Page 396 - DMTH504_DIFFERENTIAL_AND_INTEGRAL_EQUATION

Unit 23: Wave and Diffusion Equations by Separation of Variable

Notes
when 1 , 2 ...are the positive roots of the equation
J a 0
0
From (xii) and initial condition (iv) when t = 0, we have

u r ,0 A J 0 n r f r
n
n 1
u r ,0 becomes f r when independent of .

J r
Hence A must be the coefficients of Fourier Bessel series which represent f(r) in terms of 0 n
n
i.e.
2 a
A n rf r J 0 n r dr , r 1,2,...
2 2
a J 0 n a 0 ...(xv)
The initial condition (v) gives

u
B J
C n n 0 n r g r
t t 0 n 1
g , r becomes g r when independent of

Again using Fourier Bessel series, we get
2 a
c n n rg r J 0 n r dr
B
2 2
a J 0 n a 0
2 a
B rg r J r dr
n 2 2 0 n
a J a c 0
0 n n ...(xvi)
n 1,2,3.....
Hence (xiv) is the solution of the wave equation with the coefficients given by the equations (xv)
and (xvi) which is radially symmetric.

D, Alembert s Solution of Wave Equation
Given wave equation is

2 2
u c 2 u
t 2 x 2 ...(i)
Let us introduce two independent variables v and w given by
v x ct
and ...(ii)
w x ct

v w
1 and 1
x x
u u u w u
Therefore, . .
x v x x x

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