Wave Propagation [1 ed.] 3642110649, 9783642110641 [PDF]

Zitiervorschau

e,

ressa

•

ng

Iy 980

Giorgio Ferrarese (Ed.)

Wave Propagation Lectures given at a Surruner School of the Centro Intemazionale Matematico Estivo (C.I.M.E.), held in Bressanone (Balzano), Italy, June 8-17,1980

~ Springer

FONDAZIONE

CIME ROBERTO

CONTI

C.LM.E. Foundation c/o Dipattimento di Matematica ''D. Dini" Viale Morgagni n. 67/a 50134 Firenze Italy [email protected]

ISBN 978-3-642-11064-1 e-ISBN: 978-3-642-Il066-5 DOl: 10. 1007/978-3-642- I 1066-5 Springer Heidelberg Dordrecht London New York

©Springer-Verlag Berlin Heidelberg 2010 Reprint of the 1" Ed. C.LM.E., Ed. Liguori, Napoli & Birkhauser 1982 With kind permission of C.LM.E.

Printed on acid-free paper

Springer.com

CON TEN T S

c

0 U

r

8

e

8

A. JEFFREY

Lectures on nonlinear wave propagation

Psg.

Y. CHOQUET-BRUHAT

Oodes 8symptotiques .•..•••.•••••..•••

"

99

G. BOILLAT

Urti .•••••.•••••.••..•.....•.........

"

167

"

195

continui •••..•....•..•...............

"

215

Onde nei solidi con vincoli intern!

"

231

1

5 e min arB

D. GRAFFI

Sulla teoria dell'ottica non-linear.

G. GRIOLI

Sulla propagazione del calore nei Mezzi

T. MANACORDA T. RUGGERI B. STRAUGHAN

"Entropy principle" and main field for

a non linear covariant Byat••••••.••.

"

Singular surfaces in dipolar materials and possible consequences for continuUN mechanics .....•.........•••••.•••.•.•

"

275

CENTRO INTERNAZlONALE MATEMATlCO ESTlVO (C.l.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 CONTEIITS

Lecture 1.

Lecture 2.

Fundamental Ideas Concerning Wave Equations 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

Quasil1near Hyperbolic Systems, Characteristics and Riemann Invariants

2-1

1.

Characteristics

2-1

2.

WSvefronts Bounding a Constant State

2-6

3.

Lecture 3.

Lecture 4.

Riemann invariants

2-8

References

2-12

Simple Waves and the Exceptional Condition

3-1

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.

General Considerations

4-1

2.

The Initial Value Problem

Time and Place of Breakdown of Solution

4-2 4-2

References

4-9

3.

Lecture 5.

1-1

The Gradient Catastrophe and the Breaking of Water Waves in a Channel of Arbitrarily Varying

5-1

Depth and Width

Lecture 6.

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 1. Conservation Systems and Conditions Across a Shock 2. Weak Solutions and Non-Uniquenes&

6-1

6-1

6-4

10

Lecture 7.

3.

Conservation Equations with a Convex Extension

6-11

4.

Interaction of Weak Discontinuities

6-13

References

6-14

The 1l1emann Problem, Glimm.'s Scheme and Unboundedness of Solutions

7-1

1. 2. 3.

The Riemann Problem. for a Scalar Equation

7-1

Riemann Problem for a System

G1im1ll's Ilethod

4.

Non-Global Existence of Solutions

7-3 7-5 7-8 7-10

References Lecture 8.

Far Fields. Solitons and Inverse Scattering

8-1

1.

Far Fields

8-1

2.

Reductive Perturbation Method

3.

Travelling Waves and Solitons Inverse Scattering

8-3 8-6 8-9

References

8-13

4.

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, 1s that some attribute of it can be shown to propagate in space at a finite speed. In time dependent situattons, 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 which have nonlinear terms, then they also can often be regarded as describing wsve 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 liquids.

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 first 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 1s 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 1n the more general context of the variable coefficient equation

3

L

~,j·O

3

a

ij

U

i j x x

+

L

i-o

biu i x

+ cu

(2)

f

with aij' b i , c, f functions of the four dimensional vector

~

0 1 (x • x

x

2

Not all linear second-order equations of this form describe wave motion. and 00

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 classification to be adopted utilizes the coefficients of the highest-order

rl?-r!va~~ve~

and has an algebraic

~asis

but, as will be seen

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 3

I ""ijoJ i- where the GC

0,1,2,3

i

(3)

are constants.

ij

A transformation of this form gives an affine mapping of the

(xo, x1 , x 2 , x 3 )-space which Is one-one provided det I~j I

~

O.

Employing

the chain rule for differentiation we find that equation (2) may be re-wrltten 3

o•

I

(4)

i,j,k,l"O Hence the coefficients 8

1j

of the derivatives

U i

j' which are functions of

x x

position, transform to the new coefficients

of the derivatives

k l' which are also functions of position.

U

If, now,

f; f;

we confine attention to the set of coefficients a

specific point

~

z

!o

1n

o (x ,

appropriate to some ij 123 x • x • x )-space, we see that this is exactly

the transformation rule which would apply to the coefficients

8

ij

of the

quadra tic form 3

I

i,j-O

a ij nin j •

(5)

when the n i are transformed to Bit by the variable change 3

ni

I k-O

akiak'

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 attention to a fixed poine

the

8

~

the specific values

1j

8

•

!o

ij

8

ij

, let us again confine

o 1 2 3 in (x , x , x , x )-space and attribute to

• alj(~)'

This then implies that some choice of the numbers G

ij

• d

ij

exists for

which 3

I

1,j-O

where m + n < 4.

The number pair (m.D) Is called the signature of the

quadratic form (5) and, being an algebraic invariant, 1s used to classify the quadratic form.

We shall use it to classify the

partial differential equation (2) at each point

~

-

va~lable

coefficient

!o.

1n the transformation The effect on equation (2) of us1ng these numbers Q 1j (3) is co yield at .! • m-1

I 1-0

!!o

a differential equation of the form

1Itkl-1 U

3

I

tit i

U

i-m

1 1 t t

+

I

i-o

biu 1 t

+

f

Equation (6) or, equivalently, (2) is called hyPerbolic

o ( -direction

(6)

0 a~ ~

z

!o

in the

when the signature is (1,3), elliptic when the signature is

(4,0) and parabolic when m + n < 4.

o

If an equation is hyperbolic in the ( -

direction at each point of a region O. then it is said to be hyperbolic in the CO-direction throughout O. 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

2 constant factor llc , 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 lIe

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 the least of

which 1s 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 1s it possible to define the propagation speed of this feature unambiguously.

Such is the case with

a wave froDt separating, say, a disturbed and an undisturbed region and across which a derivative of the solution 1s discontinuous.

3.

The Cauchy Problem - Characteristic Surfaces Fundamental, to the study of hyperbolic equations 1s 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 determination of a unique solution to an initial value problem in which a hypersurface F 1s given, and on it the function u 1s 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 (0, (1. (2. (3 and to let the hypersurface F on which the initial data is to be specified have the equation ( to (

o

o

~

O.

In terms of the new variables. a derivative with respect

is a directional derivative normal to F so that it is an exterior

1 2 3 derivative. whilst derivatives with respect to ( , ( • ( are interior

derivatives. We now utilize this by rewritine equation (2) in a form in Which the derivative u

~o~o

is separated froa the other second-order derivatives

16

3,

L

i,j,k,.e.""O 3

Uk

i,k- 0 and such that U is

t •

o

Denote by

t

c

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

A~t), and

constant ,

introduce the curvilinear coordinates t'

constant

through the equations t'

(lOa)

t

and.

o.

(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 80 dx dt

-

~(.) along • • constant.

Thus, since equation (13) is only valid along

~ c

constant and from (lOa) we

have t' • t, equation (13) 1s identical with (13' )

which 1s 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

in1tial condition Hx,O)

x

when the wavefront 1s given by Hx, t)

0

and, in the region of constant state ahead of the wavefront, ~(x,t) >0 •

The transformation introduced through equations (10) is non-singular provided the Jacobian

• ..!... ~x

1s non-zero and finite.

(14)

The initial condition on • ensures that x. 1s initially equal to unity and so we may assume the non-vanishing of the Jacobian for at least a finite time after

t

s

O.

Let us denote by L the open region lying to the left of the

advancing wavefront C(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 t.

Then. since no characteristics enter L, U will

remain smooth in L for at least a finite time.

All subsequent limiting

operations on the side f < 0 of the wavefront will be assumed to be performed in L.

Let l(j) be the left eigenvector of A corresponding to the eigenvalue ;\ (j) then, frOID equation (3)

o .

I::

I

Employing the identities

o taken along ~ c o. limdt

Thus, integrating equation (21) with respect to t' between 0 and T and noting that X is a jump quantity defined across operation

x -

Xjust

~

c

0 we may use the limiting

defined to obtain the result

x+ ITo (~A(" u 0

ITdt' •

(22)

This equation describes the variation of X along the wavefront

~

• 0 with

advancing time and in writing equation (22) we have tacitly assumed that th~ multiplicity r. of A(t) remains unchanged. and at t' - Tl(T

l

Should this assumption not be true

< T) the multiplicity changes, then n must be re-determined

for the interval t' > T • l

We remark here that although an initial condition

O. this limiting on U may be prescribed arbitrarily by specifying lim (U) x tx x->o 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

or

x • when equation (22) becomes

x+

x

l

x +

-

+

JT (V 0

u

~(+»

0

fidt' •

However, from (14') we see that the left-hand side of ,this expression 1s

simply the Jacobian of the transformation and so is required to be finite and non-zero in order that the transformation 1s unique. critical time T • x~

Jacobian

• X

+

t

So, if there is a

at which condition (14') ceases to be valid and the

c

O. this is given by

x 0

+

o.

(23)

Geometrically the vanishing of the Jacobian 1s equivalent to the point at which the

~

• constant lines first intersect the wavefront

family of characteristics,

To determine

1 +

Jto

t

c

z

= O.

(i.e., the

constant intersect at a cusp).

in terms of U we divide equation (23) by x

c (V).(+» 0

u

~

o .

The vector n must be determined from equations (18), (19).

x.

to obtain

(24)

In the particularly

simple case that B • 0 it follows directly from (18) and (19), provided the multiplicity of value

n.

and

A'+>

is constant, that

n is a constant equal to its initial

80

n • and thus

iJx Using the definitions of X and n we see that

55

when

• x.(l + (V A('»

x

,.

u

0

Ux t')

but

and

80

U is given in L by the expression x

UIII x

+ (V A('» u

0

Ux t')

(25)

•

Thus U becomes unbounded if the denominator vanishes for some x

(V A(9»O U

=0

and

is infinite.

80 t

c

t

c

>

O.

If

it follows directly from equation (21) that X is a constant

remains finite for all

The discontinuity in this csse 1s propagated but t~.

Systems for which this property is true are

a special case of those which are exceptional with respect to the A(') characteristic field.

The general case when A, B depend on U and also explicitly on x, has been discussed in detail in [1].

t

A different approach to the problem

that involves three space dimensions and time has been described by Balilat [2,] and Chen [3].

References

{I] [2] [3]

A. Jeffrey, Quasl11near Hyperbolic Systems and Waves. Research Note in Mathematics No.5, Pitman Publishing, London, 1976. G. BoilIat, La Propagation des Ondes. Gauthier-Villars, Paris,1965. P. J. Chen, Selected Topics in Wave Prop~gation. Noordhoff J Leyden, 1914.

50

Lecture

5.

'l'be 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 1s 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. Hvungi [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

upwards, and write the equation of the bottom of the channel as

y+

hex) -

o.

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

o.

(1)

However. the equation corresponding to the conservation of mass will now be different on account of the widtb variation of the channel.

To derive

it. all that 1s necessary 1s to observe that tbe 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)

•

froll which it follows that

"t

+ [u(n+h)] x + u(n+h)(Wx/W)

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 tbe 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 1n the direction of increasing x, starting from x • 0 at

t

-

0, and that it moves into water at rest.

