Partial Differential Equations of Mathematical Physics 048665964X, 9780486659640

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S.L. Sobolev

Partial Di�erential Equations of Mathematical Physics

PARTIAL DIFFERENTIAL EQUATIONS OF MATHEMATICAL PHYSICS

ADIWES INTERNATIONAL SERIES IN MATHEMATICS A. J. L ohw ater, Consulting Editor

S. L. SOBOLEV

PARTIAL DIFFERENTIAL EQUATIONS OF MATHEMATICAL PHYSICS Translated from the third Russian edition by

E. R. DAWSON Lecturer in Mathematics at Queen's College, Dundee University of St. Andrews

English translation edited by

T.A.A. BROADBENT Professor of Mathematics Royal Nava! College, Greenwich

1964 PE R G A M O N PRESS O X F O R D • L O N D O N • E D I N B U R G H • NEW YO RK PARIS • F R A N K F U R T

A D D IS O N -W E S L E Y P U B L I S H I N G COMPANY, INC. R E A D I N G , M A SSA C H U SE T T S • PALO ALTO • L O N D O N

Copyright © 1964 P e r g a m o n P ress L t d .

U. S. A. edition distributed by A D D I S O N - WE S L E Y P U B L I S H I N G COMPANY, I NC. Reading, Massachusetts • Palo Alto • London

Library of Congress Catalog Card Number 63-19262

This translation has been made from S. L. Sobolev’s book y paenemvi Mame.uamunecnou 0 u 3 u k u (Uravneniyci mcitematicheskoi fiziki) published by Gostekhizdat, Moscow

CONTENTS T ranslation E ditor ’s P reface A uthor ’s P refaces

to the

L ecture 1. D erivation

ix

F irst

of the

and

T hird E ditions

x

F undamental E quations

1

§ 1. Ostrogradski’s Formula § 2. Equation for Vibrations of a String § 3. Equation for Vibrations of a Membrane § 4. Equation of Continuity for Motion of a Fluid. Laplace’s Equation § 5. Equation of Heat Conduction §6. Sound Waves L ecture 2. T he F ormulation of P roblems H adamard’s E xample

of

M athematical P hysics. 22

§ 1. Initial Conditions and Boundary Conditions § 2. The Dependence of the Solution on the Boundary Conditions. Hada­ mard’s Example L ecture 3. T he C lassification

1 3 6 8 13 17

22 26

33 § 1. Linear Equations and Quadratic Forms. Canonical Form of an Equation 33 § 2. Canonical Form of Equations in Two Independent Variables 38 § 3. Second Canonical Form of Hyperbolic Equations in Two Independent Variables 42 § 4. Characteristics 43 of

L inear E quations

L ecture 4. T he E quation for a Vibrating String d ’A lembert’s M ethod

of the

and its

Second O rder

Solution

by

§ 1. D’Alembert’s Formula. Infinite String § 2. String with Two Fixed Ends § 3. Solution of the Problem for a Non-Homogeneous Equation and for More General Boundary Conditions L ecture 5. R iemann’s M ethod

46

46 49 51 58

§ 1. The Boundary-Value Problem of the First Kind for Hyperbolic Equations § 2. Adjoint Differential Operators § 3. Riemann’s Method § 4. Riemann’s Function for the Adjoint Equation § 5. Some Qualitative Consequences of Riemann’s Formula v

58 62 65 68 71

VI

CONTENTS

L ecture 6. M ultiple I ntegrals : L ebesgue I ntegration

72

§ 1. Closed and Open Sets of Points § 2. Integrals of Continuous Functions on Open Sets § 3. Integrals of Continuous Functions on Bounded Closed Sets § 4. Summable Functions § 5. The Indefinite Integral of a Function of One Variable. Examples § 6. Measurable Sets. Egorov’s Theorem § 7. Convergence in the Mean of Summable Functions § 8. The Lebesgue-Fubini Theorem L ecture 7. I ntegrals D ependent

on a

P arameter

73 79 85 92 99 103 111 121 126

§ 1. Integrals which are Uniformly Convergent for a Given Value of Para­ meter 126 § 2. The Derivative of an Improper Integral with respect to a Parameter 129 Lecture 8. T he E quation

of

133

H eat C onduction

§ 1. Principal Solution § 2. The Solution of Cauchy’s Problem Lecture 9. L aplace ’s E quation

and

133 139

P oisson’s Equation

146

§ 1. The Theorem of the Maximum § 2. The Principal Solution. Green’s Formula § 3. The Potential due to a Volume, to a Single Layer, and to a Double Layer Lecture 10. Some G eneral C onsequences

of

L ecture 12. T he S olution

in an

of the

U nbounded M edium . N ewtonian

D irichlet P roblem and the

and the

§ 1. § 2. § 3. §4.

