Page 398 - DMTH504_DIFFERENTIAL_AND_INTEGRAL_EQUATION

Unit 23: Wave and Diffusion Equations by Separation of Variable

Notes
Example 1: A string is stretched between the fixed points (0, 0) and (1, 0) and released at
rest from the positions u = A sin x. Find the formula for its subsequent displacement u(x, t).
Solution: Here the variation of the string is governed by one dimensional wave equation

2 2
u 2 u
c
t 2 x 2
Boundary conditions are u 0,t 0

and u 1,t 0

Initial conditions are u x ,0 A sin x

u
and 0
t t 0

Hence, we have

u , x t C n cosn ct sinn x
n 1

1
where C 2 A sin x sinn x dx
n
0
C , C , C ,... are all zero, since R.H.S. vanish for all these values
1 2 3
1
and C 1 2 A sin x sin x dx
0

1
A 1 cos2 x dx
0
= A
Hence u x ,t c 1 cos c t sin x

= A cosc t sin x

Example 2: Find the deflection u(x, y, t) of a square membrane with a b 1 and c = 1, if
the initial velocity is zero and the initial deflection is

f , x y A sin x sin 2 y
Solution: Equation governing the deflection of the membrane is

2 2 2
u c 2 u u
t 2 x 2 y 2

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