13
, e che si abb La
(1.17) insieme a
If
,11 ,
!
, e
I
~,,(f + ,
e)•
non possano dipendere T • tr {~R l
i
= ~
:!=J-v,,'f
TR=~,,}r'i= -()ef
da
cioe
F
~ • ~ )} 0 { ...9-R • Q /? 0 ( 1.18 ) S1 r1chiamano infine le condizioni di discontinuita dovute
alle equazioni d1 bilancio quando il solido sia attraversato da un'onda di discontinuita del primo ordine; esse sono
f ~ uJ
( 1.19 )
[) :!
[~i ( 3)
u] +L~~]
uJ
= = 0
+ [~ • ~] = 0
0
[
~ E: u] + [~
•
( 3 )
~d
La va1idita di ( 1.19 )4 richiede che le sorgenti
entropia,esterne e
=0
di
intrinseche,si.ano limitate al tendere di
238
volume di b a zero. Green e Naghdi non ammettono tale condi = zione l16] .3i puo invece ammettere 1 t esistenza di un flusso intrinseco di entropia.ln tal caso esso deve essere sempre non negativo. h
e
il flusso di entropia. E' facile scrivere la versione la=
grangiana di ( 1.19 ) la quale fa intervenire la velocita dell'immagihe in
Bo
UN
dell'onda.
q/O ; se 9 e continua at= "" traverso l'onda, la ( 1.19 )4 ' tenuto conto di ( 1.19 )3' di= Osservazione
1
3i assuma
h =
,...,
[~e~u.]- [~(u.J=O
viene:
cioe [cfr. ( 1.19
)J [£-,9] = [~1 = 0
3i osserva che l'introduzione di un vincolo ha dei riflessi anche meccanici , in quanto il tens ore degli sforzi non
e piu
completamente determinato da una equazione costitutiva
come
( 1.17 )1 ' ma contiene una parte che rappresenta gli SfOTZiodi reazione dovuti al vincolo.Per un vincolo termomeccanico,anche l'entropia e
S non sono completamente determinati,mentre si
puo assumere completamente nota la forma di Osservazione
2 -
In un soliao
f
nelle ( 1.16 )
termoelastico non soggetto
a vincoli, l'essere il calore specifico a configurazione stante e
1 '
C
v
positivo implica corrispondenza biunivoca tra
co=
e
per cui s1 puo assumere come variabile termodinamica la
entropia al posto della temperatura. Tale corrispondenza viene a cadere per solidi vincolati,onde si potrebbero istituire due teorie parallele,assumendo in una ,come variabile termodinami= ca,percio soggetta al vincolo,la temperatura; nell'altra,invece l' entropia.
239
2 - Vincoli interni nei solidi Esempi di vincoli a)
Vincoli cinematici
Vincolo di incomprimibilita 2.1 )
= det
J
F
'"
Vincolo di inestendibilita Si ammette che in
Bo
=
if det
C
1
..;
esista un campo vettoria1e
~
1inee vettoriali conservano 1unghezza ina1terata in
(X) 1e cui B. Se
~
ha modulo unitario,il vincolo e tradotto in ( 2.2
Fe.. F e IV ,...,,..,
= 1
/V
cioe
2.3 Vinco10 superficia1e : Si ammette l'esistenza di una famig1ia
g.
di superfici I. , densa in
B o ' La cui di1atazione superfi=
ciale e nulla nella trasformazione
B o 4' Bl •
Poiche la dilatazione superficiale
e
cf6'" = J (~'C-1:Q con
~
norma1e nei punti
di
~
in
BoJ 1a condizione di ines
stendibi1ita superficia1e si scrive -i
J
<.:t0g £ )
per ogni punto di l.. ed ogni
= 1
r. E.. g.
b) Vincoli termomeccanici Sono estensioni di vincoli precedenti. Vinco10 di incomprimibilita ( Signorini ) J = f' (e)t"')~ )
2.4 )
e 1a
~
f
(r',7:i~) = 1
temperatura ( supposta uniforme ) in
Bo •
Vincolo di inestendibilita F~ •
2.5 )
Vincolo anolonomo A e
f(tl)'t',X); f(t')t'j.0) = 1
(Green, Naghdi,Trapp ) tr (~T
2.6)
ove
! 2
K 'V
i)
+j(! ,8) ..e=0
sono un tensore ed un vettore funzioni di!
e
240
e
e di 3i assumera ,d'ora innanzi,l'esistenza di un vincol0 della forma
r (
( 2.7 )
e ,X'"
F,
~
)
=0
( 1)
(1) II principio di obiettivita impli~a che T possa dipendere da F solo per il tramite di C = F F, ma si conserva la form~( 2.7 ) per comodita. ~ ~ ------------------------------------
Lungo un processo
F =
'"'
f (
t ),
"" di sufficiente regolarita di
tr t
ove
2.9
r .......
- +I e
1
= 0,
r
'1' •
'"J r --r3i=assuma
F )
y
Osservazione
.
,in condizioni
r e identicamente
qu.LndL
( 2.8
A
e::: e(t)
'V
r
=
r
~
(f
0
=
r
de
,e , x );
in una
trasformazione linearizzata a part ire da Bo e LX) tr (\-_'1' ELt l ) +fa ~1'J= 0 . (.il "f' (ol) (II IflT ove E e la linearizza~ione di E, 2 E - = Grad~+(Gra~~ ) (.4)....... t 0l mear ' izza t -0, e el f ) 1 a 1 an . e ar-a.za . a..v • ~ spos t amen z t.ona della variazione di temperatura;
fc
di '";),€ Te
quella di
diviene con
a
%1'.
dO e la
determinazione in
In }:articolare, se
Bb
.b = Ii,
la (II\)
div -u,L'~J = a eLf) costante.Cio accade se T' dipende da
tramite dei suoi tre invarianti
.!
per i l
lII E •
In un solido termoelastico soggetto ad in vincolo del 2.7 ),10 stress
tipo
e l'entropia risultano non completamente
determinati da equazioni costitutive : precieamente si ha ( 2. 10 )
ove
p
e un
T = - p \' + .-R ...
5'o ?r 'f,
"l = +
,
-rs:, If3
-
"8
r
parametro lagrangiano atto ad individuare la rea=
zions vincolare interna.
241
Osservazione 2-
La potenza delle reazioni vincolari
da - tr ( p ed
e
r
..
AJ
e
data
F)
.v
quindl nulla,in corrlspondenza al vincolo,in ogni trasfoE
mazlone lsoterma .Plu in generale,essendo la prodpzione speci=
5'oy + ~o"1 e-
fica interna di entropia data da _ tr (TT
....R
F_·)
la produzlone di entropia dovuta alle reazioni interne
-pT' ,-ed
e
e.E.~
50
e qUindi
I
P
ta
+ tr
zero in virtu del vincolo
Osservaeione ne assumendo
(f e•
i
(r.".!• » (cfr. ( 2.8 »
3- Una teoria perfettamente analoga si ottie= come variabile termodinamica
e
da una equazione costitutiva. Naturalmente,
ta con
g
e determina =
r
va sostitul
242
3 -
Onde di accelerazione
11 quadro delle equazioni fondamentali in presenza di un vin= colo come(2.7),e in assenza di forze di massa e di sorgenti termiche e di entropia
~ei
= Div
~
( cfr. ( 1.10' ), ( 1.15' ), ( 2.10 ») ~o
..
,q
;;R
.\ 3.1 )
~
=
= Div :£R
%-
50}'P - p S'
~
-i
=JoFq
t ~ -~
= If! ' 1'(F,e)=O
-
-
G ~
~R •
~-
3R
0
= SR ( .! ' § G = Grad
e
,
e)
31 comincia ,qui, ad esaminare le cond1zion1 per la propaga= zione di secondo ordine ( onde d1 accelerazione ). A questo scopo s1 pone
3.2
a=[X] rv
,....