of the

P otentials

171

for aS phere

N eumann P roblem

for a

180 R etardedP otential

§ 1. The Characteristics of the Wave Equation § 2. Kirchhoff’s Method of Solution of Cauchy’s Problem Lecture 15. P roperties

155 158 162 166

L ecture 13. T he D irichlet P roblem H alf-S pace L ecture 14. T he W ave E quation

150 155

G reen’s F ormula

§ 1. The Mean-Value Theorem for a Harmonic Function § 2. Behaviour of a Harmonic Function near a SingularPoint § 3. Behaviourof a Harmonic Function at Infinity. InversePoints Lecture 11. P oisson’s E quation P otential

146 148

of

Single

General Remarks Properties of the Potential of a Double Layer Properties of the Potential of a Single Layer Regular Normal Derivative

and

188

188 189 D ouble L ayers 202

202 203 210 217

vii

C ON T E N T S

§ 5. Normal Derivative of the Potential of a Double Layer § 6. Behaviour of the Potentials at Infinity L ecture 16. R eduction of the D irichlet P roblem P roblem to I ntegral E quations

and the

218 220 N eumann 222

§ 1. Formulation of the Problems and the Uniqueness of their Solutions 222 § 2. The Integral Equations for the Formulated Problems 225 L ecture 17. L aplace’s E quation

and

P oisson’s E quation

in a

P lane

§ 1. The Principal Solution § 2. The Basic Problems § 3. The Logarithmic Potential L ecture 18. T he T heory

of I ntegral

228

228 230 234 237 237 238 242 243 248 253 256

E quations

§ 1. General Remarks § 2. The Method of Successive Approximations § 3. Volterra Equations § 4. Equations with Degenerate Kernel § 5. A Kernel of Special Type. Fredholm’s Theorems § 6. Generalization of the Results § 7. Equations with Unbounded Kernels of a Special Form L ecture 19. A pplication of the T heory of F redholm E quations Solution of the D irichlet and N eumann P roblems

to the

§ 1. Derivation of the Properties of Integral Equations § 2. Investigation of the Equations

258 258 260

L ecture 20. G reen’s F unction

265 § 1. The Differential Operator with One Independent Variable 265 § 2. Adjoint Operators and Adjoint Families 268 § 3. The Fundamental Lemma on the Integrals of Adjoint Equations 271 § 4. The Influence Function 275 § 5. Definition and Construction of Green’s Function 278 § 6. The Generalized Green’s Function for a Linear Second-Order Equation 281 § 7. Examples 286

L ecture 21. G reen’s F unction

for the

L aplace O perator

291

§ 1. Green’s Function for the Dirichlet Problem § 2. The Concept of Green’s Function for the Neumann Problem

291 296

L ecture 22. C orrectness of F ormulation of the Boundary-Value P roblems of M athematical P hysics

301

§ 1. The Equation of Heat Conduction § 2. The Concept of the Generalized Solution § 3. The Wave Equation §4. The Generalized Solution of the Wave Equation § 5. A Property of Generalized Solutions of Homogeneous Equations

301 304 307 311 317

viii

CONTENTS

§ 6. Bunyakovski’s Inequality and Minkovski’s Inequality § 7. The Riesz-Fischer Theorem

322 324

L ecture 23. F ourter’s M ethod

327

§ 1. Separation of the Variables § 2. The Analogy between the Problems of Vibrations of a Continuous Medium and Vibrations of Mechanical Systems with a Finite Number of Degrees of Freedom § 3. The Inhomogeneous Equation § 4. Longitudinal Vibrations of a Bar L ecture 24. I ntegral E quations

with

R eal, Symmetric K ernels

§ 1. Elementary Properties. Completely Continuous Operators § 2. Proof of the Existence of an Eigenvalue Lecture 25. T he Bilinear F ormula and the H ilbert-S chmidt T heorem § 1. The Bilinear Formula § 2. The Hilbert-Schmidt Theorem § 3. Proof of the Fourier Method for the Solution of the Boundary-Value Problems of Mathematical Physics § 4. An Application of the Theory of Integral Equations with Symmetric Kernel L ecture 26. T he I nhomogeneous I ntegral E quation K ernel

with a

of a

334 336 339 342 342 354 357 357 364 367 375

Symmetric

§ 1. Expansion of the Resolvent § 2. Representation of the Solution by means of Analytical Functions L ecture 27. Vibrations