Then, across

the wavefront: (1)

u and n are continuous, with u(x,t) • n(x,t) • 0 ahead of the advancing wave,

(11)

the first aod 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. Usiog a superscript minus sign to denote the value of a function immediately behind the advancing wavefront (i.e. at the edge of the disturbed region) we conclude from (i) that n

U

o

(4)

Taking the total differential of equations (4) gives, just behind the W8?efront, -0

• u;dx + u~dt aoo

0

or, equivalently,

• -eux and nt

ut where c

E

(5)

-en x

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 1s 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

sex)

- nx

(7)

58 when (5) and (6) become and

u z

gale.

(8)

Now notice that the operation of differentiation with respect to x along the characteristic followed by the wavefront, behind which u; and u

t

are defined, takes the form (9)

It then follows immediately from this that

e

2

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.

Then eliminating

n ' and using (7) and (8), we find xt

c 2U - xx

2ih

U

tt

gewz]

3 2

x + - - a +~ a + (- - c W c

2

o.

(11)

Combining (10) and (11) and using (8) brings us to the required Bernoulli type equation for the amplitude a(x),

3h W] 382 da+ [ 4hx + 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/2e. 3.

The Amplitude sex) And Its Implications The standard substitution a • b-

l

reduces the Bernoulli equation (12)

to a linear first order equation, and a simple calculation then shows that (13)

in which a

l(x)

O

o•

• a(O). W

3h 3/4 w 1/2 o 0 2

W(O) and

(14)

S9 A wave of elevation corresponds to to

8

0

>

8

0




1/1(1), and at the shore line if laol ~ 1(1).

A wave of depression (8

0

>

0) in a variable width channel can only

break 1£ the depth of the water shelves to zero, and then only at the shore line provided 1(1) When we set W(x)

= We'

< -.

these general conclusions agree with the

special case of waves climbing a beach that was studied by Greenspan [3]. This is because the

one-dimensi~nal

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' he and We 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 2h

-~o

at a time t

2

c

-~o

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

(16) Result (16) also shows that when the water deepens at a constant Tace (m

< 0),

then although a wave of elevation will normally break, this will

not occur in the special case that 28

0

- m.

This was the result found in

[4] which used the transport equation approach that has been presented in a

general form in [5].

The equivalence of the method used here and of the

seemingly different one used in [4] and generalised in [5] has been established by Bol11at and Ruggeri [6].

References [1]

A. Jeffrey and J. Mvungi, On the breaking of water waves in a channel of arbitrarily varying depth and width. ZAMP (submitted for

[2] [3]

J. J. Stoker, Water Waves. Wiley-Interscience, New York, 1957. H. P. Greenspan, On the breaking of water waves of finit~ amplitude on a sloping beach. J. Fluid Mech. 4 (1958), 330-334. A. Jeffrey, On a class of noo-breaking finite amplitude water waves. Z. angew. Hath. u. Phys. (ZAHP), 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.

publication.

[4]

[5] [6]

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

m3

x

variabJ~s

t, we assume it can be written in the divergence

form

aF at + with U

&

div G

F•

U(~,t),

vectors and G •

UI

B• F(U,~,t)

G~,~,t)

and H

&

H(U,~,t)

an n x 3 matrix.

all n element column matrix

The matrix G in (1) is in

effe~t

to be regarded as a tensor so that div G has the meaning 3

div G

8-1 I

*

a CsI 8

where 8(s) is the s-th column of G. Systems of this type are of considerable importance because of their frequent occurrence in physical 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 Vet), which is itself bounded by a surface set) moving with velocity

Then the rate of

~.

change of the volume integral of F is given by

d

dt

of

FdV JV(t)

fV(t)

at

+ J

dV

F .!.d§. S(t)

where d! is the vector element of surface area. Let us now identify the column matrix F 1n Theorem 1 with the n x 1 column matrix F in system (1) and assume that a surface

a(~.t)

• const

exists acrOBS which the matrix vector U, and hence F. G and H are discontinuous. Next we choose the volume Vet) bounded by surface set) moving with velocity 90 that an arbitrary part SO(t) of the discontinuity surface divides it into the two sub-volumes V+(t) and V_(t).

o(~,t)

~

• const

Denote by S+(t) and S_(t)

those parts of Set) that bound V+(t) and V_(t), respectively, excluding the dividing surface So(t) which, we assume, also has velocity Integrating (1) over Vet) - V+(t)

I

v+uv_

~:

dv +

I

v+uv_

d1v G dV

uv_ (t)

fv+uv_

H

~.

gives 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,

of

f V uV at

+ -

dV

+f

G. d§.

(3)

S uS

+ -

where G.dS 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 1n which. it must be remembered, the dividing surface SO(t) that is part of

a(~,t)

- const also moves with velocity

~

(4 )

If, now, we subtract from (4) the corresponding expressions integrated over the separate .volumes V+(t) and V_(t), and bounded, respectively, by S+(t)USo(t) and S_(t)USo(t) we arrive at the result

63 (5)

where d.4 and dE. _ are the outward directed surface elements with respect to

the volumes V+(t) and V_(t).

This situation is illustrated diagramatical1y

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

directed

along~'

d~'

parallel to a(x.t) - const vanish In the limit as the

cylinder collapses onto the area element dS ' O

,,-

r1

Vo1ume element divided by discontinuity. surface d~

Since

a(~,t)

• const.

are both normal to the same discontinuity surface a(x.t)

but are oppositely directed so that n ..

-0

we have d.4 • ...dS

". !!dS

O

c

const.

showing

that (5) may be re-written as

o. The fact that dS

O

(6)

is arbitrary then gives an algebraic jump condition across

o (.!' t) • const of the form (7)

It is useful to re-express this result by observing that the scalar quantities

.!.t-0.!!.

d4 and dS

and .!._ o.!!. are the normal speeds of propagation of the elements

on opposite sides of, and moving with. o(~,t) - const, and as such

must be continuous across SO(t).

So writing ~ .,

.!+'E. - .!._ o!!

enables the

jump condition (7) to be expressed in an alternative form using the speed ~ normal to S

64 (8)

which 1s sometimes written

~[FJI •

[Gl~.'!.

(9)

with [Q]J denoting the jump 1n Q across discontinuity surface SoCt).

The

arbitrary nature of dS O also implies that U+ varies continuously over S. Because of the similarity of (8) to a corresponding condition 1n gas dynamics this result will be called the generalised Rankine-Bugoolat condition for system (1).

In general,

~

1s

~

equal to a characteristic speed A.

Theorem 2 (Generalised Ranklne-Bugonlot Condition) Consider the conservation system

;~ + div with F

a

G

-

F(U,~,t)1

B,

G~

G(U,~,t)

and H •

H(U,~,t).

Then, if this has a

discontinuous solution across a surface 8. on the adjacent sides

± of

S the

solution varies continuously and 1s related by the jump condition

in which n 1s the normal to S and j 1s the normal speed of propagation of S, with C+ regarded as a tensor and unit vector

~

G±.~

denoting the scale of prodyct of G and the

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

c1

solutions, and piecewise differentiable

cl

solutions separated

by shocks across which both U and its derivatives are discontinuous.

It

would be desirable, 1f 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 startins point we take the equation

au + feu) au • o.

at

(10)

b.

sub1ect to the 'initial condition u(x,O)

(11)

g(x)

and assume that feu) is a continuous differentiable function of u.

Then the

first point to notice 1s that (10) can be expressed in conservation form by

defining F(u)

(12)

If(U)dU •

to obtain (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 solutioDs 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, let us consider some strip 0


0 will be

called a weak solution of (10) 1f in this half plane it satisfies the condition

o •

(14)

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 zero 19 then known as the support of the test functions.

000-

As a general

1 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

1 Thus a piecewise C weak

classical solution, as would be

expected of any reasonable extension of the concept of a solution. Let us now show that there is a further common property shared between weak and piecewise

c1 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 oR 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 3R_ and 3R+ 1 and let the directed arcs along adjacent sides of L be oL_ and 3L+. as in the Figure.

t

L

Shock line L dividing R Then R - R_ UR+ and oR • 3R_uoR+.

The test functions w in (14) will be

assumed to have their support in R

80

on oR.

that the test functions w will vanish

Thus (14) may be written

II!:;

u

+ :: F(U») dxdt

o

(15)

67 Now multiply (13) by wand integrate over R_ to obtain

JJ

_ R

(w :~

+

W

:~)

dxdt

-

O.

which may also be written in the form

a(wu) + a(WF») [ at ax

dxdt -

II

dxdt

•

o.

(16)

R

Applying Green's theorem to the first terms in this result then transforms (16) to

I -wFdt 1aa vaL

o.

+ wudx _

(17)

However as the support of the functions w lie in R, w will be zero on itR

80

thOt (17) reduce. to

1 -wF(u_)dt

TaL

+ w u_dx

(18)

A similar result applies with respect to R+ where we find

taL-

F(U+) dt +

W

+

u+dx -

Ill:~

u + :: F] dxdt

-

o.

(19)

+

the integration along

at and 3L+ being oppositely directed, as indicated

in the Figure.

If (18) and (19) are now added. the sign of the line reversed with a corresponding replacement of aL

integral~ln

(18) is

by 3L+ and result (15) 1s

used we find (20)

where as the point (x,t) 1s now constrained to lie on 3L+ the term (dx/dt) represents the speed of propagation

~

of the shock along L.

As

w i8 arbitrary,

(20) can only be true if (21)

which 1s the one dimensional form of the generalised Rankine-Hugooiot 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

o if for all twice continuously differentiable test functions w with support

In

t

> 0 the function u Is Buch that

ff"" { Jl at:

I)

aw PCu) ] 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 au + aF(u)

at

ax

_

o.

The following results are then true: (a)

If u Is piecewise C1 in additIon to being a weak solution it is also a piecewise C1 classical solution.

(b)

1 a piecewise C weak solution satisfies the generalised Rankine-Rugoniot condition

across a discontinuity moving with speed A; (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 l 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 Deed to be imposed on classical solutions. solution is successful and,

In this respect the notion of a weak

furtbermore~

because of its method of definition

1 the class of weak solutions is even wider than the class of piecewise C

69 functions

80

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 C1 solution. a weak

solution 1s not determined tmiquely by the initial data.

Th.1e is most easily

deDJostrated by means of a simple example.

Consider a Riemann problem for an equation of the form

[1 3}

au +...!. at a,,"3

•

u

0.

with

80

for x < 0

Ol'

•

ues.O)

{

for x > 0

that in (13) we have F(u) • u 313.

11leu,

the equation

a8

'!,8

ho.:)gea.eous, when it 1s differentiable a

non-constant solution u viII be a function of x/e, and it 1s easily verified that the function

for

0 u(". t)

•

(,,It)1

1 1

1s a

cl

"It for




). - 0, .•. t-I

146

A(x,p)

et

o

sur une telle section on a

Soit n : T%X ~ X la projection canonique (x,p) l~ x de r*x sur X. On defiDit une J-forme sur TXx. appelee I-forme fondamentale ou forme de soudure par

u~T

6(x,p) (u) - p(fl' (u))

x,p

T"x

son expression en coordonnees locales est ;

6 La 2-forme

o -

d6

est fermee et de rang 2t, une telle 2-forme est dite symplectigue. La 2-forme cr munit

T*x

de sa structure symplectique canonique. Une sous-variete de T*x

qui annule a et qui est de la dimension maximum possible, c'est

a dire:

t •

est dite lagrangienne.

5i Vest une variete. de dimension t. immergee dans

T*x

par une application

f, on dit que (V. f) est une sous-variete lagrangieone [ilDl1ergeeJ de T*x si

sur V

au

£

2 Recherche dlune sous-vari€te lagrangienne (V.f) de TXx telle que A(x.p) ~ O.

Le probleme est celui de la recherche des varietes integrales (immergees

147 dans T~)de dimension t du systeme differentiel exterieur

o

(2-1)

o

(2-2)

A(x,p) '" 0

La fermeture de ce s~steme contient, outre les equations precedentes. l'equa-

T*x :

tion exterieure sur

o

dA •

(2-3)

Le systeme caracteristique de 2-1. 2-2 est le systeme associe de 2-1, 2-3. II est constitue par les champs de vecteurs v sur TXx tels que :

(2-4)

ia-kdA

avec

v

c'est

a dire

k€lR

en coordonnees canoniques (xA,p.>..) de TXX

ou

v-

A - O. • ..• 1-1 VA+!