Una notevole sempl1ficaz1one e dovuta al fatto che,in un
co~
duttore definito,tale cioe che la parte simmetrica del tensore K-
~~R
- - ()Q:
vincol1 (1)
e definita positiva, possono anche in presenza di (l),propagarsi solo onde omoterme ,per le quali,cioe
Per un solido non vincolato
V.ad es.
=
o
=
3i osserv1 ,infatt1,che pur essendo
~
6
minata da un'equaz1one costitut1va,
E =)'+je='\f- e
[B,7J
completamente
non 10 e ,ma si ha
1' +r., f e
;
tuttav1a f. e da r1teners1 continua attraverso l' onda (2) (2) 1a
(1.19)2
impl1ca
[1.e]
= 0, cioe
Dalla equaz10ne della energia s1 ha allora e percH)
( 3.3 )
[5'0 ~ t: ]
=
[3R N ,....
1
=
dete~
N 0
[p]
=
0
24 3
e La
ove ,!! Sia
G:
continua a
normale all' immagine dell' onda in
[~J ; si ha allora
discontinuQ ,peraia ~
= Grade
J; = 3R ( ~
+ ,
e
) .;a
•
dato che 9
=o(N ,
SR (~r
Bo
+0(
e
!J, () )
quindi
=(
e) -
Ef~)= L~R] . ~ SR (~- + ()(,1'4, .3R( Derivando questa identita rispetto ad~
):!J
=0
= e impossibile per l'ipotesi che il conduttore aia definito. K+ N • N
che
2- , e)
0
....
Si pua dunque asaumere che l'onda sia omoterma. Dopo di cia,dalla ( 3.1)6 od anche
a1 ha
• [t"'l'''j 1 {[ l1']= L ~ w! ,
= 0
tr l' T (
e
-- -
~ /li7 1!
=1N.a
UN
( 3.4 )
flN.a .--
,... '"
=
0:
le onde ( omoterme ) di accelerazione di un solido termoalast! co sono trasversali. Si prende in esame,ora: la ( 3.1 )1 ( [div
TRJ
ai
ha
= -!.- [TR] ! ) •
u
N
•
!] -
L~RJ = - [ p r] + f; [ tX [ g] e il tenaore quadruplo ~ ~ • Si ha poi ~ f [i] = - f (1 ~ ~ 1! ) = - 1
ove ((
UN
UN
A a ...,
---
ove ,.... A indicadn conveniente tensore del aecondo ordine,fun zione di N.•
[p·rf= [;]I+ p[d.r1'(i)J
+
p[;>e! e]
e cioe,nelle condizioni presenti ( 3.5 )
-
[P1]!
= -
[p] 1'!
+
rr:]I
A
~
=
244
(d F!( .2- ~! )
avendo posto
)N =
J~
Si ottiene infine la condizione di c0I!rpatibilita. dinamica 3. 6
[p] 1 ~
UN
+ ( Q-
t ~ 2: ) ~
=
0
ove
(3.7
e un
£.=!-p~
teneore doppio aimmetrico.
Le implicazioni della equazione dell'energia ai ottengono ra=
LS!l = 0 e
pidamente. Per eaaere
con
A N.N
K
,....
Poiche W1
k (
"'.....
) e
~~!
- 1
L
ov
-
=d.9. • ,l R
!..
~C7e[i]=A!! •
-N
=f [pJ
+
L ( ~~1!;
L~E
3.10
0{
:.0 -
~
esiate
) ={). J!. ~
• !:
UN
(~
' si ha
- P.f ) !. .~
con I
[SR] .~ =
Eaaendo
lineare di
D'a1tra parte ,se ai ricorda 1a ( 3.1)4
3.9 )
Uiv
• ....m
L
una funzione
e si ottiene,alla fine
=- L
U
U N-
tensore doppioAtale che
(3.8)
[DiV,.9-Rl
1 = - d~.E T'
5c,';);! f
Si ottiene quindi la condizione di compatibilita.
3.11 )
A!}.!.
N
-L~~· ~=
feEp]
+9(e'-
Pl)
!.~
UN
;roiche
[pJ
si puo ricavare dalla (3.6), la 0.11)
si 1imita ad esprimere e di
p.
A
in funzione di grandezze continue
Moltip1icando, infatti, la ( 3.6 ) per
r N -,.....,
si
245
ottiene
[P]
( 3.12 )
~
Sostituendo questa determinazione di
= I.!L I!!! I
nella ( 3.6 )
si perviene a (3.13)
(l-\)e'J)(Qj'U(.l)a;= "'-
0
"'\J)ON"-,....,
Si rieordi ora ehe ,... a deve essere seelto tra i vettori per: pendieolari a -v V posta la eondizione
,9,1 = (l -:i ~! 3.13) si serive
( 3.15 )
~1!t.
( 3. 14 )
II problema
'1
= ~~ UN
e quindi
)s
'
o
a .\,)
!!:,
""
ridotto a bidimensionale,ed esistono
pereio due autovettori almeno distinti e,ae i corrispondenti autovalori sono positivi,essi individuano Le direzioni di di ,:= aeontinuita e Ie riapettive velocita di propagazione. In particolare,sia per
e
si ottiene ( 3.16 )
,."
un autovettore comune, ( se esiate )
A
u = v'1. ,...
A
u
Q
~l
...11 u '"V
= PUt. U = )0 N "..,
la quale mostra che,anche se ehe
UN'"
<.
0
v
2.)
v.'L
0,
_
P -:; 2.
V">
0
)
u , puo darei
e quindi che non ai abbia propagazione.
246
4 - Onde d'urto Lo studio delle onde d'urto
e ricondotto
allo studio delle
condizioni dinamiche di compatibilita (1.19) cui va aggiunta la condizione che si ottiene dall'equazione del vincolo. Conviene scrivere la (1.19) in 1:orma lagrangiana
= 0 [!R! 1 = 0
r~ uNl
( 4.1)
+
[ ~ UN E, ]
o
• III
1
alle quali va aggiunto
N, ~ 0
(1) [I'Ll ~ 0 , una relazio=
(1) Questa equivale ad imporre ne dal significato fisico evidente •
• la condizione derivante dall'equazione del vincolo
[fJ
( 4.2 )
Lo studio di tale tipo d1 onde meno di non riiurs1
e
=0
evidentemente complesso, a
a cas1 unidimensionali.Qui ci s1 occupa
solo di onde di p1ccola inten5ita,per le qUali 5i vuole dire, sono valide le versioni linearizzate delle ( 4.1 ) e ( 4.2 ), in solidi
iso~pi.ln
tali condizioni,si introduca la differen=
za di temperatura tra la
temperatur~
attuale e quella (che si
suppone uniforme ) di riferimento, e si indichi
e
con
sua linearizzazione.lnfine,5i 5upponga naturale la
la
configuzazi~
ne di riferimento.ln tali condizioni,per il tens ore 1inearizza to degli sforzi si ha
( 4.3 ) x, ove
e
Tl'l = T('\ = - ) " t 1 + e -R p , AJ e la linearizzazione di
r.
nearizzato di deformazione,
e
=~e
=
1 + 2fJ-:!- -r(9 :l i1 tens ore 1i=
div ~,
u
line~
rizzazione dello spostamento, ~ , ~, ~ , sono cos tanti delle quali)L
e
certamente positiva.
247
Allo stesso tempo, per la linearizzazione di • 4 ) ...•
",«) t.. --
-
..k...
.!LJ' + ~o'1:-
con
~
~Q~
e
+
.s. e
fo't.
la temperatura di riferimento,mentre la condizione
di vincolo diviene
( 4.5 )
e.
u = a ...., I campi linearizzati devono soddisfare alle linearizzaeioni
delle
div
(4.1) e ( 4.2) -
• ma [ ~ 1
: precisamente,posto =
r ! tfl !!I,1
ma C~(4 + Cg~).N] 't.