327

R ectangular P arallelepiped

376 376 378 382

L ecture 28. L aplace’s E quation in C urvilinear C oordinates. E xamples of the U se of F ourier ’s M ethod 388

§ 1. Laplace’s Equation in Curvilinear Coordinates § 2. Bessel Functions § 3. Complete Separation of the Variables in the Equation V2u = 0 in Polar Coordinates L ecture 29. H armonic P olynomials and Spherical F unctions § 1. Definition of Spherical Functions § 2. Approximation by means of Spherical Harmonics § 3. The Dirichlet Problem for a Sphere § 4. The Differential Equations for Spherical Functions L ecture 30. Some E lementary P roperties of Spherical F unctions § 1. Legendre Polynomials § 2. The Generating Function § 3. Laplace’s Formula I ndex

388 394 397 401 401 404 407 408 414 414 415 418 421

TRANSLATION ED ITO R ’S PREFACE classical partial differential equations of mathematical physics, for­ mulated and intensively studied by the great mathematicians of the nineteenth century, remain the foundation of investigations into waves, heat conduction, hydrodynamics, and other physical problems. These equations, in the early twentieth century, prompted further mathematical researches, and in turn themselves benefited by the application of new methods in pure mathematics. The theories of sets and of Lebesgue integration enable us to state conditions and to characterize solutions in a much more precise fashion; a differential equation with the boundary conditions to be imposed on its solution can be absorbed into a single formulation as an integral equation; Green’s function permits a formal explicit solution; eigenvalues and eigenfunctions generalize Fourier’s analysis to a wide variety of problems. All these matters are dealt with in Sobolev’s book, without assumption of previous acquaintance. The reader has only to be familiar with element­ ary analysis; from there he is introduced to these more advanced concepts, which are developed in detail and with great precision as far as they are re­ quired for the main purposes of the book. Care has been taken to render the exposition suitable for a novice in this field: theorems are often approach­ ed through the study of special simpler cases, before being proved in their full generality, and are applied to many particular physical problems. Commander Dawson has taken pains to render his translation idiomatic as well as accurate, thus assisting the English reader to avail himself readily of the vast amount of information contained in this volume. The

T . A. A . B roadbent

A U T H O R ’S PREFACE TO THE FIRST E D IT IO N book is based on a course of lectures given in the “Lomonosov” State University in Moscow. The author has therefore retained the name “lectures” for the various sections. The same circumstance also explains the selection of material, the extent of which was limited by the number of lecture periods. The author expresses his deep gratitude to Academician V. I. Smirnov, who read through the manuscript, and also to Professor V.V.Stepanov for his useful comments. _ _ S . Sobolev T h is

A U T H O R ’S PREFACE TO THE THIRD E D IT IO N third edition of this course on “ Equations of Mathematical Physics” differs little from the second edition, which underwent extensive revision. In the second edition the lecture on the Ritz method was omitted since that subject lies somewhat apart from the rest of the course. The theory of multiple Lebesgue integrals and of integral equations has been simplified somewhat, and the proof of the Fourier method has been made more precise. As in the second edition, various improvements in style and clarifications of the presentation have been made. Moreover, in this third edition the lecture on the dependence of the solutions of equations of mathematical physics on the boundary conditions has been developed in greater detail by the editor, V. S. Ryaben’kii. The author expresses his gratitude for valuable comments made by var­ ious people when the second and third editions were being prepared. Partic­ ularly valuable comments were made by Academician V.I.Smirnov and the editor of the third edition, V . S . Ryaben’kii. __ S . Sobolev The

x

LECTURE 1

DERIVATION OF THE FUNDAM ENTAL EQUATIONS theory of the equations of mathematical physics has as its object the study of the differential, integral, and functional equations which describe various natural phenomena. It is somewhat difficult to define the precise limits of the subject as it is usually understood. Moreover, the great variety of problems relating to the equations of mathematical physics does not allow them to be dealt with at all fully in a university course. The present book contains only a fraction of the whole theory of the equations of mathematical physics: it includes only what seemed to be most important for an intro­ duction to the subject. The course is devoted for the most part to the study of second-order partial differential equations with one unknown function; in particular, we shall deal with what are usually called the classical equations o f mathematical physics, namely, the wave equation, Laplace’s equation, and the equation of heat conduction. We shall develop the necessary theory of related problems as we go along. T he