(2-5)

- aA/a/'

Un champ de vecteurs VA possedant la propriete 2-4 est dit hamiltonien pour la structure symplectique a et l'hamiltonien A. On remarque que VA est tangent

a

Jt (sous-variete

A(x,p) • 0 de T*x)

Une trajectoire du champ de vecteurs hamiltonien VA est appelee une (courbe)

bicaracteristique de Itequation aux derivees partielles A(x. On

• O.

suppose que I'hamiltonien A nta pas de point critique sur cf};(c'est

dire que dA sur

~x}

cfC.

Theoreme

~

a

0 quand A • O}. Ie champ hamiItonien VA n'a alors pas de zero

et on demontre Ie theoreme fondamental suivant (cf par exemple

Ill):

Soit Y une sous-variete compacte de dimension t-I de T*x verifiant

o • 0 et A • O. transversale en chaque point au champ hamiltonien VA" Soit Ie flat du champ de vecteurs VA' alors (y x ~.f)

au

f : Y x:R ~T*x

par (y.t) .... ft(y)

f

t

148

est une sous-variete lagrangienne immergee de T~.

3. Determination de 18 phase.

Etant donnie une sous-variete. lagrangienne (V,f) immergee dans X, il existe dans chaque ouvert U C V simplement connexe une phase $ • de.terminee a une

constante additive pres, satisfaisant a l 1 equation d ~ •

(3-1 )

f'to

de

puisque l'on a. sur U. ft:

f"

e

_ o. v

Remarque : II existe toujours, globalement. sur Ie recouvrement universel V v

de V. projete sur V par

n.

v

une fanction v d ~

On de-duit de

IT

0

f

:

W•

COnDU

(f

$ telIe que 0

v IT)"

dans U C. V. une phase

f

e dans

nC

X si l' application

U + nest inversible.

L1 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'applica-

tion

n0

fest alers inversible sur U si 1a projection IT : T*X ~ X restreinte

a £(U) est inversib1e : i1 en sera ainsi au voisinage de tout point OU f(U) n1est pas tangent

a

1a fibre de T*X, c 1 est

a dire

n 1 a pas un plan tangent

"vertical". Soit x Eo

n c.

X. tel que

n-I (x)

ne soit pas tangent :it f(U). Soit Y1 "··· Yk

les points de feU) tels que n(Yi) ex:

149 Le point x admettra un voisinage dans X, encore note Q tel que IT

-)

(0) soit

l l union disjointe d'ouverts de feU) : f(U )

U

(3-2)

i

i • I ••• _,k

A chaque U, pour une meme phase $ sur V. correspond une phase

f

i sur

n donnie

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 (dans l'espace des phases) dans les problemes d'optique. les trajectoires des particules materielles dans d'autres problemes. Nous allons considerer Ie cas OU la p~ojection de V, supposee sous-variete de T*x pour simplifier, sur

n c:

T~X n'est pas bijective, mais

ble I: de V (son "contour apparent") tel que (3-3)

-I

n

au

(0). V

U

i • 1, ••• k

au

chaque restriction

J':i de n a

il existe un sous-ensem-

,1:

soit de 1a forme

U.

•

U i

u..... n

•

est un diffeomorphisme.

4. Solutions asymptotiques.

II est naturel de chercher une solution asymptotique du systeme differentiel

150

d'equation eikonale A(x.

~x)

a

- O. correspondant

une variete lagrangienne du

type 3-) sous la forme : k u(x)

(4-1 )

r

i ou les

~i

sont des phases. sur nc x. correspondant a la variete V. On sera

aide dans ce caleuI par la methode de la phase stationnaire qui montre (ef VII)

comment 4-1 liee a l'evaluation asymptotique d'une integrate. Des developpements du type 4-1, et les equations de transport correspondantes. sont utili-

sees pour determiner l'intensite lumineuse en presence de caustiques (enveloppe des rayons lumineux en projection sur l'espace-temps X). Remarque : Chaque phase

"Pi

n'est connue quia une constanCe additive pres.

puisque la variete lagrangienne V

De

determine

~

qu'a l'addition pres d'UDe

constante. depourvue de signification physique. La theorie des integrales asymptotiques et de ltincide de Maslov permet de determiner ces

constantes~

tification de

et

[51

relations entre

puis des conditions sur la variete V. dites conditions de quan-

Maslov~

pour qu'il corresponde

globale. avec une phase

[31

d~s

de VIII).

If

a v une

solution asymptotique

determinee globalement sur n • fi(V) (cf references

151

VII PHASE STATIONNAlRE. PARAMETRlSATION

D' UNE VARlETE LAGRANGlENNE.

1. Methode de 18 phase stationnaire (one variable).

On

se propose d'evaluer. pour w grand, une integrale de la forme l(w) m

au a et f sont des fonctions C , et a est )0)

On suppose que ~~

a

support compact.

ne s'annule pas sur Ie support de a; on deduit alers

de

a ( iwf)

aa

e

af

-iwaae

iwf

que l(w) -

1 iw

-

iw

l'integrale etant bornee par un nombre M independant de w on a

Mw

-1

152

Par iteration du

prod~de

on trouve, pour tout n €.

~

2°) On suppose que f s'annule en un point et un seul 0

if

a dire

que ---2 ' 0 pour a • a ' c'est o

aa

0

du support de a et

que f a un point critique, et un seu!,

non degenere sur Ie support de a. On montre que f peut alers s'ecrire dans un voisinage de ment de variable a --

t (0)

0

0

,

par un change-

tel que t (0 ) - 0, sous la forme 0

g( t) _ f(a(t» • g(o) +

f

t

2

e:

K

sign

done

I(w)

I

+

~

0

.!! )

pourt>O

4

gTand

•

.gTand

(peut etre obtenu par 18 mithode de 18 phase stacionnaire).

On volt que pour p > 0 et w2/3 grand on a pour Ie premier terme du developpement :

163

eil.l.lO(X) u (x)

2

2/3

cos(~

~ -tiI7"'W~IF{.1l1·(W""'2"J'l'3p·)"T07'T.4

p

J /2

. (2wp2/3

s~n

3

IT ]

-"4

cette expression coincide avec celIe obtenue dans Ies 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 nus en faisant

£ •

+ 1 ou E

a l'ensemble

des deux systemes lineaires, obte-

I dans

00 en conelut, en accord avec les resultats precedents, que les fenctions

If••

£ •

doivent etre solutions de I'equation eikonale

+ I

ou

165

Un calcul analogue

a celui

fait en l'absence de caustique (cf II) peut alors

etre fait pour determiner les equations de transport de b

'[lJ

o

et co'

Choquet-Bruhat Y., De Witt-Marette C•• "Analysis manifolds and physics"

nd 2 edition. North Bolland 1980. [2]

LUdwig D. Uniform asymptotic expansion at a caustic. Comm. pure and app.

Math•• Vol XIX p. [3J

Leray J'

t

215-2~O.

1966.

Solutions 8symptotiques des equations aux derivees partielles

(une adaptation du traite de V.P .• Maslov). CODvegno International. Metodi valutativa della fisLes matematic8, Acad. Naz. dei Lineei 1972. [4J

Leray J •• Seuunaires du College de France 1976-1977 et livre a parattre

aux H.I.T University press. [5]

Gui11emin V. et Sternberg S. "Geometric Asymptotics" A.M.S., Providence

1977 (Math. Surveys nO 14).

CENTRO INTERNAZIONALE MATEMATICO ESTIVO (C.I.M.E.)

U R T I

GUY BOILLAT

URTI

Guy BOILLAT

Introduciamo un vettore che dipendenti

x•

e

un inBieme di N funzlonl delle variabili In-

(CI. 1, 2, ••• In)

u'

•

u(x )

u' c

La varlabl1e xO • t rappresenta usualmente 11 tempo mentre xi (1 n) 80no variabili di spazio.

1,2 •.•. ,

Scrlvl~o

A"'Cu, xlt)u. • fCu.

x~

(1.01)

dove

e Ie AII.

80no matrici NX N generalmente dipendenti dal campo u e delle varia-

bili x~ • La somma ~ sottointesa sugl! indict rlpetutL .• Un tale sistema 61 chiama quasi lineare. 5e Ie matrici

60no

indipendenti da

semi linearei se poi anche la fUnzlone sorgente

U

61 he un sistema

r non dlpende da u il siste-

ma s1 dice lineare. Dimentichiamo ora la dipendenza esplicita de ~ .11 sistema pub easere ri.&cri tto coal

170

o A (U)U

t

1 + A (ulu 1

= £(u)

(1.02)

Definizione di IperbolicitA. Gli autovalori di A

n

Ain. 1

(ir veraore

delle

spazio) rispetto a AO 80no tutti reali per egoi ~ ed estate una base di autovetteri delle spazio di u. Questa implies Ie regolarita di AO e pertanto 11 sistema (1.02) s1 puo mettere Botto Ie forma 1

+ A (U)U

1

All'autovalore

= £(U)

(1.03)

(i)

di molteplicitA m

(1 )

devono corrispondere m

Butovetto-

ri (destri e sinistri) linearmente indlpendenti cosl definiti = 0, (1.04)

J '"' 1,2, ••. ,m

che denoteremo

anche

se~pllcemente

IJ

(j)

dIe

In particolare se tutte Ie metric! di (1.01) sono simmetriche,

ed inoltre AO

e

cio~

definite positive, (1.01) viene chiamato sistema di Friedrichs.

E chiarO che tali sistemi 80no Iperbollci. In generale un sistema qualunque non s1 puc mettere nella forma 61mmetrica; perc 1 ~tica

616~eml

della Fisica mate-

si possono rieondurre. come vedremo • ad una tale forma.

2~

Sistemconservativl

Questi hanno una (orma speciale nel senso ehe si Bcrivono come divergenza nello spazio-tempo di certi vettor! t~(u)

1 £"(u) •

£(u)

(2.01)

oppure. 'con l'introduzione del aradiente rispetto a1 campo u.

(2.02) che corrisponde 8 (l.Ol) con

(2.03)

171

Poiche AO non ~ 8~ngolare s1 pub quindi scegliere f

O

come campo u ed allore

(2.04)

Per un fluida, ad esempio.

('U u*

1 k u

rik + p, ~u

i

(f+ p)u

dove,f

=eu2 /2

+

e

e

(2.05) i

l'energia totale, e l'energia interns, 1

• +

pit

l'entalpia libera legata al1'entropia da di • TdS + dr/e

(2.06)

Come conseguenza segue 18 legge di conservazione dell'entropia

(2.07)

Abbiamo qui un sistema con un equazione in piu; l'ultima e pera una conseguenZB delle altre. In generale data un sistema conservativo

(2.08)

e un'equazione scalare canseguenzB (2.09)

se s1 fa 18 derivata rispetto a t e 81 sostituisce (2.08) s1 he l'identlta

che deve essere vera per ogni U da cui 1 i

1,2 •••• to

(2.10)

Friedrichs e Lax hanna fatto vedere [1] che. definita 18 matrice hessians ~

Hor;:YVh, la matrice HAL ottenere

Wl

• slmmetrica. Basta allors moltLpllcare Is (2.08) per H per

s1stema del tipo

172

(2.11)

HC

che e un sistema simmetrico nel sensa di Friedrichs purche Is funzione h(u) sia una funzione convessa di u, cioe H

definite positiva.

Quando s1 moltiPlic8 (2.08) per H s1 perde 18 forma conservativa, pero con

&.

un cambia di variabili s1 pub ritrovare e si perviene ad un sistema conservativo e simmetricQ. S1 introduce 11 campo nuovo data da

u'

e

quat~ro

C2.J

.-

=Vh

(2.12)

funzioni scalar! (2.13)

In particola~ per ~ : 0 h' = u·.u _ h.

e una

h'

O.

h'.

(2.13' )

trasformata di Legendre.