( 4.6 )
=
ma .=
~
UN
[ ,.,'1 [ ma €'.l = ~~'.!, 1 0
(div u,...,1 = areJ. Conviene distinguere due casi :
(eJ = 0
a)
( onde meccantche)
a il parametro caratteristico delle discontinui= "0 il vincolo impone ta delle derivate prime di 58
~,
-
~o • N =
( 4.7 )
0
l'onda a trasversale (1). Dalla prima delle
Propriamente percH) non a piU un' onda d'urto
(1)
allora (4 ' •7 )
e cioa, (div ~1
[Bl = - ~ ~o )
m • J = ,.. tenuto canto di ( 4.3 )/della continuita di e [ T (t)
=0 [x]
Ma
( 4.6 ) si ha
e
e
I""
N
0
=
=!
2
e infine Segue dunque
r e N 1 .N ~......
= 2" 1 "_0 • ,....., N= 0
e di
248
( 4.8 ) e,daJ.la ( 4. 9 )
=
0
C,,(4'1 =
0
[7"C.]
(4.4
Ancora dalla ( 4.6 )1 ,molt1plicando scalarmente per
)~
si ottiene
e perc10.
=
( 4.9 )
che assegna la veloc1ta di propagaz1one dell'onda.Infine,po1che cc,(4) = ()~) e +f~f.) [£(')J = 0 la(4.6)2
fJIt;
0
~'O'
0
e soddisfatta da
e
'
(~' •
.!! 1 '"
soddisfatta anche 1a ( 4 .6 )3 (2)
0, che rende
subordinatamente
=
0
Gtll • [ ,,Se vale la legge di FOURIER , r<;. qiente di temperatura normale deve es~ere
( 4.9 ):il flusso normale di calore
alla
! 1 =
0:11 gra= continuo.
e continuo.
249
[& J
b)
-I
0
(onde termiche ) ,
a
-I
0
In questo caso,il vincolo essendo rappresentato da
ha
"0.
( 4.10 )
N
-.
~
( 4.11 )
IV
=a®
da" ora,moltiplicando Bcalarmente per
La ( 4. 6 )1
U;
a @
[ - ';C.
=:
+
A
~
e -
[~l.N =
~ UN ~ oJ
Se
~0
non
e
la ( 4.12 )
e
-..I
.! =jJ. ~
0
~
parallelo ad
UN la determinazione
N.
N
2p..a G)
+
t( ~o i(~o· ~) !! ) 0
•
N
=a
e.
t
mentre,moltiplicando per ( 4 .12 )
aJ
! 1.! =~
[~ 2..
(4.5 ),si
~
•
, si ottiene ancora per
(4.9). Se invece l' onda
una identita. e
Per onde non longitudinali
u';(e
e
e
longitudinale
indeterminato.
neanche trasversa11, v , (4.10)
s1 ottiene da11a ( 4.11 ) per 1a ( 4.9 ) ( 4.13 e ( 4.14 )
[)tJ
=
a
e
(~
+)J. -
~
n>
[I'()] = ( .!L ( 2'1 -a(l\+p» ~'t
}@. + L e'o't;
D'a1tro canto,la ( 4.1 )4 imp1ica che tr sia continua cfr.n.1 Oss .1 ) e quindi che sia continua anche a..r(1). Segue,po~endosi
4 • 15 )
assumere
e quindi, da ( 4.6 }2 ( 4.16 )
~
=0
[t;C1'l = L\f>C4J] (q~'
1.
N.
=
mo'"C
(1Z. (4)J.
4.6 )3 • Le condizioni di compatibilita che coincide con 2termomeccaniche aono qUindi atte a determinare UN ed inaie=
250
me le discontinuita di
e
e .....
di ~ (I) in funzione di @ •
5i ha anche ( cfr. ( 1.19 )1 )
c~ con
~:
uJ
0
=
U
= un
- vn
velocita di avanzamento ( locale di propagazione )
dell'onda. Linearizzando
(
[~1
=
(~J.
N
= - UN ~o
= - UN a
• N
e. )
e quindi
[~(t)J = - ~o a a @ > l'onda e compresaiva se 4.17 )
e: 0 , espansiva nel
caso
opplilsto (3) (3) Si osserva che ( 4.17 ) vale anche se 1'onda e 1ongitudi= nale, mentre ,per onde meccaniche (e = 0 ) implica La continuita di ~u;
Per onde Longd, tudinali,
UN
non puo essere determinato da1=
le sole condizioni di discontinuita,mentre per le discontinui= ta di JC , di '1.{')
[lel ( 4.18 )
si ha
=
a @ ( ? + 2}J- -
~ -~" a
u; )
251
mentre
e
anche
( 4.19 )
BnlLIOGRAFIA A
Vincoli
1 - J.E.
Adkins
: A three-dimensional problem for
highly
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: The
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nics.Encycl.of Phys.Vol
foundations of solid mecha=
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Formal structure and
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H.H. Erbe
W~rmeleiter
mit thermomechanischer in=
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76-79 (1975)
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) Omie.
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1
G.B.Amendola: Acceleration waves in incompressible rials. Atti
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24
mate~
(1975) 381-395.
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(l~13)
in
a
theory of thermoelasticity,Acta Mech. 20(1974)
67-'19. 5 -
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6 - P.Chadwick - P.Powdrill:3ingulnr surfaces in_linear thermo=
254
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49
(1972) 137-158.
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: On wave propagation in inextensi=
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: One dimensional shock waves in heat conduct!
ng materials with memory : 1- Thermodynamics-Arch.Rat.Mech. An.~7
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255
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wave~
in
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20 -
Cfr.anche A - 15.
R.W.Ogden:Growt~
and decay of acceleration waves in incom=
pressible elastic solids,Q.J.Mech.appl.Math. 27 (1974) 451-464. 21 - N.Scott:Acceleration waves in constrained elastic materials Arch. Rat.Mech.An. 58 (1975) 57-15. 22 -
N~Scott:Acceleration
waves in incompressible elastic so=
lids,Q.J.Mech.appl.Math. 29 (1976) 295-310. 23 - N.Sc_ott-M.Hayes:Small vibrations of a fiber-reinforced composite,Q.J.Mech.appl.Math. 29 (1916) 467-486. 24 - C.Trimarco:Onde di accelerazione in materiali termoelasti= ci con vincolo di inestendibilita,in pubbl. su Atti Acc. Sci.Modena. 25 - C.Truesdel1:General and exact theory of waves in finite elastic strain,Arch.Rat.Mech.An. 8 (1961),263-296.
CENTRO INTERNAZIONALE MATEMATICO ESTIVO (C.I.M.E.)
"ENTROPY PRINCIPLE" AND MAIN FIELD FOR A NON
LINEAR COVARIANT SYSTEM
TOMMASO RUGGERI
INTERNATIONAL MATHEMATICAL SUMMER CENTER (C.I .M.E.) 1" 1980 C.1.M.E. Session: "Wave propagation".
Bressanone 8-17 giugno 1980 . "ENTROPY PRINCIPLE" AND MAIN FIELD FOR A NON LINEAR COVARIANT SYSTEM
by TONNASO RUGGERI
Istituto di Matemat ica Applicata -
Universita di Bologna
Via Vallescura 2 - 40136 Bologna (Italy).
I.Introduction My lecture is complementary to the lectures given by G.Boillat i n the first part of this course. In
part~cular
I am shall deal with some problems concer-
ning quasi-linear hyperbolic system compatible with a supplementary conservation law; relativistic theories will be considered with special emphas i s . I start with a brief bibliographical introduction to the subject I shall be concerned with. In 1970-71 1. MUller, in some works [1] on "rational" thermomechanics of continuous media, proposed the "entropy principle" as a criterion for selecting the constitutive equations. This author considers the equation governing the evolution of a thermomechanic system : a) balance of momentum, b) balance of mass and c) balance of energy equations . Adding the constitutive equations to the prev ious system one gets a system of 5 equations in 5 unknowns. Each solution of this system is called a "thermodynamic process" . Then MUller postu- lates the existence of an additive function cr (entropy) such that : +
d dX i
(PSv. + ~
~ .) ~
= cr
o
lJ
t her modynami c process . (l)
Furthermore he supposes that both the entropy density S and the flux ~i are
....
constitutive functions (p and v are respectively the mass density and the velocity). Hence, from (1) further constrains arise for the constitutive rela-
260 tions,
besides
the
usual
ones
which
can be
imposed
according
to
the
principle of material objectivity. In particular the author shows which
he
identifies
with
the
the existence of a universal function,
absolute
temperature ;
hence,
he
deduces
the first principle of thermodynamics . In a different conte>ct in 1971 K.O.Friedrichs and P.D.Lax [2] and later the
former
in
1974
[3J
examined
a
similar
problem .