§ 1. Ostrogradski’s Formula f Before we undertake the derivation of those equations of mathematical physics with which we shall be concerned, we recall a formula of integral calculus dealing with the transformation of surface integrals into volume integrals. Let P{x, y, z), Q(x, y, z), R(x, y, z) be three functions which: (i) are specified in a certain domain D of the variables x, y, z; (ii) are continuous right up to the boundary of D; and (iii) have continuous first-order partial derivatives with respect to x, y, z throughout D. Consider within D some closed surface S consisting of a finite number of pieces for each of which the tangent plane varies continuously. Such a surface f Mikhail Vassilievich Ostrogradski (1801-1862). See Mem. Acad. Imp. Sci., St. Peters­ burg (6), 1, 130(1831). The result is otherwise known as Green’s lemma, Gauss’s theorem, or the Divergence theorem. —Translator.

1

2

D E R I V A T I O N OF F U N D A M E N T A L E Q U A T I O N S

L.l

is said to be piecewise smooth. We shall further suppose that any straight line parallel to any of the coordinate axes either intersects S in a finite number of points or has a whole interval in common with it. Consider the integral J J [Pcos(ra, a ) + Q cos (n,y) + R cos (n, z)] dS.

(1.1)

where cos (n, a) , cos (n, y ), cos (n, z) denote the cosines of the angles formed by the inward-directed normal to the surface S at the point ( a , y, z), and dS is an element of the surface. Using vector notation, we can regard P, Q, R as the components of a certain vector T. Then P cos (n, a ) + Q cos (n, y) + R cos (n, z) = Tn, where T„ is the projection of the vector T in the direction of the inward normal. A classical theorem of integral calculus enables us to transform the surface integral (1.1) into a volume integral over the region D bounded by the surface S. This theorem asserts th a t:

Jf,

[P cos (n, a ) + Q cos (n, y) + R cos (n, z)] dS dP dQ dR ---- + + ----_ dx dy dz

dA dy dz

or in vector notation Tn dS = -

div T di?

( 1. 2 )

where du denotes an infinitesimal volume-element and ,. „ dP dQ dR div T — ---- + — + dy dz dx

(1.3)

The formula just obtained is valid under rather more general assumptions with regard to S. In particular formula (1.2) holds for any piecewise smooth surface bounding a certain region D. We shall in future take the word “surface” to mean a piecewise smooth surface unless a further restriction on its meaning is made.

§2

3

VI BRATI NG STRING

An important result follows from formula (1.2): L e m m a 1. Let F be a continuous function defined in some domain in threedimensional Euclidean space. Let S be any closed surface within the domain over which a vector function T is specified, and let S bound the region f t . Then the necessary and sufficient condition for the equality F dv = 0

Tn d S % )J s

(1.4)

ft

to hold good is that div T + F — 0. For, using formula (1.2), we can put the equality (1.4) into the form (div T + F) dv = 0, Jft and then the sufficiency of the condition in the lemma becomes obvious. It is also necessary. For, suppose, if possible, that the function div T + F is different from zero, say positive, at some point A; then because of continuity it would also be positive in the neighbourhood of A, and the integral /%

(div T

F) dv

J J J CO

taken over a small region co round A would be non-zero, and so the left-hand member of (1.4) would also be different from zero. Hence our supposition contradicts (1.4), and the necessity of the equality div T + F = 0 is proved. The analogous lemma for a two-dimensional region lying in a plane can be proved in a similar way. § 2. Equation for Vibrations of a String Consider a string stretched between two points. By string we mean a rigid body whose other dimensions are small compared with its length; and we also suppose that the tension in it is considerable, so that its resistance to flexure can be neglected in comparison with the tension. We take the jc-axis to be along the string when it is in equilibrium under the action of the tension only. When transverse forces act on the string it will assume some other form, in general non-rectilinear. Imagine the string cut into two pieces at some point x and consider the interaction between the two parts. The force which the right-hand part exerts