Deriviamo la (2.13) rispetto 8 u'

e da (2.10) risulta

'f-. V'h.tl

(2.14)

~ • "'h'

(2.15)

In particolare

Ne segue

"/'(UI)U~

(2,]6)

=C,

cioe Ie nuove metric! non solo Bono simmetriche rna anche hessiane. La forma (2.16) estate introdotta de Godunov

Notare 18 differenza

in (2.11) • (2.16).

131

con tre esempi.

Ie matrici simmetriche sono

ri8pettivam~nte

173 3. Eguazionl di Eulero Applicando un principia variazionale alIa lagrangiBna L = L(q: ' qB)

(3.01)

dove Ie q B (x" ) Bono funzioni delle spazia-tempo e \ B

:z::

" -UllI(q.

31 arriv8 aIle

equazioni di Eulero (3.02)

che s1 possono mettere nella forma conservativB (2.04) con 'ClL;)qB

,r!

B qi

u

q

•

~L~·

')L/)q;

0

=

-qB ~ j o i

, f =

0

(3.03)

0 q

• 0

Se L non dipende esplicitamente de qS basta eliminare 18 terza riga. C'altra parte s1

pu~

definire una quantitA con due indici (3.04)

tale che

,..

')T"-O -

(3.05)

se Ie equazioni di campo (3.02) sono soddisfatte. Le (3.05) rappresentano quattro equazioni supplementeri. La conservazione dell'energia corrisponde a ~

= 0 che seeglieremo

come equazlone (2.09) con (3.06)

51 deve ricavare u'. 11 che signifies che dobbiamo valutare Ie derivate par-

z1a11 di h rispetto aIle component! della u. Indichiamo le componenti di u nella maniera seguente U

o B

u.

U

i B

u·

174

s1 ricava u· in

questo modo

• q

r

o

e similmente per Ie a!tre componenti. Infine s1 ha

u' u'

u'

S

S

0

• qo

1 • _ ')L!)qs 1

(3.07)

S

_ ')L/~s

u'

S

(51 taIga la terza riga se L 01 conseguenza - u

,j

- u'

•

S

o

f

u'S

o cio~ ~,

f

(3.08)

o

sono funzloni linear! di

U'j

Ie equazionl di Eulero prendono la

forma (4J • (5J, 1 H'(u')u' + AI u' "" B'u' t 1

(3.09)

L'uniea nonlinearita e dovuta a1 coefficiente della derivata temporale. Le 1 matric! A' • 5' 50no costanti, Ie prime simmetriche, l'ultima emisimmetrica

.-

B'

:c _

S'

Se L non dipende esplicitamente da qS, f e il secondo membra della (3.09) sono nul11; un caso gill considerate da Godunov [3j .

Studiamo ades60 la convessitA di h.

Con (3.03.07) s1 ottiene tornando aIle variabili 101z1ati

175

'1>u.h' (3.10)

somma di tre forme quadratiche ciascuna delle qual! deve essere definita po-

sitive. Se cerchiamo Ie velocitA caratteristiche basta far corrispondere

e 81 ottiene

Quando la velocita

e

nulla segue Bubito de (3.08)

n'lu,j J r

'" O.

Se invece supponiamo ~ ~ 0

'i u tr

:: O.

(3.11)

0

de (3.02) 81 he

o. Deve risultare nullo 11 determinante delle quantitA tra parentesl. Le sue radiei

~ (1)

usualmente sono diversi de zero. Ad ognt autovalore

~ I:

0 e asso-

ciato un certo 8utovettore. Supponiamo che per un certo ~alore di ~ Is veloetta diventa nulla, signifies che la molteplicita dell'autovalore ~= 0

menta, rna 11 numero di 8utovettori associati

e

8U-

sempre 10 stesso. date de (11).

51 perde cod l' iperbolici tAo 51 deve dunque evl tare che

c\ (;r)

= 0; Is matri-

ce

deve essere regolare. Una condizione ovviamente soddisfatte se vale la con-

vessita (vedi (3.10).

176

4. Urti Supponiamo che attraverso una superficie ~(x~)

o che s1 muove col tempo,

di normale,

e veloci til

il campo u

~

Itt /19'(\

~ -

6

discontinuo

[uJ • u -

U

~

o

O.

(4.01)

Al sistema (2.01) 51 applies i1 teorema di fluBso-divergenza e 81 arriva alle equazioni di Rankine-Hugoniot che s1 scrlvono (6)

[r16[U)= n

r

0,

ri

n

(u)n.

•

i.e .. f (u) n

B U

r

=

n

(u ) 0

IS U

0

e

trovare i1 campo dopa l'urto in termini del campo U prima o l'urto e di s (velocitA dell'urto) Il problema

U

:::: U

(4.03)

o

Derlviamo (4.02) rispetto 8 8 e u o (A

(A

n

- 01)& - h

- 61)(1 +

(4.04)

• 0

V h)

• A

noon

- 61

Supponiamo che l'urto sia debale. Facciamo tendere b a zero; s1 he (A

on

II che signifies che s

6

- 61)&

O.

0

e

autovalore di A oo

& =rJ.d(u~). o

0

(4.05)

177

Os (4.03) segue

h(u •

',n) ~ 0

(4.

I

0

0

(4.06)

e, facendo 18 derivata rispetto a u

o

(4.07) Prendendo 11 dterminante della

(4.05) s1 ottiene

Voh)

dot (1 +

• dot (A

- sI)/ dot(A

on

n

- .1).

L'equazione precedente d~ Bubito 11 valore del primo membra

dot(I + V h) • 1 - V ~ .Ii • 1 - ~. o

0

11 secondo membra s1 presents -! II (~ o

-

1 .)/II(~

-

'0 una

0

0

0

forma indeterminate

s)

11 cui lim! te - 1/(~' - 1)

o

derive facilmente della regale di L'Hospital. Pertanto

1-/\ So ~:

o

o

=l/(l-~·). 0

V>. o

o signifies che

vo ~.d 0

0

•

0

.Ii

0

• 0

li(~ 0

0

0

-2).0.

(4.08)

qulndi

(4.09)

O.

Torneremo eu questa condizione di "eccezionalita. ". cosi chiamata da Lax e supporremo per 11 momento

i.. o

= 2,

•

(~+

che non

verificata (urto normale). Allere

~ = ~ + ( (. - ~ ) + •••• o "'0 0

l- o )/2

+ 0(. _). )2

cioe 18 velocita di un urto debale ristica.

e

(4.10)

0

e

11 vel ore medic della velocita. caratte-

178

Questa mostra che s u

e

compresa fra

~

o

~. 5i pub quindi sempre chiamare

e

10 stato tale che

o

(4.11)

escludendo 11 caso ~eccezionale" (4.09).

Unatale disuguaglianza insieme aIle equazioni (4.5) conduce aIle condizioni di Lax [71

: s non raggtunge rna! una velociU. cratteristica rna

e

compreso

fra due autovalori consegutlvi sia per 11 campo u che per 11 campo u

o

(4.12)

(4.13) per ogni

1'.

in modo tale che (11) viene verificata qua Ie che sia k. Cosl. gl1

urti vengano classificati secondo 11 valore dl k.

5. Entropla. Funzione generatrice

Supponiamo adesso l'esistenz8 di una legge di conservazione supplementare (2.09) e, in analogta con Ie equazioni di Rankine-Hugoniot consideriamo la funzione dell'urto (5.01)

In condizione di differenziabllitA (2.09)

e

ebbe pensare che ~ o

"J.

e

conseguenza d1 (2.08) e 81 potr-

nulla se Yalgono (4.02). Inyece, di solito, non e Ye-

Se s1 inserisce (4.03) in (1) 51 ottiene (5.02)

Deriviamo rispetto a s

i

=

b.v

v/ ji

con 18 conservazione dell'entropia

~S/~ = 0 . Da h

~

- S e (6.06) ai deduce ~

v

u· •

T

-1

(6.07)

- T - 1

51 dimostra come nel caso del fluido che -5

che questa sia vera

~e(s.

18 velocita dell'urto

e

e

una funzione ccnveSS8 di u pur-

F ). Ne segue la crescenza dell'entropia quando ij

positiv8

£12 J .

11 quadrivettore (2.13) he Ie componenti

182

7. Urti caratterlsticl Floora a1

e

conslderata una soIuzlone (4.03) delle equazioni di Rankine-

Hugoniot (4.02) dipendente da un solo parametro s. Adesso ci chiediamo se e possibile di trovare una Boluzione = u

u

o

+ h(u

• u

I

~

(7.01)

,n).

0

ehe dipende da piu parametri u

I

(1::l:" 1. 2 •..•• p). 5e p : 1 posslamo sce-

gliere generalmente s come parametro. Deriviamo (4.02) rispetto a u (A

poi rispetto a u

I'

n

-"r)')

I

h -

1 (7.02)

')1" h • 0

e eliminiamo h

':lI" ~,h) = O. Se

p.., 1 esiste

almena uno de! vettori che non

sarlamente un autovalore

e nullo.

Ne segue che B

e neces-

di A

n

Studiamo, in generale, questa possibilita. Supponiamo per primo che 6

...

= stuo ,n),

autovalore di A on

Sostituendo nella (4.02) e facendo la derivata rispetto a

e

u

I

viene

pertanto s deve essere soche autovalore di A . Scriviamo dunque Ie equazioni n

(4.02) con

(7.03) per ottenere

(7.04) e d'altra parte. facendo Is derivate rispetto a u

o

(7.05) Mol~lichiamo

queste equazioni per un autovettore corrispondente all'autova-

lore ~ di mol teplici tA

(il

m

183

(7.06) (I ,·h)Vc\(I + 'loh) I

-

I'=1.2 •.•• ,m Se l ,.h I

l ,(A -~I), on I

(7.07)

(i)

~I ~

"0, dalla prima segue

O. Se invece vale l'uguaglianza.

18 seconds equBzione d~ 1

cioe

r.

I'

(A

on

- ~I) = 0,

e autovalore anche di A

• dipende soltanto da U e dunque abbiamo'

o

on

encore

(7.08) e d.

(7.04) (7.09)

i. e. h puc dipendere dB tanti parametri quanti pOBsono essere i vettori (1)

f)Ih indipendenti. vale a dire m

(1)

leI. 2,

.'Of

(7.10)

m

La (8) s1 scrive

oppure tenendo conto di (9) (7.11)

Questa uguaglianza, ehe deve vel ere qual~he sia u

erie

la condizione di ec-

cezionalita di Lax.

La (8) fa vedere ehe ~ zione nulla u I

=u

o

e

indipendente di u

I

e siccome ammeUlamo la 601u-

(corrispondente per esempio alIa nullita di tutti i para-

metri u ), segue

...

...

~(u,n) = ).(u ,n).

o

(7.12)

La condizione (11) 51 incontra speaso in Fisica matematiCBj b3sta citare

184

Ie oode di materia, di Alfven, gravitationali, di Born-Infeld, della corda relativistica, ecc. [13]

• E'sempre verificata per Ie oode moltiple di un sis-

tema iperbollco conservativo (14) • Deriviamo (A-~I)d_O n 1

nella direzione dell'autovettore d

I'

Scrivlamo la steese cosa cambiando gIL indict e facciamo 18 differenz8. Perch~ 11 sistema ~ conservativo An ..

Vfn e

VVtndI,d

J

"VVCndrdII .Resta

Ne rlsulta necessariamente (11). Nel caso di motepllcltA esistera dunQue sempre un urto che 81 propSSB con una velocitA caratteriatlca e che chiameremo

urto caratterlstico. Nel caso di un autovalore aemplice la condizione (11) di ortogonali tA del gradients dl ~ edell' autovalors corrispondente pu~ anche

essere noddisfatta. p. e. per ali urti di Alfv6n. 5ia ~ (~)

=0

l'equazione del frente d'urto che verifiea l'equBzione carat-

teristica

If' (u,'fill ) - \flu0 •'I) ill

(7.13)

0,

'f t +\''fhl. (u, 1)

\f'(u,'t.t) -

(7.14)

51 ha (7.15) e derivando rispetto a

1/'

':1 +'#''$\

MoitipUch1amo Ie equaz10nl del campo per 1 1 AJ. 1

Me lIA~).

e

u

fI

(7.16)

1 (7.17)

- lIt

un operatore di derivata tangenziale aIle superCicle d'urto come

a1 vede da (l3,lS). 01 conaeguenza a1

pu~

aostituire nella (17) il velore

185

(1) di U Bulla 8uperCicie

~(u0 + h( u0' UI',n») "" II' f ; I, I 1 I A" ~ I~un

I

(I ) "" 1,2, ...• m

I

sistema di equazioni differenziali ordinarie per ali u . Infatti, easen-

dO')rh un autovettore destro di An

(vedi 9) Ie derivate de! coefficienti ap-

tramite termini del tlpo

patono

(7.18)

che tenendo conto di (16) 81 mettono Botto la forma (7.19) dove

dId . . .