In
particular
in
[3] Friedrichs, in a covariant formalism, considers a conservative quasilinear hyperbolic system of r first order equations of the type
aa
~
o
(U)
= .2(U)
(*)
(2)
In (2). N eqs. may be identified with the field equations. while the remaining r - N are supplementary conservation laws. Then
comp~
tibility conditions are required in order that the system has a solution. In particular, when r=N+l (one supplementary law), as the system is quasi-linear, compatibility is ensured by the existence of an r-vector
z(U).
such that: a
= 1-.2 I,l V = a/au we have =0
I·aa~
Introducing the operator (condition I)
U,
I,l
aa u. (3)
Moreover Friedrichs supposes another condition holds: it exist at least a time-like covector
{~
a
}, independent of the field, such that the quadratic
form ( condi tion II) Here
oU
is positive definite .
(4)
is an arbitrary variation of the field and
o2~t
= su - VV ga su .
Using condition I and II Friedrichs shows that the system of the field equations is a hyperbolic symmetric system . (*) To avo id misunderstanding the vectorErn
r
are underlined .
261
Later several authors [4J. [5], [6J provided further contributions on th is subject. especially concerning shock waves in non-covariant formalism. Now we shall obtain the above mentioned results in an explicitly covariant formalism. dealing with
the physically relevant case of one supplementary
law. The covariant formulation allows to apply the results to explicitly covariant theories and. moreover. to emphasize some conceptual aspects that, in our opinion. have not yet been pointed out. 2. Main field and Covariant convex density. Let V~ be a C·, 4-dimensional mani f ol d and x a point of v~. x a being local coordinates of x. The manifold is supposed to endowed with a pseudo-Riemannian a
metric . In the local coordinates x • g
a8
represents the components of the
metric tensor of signature (+ - - -) . On V~ we consider a quasi-linear conservative system of N first order partial differential equations for the unknown N-vector
U(xa)E~N (5)
the components of Fa and U are contravariant tensors and da is intended as a covariant derivative operator. We suppose that the system (5) is hyperbolic. Le. :
.:J
a time-like covector
{Cal. such that the following two statements hold : i)
(6)
ii) V covector{~a} of space-like the eigenvalue problem a (~ a- .llCa ) A - d = 0
(7)
has only real proper solutions ll(k) and a set of linearly independent eigenvectors d The
(k)
(k=I.2 ••••• N). • where II is solution of (7) are called "characteristic". a} fUlfilling L}, ti) are said "subcharacteristic".
covectors{~a
while the{c
a}
- llC
262
When a differentiability conditions holds, let us suppose that (5) is compatible with a supplementary conservation law
aa h
a
a
(8)
h (U) = g(U),
being a contravariant vector and g a covariant scalar. In this case we may write the conditions I and II of Friedrichs in a more
convenient form. We have:
ga -
(J
E - (: )
Since by (3) I is defined up to a scalar factor, we may write
I
='
then Friedrichs conditions lock like U' • VFa
uv-r
=
V h a,
(9)
= g,
(10)
We remark that (9), multiplied by 6 U, can be written equivalently: V
s u.
The identity (12) show the first important result : ,
U' i s invariant with respect to field
-it
transformatio~:in fact
(12}
U' depends only
and then it does not depend on the choice of the field variables.
By applying the operator 0
to (12) and replacing into (ll} we get a( OF >0
\l 6U (13) a Hence (13) too is independent of the choice of the field; we may then choose
Q = 6U"
the field in the convenient form (14)
263 We put also a h = h I;a '
(15)
and contracting (12) with I;a we have
U' -OU =oh
+~
U' =Vh
(16)
For the particular choice (14) of the field variables, U' is espressed by the gradient of a covariant scalar function h only. In the case of continuum mechanics, the expression (16) is equivalent to the first principle of thermodynamics . We point out that the components of U' play the same role of the Lagrange
multipliers introduced by I-Shih Liu [7J
in the context of entropy principle of MUller. Condition (13) is now equivalent to Q
= oU'
- tSu
= tS 2 h
(17)
> 0,
i.e . to the convexity of the covariant scalar density h=hal;a with respect to a the field U = F I;a'
If for a system (5) there exists a vector U' and at leaet: a eooeetor (I;a}suah that (12) and (17) ho ld, we say that the system is a aonvex aovariant density system. Conditions (16) and (17) ensure also that the mapping
U'+~U
is globally
invertible , becauseVU' =VVh and th is gradient matrix is symmetric and positive definite; then for a theorem about globally univalence ([8J) U'
+~
U is glo-
N bally univalent in every convex open domain D ~ R Therefore it is possible to choose the vector U· itself as field variables and prove that in this case system (5) has the form L {U'}
where the operator
~
f(U' )
(18)
is given by A' a ~
(t)
(t)
aa
(19)
For the proof of the statements proposed in this lecture one may see [9J •
264 and (20) System (18) is symmetric hyperbolic : in fact a system A,a a U' = f is sima a aT a metric hyperbolic if ~' A' and A' ~ is positive definite, and in our a a case we have A' ~ a ='J' 'J'h', but from (20) h ,a ~ = h' = a = U'·U - h is the Legendre conjugate function of h and then it is a convex ~
function of U' • We remark also that the differential operator in (19) depends only on onefour-vector h ia and this justifies our definition of "four veator generating funa-
tion" for the symmetric system . We have seen that any convex covariant density system is endowed with a vector U' that may be expressed as a function of the field variable and is invariant with respect to transformations of field. In fact it is determined completely
only
by
the
conservative
system
law
(8).
the
system assumes a symmetric form,
is
well
Moreover we pointed out
posed.
Such
remarkable
that, so
(5)
and
when U'
the
supplementary
is chosen as
that the local Cauchy
properties
suggest us
to
call
field, problem
U'
the
"main field" of the system. We remark that not only on the mathematical point of view U' and h,a possess a special role with respect to other quanti ties, but also from the physical point of view, they are privileged, since they are related to the "observables" of the physical system, as we shall see later. System
(5),
compatible with
(8),
is
riducible
to the form
a suitable choice of the field variables and viceversa;
for
(18)
in fact it is
easy r:o prove that the system (18) provides always a supplementary law (8) :
a let h
= U'
• 'J'h'Cl
_ h,a, we have
U'·f =g. Finally we have shown that
a
a
h
Cl
'J'h
Cl
•
ac
U'
U' •
f { U'
}
265
A necessary and sufficient condi t i ons f or the sys t em
( 5)
to be compat i ble
with a supplementary conservation law (8 ) with h ( U) convex fun c tion wi th r es a pect to U = F ( (con vex covariant dens ity systems) , is t hat t here exists a a
choice of the f i e ld
(invariant r espect field transformations and indepen-
U'
dent of t he congru ence defined by the time-like covector {(a })' so t hat t he system (5) as sumes the syrmzetric f orm ( 18) with h'=h, a ( Th is
proposition
is
a
f irs t
c ontr ibut ion
of
the
a
convex fu nc t i on of U' •
que s t l on
proposed
by
LMUller ("a c hallenge to mathemati cians" [lJ ) . At
least we point out t hat if in (8 ) we impo se the c on di t i on g
> 0,
t h e n , by (10) all solutions of (18) satisfy
3 . Shock Waves Theory for Convex Covariant Density Systems.
i) Entropy growth across a shock wave. Let
a
Il
connected
open
se t
i n t o two open subset Il l. Il z. of
r :
we shall
Let ~( x
r
ind en t ify
v"
of
a
and
a hypersu rface cutting Il m (m> 2) , be the equation C
r
) = 0, ~
E
with a shock hypersurf a ce for the fie l d U.