4

D E R I V A T I O N OF F U N D A M E N T A L E Q U A T I O N S

L .l

on the left-hand part is directed along the tangent at x to the curve represent­ ing the string and is denoted by T(x) (see Fig. 1). To simplify the discussion, let us suppose that the motion of the string takes place in one plane, and let u denote the displacement of the string from its rest position. Let u — u(x, t) be the equation of the curve assumed u

by the string in the plane xOu. Let q ( x ) denote the linear density of the string at the point x, i.e., the limit of the ratio of mass to length for a small part of the string. We consider first the equilibrium position of the string under the in­ fluence of a transverse loading p(x)\ by which we mean that the part of the string defined by x t ^ x ^ x 2 is acted on by a force, directed along the w-axis, of 'X2 magnitude p(x) dx. Let a(x) be the angle formed with the x-axis by the V^1 tangent to the string at the point x; then the component along the u-axis of the tension acting at the point x 2 is given by |j(x 2)| sin a(x2) = T(x2) sin a(A"2) where T(x) is the absolute magnitude (the length) of the vector T(x) . Similarly, the component along the w-axis of the tension at x t is given by - I^OOl sin «(.\t ) = - T(at) sin a f o ) . Now du dx

and if we take dujdx to be so small that its square can be neglected, we obtain

§2

5

VI BRATI NG STRING

as the equilibrium condition for the string p(x) dx = 0

+

(1.5)

*i

But, clearly,

*, dx \

dxj

so that the condition (1.5) can be written as ’*2 ' d J *1

' T - ^ - N) + P(x) dx = 0. dx \ dx J J

( 1. 6 )

Since (1.6) holds for any values of x t and x 2, the integrand must be identic­ ally zero, i.e., - ( r - )

+ PW = °

(1.7)

and this is the required equation for equilibrium of the string under the trans­ verse loading p(x). Next we pass from statics to dynamics and consider vibrations of the string. To do this we apply d’Alembert’s principle and include in the equation for equilibrium the inertial forces for the string as well; these take the form *X2 e( x) *1

d2u dt2

dx.

The condition for equilibrium becomes + p(x)

dx = 0,

and the equation for vibrations of the string will be d_ dx

- Q(x)

d2u + P(x) = 0. dt2

( 1. 8)

We suppose that the vibrations of the string are transverse. The com­ ponents along the x-axis of all the forces acting on any element of the string must therefore add up to zero. Since the loading p(x) is also transverse, we have [T cos a]X2 — [T cos oc]Xt = 0 for any values of x x, x2.

6

DERIVATION

of

fundamental

equations

L. 1

Hence, neglecting a 2 and higher powers, we get [T]X2 = [T]Xl i.e. T is independent of x. Hence T can be taken outside the differential co­ efficient with respect to x in (1.8). If we also assume that T is independent of the time, i.e., that it is constant, and also that the density q is constant and the loading p(x) zero, then equation (1.8) takes the form dzu , dzu ----- = a ----dtz dxz

(1.9)

where a2 = T/q = constant. The equation (1.9) was discussed by Daniel Bernoulli, d’Alembert, and Euler as early as the 18th century. § 3. Equation for Vibrations of a Membrane Consider a film, i.e., a very thin rigid body stretched uniformly in all directions. We suppose that the film is so thin that it offers no resistance to flexure. Such a film is called a membrane. Let its rest position be in the plane xOy and let u = u (t,x ,y ) be its equation when displaced. Considering any piece .S' of the membrane, we suppose that the rest of the membrane exerts on it a uniformly distributed tension T which, at any U

F ig . 2.

§3

V I B R A T I N G MEMBRANE

7

point on the boundary of S, is directed along the normal to the boundary and lies in the tangent plane to the membrane (see Fig. 2). We shall establish the equation for the equilibrium of the part S of the membrane which is bounded by the curve s under the action of transverse forces. The component along the u-axis due to the tension is given by the integral j T cos (/, u) ds (1.10) where T is the length of the vector T and

/ is a vector directed along the line of action of the tension.

We next evaluate cos (/, u) . By hypothesis, the vector / is perpendicular both to the boundary s and to a vector v along the inward-directed normal v to the surface u = u(t, x, y). Again, any vector s directed alonga tangent to theboundary s isperpendicular to v and to a unit vector n along the inward normal (at the corresponding point) to the projection of the boundary s on to the plane xOy (since the tangent vector s and the tangent to the projection of s on to the plane xOy lie in a plane touching the projecting cylinder). Hence as the vector s we may take the vector product n Av, and the vector/! defined by ^ = s Av = (nAv)Av can also be written as lx = — rtv2 + v(/iv). Since n has components {cos (u, x), cos (n, y), cos (u, z)}, and v has components

j—

lj

, we find for the components of /j the ex­

pressions ( —cos (u, x), —cos («, y ) , -------cos («, x ) --------cos (n, y) , { dx 8y J r , , , , / 8 u \ 2 ( d u \ 2 du du .. from which we have dropped ---- , — , -------- as being quantities of \ 8 x J \ 8 y J dx 8y the second order of smallness. To the same order, the length of vector ly is unity, and so we may now regard l± as the unit vector / directed along the line of action of the tension, and thus du du cos (/, u) cos (n, x ) -------- cos (n, y ) . dy dx The equation for equilibrium of the membrane has the form

11

* T cos (/, u) d j = 0

p(x, y) dx dy + s

where p(x, y) is the magnitude of the transverse loading per unit area, and (o is the projection of S on to the plane xOy.