~"'f '"

A la derivate lunge i raga! dell'urto (7.20) Po~eh~

11 sistema

~

iperbolico la matrice dl coefficient! II.dr' i invertibile

e (14) pub essere rfsolto rispetto aIle derivate. Le quantitA ~

mente

~i~ Bono Ie component! della velocltA radiale. La ~ chiars-

una funzione

omogent~del

primo grade rispetto aIle

f•.

Pertanto

del tearema di Eulero

oppure

E'interessante di notare che mentre Is velocita normale ~

e vero, in generale,per 1a ve1oc1tA rlspetto a 'f.c. tenendo U costante s1 he o

questo non (13)

~~ r =

e continua (12), radiale [141. Derivando

'4"rr +V'f'')~h

pl'/'1. -lV'fIVc\.')loh e i1 secondo membro non

e.

di solito, nullo

Per esempio, 1n un fluido esia-

te un urto. cosl chiamato di contatto, che 61 propaga con la veloc1tA carat-

186

teristica

continua s "" -:.n

"t.n. Invece it bene conosciuto che, per questa o urto, la velocita radiale ~ e discontinua [~1 ~ o.

8. Soluzione esplicita Abbiamo

gta

debole, il saIto di u appartiene al-

~ • Quando l'urto non

10 sottospazio degli autovettori di pu~

e

vista che, quando l'urto o

e piu

debale 51

encore dare una forma espllcita del saIto. non di u rna di u'.

1 . Derivando

alIa funzione aeneratrice

r.>"'.';;'(A r t n

Ie (5.01) con s

grazi~

=~ .

.0,

-l-I}?h

I

in virtu di (7.09). Siccome ~ ~ nulla per l'urto nullo ne risulta '" (u

I

0

,~ ,~) !! 0

(8.01)

0

Pertanto, la sua derivate rispetto a u

o

e

anche nulla e tenendo conto dalla

(5.03) viene

"'1 '0

.-wV~ 0 0

che inserita nella (5.08) foroisee

= u'

h'

(8.02)

- u·

o

Adesso introduciamo un vettore a(u.~) cosl definito I • 1,2 ••.. ,m

(i)

(8.03)

La soluzione estate proprio perche vale la condizione di ecceziona1ita (7.11),

e

unica perche

e

ortogona1e ag11 autovettori d

puo mettersi sot to la forma

r .

11 saIto del campo princi~~

[151 (8.04)

Nel caso lineare g sarebbe nullo. 11 secondo termine rappresenta dunque 1a parte non lineare dell'urte caratteristico. I due vettori dipendona sol tanto I

delle state prima dell'urto e 90no cono9ciuti. Resta da determinare ..... (u ,u ). o I Deriviamo (5.04) rispetto a u =

';l

I

'"h'.h

187

cio~

• w• 1 I

10

.h/(l - g .h)

(8.05)

0

purch~

(8.06)

Per la derivata seconda

v +') h'H'~ hi I I'

•

(8.07)

9. Stabll! ti. dell' urto caratteristlco Sia (9.01)

01 = cost .•

una soluzione ovvia di (7.13). Scegliamo come parametri u sono del tutto arbltrari ) delle funzioni lineari di

Se supponiamo che

If.

1

(chef a priori,

Segue da (B.05. 07)

quat!he sia l'urta, (9.02)

l-gh>O o

come 10

e

quando l'urto

e

debale, ailars

la derivata prima cresce e siccome

e

nulla quando l'urto

Ne risulta che w tende all'infinito con h.h' ~ w

't .

e

nullo,

e

positiva.

Ma s1 vede facilmente che

188

e

e dunQue anche l'urto non rappresentata ad

e~empio

f e Questa

e

- '(

limitato appena s1 sposta 18 superficie d'urto

da un'equazione del tipo (1) i

'no.

)I·0 t

= x ",, -

i l caso dell'urto di contatto gil citato. In un urto stabile Is quan-

tita (B.D6) cambia di segno [161 . Vedremo come 51 traduce Questa condizione per Ie equazioni di Eulero.

10. Evoluzione dell' urto earatteristico di Eulero

DaIle equazioni di Eulero scritte nella forma

U

t

(vedi §3)

+

AIlu'

= Btu·

i

(10.1)

derivano Ie condizioni di Rankine-Hugoniot A'h' -I-h

(10.2)

= O.

" Da cui segue, tenendo conto del legame (§ 2)

e di (8.04). h = H'h' -

o

wV')./ 0

0

At

(10.3)

Definiamo gIl Butovettori 1'(A' - ~H')= O. I "

tale chef I' =- 1 I

.-

d'

I

I

(A' - ~H' )d' n I

H'd' I

(10,4)

0,

~II'

.

Allers da (3) viene semplicemente,

mentre g .h =

o

dove 0(.0 =o«u ' o

"1) e

0( w ,

(10.5)

0

una ftJn:z1one conosciuta del campo prima dell'urto

189

(10.6)

La (8.05) diventa u 1(1 -0( w)

'I

(10.7)

0

che s1 integra subito -' 2 (1 -",w) = - 0(

o

lui 2

0

+

a(u • ~). lui o

La costante di integrazione s1 determine sapendo che l'urto I

que anche g11 u • vedi 6.04) quando w

~

nullo

(5.05) e

~

nullo (e dun-

r17] (10.8)

Vediamo dB (5) che 1-ah-1-o(W o 0 L I instabi!i tlt corrisponde

guenzB l'urto

Non

e

e

8

- o. Iu I

e di conse-

~

continua. Derivando (4)

HI )V'd' - H'd' V'~ -~V'H'd' = O. I I I

Moltiplichiamo a slnistra per

vata del autovettore

W

cambia allore di segno .

difficile di vedere che Ie velocita radi31e (A' n

t

Ii

e a destra per ')i

di

che rappresenta Is deri-

rispetto a ~i

V't, ~id'

• -l ~

Eo Eo

- -

, (1.4) ~ = .. '" ./

1'"

dove E. '.Y' SOIlO rispettivamellte la cost ante dielettrica, la peraeabilitA magnetica, la conduttivitA del mezzo ael punta

X in

cui si cOllsiderano i vettori che cOlllpaiollo rispet-

tivamente in (1.3),(1.4),(1.5). Be poi il mezzo e anisotropo Ie £ ,

r-

Poiche E ,

' f vanno sostituite COil tensori -doppi.

r:J'"

dipendollo solo dal mezzo e nOll dal campo e-

lettromagnetico, le (1.1),(1.a),(1.3),(1.4) e (1.5) costituisCODO un sistema 1iaeare, perci6 l'elettromagnetiSJllo ordilla-._ rio si puO chiamare anche e1ettromagnetismo liaeare. Per6 in quei dielettrici (ai quali ci riferiremo sempre ill seguito) dove si manifestano i fenOllleni dell' ottica noa lineare J mentre restano val ide Ie (1.4) e (1.5). 1a (1.3) va sostituita

COD

UJla relazione non lineare ira 0 ~

E

che scriveremo:

(1.3') sicche i mezzi in cui vale (1.3') si possono chiamare dielettrici non 1ineari. Ia questa lezione cercher6 di.stabilire alcune proprietA

del~e

onde elettromagnetiche che si propa-

,,\\--..t.o

gano ne1 die1ettricorcosi da interpret are qualche fenomeno del1'ottica nOll lineare.

2.

Riferiamo i punti della spazio a un sistema di coordinate

cartesiane ortogonali (O,x,y.z) e supponiamo il dominio coincidente con una 1amiaa di spes sore

5

riempita da Ull

~

199

dielettrico non lineare omogeneo. Porremo ~

l'origin~

0 e l'asse

del sistema di assi in modo che le facce della lamina abbiano

equazione z=O, z=s. All'esterno della lamina supporremo il vuo-

to che, dal punto di vista elettromagnetico, si pu6 identifica_ re con I'aria. Indicheremo con £.la 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 vista fisico)

ci~ ammetteremo~ identica

i. tutto 10 spa~io.

Nel semispazio z < 0 sia posta una sorgente che generi un' onda

elettromagnetica piana con all'asse

~.

dire~ione

di

propaga~ione

parallela

Supponiamo la lamina tagliata e disposta in modo

che il campo elettromagnetico dipenda solo da

e t; anzi, con

~

un'opportuna disposizione degli assi x e y si possa scrivere, per ogni punta dello spazio, : (2.1)

"E.

E. (l,lo)L"

laoltre supporrenioIi parallelo ad

E ci~; avremo:

(2.3) Allora la (1.3') diventa (sottintendendo le variabili z e t)· l'equazione scalare:

(2.4) e la fUnzitne D(E) verrA supposta di classe

c~

in qualunque

intervallo limitato dell'asse reale. Le equadoni di Maxwell nella lamina si riducono a:

200

La. (2.5) e (2.6) valgol1o aache all'esterno della lamil1a purchi!

r

si pOl1ga .0, e. ill luogo di ~ ~ • Ammetteremo inoltre, conPorme llesperienz~: (2.4')

cioi! D Pul1zione crescente di E e D(O)_O. Stabiliamo ora alcune condiziOl1i sui pian! che limitaao la lamina,pi~

precisamente sui piani z=+O, z=s-O ; si i! scritto +0 e

s-O per identi£icare Ie Pacce dei piani rivolte verso l'interno della lamina

o,pi~

brevemente,£acce interne.

Ora, nel semispazio z < 0 s1 avranno due onde, una che diremo

diretta, emessa dalla sorgente e che si propaga nel verso positivo dell'asse z, l'altra ri£lessa dal1a lamiaa e che si pro-

paga nel verso negativo dell'asse z. Detti E~(z,t), ~(z,t)

g~(z,t),

HdJz,t),

rispettivamente il campo dell'onda diretta e

il campo dell'onda rifles sa, si hal (2.7 )

Ora,

} c~e

e

noto, su un piano che separa due mezzi divers! so-

no continue Ie cOlllponenti taagenziali al piano del campo elettromagnetico (ovviamente z=-O, z=s+O sono Ie facce della lamina rivolte verso l'esterno

0

facee esterne). 5i ha cos!:

(~.:l)

£.olJ_O,!;)+fo",(-o,~). E (d,t)

(2.9)

Hd.(-o,~) .. I-1,,{_O,~). 1-1 (+o,~).

Ora, per note proprietA delle onde elettromagnetiche piane si ha:

(2.10) l4.l(-O,!;).~ Eol(-O,~) ./

Sostituendo (2.10) e (2.11) in (2.9) e sommando con (2.8) moltiplicata per \;~

.r

si eliminano E,.,. e HJ(... • Allora, riservando

201

il simbolo E{z,t), H{z,t) al campo entro la lamina ed ometteado, per semplicita di scrittura e perche ora non vi

e

luogo ad

equivoco, i segni + e - davanti allo 0, si hal

Hel semispazio z;> s si ha solo un'onda che diremo trasmessa e che si propaga nel verso positivo dell'asse z (non si possono avere riflessioni perche per z >s il mezzo e omogeneo) i cui campi indicheremo con

E~(z,t), H~(z,t).

Per la continuita del-

le componenti del campo elettromagnetico sul piano z=s (ora si possono evitare i simbcli +0 e _0) si hal

e poiche "~vale

Viii- If~ si ha subito:

(2.14) Le (2,12), (2.14) in cui

E~(O,t)

si suppone assegnato, cost i-

tuiscono condizioni alla frontiera per (2.§) e (2.6). Ad esse si possono eventualmente associare opportune condizioni iniziaIi, sicche i1 campo entro la lamina resta determinato.