It is known that the Rank ine-Hugon iot c ondit ions mu s t h o ld [ F a]
~
= 0, on
a
r
where brackets denote t he jum p a c ros s Formally
t he
Rank i n e-Hug on io t
r,
(21) = aa ~ .
~a
and
equations
a re
ob t a i n e d
from
the
field
e q s . (5) through the correspond e n ce rule
a
...
a
However
this
r ul e
[ J
~
a not work
does
(22)
wh en
a pplied
to
the
s up pleme n tary
equation (8); in fact
n does
not,
n is
non
in general , negative.
[4J
by P .D.Lax
[h a ] ~
=
van i sh .
Th i s
n for
dyna mic n
>
0
,
on
( 23)
r ,
Fu rthermo r e
r esult wa s
pr oven
1
t
is
in
to
show tha t
a non c ova r i a n t
possib le
f ormal i sm
introducing an ar tifi c al vi scosity in the f i el d e qu ations;
a different proof was given in of
a
[51.
It is know that th e pos i t i v e s i g n a t u r e
t he n on r e l a tiv i stic perfect fluid implies the g r owth of t hermo e ntropy is
a cros s
o f t en
t he
c alle d
in
s h oc k. the
That
is
the
literature
reason why the c ond it i on
" entr op y
gr owth
c ondi tion"
266
a nti i s
assumed as a criterion to pich u;> the physica l
shocks a mong the
solutions of the Rankine-Hugoniot equations. In this section we suggest the main steps of an explicitly covariant proof of the fact that n is non negative on Let U; }
r. 1 and a a
be a subcharacteristic covector such that
a
covariant scalar defined as :
a = -
(24)
f;a4>a
thm there exists a space-like covector {l; } a
such that f; l;
a
a
°
be
= 0.
(25)
the equation of a characteristic hypersurface which
locally has the same "direction of propagation"
l;a
of the shock surface.
i.e. (26)
0,
(27)
where
(k)
(k=1,2 •..•• N) are
IJ
the solution of (26);
these eigenvalues
are real by the hyperbolicity condition. Now
we
consider
a
solution
of
the
Rankine-Hugoniot
equations
(21)
(shock) : (28)
u,
U* being
the
perturbed
and
the
unperturbed
fields
respectively
on
r (in the following * will denote the values of any function of the field
computed
for
U
=
U*) .
Here
we
take
only k-shocks
according to
the following
De f i ni t i on of k-shock
We
shall
say
that
a
shock
is
a
k-shock
if
there exists a number k (=1,2 •••. ,N) such that lim
O-+-\J(k)
*
U = U·
(29)
267
Roughly speaking a k-shock is a shock that vanishes when the shock speed approaches to a characteristic velocity weak shocks when We suppose
0
~~k)
is near to
to know the
(of course,
these shocks become
).
solution
(28)
for
into (23): then we get n as function of u* and
a
k-shock
and
replace
it
~a
(30) By differentiating
(30)
respect
to
~a
and
taking into account
(9),
after some calculations we obtain
[h
a
J-
Vhf
a
[F ]
Thus (31) Since h is a convex function of U, defined in a convex domain D, we have: w(U,U*) = -h(U) + h(U*) + Vh'(U - U.. ) > 0,
So the r.h.s. in (31) is equal to
-\!T,
F;a an/a~a
Furthermore,
<
restricted to
l/- U
r.
ot U.. '" D ,
Hence (32)
0
in the frame S • in which F;
0
i
=1,
F; =0,
locally, cond i t i on
(32) can be written
ani ao
>
(33)
0
As an/ ao is a scalar quantity, inequality (33) is independent of the frame; So n is a strictly increasing function of
in any frame.
0
Since our shock is supposed to be a k-shock we have lim
o~(k)
hence we get
n
=0;
*
Fop a aonvex aovaPiant density system and a k-shoak one has when
0
(k)( on
.::. Jl*
r.}
268
ii} 'I
If
as generating function of the shock.
i8 a k1101J function of u· and 'Po' it is easy to prove that the
'I
r
fol'LoLn.ng relations holds on
v· 'I
(34)
"'here V· .,
Eq.
(34)
means
that
a function of U· jump
of
U'
and
and
-. o
a/au-,
o
~ (U·)
A
if we know only
'P a' wi th
~a
therefore of U;
the scalar function
'I
(which is
non characteristic) we may find the behaves
'I
like a
"potential"
for
the
shock. f\(U·. 'P } is computed when the shock is known a as a solution of the Rankine-Hugoniot equations. However it is interesting Of course,
the
fact
in practice,
that,
were
it
possible
to
determine
'I
through
experimental
tests, we should be able to have all information of the shock. iii} Relativistic bound of the shock speed. The Rankine-Hugoniot equations (35) provide N equations for the perturbed field U if U· and 'P
a
are known.
Eqs. (35) are equations of the kind (36)
which always possess
the trivial
have also non trivial solutions U .s o t u t aon )
which
in turn are
solution U ., U* for any ~
~a
They may
U· (branching solutions of the trivial
physically acceptable only
if gaa ~ a ~ a
~
0,
so that the speed of the shocks does not exceed that of light, according to relativity theory. We put now the following question: ~a
such
U for a
function
f
is
does it exist a set of values of
that
the
globally
invertible
with
respect
to
fixed
'P a ? If the answer is affermative, then only the trivial
269 solution U The
[6J
= U* i s
allowed.
problem has be e n e xami ne d by G. Soillat and T.
Rugger i. who proved
that non van i s hi ng schoks t ake place only if their speed is greater
than
the
smallest
c ha r ac t e r i s t i c
speed
and
smaller
than
the
greatest
one. It
is
possible
proof given in
to provide an explicitly
[6J
covariant formulation
of the
and show that:
For hyperboUa convex covariant density systems the speed of the non vanishing shook fulfiZ the condition: m<
M
<
0
(37)
where
m
. (k) mln{lJ },
inf
UeD
As a aonsequence the
sup Max {lJ UE: D k
M
k
sh~ak
(k)
}
manifolds are time-like or light-like if so are
the aharacteristia manifolds. In fact
if
(37)
holds,
and the characteristic manifolds are time-like
or light-like we have : Cla
sup
<
{ Maxg k
U€D
I.
Relativistic
Hydrodynamics
-
(k) (k)}
lil
qJ
Cl
Existence
of
<0.
a
a
Convex
Covariant Density.
The equations of relativistic hydrodynamics are (38)
aCl
Cl
(ru
)
=0
(39) Cla
d
denoting the covariant derivative operator and T
Cl
tensor T
Cla
= rfuc ua
ClB
- p g
the energy-momentum (40)
Here r is the matter density, f the index of the fluid, uCl the 4-veloci ty Cl
(u
u
one.
o
1)
and p the pressure.
The speed of light is assumed equal to
270
By (38) and (39), taking into account the thermodynamic relations,
a dS rf = p+
r
rdf - dp
(41)
p
(42)
we can prove the existence of the supplementary conservation law (43) where S is the specific entropy (entropy of the unit mass), dynamic
absolute
temperature
and
p
the
a
the thermo-
proper energy density of the
fluid. The system (38), (39) may be put in the compact form (5) on choosing:
f
_ 0; (Il=O,l.2,3).
The supplementary law (43) coincides with (8) when g
O.
It is possible to show that the system of relativistic fluid dynamics possesses a convex covariant density. In fact: i) there exist the main field satisfying (12) 1
U' where G = f
-
eS
- 1 is the free enthalpy (it is remarkable that the
components of the main field, elle velocity,
(44)
a
all independet,
the absolute temperature
essentially coincide with Q
(because u uQ
=1)
and the free
enthalpy, Le. "observables" of the system); ii) it is possible to prove that the covariant density
is a convex function of the field
271
u
(45)
for any unity time-like covector{(a}
oriented towards the future, provided
the usual thermodynamic conditions 0;
are
satisfied
and
_
the
Sound
{G
ap
velocity
(G =3G/oa. G =oG/op)
}2> 0
is
a
p
smaller than the velocity of
light in vacuum. Hence
the
system
of
hydrodynamics
equat.ions
is a
particular case of
the general theory. As a consequence we have:
The system of re 1Ativistia hydrodynamics equations is syrrunetric hyperbo ldc
1)
in the fieUl. U' (44) with the four-veator generatirlfJ functions simpLy given by
This is a
r,
on
2) [ 5 ] > 0
when
conseq~ence
of i) since it is possible to show that
a n = _r*u ~
* a
where
)1*
is a
a
[5]
= r*(o_)l*)u
(
* a
[5] •
(46)
characteristic speed of the corresponding material wave: IJ*
= (u o *
1;) / (u 0
a( *
) •
a
3) The knauledqe of [5 ] as a function of u* and ~ 0 detierminee the shook, This is a consequence of ii) and (46).