L .l

D E R I V A T I O N OF F U N D A M E N T A L E Q U A T I O N S

8

Substituting for cos (/, u) we get p(x, y) dx dy -

' f du du ( ---- cos (n, x ) 4------- cos {n, y) ) T ds = 0, s \ dx dy )

or by virtue of Lemma 1, d /_ du\

d /

du\

,

.

.

(1 .1 1 )

n { T ^ ) + ^ { T ^ ) + p{x’y) = 0 The equation for vibrations of the membrane has the form

/ d2u \ f / du . . du \ ( p(x, y) — Q------ dx dy — T [ ---- cos («, x) H------- cos (n, y) ) dj = 0 dy dt2 \d x where q = g(x, y) is the density per unit area of the membrane, or, by Lemma 1, d f

du\

d f

du\

d2u

a l y a j ) + H y \ l i y ) + * x-y) _ e('v"r)W If T and

q are

= °' (U2)

constant, we get from (1.12) . d2u + \d x2 dy2

+ f(x , y) = Q

d 2u dt2

(1.13)

d2u d2u d2u d2u d2u The sum — - H------ or, in three-dimensional space, — - -I----- - -I------ , d x2 dy2 F dx2 dy2 dz2 is often called Laplace's operator and is denoted by V2u. Using this notation we can write (1.13) as d2“

=

v2« + Afiii

dt2

(1.14)

where T = a constant. a2 = —

§ 4. Equation of Continuity for Motion of a Fluid. Laplace’s Equation Before deriving the equation of continuity, we establish an important formula. Consider a closed, piecewise smooth, time-dependent surface S(t) en­ closing a variable volume & (/). Let g(x, y, z, t) be some function of the co-

§4

fluid

motion

:

9

l a p l a c e ’s e q u a t i o n

ordinates and the time t, and consider the integral q dx

QO) =

dy dz:

Q(«)

our aim is to calculate the time-derivative of Q. Consider first the special case when the volume is bounded by a cylindrical surface with generators parallel to Oz and having a fixed base SI x in the plane z = 0 and with an upper surface z = cp(x, y, t ) . Suppose that z = cp(x, y, t) is the equation of a piecewise smooth surface S(t). Suppose dcp . dcp also that the derivative is bounded: < M. dt dt Then e(x,y, z, t) dz \ dx dy.

Q(0 =

fit1 where the plane region SI i is the part of the boundary of the volume SI which lies in the plane xOy. To calculate dQjdt we first set up an equation for the ratio of increments. We have: + A t)

q(x, y, z, t + At) dz dx dy

AQ =

*

r(x.y.t)

q(x , y , z , t )

A Q = ± rr At At

dz dx dy

C(x,y,t) 4>(.x,y,t)

+

[j?(x, y , z , t + At) -

q(x , y,

z, 0] dz I dx dy

At Acp n i At

’*+A$ q{t + At) dxV cl dxdy < t> J

1 Acp

r +

\

+ At) ~ gQ d z l dx dy.

Passing to the limit as At -> 0, we find lim — = lim at-»o At j t ->o + lim At-*0

Sh

Acp

1

At

Acp

SI i ^

o

g(t + At) d z l dx dy

+ At) ~ e- ^ d z l dx dy At I

10

L .l

D E R I V A T I O N OF F U N D A M E N T A L E Q U A T I O N S

or r aq lim — = A t = 0 At

Um

ft ( At-*o

. -L A(p

At

r lim p e p +

+

% < f>+A p(t + At) d z l dx dy < t> Ai) - e(t) d J dx dy _ At

fti 1At-*0 I n [this change in the order of the limiting processes can be justified], - - - e(x.y. =

f(x, y, z, n, t) dS +

d(coT) du dt q du

(1.18)

where the vector n is directed along the outward normal. Equation (1.18) holds good for any volume D. We apply it now, taking as D the tetrahedron SI shown in Fig. 3, having a vertex at the point

