pde poisson thry sound

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A Partial Differential Equation Associated with Poisson's Work on the Theory of Sound Author(s): H. Bateman Source: American Journal of Mathematics, Vol. 60, No. 2 (Apr., 1938), pp. 293-296Published by: The Johns Hopkins University PressStable URL: http://www.jstor.org/stable/2371293Accessed: 22-04-2015 21:49 UTC

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A PARTIAL DIFFERENTIAL EQUATION ASSOCIATED WITH

POISSON'S WORK ON THE THEORY OF SOUND.*

By H. BATEMAN.

Introduction. In his famous memoir of 1808 on the theory of sound, in which he discussed the theory of sound waves of finite amplitude, Poisson' also made some important advances in the theory of sound waves of small amplitude. In particular he attacked the problem of solving the equation of wave-propagation in three dimensionis by usilng the mean value of a function over a sphere; a method which eventually led to a general solution and proved very fruitful in potential theory. This method has been supplemented by the consideration of mean values around circles on a sphere.2

To determine the velocity components of the inidividual particles of air Poisson tried to solve the wave equation by means of an infinite series of powers of the inverse distance in which the n-th coefficient is an integral of order n. The integrands of the various integrals are connected by a recur- rence relation of the first order involving partial derivatives of the first two orders. Poisson found that these integrands could be obtained from a generat- ing function which in turn satisfies a certain partial differential equation, namely, the one designated as equation (1) below. Incidentally, this differential equation also occurs in the above mentioned theory of mean values around circles on a sphere.

Here it is pointed out that the same partial differential equation may be derived from the wave equation by a simple transformation suggested by Poisson's work. The general problem of obtaining a wave function from a solution of (1) is considered. It is found that (1/ri) U (w, u, 4), where M = cos 0 and r, 0, p are polar co6rdinates, is a wave function provided wv is clefinei bv equation (6).

Equation (1) has particular solutions represented by products of Legendre functions. By comparing these solutions with another solution of the wave equation expansions involving Legendre functions are suggested. The determi-

* Received December 6, 1937. 1 S. D. Poisson, Bull. Soc. Phitlon., vol. 1 (1807), p. 19; Journal de t'k'cole Poly-

techniqute, t. 7, Cah. 14 (1808), pp. 319-392. 2lH. Bateman, Proceedings of the National Academy of Sciences, vol. 16 (1930),

pp. 205-21 1; Annals of Mathematics (2), vol. 31 (1930), pp. 158-162.

293



294 H. BATEMAN.

nation of the coefficients in these expansions requires the evaluation of a class of integrals discussed in an accompanying paper.

Relation to the wave equation. The differential equation which forms the subject of this paper is

(1) -0 [(i ,W2) ] U.

Poisson's work indicates that the wave equation

(2) ~02W 0-2W +02W 02w 2t2 Ox2 +y2 + az2

has particular solutions of the type

(3) lF - 1U (w, tu, r

where U is a solution of (1), % = cos 0 and r, 0, ( are polar co6rdinates, anld iv

is defined by t

(4) w - r

This may be verified by transforming the wave equation by the substitution z = px, r2 = x2 + y2 + Z2, y x tan ( after which it takes the form

(5) 2(rW) 2(rW) 1 l8 (12\O (a\ 1 02

rrr (p

Direct substitution of (3) and (4) ill (5) thell results in equation (1). When we consider the general problem of finding a function w = iv (r, t)

such that W = (1/r) U(w, jk, () satisfies equation (5), and hence is a wave

function, we are led to the two equations

02w 52w 2w (aw2 (W 2 w2 1- w2

Ot2 Or2 r2 - ____2

00

If we seek a solution of the type w = N An (r)tn we find that n=O

(6) rw C(r t D +t(l 4CD)'

wliere C and D are arbitrary constants. When C = D = 0, Poisson's expression w = t/r is obtained, while if C = 1/2a, D = a/2 the value 3

3H. Bateman, loc. cit.



DIFFERENTIAL EQUATION ASSOCIATED WITH POISSON' S WORK. 295

-2 t2 + a2 (79) I) 2ar is obtained.

An expression suggested by a solution of equation (1). The differential equation (1) has solutions of the type

U = Pn (w) P.m ( u) ein,0

(8) U P n (W) Qnm(A) eimO

where m and n are arbitrary constants. We shall take them to be non-negative integers. When we replace pt by tanh v, the corresponding solutions of the wave equation are

W - Pn (w) P.m (tanh v) eimO r

(9) IV = P E p (w)Q.m (tanh v)ei-1n.

Another solution may be obtained directly from the wave equation by assuming that W is independent of z. Upon setting Z = t, X = ix, Y = iy the wave equation reduces to Laplace's equation which is known to be satisfied by

W -_ 1 F (Z + R R R2 X2 + Y2 + Z2.

The values of X + iY and R2 expressed in terms of r, w, v and p, where we nlow take w = t/r, are

X + iY ir sech veiO

R12 =t2 2 y2 r2(wV2 - sech2 V)

and hence the expression above for W assumes the form

W V 1 sh f [ei{w ch v + (w2ch2v - r -/ w2-sech 2 1)

where F(- is) = f(s). The argumenit of the function f may be simplified by writing ch u = w ch v. Theni

IV ~ 1 f(eio+u). r Vw2 -sech2 v

The conjugate complex of this expression is also a solution of the wave equation.



296 H. BATEMAN.

IV == 1 fl(ei0+U). r / w2 - sech2 v

Taking f1 (s) = sm/2 and f (s) - sm/2 and adding shows that

eim e im(T. ,(w ch1 v) (10) w mq= ch mu=TL

rV/w2 - sech2 v r/Vw2 - sech2 v

is also a solution. T,, (z) is Tchebycheff's polynomial defined as

T... (z) == ch (ni ch-lz) .

The solution (10) may be expanded in terms of the simple solutions (9). For example, when m is even, say m = 2k,

E (4n + 1 )P,2 (W) Q2k (tanh V)/Q 2k+1 (0) or - v) 0 n=o _Vw -sech2 v

1 < IV < 1

the first or seconld value on the right being taken accordingly as w2 - sech2 v is positive or negative. The determination of the cofficients in this series requires the evaluation of the integral

(IV (~chu\ J(du P2 ch v ch(2klu)d'u

which is discussed in an accompanying paper.

CALIFORNIA IN STITUTE, OF TECHNOLOGY.



pde poisson thry sound

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