The veLocity of propagat.ion of the reZativistia hydrodynamias shocks never
4)
exceeds the speed og Light. REMARK
We have seen above that the main field U' of a convex covariant density system is may
be
invariant
expressed
as
under
transformations
of
the
field
U.
and that it
the gradient of the convex covariant density h = a = hat • with respect to the field U = F (0 here h represents the a a component of the proper density of the conserved quantity h in the
272
congruence defined by the time-like covector It
is
another {u} a
remarkable convex
that,
density
in the fluid
h,
relative
to
}. a case, it is possible to define {~
the
field-dependent
congruence
with the same properties of h
h = ~u - rS , a whose gradient with respect to the field
(:' ) is still equal to the same main field U' (44)
u'
dh
Le.
Vh, (V -d{rS)
=
waul
= -(ua/e)d(pu a)
+ {f/e-S)dr
as it immediately verified taking into account (41) and (42). Moreover it is possible to prove that wi th respect to (il pi ilp)S ~ 1.
U and
h = -rS
is still a convex function,
the prove does not require the auxiliary condition
Hence the mapping U' ..... U is globally univalent.
REF ERE N C E 5 [lJ I.MULLER, Habilitationsschrift an der RWTH Aachen (1970). Arch.Rat.Mech. Anal. 40, 1-36 (1971). (See also "Entropy in non-equilibrium - a challenge to mathematicians" in Trend in Ap~lication of Pure Ma~1ematics to ;.Iechanics, Vol.II; Ed. H.Zorski, Pitman London, 281-295 (1979)). and
P .D.LAX,
Proc.Nat.Acad.Sci.U.S.A.
68
1686-1688
[2J
K.O.FRIEDRICHS (1971) •
(,]
K.O.FRIEDRICHS, Comm. Pure Appl. Math. 27, also Comm. Pure App1. Math. 31, 123-131 (1978)~
[4J
P.D .LAX, ~Shock waves and entropy" in Contribution to non linear functional analysis Ed. E.H .Zarantonello, 603-634 New York; Academic Press (1971).
749-808
(1974).
[5) G.BOILLAT, C.R. Acad.Sc. Paris 283A, 409-412 (1976).
[6J
G.BOILLAT and T.RUGGERI, C.R. Acad.Sc. Paris 289A, 257-258 (1979).
[7] I-SHIH LIU, Arch. Rat. Mech. Anal., 46, 131-148 (1972) .
(See
273
(8J
M.BERGER and M.BERGER , Perspectives Inc. New York, p.137 (1968).
in
nonlinearity . . W.A.Benjamin;,
[9]
T.RUGGERI and A.STRUMIA, "Main field and convex for quasi-linear hyperbolic systems. Relativistic (to appear).
densi ty dynamics".
covar~ant
fluid
CENTRO INTERNAZIONALE MATEMATICO ESTIVO (C.I.M.E.)
SINGULAR SURFACES IN DIPOLAR MATERIALS AND POSSIBLE CONSEQUENCES FOR CONTINUUM MECHANICS
B. STRAUGHAN
CIME Session on Wave Propagation Bressanone, June 1980
Department of Mathematics, University of Glasgow University Gardens, Glasgow G12 8QW
Singular surfaces "i n dipolar materials and possible consequences for continuum mechanics. B. Straughan, University of Glasgow.
1.
Introduction In this paper we study the evolutionary behaviour of a
propagating singular surface in two types of nonlinear dipolar materials;
a compressible inviscid dipolar fluid and an elastic
dipolar solid. The basic theory we use was introduced by Green and Rivlin
[1] and from the constitutive theory viewpoint essentially extends classical continuum mechanics by including gradients of the independent variables present in non-polar theories.
Gradient type
theories were suggested earlier by, for example, !1axwell and by Korteweg, see Truesdell and Noll [2], §125;
in particular,
Korteweg developed an interesting theory of surface tension by allowing for the possibility of rupidly changing density gradients in an interface.
Since in a singular surface quantities such as
density and its gradients of various orders may change very rapidly a study of wave motion in multipolar materials may prove of value. for an elastic dipolar material the theory we empl oy was derived by Green and Rivlin
[1], whereas the
con s t it ut iv ~ develop-
ment for dipolar fluid theory is due to Bl eus t ei n and Green [3]
278
(later modi f i ed by Green and Naghdi [4J). This theory allows for an additional dipolar stress as well as the normal one . In contrast with Newtonian theory the constitutive variables include temperature. velocity and density gradients. This is in one sense a generalization of the Maxwellian fluid of Truesdell (see [2]. §125 and the references therein) in that a dipolar stress is included from the outset. although Truesdell's Maxwellian fluid involves a constitutive theory which includes density. temperature and velocity gradients of arbitrary orders. We pay particular attention to the compressible dipolar fluid since as Truesdell and lIoll [2J point out. the Maxwel lian theory.
• •• "is set up in such a
~Jay
as to emphasize effects of
compressibility" • The dipolar fluid equations given by Bleustein and Green The equations of morner.tum and continuity
[3] are now reviewed. are (1.1)
0 • •• J1-~J
+ pf., 1-
where standard notation (see ego [3)
is employed throughout.
However, the energy equation takes form (1.3)
PI' -
peA
+
70s
i-
TS) - q • . + 1-~1-
.. + r.(")kA~ .. J1-d1-J 1-J :;)1.
t ..
= 0,
where Akj' = 1Jk •• and T •• and r. i j k are a symmetrr-Lc tensor and 1~J1J1the dipolar stress tensor, respectively, related by the equa t Io-.
(1.4)
'ij
= °ij
+ r.kij~k + p(Fi j -
ri j
)·
Here Fi j are components of a dipolar hody force and
ri j
is th e
dipolar inertia which has form (see Green and Naghdi [4])
(1.5)
rj i = d2(Vi~j
- 1Ji~k1Jk~j - 1Ji~k1Jj~k + 1Jk~j1Jk~i)'
d 2 being the inertia coefficient.
Furthermore, the entropy
inequality takes the form (l.6)
- peA· + S,,-) l'
q.T .
.:!:.-t..:!:.. T + 'ji d ij + r. (ij)kAkji -> 0 •
279
Bleustein and Green [3] develop the equations for a compressible, viscous dipolar fluid.
However, as we are
primarily interested in wave motion we shall derive the constitutive equations for a compressible inviscid dipolar fluid, following the procedure given by Bleustein and Green. Suppose then, that A, S, qi' variables
T
i j and L(ij)k depend on the
(1.7)
Using (3.4) of Bleustein and Green [3], the entropy inequality (1.6) may be written as aA
(1.8)
_.-
p--T . . aT •• ,tJ ,tJ
t
p
q.T . -
tT,t
t
T.-do. Jt 1-J
L( •• )kAY,.,. . ~ 0,
t
tJ
, "-
and the constitutive relations may then be reduced with the standard Coleman-Noll procedure whereby t he terms appearing linearly in (1.8) may be chosen arbitrari1y, the momentum and energy equations being balanced by suitable I i or r; coefficients of these terms are zero .
Hence,
(l.9 )
A
thus the
= A( p , p , 1-.,T).
Since, as in [3], A must be a hemitropic funct ion of i t s arguments, (1.9)2 reduces to (l .10) where (1.11)
A
= A(p,v,T),
280
What remains of the entropy inequality may be written as
[3], (3.llf»
(cf ,
(1.12 )
2p 'aA " (\10 •• 0\1
1,J
+
P .p
.)] d • .