Le (2;12) e (2.14) si devono al Frof.Cesari (11 (2) [3] [4] il quale ha dimostrato importanti teoremi di esistenza, di unieita, di dipendenza continua dai dati per Ie soluzioni delle

equazioni

(2.~)

e (2.6) corredate da (2.13) e (2.14), qual ora

sia noto Eo\,{O, t) per ogni t (positivo " negativo). Nel caso, importantissimo per le applicazioni, in cui co rispetto al tempo e con periodo T,

~nche

E~{O,t)

e periodi-

i1 campo entro la

lamina risulta periodico con 10 stesso periodo. Noto i1 campo

entro la lamina, mediante (2.8), (2.9) e (2.13)

e

colare i1 campo ri£lesso e trasmesso dal1a lamina.

facile cal-

.202

I teoremi di Cesari sono stati dimostrati per valori dello spessore s della lamina non trappo elevati. Torner6 in seguito su

[5]

questi risultati, per ora noterO che il Prof. Bassanini

ha dimostrato che i valori di s per cui sono validi i teoremi ora citati risultano superiori allo spessore delle lamine usate in pratica_

3.

Passiamo ora a

ricerca~

una soluzione di notevole interes-

se di (2.5) e (2.6) supponendo (come faremo sempre in seguito)

fhO. A questa scopo poniamo, ricordando (2.4'), (3.1) ~

Nel caso lineare (si ricordi (1.3»

= E.

(cost ante die-

lettrica) ed esiste una soluzione delle (2.5) e (2.6) per cui i1 campo elettrico ha l'espressione:

E(Z,t) dove G(u)

~

= G(u)

,

una fUnzione di classe C. 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,

sicch~

Ie proprietA della ru,zione di t,E(O,t), sono Ie stesse di G(t) o G(u).

Ora. nel caso lineare p(E)=

J8y' ;

viene perciO naturale con-

getturare valide Ie (3.2) anche nel caso generale sostituendo perO nell'espressione di u a V~, peE) come definita do. (3.1) e con seguo positivo. Si ha cosi:

(3.3)

E .. G(u)

(3.3')

u .. t - p(E)z.

203

~rimo m~mbro

di (3.3) si ha l'~quazion~ implicitam~nt~ E in £unzion~ di t ~ z :

Ora, portando G(u) al ch~ d~fiftisc~

E _ G(

(3.4)

t -

p(E)z ) • 0 z=o.

Qu~sta ~quazioft~ ~ ovviament~ risolubil~ p~r

risolubil~ p~r z~O ~ sufPic~ftt~, p~r ch~

implicite,

il

£Unzioni

)

condizione

c~rtam~nt~

discuter~

Cal n.5) di ~

t~or~ma d~ll~

sia

sin I

(3.5)

sto

AffiRCh~

intuitivo, che

z e (O,h), t

E:

soddisfatta

z=O. Ora,

p~r

amm~ttiamo, com~ d~l r~­

meglio la (3.5), ~sista

(-'J;,T) ( T,

~

Uft

h> 0

ftum~ro

~

T positivi avv~rt~nza

sia valida (3.5). Fino ad

ris~rvandoci

tal~

del

che

r~sto

p~r

ogni

arbitrari)

ift contrario,

amm~tt~r~­

i1

campo magne-

mo z e t Dei limiti ora indicati.

Cic

prem~sso, v~diamo

di

d~t~rminar~

tico H che, associato al campo

~lettrico ~spr~sso Maxv~ll

(2.5)

A questo scopo ricordiamo che i l Prof.

J~ffr~y

ha dimostrato,

n~lla

da

prima

soddisPa

d~ll~ Su~ l~ziofti, ch~

all'~quazion~

fi

(3.6)

soddisfar~ l~ ~quazioni

da (3.3),

di

sia

ta1~

valor~ d~l

~ rL~)

a

d~rivat~

%f "

E,

com~ ~spr~sso

~

(2.6).

da (3.3),

parziali:

0

che ora veri£icheremo direttameate.

A

qu~sto

a z

~

(3.7)

scopo

poi ';IE

III

oss~rviamo

risp~tto

che, derivando (3.3) prima

a t, si hal

(-L~ c;.'(I. 0 si avrA un f'ronte d' onda, n0.lche si sposta col

temp~ di

ci~

Ull pia-

ascissa z. s z.(t) tale che per

z >z., E(z.t)=O. per z 0 e del resto qualsiasi. Poich~ E ~ uguale a zero per ogni t suI f'ronte d'onda. la sua velocitA sarA la velocitA del campo nullo,

ci~

il fronte d'onda si sposta con velocitA l/p(O).

5. Passiamo ora a discutere la (3.5). Anzitutto se G' (u) e 'df>P/; haJUlo (se diversi da zero) per ogni u e per ogni E 10 stesso seguo (per esempio G(u) e peE) sono

206

Punzioni crescenti, la prima rispetto a u, l'altra rispettG a E), la (3,S)

e

sempre soddisfatta e It

zOO

per ogni t, ~O

IB questa caso, se Ie condiziOBi iBiziali sonG Bulle per z • suI piaao z=O

e

assegnato per ogni t positive il campo elet-

trico, per UB teorema di unicitA del campo elettromagnetico, (3.3) e (3.10) (purche si asswu G(U)=O per u ... 0) rappreseJltaJlO 11 campo elettromagnetico caapatibile con Ie condiziOBi iJliziali e alIa frontiera e che si propaga nel verso positivo del-

l'asse

2..

Tornando al caso geJlerale, cerehiamo di dimostrare l'esisteJl_ za del Dumero h;> 0 di cui si

e

accennato al B.3.

A questo scopo, aggiungeremo un'ipotesi

pi~

che plausibile dal

punta di vista fisico. Ciee la funzione G(t) (0 che e 10 stesso G(u»

che rappresenta il campo E(O,t) sia limitata assieme

alla sua derivata G' (u) per t" (-- ,T); ia altre parole esistaao dUe lNIIleri positivi: H e H' tali che per ogni u .. (-..., ,'1') sia:

(61(....) I ~ M

(S,l)

Inoltre per Ie nostre

I g I'(E¥c>E \

Ci~

I ~ '(.. . ) I E; M'. ipotesi -fi.. o a c-k...

liJlitata da

U1l

I B \ .. H

sarA

numero II.

premesso, fissato un istante t, esisterA un numero positi-

vo h(t) tale che per z £ [O,h(t») , (3.S) e verificata e quiadi (3,3) risolubile. Allora per questi valori di z, t, che

I~

1< N,

I E(z,t)l::: IG(U)\ ~ H,

inoltre IG'(u)l~ H'.

Dimostriamo ora che esiste un numero h o tale che h(t);;. h o t E. (to che

0

e

sic-

'

,T). Infatti sostituendo h(t) in (3.S) e tenendo COnsoddisfatta se G' (u)

?lff) ~,= ~~

cCllle si era a1'i'e1'lllato. Assumeremo h~h. l'estremo ini'eriore delJli h(t) per te(-oc.T). T puc) essere anche ini'inito Perc) nel caso G(t)-O per soddisi'atte) e t non

~

purch~

t~

0

sia soddisi'atta (5.1).

(sicch~

(5.1) sono certamente

molto elevato, segue h-

sia p_ ~ 0 i l minimo valore di peE) per gono

le relazioai:

(5.4)

t .. T

t < P.

.-..

ho =

00.

I III ~ M,

lni'atti

aHora se val-

n"",..

N M'

i l valore di u che CClllpare- nella (5.2) ~:

-

u = t - p(ll)h(t) (; t - " h < P.

(5.5)

....

Ma allora il G'(U) della (5.2)

~

-

h - p h - O.

nullo e questa equazioRe noa

puc) essere soddisfatta per h(t) finito. Deve essere h= 00,0, che

~ 10

stesso, la soluzione (3.3)

~

valida,per valori di t

soddisfacenti (5.4), per ogni z, ed essa rappresenta il campo elettromagnetico in tutto il semispazio. Si noti che. come ved~~o

nel

num~ro segu~ntet

tempo in cui la (3.3)

~

N e malta piccolo; l'intervallo di

valida puc) essere sui'ficentemente gran-

de per le applicazioni pratiche. Hel caso in cui non siano soddisfatte Ie ipotesi ora esposte,

fissato t puc) esistere un valore z di z per cui la (3.5)

~

nul-

la, e se G(t - p(E)z) risulta diverso da zero. da (3.6) e (3.7) segue che

IH.'

te di E per z

:;l~J:E -to

z

1-

-1" DO

•

n.' BE 1~IM;lt' - ~~ --

,cioe Ie deriva-

tendono a di vent are infinite. 51 ha cioe,

conforme a una locuzione del Prof.Jeffrey, una catastroi'e. Si puc) cosi interpret are l'accennato risultato di Cesari per qui i suoi teoremi sono validi solo mina

~

sufficentemente piccolo.

~e

10 spes sore della la-

208

In seguito comvaque ammetteremo che (3.3) e (3.10) rappresentiao il campo elettromagnetico, almeno per valori di t e z suI£icentemente grandi per Ie question! pratiche.

6.

Nel casO s _... aotiamo che, mentre (2.12) rimae valida,

(2.14) non ha

pi~

signiEicato e si

pu~

sostituirla con la con-

dizione che il campo sia nullo all'inEinito, 0 meglio che il canapo rappresenti un'onda che si propaga nel verso positivo del-

l'asse z, condizione questa, come si

~

osservato al ••4, sod-

dis£atta dalle (3.3) e (3.10). Supponiamo ora I' onda Eot incidente suI piano z=O, col campo elettrico' (e quindi allChe i l campo magnetico) per t

a.. seali>t

Ie a

( p[d!'l'(i )J

e cloa,nelle condlzlonl presentl

( 3.5 )

ha

el

- [P!]!!. = -[pll'!

+ L

\r:' If

244

avendo poato

(

d P.2'( if 18l!J )

)N =

J~

Si ottiene infine la c-ondizione di c~patibilita dinamica

UN[pJ2'!+(Q-t~l)~ =

(3.6)

0

ove !!=~-p~

(3.7)

e un

tens ore doppio simmetrico.

Le implicazioni della equazione dell'energia si ottengono ra=

[2}

pidamente. Per essere

N

= 0 con

l.... [sR1 .Jl

[DiV SR1 = -

e

1lj.,

:j

=

a.9. 1. R . Essendo

L =

~

ottiene

-- aeN ) • JJi

Poiche

un tens ore

-

e una funzione doppio~tale che N L ( a 8 N )

-

e si ottiene,alla fine

~oe[iJ = A!J!.! -

( 3.8 )

lineare di L ( a

__

l....~!J

=f [pJ

con I

~

3.10

+

" - 5. d~! 't'

-

esiste

=.11. N.

a

"-"",,o.J

•~

UN

D'altra parte ,se B1 ricorda la ( 3.1)4

3.9 )

liP'; )

~

, s i ha

(~- p.f) .!!-.~

1 = - 1:! T'

Si ottiene quindi la condizione di compatibilita

3.11 )

"'!J!.J! -l....!!-.!!. ~= feEpJ

+fJ(e'- Pl) N.a ~

~

UN

~oiche

[p J

si

a1 limits ad esprimere

e di

p •

puo

ricavare dalla ( 3.6 ), la (3.11 )

A in funzione

di grandezze continue

Moltiplicando, infatti, la ( 3.6 ) per

T'

N

-~

si

245

0]

ottiene

G1

( 3.12 )

=

J.JL

~ =

50stituendo questa determinazione di

IT!I

nella ( 3.6 )

a1 perviene a ( 1 - \I ClII ) ( Q _~ U..

3.13 )

...........

5i rieordi ora che pendicolari a

a

( 3.15 )

-

Q a l

~

II problema

=

0

posto

'"

( 1 _::!Ilf~ .8.1 = la condizione ( 3.13 ) a1 scrive ( 3.14 )

a:

deve essere Bcelto tra 1 vettori per=

~

\I

1 )

"\J)ON---"",

~

e quindi

) ,g ,

= ~ U'l a "

N

-

-

a • \l =

~

0

ridotto a bidimensionale,ed esistono

pereio due autovettori almeno diatinti e,ae i eorrispondenti autovalori sono poaitivi, easi individuano Ie direzioni di di' ,= aeontinuita e Ie rispettive velocita di propagazions. In particolare,sia per

--

e

A

u

A

A =

v'l.

u

un Rut ovett ore comune, ( se esiste )

11 ~

!

u

=

:;2.