A (x0, y 0, z0) and the three faces through A parallel to the coordinates planes. Let Sx, Sy,S z denote the faces perpendicular to Ox, Oy, Oz respectively, and S 0 be the inclined face; let ox, a y, a z, cr0 be their areas. We then have ax = o0 cos (n0, .v),

ay = o0 cos (n0, y ) ,

az = o0 cos (n0, z ) ,

§5

15

HEAT C O N D U C T I O N

where cos(n0,x ), cos(«0,.J;)> cos (n0,z) are the direction-cosines of the outward normal to the face S 0. Applying (1.18) to the tetrahedron Si we get i

~ (cqT) do dt

Q

/% /%

qdv =

* J sx

f{x, y, z, i, t) dax

/% /%

f(x,y , z j , t) day +

+ sy

f(x,y, z, k, t) da2 Sx (* (%

v

A x , y , z , - n 0, t) da0

+

J *1 S0

where i,j, k are unit vectors parallel to Ox, Oy, Oz. If the volume of SI is u>, then by a mean-value theorem, (o — (cqT) — (oq = axvx + ayvy + azvz dt + a0A x , y , z t — n0, t) where the bars denote mean values, and we have written for brevity

o, = A x , y, z, i, t),

vy = A x , y, z, j, 0 ,

vz = A x , y, z, k, t) .

If h is the perpendicular distance of A from S 0, the above equation can be written \haQ— {coT) - \h a 0q = a0\vx cos (n0, x) + vy cos (n0,y) + vz cos (n0, z) dt + A x , y , z , - n Q,t)). We divide both sides by cr0 and, keeping the direction of rt0 constant, let h -»• 0. Then the left-hand member clearly vanishes, and so, noting that Ax, y, Z, - n,t ) = - A x , y , Z, n, t), we find Axo, yo, zo, n, t) = [vx cos (n, x) + vy cos (n,y) + vz cos (n, z)] (*0>yo>,o) [^n] (A:0,yo«zo) where n is the direction of the outward normal to a side S of the tetrahedron Si, and concides respectively with —i, —j, — k, n0 for the sides Sx, Sy, Sz, S 0. We thus see that the function /(x , y, n, t) is the projection of a certain vector v on to the direction of it.

16

L .l

D E R I V A T I O N OF F U N D A M E N T A L E Q U A T I O N S

/ Hence, for an arbitrary volume D, (1.18) becomes r 'f % )

J

d

—

D dt

+ r r r q d V. JJJD

(QcT ) S

This formula is somewhat similar to (1.15). The vector v, which is analogous to the velocity of a fluid flow, we shall call the heat flow. The heat flow existing in a medium is bound up with the temperature distribution in the medium. Under natural conditions heat always flows from the parts with higher temperatures to those with lower temperatures. We shall select some small area d S in the medium and investigate how the tem­ perature varies at points close to this area. The increase of temperature is characterized by the quantity dT dn

it grad T.

Suppose the medium is isotropic, i.e., it has the same properties in all direc­ tions. It is natural to assume that, if the temperature increases in a direction normal to a surface S, then the heat flow across S will be negative. In other words, the quantities n grad T = grad„ T

and

vn dS

must have opposite signs: and this must be true for any direction n. Hence the projections of the vectors gradT and v in any direction must be opposite in sign, and this is possible only if these vectors have opposite directions. That is, v = —k grad T where A:is some positive scalar quantity, which may depend on the properties of the medium, on the temperature, on the way the temperature changes, and so on. If the temperature does not change very sharply, then as a first approximation we may assume that A: is a function of position in the medium only. This assumption agrees very well with experiment. Substituting this expression for v in the last equation for the heat balance, we obtain c * dT q dy. k dS + (qcT) du dt s dn D —

Applying Lemma 1 and taking into account that when the direction of the outward normal is replaced by that of the inward normal the sign of the derivative dTjdn — grad„ T changes, we get -L fe c T )

— (k — dx \ dx

+ ± ( k ?L\ + ± \

dy J

dz

k

dT dz J

+ q.

(1.19)

§ 6

17

S O U N D WAVES

If we assume q, c and k to be constants, we get ---- = a2 V2T + x

=

------

dvx

—

------

dt d 2y dt2

-

+ Vr

dvx dx

+ Vv

dvx

dvx + Vz ■ dy dz

dvy dVy dVy •+ + Vr ----- + Vv dy dt dx

dVy V

z

(1.21)

dz

dv7 dvz dvz dvz d2z = ------ + vx + vz ■+ Vz dy dt2 dz dt dx Suppose that at each point of the fluid there is a force acting which, per unit volume, we denote by F with components X, Y, Z along the axes. If p(t, x, y, z) denotes the pressure at any point, then the force acting on a surface S enclosing a volume SI will have a component along the x-axis given by p(t, x,y, z) cos (n, x) d S . s

18

L .l

D E R I V A T I O N OF F U N D A M E N T A L E Q U A T I O N S

We obtain the equation of motion by applying d’Alembert’s principle; in the x-direction, p cos (n, x ) dS +

Xdv

Js

SI ' dvx ~dt

SI

+

vx

dvx dvx dvx dv = 0. + vv + vz dy dz dx

where du is an element of volume. Applying Lemma 1 the equation of motion becomes dv. ' dvT dvx dvx + vz -------1- vx ----- + vv dy dz dt dx

+

dx

- X = 0.