• 'I-.J
'l-J
q.T • ~>O T - •
Next, since
Ii
and r are at our disposal in (1.1) and (1.3) we
may choose a motion such that at a particular
~,t
only the second
term in (1.12) is non-zero. Moreover, A" appears only linearly '1-,1 k and so we may select the motion in such a way as to invalidate the inequality.
Like-
Thus, the term multiplying A • 'k must be zero. 1,J
wise, we may argue that the term multiplying d • • is zero. 1,J
leads to the constitutive relations
This
0.13)
=-
0 .14)
p
2 aA '0\1 " (p .1,'O'k J + p.J'O1,'k)' '.
For a dipolar elastic material I find it easier with wave motion problems to use the equations as referred to a reference configuration, here taken to be a homogeneous one.
In this form
the equations are ([lJ, p.342) for the isothermal case, (1.15)
(1.16 )
IT(BA).· u
au
= Po aXi-• AB
(1.17)
where U
•
~s
.
the Lnternal energy, x i • A
stress tensor,
lT
1,
Ai represents a
lT
Ai is given by B4i is the dipolar stress and P
(see Green and Naghdi [4,5]) (1.18)
= ax./ aXA'
F Ai
= FAi
• f
Ai,
281
Ai components of dipolar body force and rAi the dipolar
with F
intertia defined by ([4], ego (11» (1.19 ) M AB
being a constant tensor.
In these equations the internal
energy is a function of x. A and x. AB' i .e. 'Z.~
'Z.~
u = U(X 'Z.~. A'
(1.20)
x , AB). 'Z.~
We should remark at this stage that the constitutive functions in (1.14) and (1.16)
ar~
given only for the
symm~tric
part with
respect to the first two indices of n~i or Ei j k • This does not make any difference to the equations of motion, as the skew symmetric part is left unspecified.
However. it is important in
considering boundary conditions as is discussed in
[1,3,5].
The
skew-symmetric part also plays an important role in three-dimensional wave motion studies (see [6]>, but in this article we shall not consider these matters an~ only investigate the novel effects of the theory present in a one-dimensional analysis. It is pertinent to draw attention to a paper of Mi ndl i n [7] who discusses other ways of describing elastic materials allowing for micro-structure effects and also includes a detailed account of linear wave motion and a critical comparison with results from discrete-type lattice theories.
2.
Dipolar stress waves Restrict attention now to the one-dimensional form of equations
(1.1). (1.2). (1.4). (1.5). (1.9). (1.13) and (1.14) with
f.'Z- = F'Z-J•• = 0
and suppose P. v, the density and velocity. now depend
only on x and t.
The one-dimensional form of the stress tensor.
dipolar stress tensor and dipolar inertia are denoted by a, E. and satisfy the relation a
=
T -
Ex .. prJ
r
282
where the notation JjJx signifies ut ive equations become
~~,
al!J ~ = at + v 1JJ;:; .
The constit-
Px p - 2p-q,
<2.2)
T
=-
(2.3)
1:
= - q,
(2.11 )
r = d2 ( iJx _ v 2 )
(2.5)
A
x '
= A(p,px)'
where t he pr es s ure p and dipolar pressure q are defined by
= p 2 ap' aA
( 2. 6 )
p
( 2.7)
q =
p2
2! . dP X
The cont i nui t y and momentum equations are ( 2. 8 ) <2. 9 )
Suppos e (2.8) and (2.9) hold on ~2 and p and v are continuous l y differentiable functions of x and t on P2. moreove r , that t her e is a surf ace 1: x {t } for each t gating in the :::-direct ion with speed
v, iJx '
un(~
0).
Suppose, ~
0 propa-
The quant it ies
vxx ' ~ , Px' Pxx are assumed cont i nuous f unct i ons of x , t on (R , 1:) x R and hav e at mos t jump di s cont inui ties across 1:. The jump in a quantity P i s defined by ( 2. 10)
where a supers cr i pt + s ign denotes t he region ahead of t he wave, a negative s i gn denot i ng t he r egion beh ind t he wave. A s urface as defined ahove is s a id to be a one-di mens iona l di polar stress wave and the corresponding wave amplitudes are de fined by
= [pxJ.
( 2.11 )
B
<2.1 2 )
c =
.rv l :::
283
To obtain the first discontinuity equation, (2.8) is differentiated with respect to x and limits either side of
r
are taken
to find (2.13) To further analyse this equation we observe that for a function W it can be shown (see Truesdell and Toupin [8]) (2.14)
where ~/~t is the displv~ement derivative defined by (2.15) and the relative wavespeed is (2.16) With the aid of (2.14), (2.13) may thus be rearranged as (2.17)
- UB + pC
= O.
Until this point our analysis has proceeded as for a third order wave for a classical perfect fluid;
it is now things change.
For, if we apply the jump conditions to (2.9) and use (2.14) we find the resulting equation already involves the displacement derivative of C.
Thus, it is necessary to adopt another approach
to find a suitable equation to employ in conjunction with (2.17). We instead adopt the approach which is us ed with shock waves in a classical perfect fluid and use the discontinuous equation of balance form of (2.9) (see Green and Naghdi
[9]).
For this
reason we refer to the third order waves s t udi ed here as dipolar stress waves. The appropriate equation of balance is (2.18)
- U [w]
= [a].
Of course, for the waves considered here, the left hand side of this equation is zero, and so (2.19)
284
By the assumed regularity conditions at the wavefront, (2.2) allows us to deduce that
[T] = 0
and similarly, with the aid
of (2.4), (2.20 ) Also, employing (2.3) and (2.7: (2.21)
Inserting these expressions into (2.19) leads to p2
2
aA
3P;
B
= O.
finally we solve for U between (2.17) and (2.22) to obtain (2.23)
U2
= .!... ~ cJ.2 3p
t
:x:
(cf. the corresponding expression for sound waves in a classical perfect fluid, ego Truesdell [10].) To determine the behaviour of the amplitude as time evolves we differentiate (2.8) first materially with respect to t, and then with respect to :x:, and then take jumps across the wave; also, we take jumps of (2.9) to find a second equation.
The
details of these calculations are contained in [6] and so we simply describe the results here. (2.24)
6B + iIT
2V:x: B + C
(2U PV:x:
The relevant equations are
P)
+ P:x: - U - UE + pG = 0,
and t
Although the principal part of (2.8) and (2.9) suggests these equations may be written as a hyperbolic system this is not evidently so.
While we would have two real wavespeeds as
in (2.23), the remaining two would be complex.
I am indebted
to Professor T. Ruggeri for pointing this out to me.
285
(2.25)
2 + .;....;o.+-"'""----V aZn 1 an W 2 an) = 0, ~ U ~ apZ U3 ap 6t UZ x ap x x :t:
--p
where
Finally, equations (2.24) and (2.25) may be combined together to obtain
2!. + _1_ ~
(2.27)
6t
a
2UZ 2 apZ
BZ + KB
= O.
s:
where K is given by 5 (2.28)
K=-V 2 Z
+
UP:t: P:t: a q p~ a q 1 6U --+---+-----. 2p uaz ap ap ua z apZ 2U 6t Z
:t:
2
:t:
The amplitude equation (2.27) is of Bernoulli type and can be solved explicitly. see Chen [11] who also includes several theorems describing the behaviour for various initial data ([11], pp. 387- 395). Of course. once B(t) is known C(t) may be found from (2.17). Rather than describe in detail the solution to (2.27) we suppose the reginn ahead of the wave is one of constan! density for which K = O. For this case (2.27) has solution (2.29 )
If the initial wave amplitude is such that sgn B(O) then the amplitude becomes infinite as
286
dipolar stress wave in some sense breaks down. (Iihile the treatment of the continuity and momentum equations is different to the usual procedure for acceleration waves in a perfect fluid. we have essentially used the compatibility relations in the manner employed by Chen and his associates. see ego [11].