U

si ottiene

~l ~ =l]oU; ~= (v.. _pV2.)u

(3.16)

la quale mostra che,anche Be

ehe

U 'N

-

Segue dunque

2fl[~

.!! 1.

N

T

l)

e infine (e N 1 .N

=

i grad ~ +

( grad

,... -

l::. )

-

L~ J= ~~ ~411 ~ 1 ;>. • N = 0

2" .... 0

,...,

+

!!;illI~,}

248

( 4.8 )

frtl =

0

C"L(" 1 =

0

(4.4 )

e,da~la

( 4.9 ) Aneora dalla

4.6 )1 ,molt1pl1cando scalarmente per

iI"

a1 ottiene

e percH. ( 4.9 ) che aasegna la 'i,(')

=

la ( 4.6 )2

d1 propagaz1one dell'onda.Infine,po1che +/f1l.'i) [t e.)] = 0

veloc1t~

(f)~)

"'It: •

e

~~ •

e

e aodd1afatta da

'

[~'.!! 1

sodd1afatta anche La ( 4 .6 )3 (2)

=

0, che rende

aubordinatamente

1

["It GlJ.1 • N = 0'11 gra= temperatura normale deve es~ere continuo .•

Se vale 18 lagge d1 l'OURIER , 4iente

di

alla

( 4.9 ):11 fluaao normale d1 calore

e continuo.

249

[e 1 f

b)

0

(onde term1che ) ,

a

~ 0

In questa caso,11 vincolo essendo rappresentato da (4.5 ),s1

ha

-

( 4.10 )

".'

~

( 4.11 )

= a@

N

~

da'ora,molt1pl1cando scalarmente per

La(4.6)1

u;

= [- 'It +

a @

A

[~1.1I

Se

~.

non

e

=,.u.~.

parallelo ad

~

UN la determinaz10ne

la ( 4.12 )

e

~

-(~.'.!!)!i.)

=~ •• ! = a e .

t

.. " ~UN~.·~

mentre,molt1pl1cando per (4.12)

+ 2)LA Gl

= t(~.

!!.].

[!.

eJ

~

e -

N,

~

t

,s1 ott1ene ancora per

(4.9). Se invece l'onda

U:

una 1dent1til. e

Per onde non longitudinal1

(e

e

e longitudinale

indeterminato.

neanche trasversal1, v. (4.10)

s1 ott1ene dalla ( 4.11 ) per la ( 4.9 )

[}GJ

( 4.13

a

=

e (;\

+)J. -

["l"11 =

ofr.n.l Oss .1 ) Segue.po~endos1

4 .15 )

-a(~+).l»

( .!L. ( 2'{

r.'t-

D' altro canto,la ( 4.1 )4

)

~

e ( 4.14 )

~

+!!...

implica

che

1f

sia continua

e quindi ahe sia continua anche assumere

(€("] =

)@.

~.'t;

"!'('J .

=0

~

L'P '!] l

e quind1, da ( 4.6 )2 ( 4.16 ) che coincide con

(q~11.

N,

=

m. 'l:

['1("].

(4.6)3' Le cond1zion1 di comp~tibilita

termomeccaniche Bono qUindi atte a determinare

UN ed insie=

250

-

me le diseontinuita di

e

e

di '7.. (Ii in funzione di ® .

3i ha snehe ( efr. ( 1.19 )1

c~ con

uJ

=

U =

0

u

n

- v

n

u;..c veloeita dl avsnzamento ( locale dl propagazlone )

dell' onda. Linearlzzsndo

quindl

[~l"

UN' =

-t, (#lJ

=

~o[l\J

= -

go af> UN

e quindi

[ ~ ('ll =

4.17 ) l'onda

e

compresaiva

se

-\,oa6:

>

a @

O. espansiva nel

caso

opplilsto (3) (3) 31 osserva ehe ( 4.17 ) vale snehe se l' onda e longltudl= ( ® = 0 ) impliea lOa nale, mentre,per onde meccaniche cont inulta dl ~(". Per onde longitudinall,

UN

non pub eSsere determinato dal=

le sole eondlzlonl di diseontinuita,mentre per le dlseontinul= ta di )C , di

"'L 1') [}Cl

( 4.18 )

sl ha

=

a@(?I+21J-1l'

r

-

a

-oU") I~ N

251

mentre

e

anche

BDlLIOGRAFIA A

Vincoli

1 - J.E.

Adkins

: A three-dimensional problem for

highly

elastic materials subject to constraints,Quart.J.Mech.Appl. Math,11,88-97 ( 1958 ) 2 - J.E. Adkins - R.S.Rivlin : Large elastic deformation of isotropic materials : X • Reinforcement by inextensible cor: ds, Phil.Trans.Roy.Soc.London A, 248 201-223 ( 1955 ) 3

T.Alts -

Termodynamics of thermoelastic bodies With

kinematic constraints : fiber reinforced materials.

Arch. Rat.Mech An. 61 (1976) 253-289. 4 - G.B. Amendola

A genaral theory of thermoelastic solids

restrained by an internal thermomechanical constraints,

Atti Sem. Mat. Fis. Univ. Mod. 27 (1978) 1-14. 5 -

J. Bell

: The experimental foundations of solid mecha=

nics.Encycl.of Phys.Vol

VI a/I ,Springer Verlag ,Berlin

1973. 6 - G.Capriz - P.Podio- Guidug1i

Formal structure and

classification of theories of oriented materials,Ann.mat.

pura appl.(4) 115 (1977)17-39. 7 -

H.H. Erbe

W~rmeleiter

mit thermomechanischer in;

neres Zwangsbedingung,ZAM,Sonderheft

76-79 (1975)

252

8 - J.L. EriksGn- R.S.Rivlin

: Large elastic deformations of

homogeneous anisotropic materials, J.Rat.Mech.Anal.3,281361, '1954) 9 - A.E.Green-J.E.Adkins : Large

elastic deformations. and

non -linear continuum mechanics,. Oxford,Clarendon l'ress 1960 ~O-

A.E. Green- P.M.Naghdi-J.A.Trapp : Thermodynamics of a con= bnuum with internal constraints,lnt.J.Enw1g.

.Sei.8 (1970)

891-908. 11- M.F.Gurtin-P.Podio-Guidugli : The thermodynamics of constrai ned materials, Arch.Rat.Mech An.51 (1973) 192-208. 12 -E. Hellinger , Die allgemeinen Ansatze der Mechsnik der Kon= 4 tinua,Enz.math.Wiss. 4 , 602-694 (1914). 13 -S.Levoni:Sulla integrazione delle equazioni della termoela= sticita di solidi incomprimibili,Ann.Sc.Bbrm.Sup. Pisa (3), 22,(1968),515-525.. 14 -T.Manacorda:Una osservazione sui vincoli interni in un soli= do,Riv.Mat.Univ.Parma (3),3,(1974),169-174. 15 -T.Manacorda:Introduzione alla termomeccanica dei continui, QUaderni UMI,n.14,Pitagora Ed. ,Bologna 1979. 16 -S.L.Passman:Constraints in Cosserat surface theory,lst.Lomb. Sc.Lett. Rend. A,106,(1972),781-815. 17 -A.C.Pipkin:Constraints in linearly elastic materials,J.of El. 6,(1976),179-193. 18 - H.Poincare:Le90ns sur la theorie mathematique de la lumie= re,Paris, 1889. 19 -H.Poincare:Le90ns sur la theorie de l'elastcite,Paris,1892. 20 -S.Rivlin:Plane strain of a net formed by inextensible cords, J •Rat.Mech.Anal. 4, (1955) ,951-974. 21 -A.Signorini:Trasformazioni termoelastiche finite,Mem.III, Ann.Mat.pure app1. (4),39,(1955),147-201.

253

22 - A.J.M.Spencer:Finite deformations of an almost incompres= sible elastic solid,Second-order Effect in Elasjicity, Haifa 1962,Oxford,Pergamon Press,1964. 23 - J. A. Trapp: Reinforced materials with thermo-mechanical

COIl=

straints,Int.J.Engng.Sci. 9,(1971),757-778. 24 - L.R.G.Treloar:The physics of rubber elasticitY,Clarendon Press,Ox~ord,1949.

25 - C.Truesdell - W.Noll:The non-linear field theories of Me= chanics,Encycl.of Phys. Vol.III/3,Springer Verlag,Berlin, 1965.

Bl

) Omie.

1 - A.Agostinelli:Sulla propagazione di onde termoelastiche in un mezzo amogeneo e isotropo,Lincei Rend. (8),50 (1971) 163-171,304-312. 2 -G.B.Amendcla : On the propagation of first order waves in incompressible thermoelastic solidS, B.U.M.I. (4) 1 267-284. 3 -

G.B. Amendola rials. Atti

Acceleration waves in incompressible mate.

Sem.Mat.Fis.Modena

24

(1975) 381-395.

4 - C.E.B.evers : Evolutionary dilational shock waves generalized

(1~73)

in

a

theory of thermoelasticity,Acta Mech. 20(1974)

67 -'19.

5 -

B.R.Bland: On shock waveS in hyperelastic media,IUTAM, Second order effects in Elasticity,Plasticity and Fluid Dynamics,Pergamon Press,1964.

6 - P.Chadwick - P. Powdrill:3ingulllr surfaces ilL linear thermo=

254

elasticity,Int.J.Engng Sci. 1 (1965) 561-596. 7 -

P.Chadwick-P~.Currie

, The propagation and growth of acce=

leration waves in heat-conducting elastic materials,Arch.

Rat.Mech.Ah 8 - P.J.Chen

49

(1972) 137-158.

Growth and decay of waves in solids,Enc.of Phy=

sics, VI a/ 3 Springer Verlag,Berlin, 1973. 9 - P.J .Chen - M.E.Burtin

: On wave propagation in inextensi=

ble elastic bodies, Int.J.Solids Struct.1Q

(1974) 275-281.

10 - P.Chen-J.W.Nunziato :On wave propagation in perfectly heat conducting inexteneible elastic bodies,J.of El.5(1975) 155-160. 11 - B.D.Coleman, M.E.Gurtin,I.Herrera,C.Truesdell : Wave prop! gation in dissipative materials,A reprinj of five memoirs, Springer Verlag, New York 1965. 12 - WEiD.Collins : One-dimensional non linear wave propagation in incompressible elastic materials,Q.J.Mech.appl.Math.19 (1966) 259-328. 13 - J •Dunwoody

: One dimensional shock waVeS in heat conduct!

ng materials with memory : 1- Thermodynamics-Arch.Rat.Mech. An.~7

(1972) 117-148; 2-Shock analysis, ibidem, 192-204;

3- Evolutionary behaviour, ibidem 50;278-289. 14 - J.Dunwoody:On weak shock waves in thermoelastic solids,Q. J.Mech.appl.Math.30 (1977) 203-208. 15 - J.L.Eriksen:On the propagation of waves in isotropic incom= pressible perfectly elastic materials,J.Rat.Mech.An. 2 , (1953) ,329-338. 16 - A.E.Green-P.M.Naghdi,A derivation of jump condition for entropy in thermomechanics,J.of Elast. 8 (1978) 179-182. 17 - E.Inan:Decay of weak shock waves in hyperelastic solids, Acta Mech. 23 (1975) 103-112.

255

18 - T.Manacorda:On the propagation of discontinuity wave~ in ther-moelastic incompressible solids,Arch.Mech.Stos.(2),

24 (1971) 277-285. 19 - T.Manacorda: Zagadnienia Elastodynamiki,Ossolineum,Varsa=

via,1978; Cfr.anche A - 15. 20 -

R.W.Ogden:~rowth

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) 57-75. 22 -

N~Scott:Acceleration

waves in incompressible elastic so=

1ids,Q.J.Mech.appl.Math. 29 (1976) 295-310. 23 - N.Scott-M.HaYes:Small vibrations of a fiber-reinforced composite,Q.J.Mech.app1.Math. 29 (1976) 467-486. 24 - C.Trimarco:Onde di accelerazione in materiali termoelast±:

ci con vincolo di inestendibi1ita,in pubbl. su Atti Ace. Sci.Modena. 25 -

C.Truesdell:~enera1

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.) 10 1980 C. LM.E. Session: "Wave propagation".