(1.22)

Similarly we find dty e

dt

+ vx

dVy

dVy dVy + Vy — L + vz dy dz dx

dv. dv. dvz dv. + vx + Vy ----- + V, dz dt dx ' dy

+

dp_ dy

- 7 = 0

dp + — - Z = 0. dz

These three equations contain five unknown functions :vx, vy, vz, p and g. Two more equations are needed to make the system determinate. We have already derived one other equation relating these quantities, namely, the equation of continuity. So, in the general case, we still have to seek one more equation. First, however, we consider an incompressible, homogeneous fluid, for which we can put g — const., and then we immediately have sufficient equations. We can now verify what was said earlier about the potential flow of an incompressible fluid: namely, that v - grad V, V2V = 0, do actually satisfy the complete system of equations, if the function g is defined correspondingly, and if further X =

7 =

dU dy

i.e., if the external forces have a potential.

Z =

dU dz

§6

19

S O U N D WA V E S

It suffices to show that if we take dV

vy =

dx

dV

dV

dy

~d7’

then the equations (1.22) allow the function p to be constructed. When the expressions for vx, v y, vz are substituted, these equations yield explicit ex­ pressions for dp dp dp dx

dy ’ dz

And it is known from the theory of partial differential equations of the first order that the equations will be compatible provided that the mixed secondorder derivatives d2p d2p d2p dx dy ’ dy dz

dz dx

determined from the different equations have the same values. It is left to the reader to verify this. Returning to the general case of a compressible fluid, it is known from physics that the density and pressure in any fluid are related by a so-called equation of state, into which the absolute temperature T also enters. For an ideal gas, for example, the equation of state is

where R is the gas constant. Since this equation has introduced another unknown function, T, it may be necessary in some cases to bring in yet another equation for the inflow of heat. However, in a number of cases we can suppose that there is a functional relation between density and pressure: Q = A p) (1.23) w h ere/is a given function. Such a circumstance holds good if, for example, we consider processes occurring so rapidly that there is no time for heat to be transmitted from one particle to another. Such processes are said to be adiabatic. In order to obtain from the general equations (1.16), (1.22), (1.23) the required equations for the transmission of sound, we now make certain simplifying assumptions. We shall suppose that the motion of the fluid con­ sists of small vibrations about an equilibrium position. In the equilibrium condition the pressure p 0 and the density q0 are constants. The deviations p — P o and Q — Qo and also the velocity will be supposed small and, in

20

L. 1

D E R I V A T I O N OF F U N D A M E N T A L E Q U A T I O N S

particular, we shall suppose that terms such as vx —- , neglected. We then get o

dp_ dvx + dx dt

dVy

X,

o

—

do, o—dt

Y,

-

" dt

in ( 1 .2 2 ) may be

= Z.

Working to the same degree of accuracy, we may write these equations in the form f w dt

+

f - r , dx

±

X (sVr) + l t = y , dt dy

(e 0

dt

+ 2c dz

=

z.

Differentiating these equations with respect to x, y, z respectively and adding, we get d2p + — [div (e *01 + 'dx2 dt

d2P 'dz2

dX dY dZ ---- + ----- + ---dx dy dz

(1.24)

Using the equation of continuity, we can write the left-hand member of (1.24) as d2p , d2p t d2p d2Q d2p t d2p ( d2p fa* + Hy2 + ~dF ~ ~dF ~ ~ d ^ + Hy2 + H z2 - / 'W -dt2 r f - f " (p)

dt

Finally, regarding/'(p) as constant for a small change in p, so that f ( p ) = f'(Po)

8X 8Y 8Z and denoting the right-hand member of (1.24), i . e . , ------1--------- 1-----dx dy dz by 0 we have 1 d2p d2p [ d2p i d 2p -----------i- = 0 , (1.25) dy2 dz2 a2 dt2 dx.2 i

j

■