This approach would initially appear
~ifferent to that of Jeffrey. see ego [12];
but. the two methods are in fact equivalent as was shown by Boillat and Ruggeri [13].)
3.
Elastic dipolar stress waves. While it would be possible to develop a theory of three
dimensional elastic dipolar waves. in the spirit of the work described by Chen [11]. it seems more revealing at this stage to concentrate on the novel effects caused by the presence of strain gradients.
As these effects are present in one-dimen-
sional wave motion we restrict attention to this case.
To
this end, therefore. let us rewrite equations (1.15) - (1.20) in their one-dimensional form. Let v and F = aX/ax be functions of X and t and supposing the body forces are zero the equation of motion is then
where (3.2)
11
= poU:tX
where in this and the next time derivative. F
= aX/ax.
§
+
Po
au
aF -
Po
au
W. X
a superposed dot denotes the partial
M is the dipolar inertia coefficient
and
In the form (3.1) it is clear that the dipolar theory leads to a differential equation of fourth order.
287
An elastic dipolar stress wave is defined to be a surface across which the third derivatives of x are finite discontinuities, i.e. x €
c2 aR 2 ) .
~he firs~ ~o
~
have
Because of ~he
order in (3.1) it is clear that to obtain the wavespeed of such a wave we must consider the discontinuity relation given by Green and Naghdi in Chen [ll]).
[9] (cf. the shock wave equation
(5~16),
If we deno-te the wavespeed by V the appropriete
halance relation is
POV[.i:] = [11] Subsequent analysis makes use of the relation (equivalent to (2.14»
~t
(3.5)
[I] = [I]
+
v[if]
In view of the assumed regularity of z , (3.4) reduces to (3.6 )
[F ] -a2u = M [.. "x] . XX aF2 X
From use of (3.5) this may be changed to
a2u = H
(3.7)
When
V2
aF2 X
[FXX] t
0, which we are implicitly assuming, <3.7) leads
immediately to an expression for the wavespeed, V, (3.8)
V2
2u
= .!. a
M aF} It is worth comparing (3.8) with the analogous expression for an acceleration wave in a nonlinear (ordinary) elastic material, see ego Chen [11], equation (6.12). To derive an expression for the wave amplitude we now return to equation (3.1) and take the jump of this. (3.9)
0
= r.L(l£)l -
Lax
sr
rl:....(.2!!...)]
J lax 2
aFx
The result is
+
n [~xy] -
288
Expand ing the first two terms on the right gives
(3.10)
(3.11 )
a2y [:X (:~)] = 'ilFX~F [FXX] (::2 (:;X )]
[ f a2y
= F
XX]
--10
afaF
X
+
3y} a3y 10-aF 2aF aF3 X X 2y a [F ] aF2 XXX 2 -a -
[Fh]
X
These expressions are inserted into (3.9) and together with (3.5) and the relation
(3.12) we may derive the following ordinary differential equation for the amplitude, aCt) = (3.13)
[FX]
;
cSV 2 a3y FXX a3y} 2M~+ a { - -M -1o-F - - - + 2 - - 10
V X aF2aF X
V cSt
cSt
1 a3y 2 +-- a V2
aF3X
Again, th is equation may be so lved as in Chen
F;
ahead of the wave is in constant strain,
= a(O)/
aF3 X
= O. [11] •
= F+
XX
constant and then aCt)
V
= 0,
If the region
V is
(1 + t ~ a mv aF3
3y).
2
X
a3 y
Obviously, under the right conditions on a(D), M, V and - - there will be breakdown of aCt). In both th is and the preceding
aF3X
it is eviden t t hat hi gher
order waves , i. e . of one order higher than what i s usually def i ned as an acceleration wave, suffer a type of breakdown ana logous to that observed in acceleration waves in classical theories, usually thought to be associated with formatio n of a wave of lower order. If the behav iour encount ered here i s not to be acc ep ted, t hen one
289
may argue: the dipolar theories should not be used as exact
(a)
theories for wave propagation problems. the classical case is the first order approx-
(b)
imation and dipolar terms should be included in an expansion procedure. If (a) is thought to be the case then perhaps one has to look at a viscoelastic type theory in which gradients of all orders are present.
Certainly, it would appear necpssary to invest-
igate higher order theories. If we try to study shock waves or acceleration waves in the theories considered here, we find that the basic equations are not in a
suita~le
form.
gov~rning
For example, equation
(3.1) contains terms of second, third and fourth orders and it is not obvious how to proceed in that case.
It is perhaps
likely that the higher order nature of the equation prevents the formation of waves of lower order.
Certainly the phen-
omenon of focussing (see ego Koops and Wilkes [l~J) is not likely to occur.
a linear
To see this and complete this work we consider
dipolar elastic material and show that, unlike classical
linear elasticity, stability in the CO norm is possih1e.
(This
possibility was pointed out by Koops and Wilkes [14J and Koops and Straughan [lSJ).
4.
Uniform stabil;ty of a linear d;polar elast;c mater;al. The appropriate equations for an anisotropic l inear dipolar
elastic material, which may be derived from the cited papers of Green and Naghdi and Green and Riv1in, are (see [l SJ ) pU. - pf ••• 1-
where u
J1..J
T •••
J1..J
+ l:(k.).
1
•
::; 1.. ,<J
=-
pF .•• + pr·, 1-J.J
are the components of displacement about a reference
i configuration and the other terms are named in §1.
inertia coefficient satisfies
Here, the
1.
290
r j i = mj k
(4.2)
ui • k '
(1$)
where mi j is a symmetric tensor which defines a non-negative bilinear form. The constitutive equations for Tij and L(kj)i are
(4.4)
E(kj)i = hpqi j k Up~q + CpqmiJk Up~qm'
where the "elasticities" satisfy the following symmetries
Ci j khmn = chmnijk We suppose (4.1) are defined on a bounded region 0 of R3 Say with U.1. and aU i prescribed on the boundary of o. r.
au
Ui = qi
(~.t)
hi
(~.t)
au.'l. = __ an
on
r )(
[0 ...).
(Green and Rivlin [1] suggest that (4.6)2 should be replaced by conditions on the dipolar traction.
However. we have chosen
these conditions so that we may apply the appropriate Sobolev inequality for illustration of the new stability effect.) Let Vi and
V~
be two solutions to (4.1) in 0 with the same
body forces and same boundary data (4 .6).
Define
u.
'l.
= v*i -
and then a routine calculation enables us to show the total
V.
'l.
energy E(t) is constant in time. i.e. def (4.7) E(t) = ~ (pll .ll. + pm'k U. .U. k
f
p
+
'l. 'l.
J
'l.. J
'l.• •
a kmij Up..mUi.j + Cmnqijh Um.nqU i• j k
+ 2 b1'8tij U1'.8 tU''l..J. )dx
=
E(O).
'V t
~ 0.
If now, the potential energy is positive definite in the following sense
291
(4.8)
J (a'_'J' Uk,mu,1-, J,+2b 1<1111-
t"
1'8 1-J
U
1',8
.».
, ,+cT11I1lj1-J t U1-,J '. 'k U171,,",! . 1-,J'k)dx
Q
(A > 0).,
we deduce 'from (4.7) that (4.9)
E(O)
~
A
, 'k liz. J u,1-.J'k u1-,J fl
However, for
Q
C R3, ~,2
(Q)
C CO(Q), see Gilbarg ~nd Trud~nger
[16], corollary 7.11, and so there is a cons rerrt u depending on Q, A such that
(4.10)
,
[E(O»)' ~ \l sup fl
IY(lS,t)l.
Thus. by making F(O) arbitrarily small
lui
is likewise a~bi~rarily
small and so the solution is stable in the CO norm.
Acknowledgments I should like to express my s incere thanks to Professor D. Graffi for his kind invitation to study in Italy in June - July 19BO and to both Professor Graffi and Professor G. Ferrarese for the opportunity to hold a seminar in Bressanone.
finally, I
wish to express my appreciation to Professors D. Graffi and M. Fabrizio and to Dr. F. Franchi for many stimulating discussions.
292
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