Bressanone 8-17 giugno 1980. "ENTROPY PRINCIPLE" AND MAIN FIELD

FOR A NON LINEAR COVARIANT SYSTEM by

TOMMASO RUGGERI Istituto di Matematica Applicata -

Universita di Bologna

Via Vallescura 2 - 40136 Bologna (Italy).

1.Introduction My lecture is complementary to the lectures given by G.Boillat in the first

part of this course. In

part~cular

I am shall deal with some problems concer-

niog quasi-linear hyperbolic system compatible with a supplementary conservation

law; relativistic theories will be considered with special emphasis. I start with a brief bibliographical introduction to the subject I shall be

concerned with. In 1970-71 1. MUller, in Bome 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 eqlJations. Adding the constitutive equations to the previous system one gets a system of 5 equations in 5 unknowns. Each solution of this system is called a IIthermodynamic process". Then MUller postu- lates the existence of an additive function cr (entropy) such that: +

(PSv

i

+

~

i

) = cr

>

o

¥ thermodynamic process.

(1)

Furthermore he supposes that both the entropy density S and the flux .1 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

which

he

the

author shows

with

identifies

the

the existence of' a universal function,

absolute

temperature;

hence,

he

deduces

the first principle of thermodynamics. In a different conte>d: in 1971 K.O.Friedrichs and P.D.Lax the

:former

in

1974

[3] Friedrichs,

[3J

examined

a

similar

in a covariant formalism,

problem.

In

[2] and later particular

in

considers a conservative quasi(0)

linear hyperbolic system of r first order equations of the type ,

a

3

a

(2)

(U) = .2(U)

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 l"a!!

Introducing the operator V

• U,

= l".2

= a/au

• , U•

a

we have = 0

(3)

(condi tion I)

Moreover Friedrichs supposes another condition holds: it exist at least a time-like covector

{~

a

l, independent of the field, such that the quadratic

form (condition II)

is positive definite.

Here 6U is an arbitrary variation of the field and

a

=6U,VV!!

6U.

Using condition I and II Friedrichs shows that the system of the field equations is a hyperbolic symmetric system.

(*) To avoid misunderstanding the vectorErn

r

are underlined.

261 Later several authors [4J,

[5].

[6J provided further contributions on this

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 opi-

nion, have not yet been pointed out. 2. Main field and Covariant convex density.

Let V~ be a C·, 4-dimensional manifold 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

aB

represents the components of the

metric tensor of signature (+ - - -). On V4 we consider a quasi-linear conservative system of N first order partial differential equations for the unknown N-vector U(x

Q )

ERN (5)

a

the components of F

and U are contravariant tensors and

~Q

is intended as a

covariant derivative operator. We suppose that the system (5) is hyperbolic. i.e.: ~ a time-like covector (tal, such that the following two statements hold: det(Aat ) ~ 0

i)

•

a

(A.a •• Fa: • • a/aU)

ii) V covector{t a } of space-like the eigenvalue problem (t-."t)Aad.O a a'

(7)

has only real proper solutions p(k) and a set of linearly independent eigenvectors d

(k)

(k=1.2 ••••• N).

The covectors{C

- pta} • where p is solution of (7) are called "characteristic", a while the {tel} fulfilling 1). 11) are said "subcharacteristic ll •

262

When a differentiability conditions holds, let us suppose that (5) is compatible with a supplementary conservation law

a. h

G

(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:

Since by (3) 1 is defined up to a scalar factor, we may write

l

-

then Friedrichs conditions lock like U'·VFO= Vho,

(9)

U'·C

(10)

g.

(11)

We remark that (9)

I

multiplied by 15 U, can be written

equivalently~

(12)

"6 U.

The identity (12) show the first important result:

u'is invariant mth respect to field transformatiors:in fact d!'! fa and

{t and then

U' depends only

it does not depend on the choice of the field variables.

By applying the operator li

to (12) and replacing into (11) we get

Q "" OUI • oyQ

~ >0

•

(13)

Hence (13) too is independent of the choice of the field; we may then choose the field in the convenient form U"" yQf;

•

(14)

263 We put also (15)

and contracting (12) with

to we

U' ·6 U

have

= 6h

+4

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

mul tipliers introduced by I-Shih Liu [7]

in the context of entropy principle of MUller. Condition (13) is now equivalent to (17 )

i.e. to the convexity of the covariant scalar density h=ha~ with respect to a the field U ::: F

a

to'

If for a system (5) there exists a vector U' and at least a caveator (£';a}such that

(12)

hold~ we say

and (17)

that the system is a convex covariant density

system. Conditions (16) and (17) ensure also that the mapping U' +-+U is globally invertible, becauseVU' =VVh and this gradient matrix is symmetric and positive definite; then for a theorem about globally univalence ([8J) ur +-+ U is gloN 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

~

(t)

(18)

flU' )

is given by A' ().

-

a

a

(19)

(t) For the proof of the statements proposed in this lecture one may see [9]

.

and (20)

System (18) is symmetric hyperbolic: in fact a system A,a a U' :z: f is sima a aT metric hyperbolic if ~' is positive definite, and in our ~' a case we have ='Q' V'h', but from (20) h,Q t = hi = •A' I;a 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 oneiQ

four-vector h

and this justifies our definition of "four vector generating func-

tion II 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 law

(8).

the

system

is

only

by

Moreover

well

assumes

posed.

the we a

conservative

pointed

out

system

that.

symmetric form,

Such

remarkable

so

(5)

and

the

supplementary

when

V'

is

that

the

local Cauchy

properties

suggest

chosen

us

to

as

field, problem

call

VI

the

"main field" of the system. We remark possess a the

that

not only on

the

mathematical

special role with respect

physical

point of view,

point of view U'

to other quantities,

and h I a

but also from

they are privileged, since they are related

to the Itobservables" of the physical system, as we shall see later. System a

(5),

sui table

easy

;-.0

compatible

choice of

prove

that

with

the

the

field

system

(8),

is

riducible

variables (18)

to

the

and vicevers8;

form

(18)

in fact

it is

provides always a supplementary law

(8):

let h

Q

= U'

• V'h lQ

_

h l o, we have a

U'·!' = g.

Finally we have shown that

for

a

h

Q

aa u'

U"f{U'}

265

A necessary and sufficient conditions for the system (5) to be compatible IJ'ith a supplementary conservation Z 2). be the equation

with a shock hypersurface for the field U.

It is known that the Rankine-Hugoniot conditions must hold

[Fa]~

=O,onf a where brackets denote the jump across rand q. a = 0a q. • Formally

the

Rankine-Hugoniot

equations

are

(21)

obtained

from

the

field

eqs. (5) through the correspondence rule

aa However

this

rule

does

+

~

[

(22)

]

a not work

when

applied

to

the

supplementary

equation (8); in fact (23)

does

not.

in general,

n is

non negative.

vanish.

This

Furthermore

result was proven

1. t

is

possible

to show

that

in

a non covariant formalism

by P.D.Lax [4J introducing an artifical viscosity in the field equations; a different proof was given in [51. It is know that the positive signature of

n for the non relativistic perfect fluid implies the growth of thermo-

dynamic

entropy across

n

is

> 0

often

the

called

in

shock. the

That

is

the

literature

reason why the condition

"entropy growth condition"

•

266

aoo

is

assumed as

a criterion to pich ur> the physical shocKs among the

solutions of the Rankine-Hugoniot equations. In this

section we

suggest the

proof of the fact that Let It

I

main steps of an explicitly covariant

r.

is non negative on

Tl

1 and a a

be a 6ubcharacteristic covector such that

" covariant scalar

defined 8S:

r. a +a

a = -

(24)

th8'l there exists a space-like covector {t )

such that

"

(25)

Let

cp(x

Cl

)

= 0 be the equation of a characteristic hypersurface which

locally has the same Itdirection of propagation" Le.

det(A" -

(jl )

det{A"(~

•

a

-

a

t )}

-"

a

to of the shock surface. (26)

0,

(27) where

Ik)

are

(k-=l,2 •...• N)

~

the

solution of

(26);

these

eigenvalues

are real by the hyperboliclty condition. Now

we

consider

a

solution

U U.

r

U* (in

field

being the

the

U(U',9),

perturbed

following

computed

~

for

*

"

and

will

U ::

of

the

denote

U*).

Here

the

U

Rankine-Hugoniot ~

U'

we

values take

(21) (28)

unperturbed the

equations

fields of

any

respectively

on

function of the

only k-shocks

according to

the following

Definition 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 C"'Il (k)

•

U

:: U·

(29)

267

Roughly speaking a k-shock

approaches

is a

shock that

to a characteristic velocity

weak Sh DC k 5 Wh en a

We suppose

is near to

to know

the

(k) )

~*

solution

vanishes

(of course,

(28)

for

a

shock

speed

these shocks become

k-shock

and

replace

it

a

a

differentiating

the

.

into (23): then we get n as function of U* and ¢I

By

when

a

"(U*'.a) • h (U(U*'.e».a - h (U*).a

(30)

to

ifa

(9).

a

J -

(30)

respect

and

taking

into account

after some calculations we obtain [h

Vh· [F

a

)

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·CU - Ufo) > 0,

lJ. U

f. U* oS D

So the r.h.s. in (31) is equal to -w, restricted to r. Hence 3 n Ja¢la

(a

Furthermore.

in

the




(33)

0

As an/CIa 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

n

= 0

0"" (k)

•

hence we get

For a convex covariant density system and a k-shock one Iuw when

0

;:.

(k)

~*

(on rJ.

268

1i) Jl

If

as generating f'unction of' the shock.

is a know function of u· and

1)

+" a

it is easy to prove that the

to llo»iYl{/ re lations ho lde on r

v· '1

A~ • a

J

[U'

..

(34)

whsre

EQ. a

(34)

that if ve kno... only the scalar function

D'le8nS

function oC U·

Jump

of

U'

and

and

• a ' w1 th

(which 1s

non characteristic) we may find the

•Q

therefore of U;

Tl

"behaves

like a

"potential"

for

the

shock. Of course,

in practice,

ll(U·, +0)

is computed when the shock is known

8S a solution of the Rankine-Hugoniot equations. However it is interesting the

fact

that.

were

it

possible

to

determine

n

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 Fa (U)+

(35)

a

provide N equations for the perturbed field U if U· and fa are knaYn. Eqs. (35) are equations of the kind

which always possess

feu. +) = f(U·.+ ) a a the trivial solution U

have also non trivial solutions U .solution)

which

in turn are

~

(36) :=

U"

for any

~a

They may

U* (branching solutions of the trivial

physically

acceptable only

aB

if g

tate

~

o.

so that the speed of' the shocks does not exceed that of light, according to relativity theory. We put no'"

the followin&; question: function

f

is

does it exist a set of values of

+0 such

that

the

globally

invertible

U for a

fixed

to? If the answer is affermative,

with

respect

to

then only the trivial

269 solution U

= U·

is allowed.

The problem has been examined by G.

[6]

SoHIat and T. Ruggeri

I

who proved

that non vanishing schoks take place only if their speed is greater

than

the

smallest

characteristic

speed

and

smaller

than

the

greatest

one. It is possible

to provide an explicitly covariant formulation of the

[6J

proof given in

and show that:

Fop hyperbolic convez covariant density systems the speed of the non vanishing shoek fuLfa the eondition: (37)

where m

mini \l

in!

UfD

As a consequence the

(k) }

I

M

sup

Max {\l

UfD

k

8h~ck

(k)

k

manifoLds are time-like or light-like if so are

the characteristic manifotds. In fact

if

(37)

holds,

and the characteristic manifolds are time-like

or light-like we have: .B (k) (k)} ( gcp~ Ma x