Page 394 - DMTH504_DIFFERENTIAL_AND_INTEGRAL_EQUATION

Unit 23: Wave and Diffusion Equations by Separation of Variable

be transformed to polar co-ordinates, we may write the basic equation of motion of the membrane Notes
in the form.

1 2 u
2
u where 2 is Laplacian operator in two dimensions. ...(i)
c 2 t 2
Transforming this equation to polar co-ordinates, we have

2 u 1 u 1 2 u 2 u
c 2 ...(ii)
r 2 r r r 2 2 t 2

Let f r , be the initial displacement and g r , the initial velocity of the membrane. Therefore
the function u r , ,t is required to satisfy (ii) and all the boundary and initial conditions, i.e.

Boundary Condition

t
u a , , = 0 ;l 0 ...(iii)
Initial Condition

u r , ,0 = f r , ...(iv)

u
and g , r 0 r , a ...(v)
t
t 0
since u is a function of r, and ,t we suppose the solution of equation (ii) as

u r , ,t R r T t

or u r , ,t R T say ...(vi)

Using the solution (ii) we have
2
2
1 1 d T 1 d R 1 1 dR 1 1 d 2
. .
2
T c dt 2 R dr 2 r R dr r 2 d 2
L.H.S. is a function of t and R.H.S. is a function of r and , hence both sides will be equal only
when both reduce to a constant.
Hence

2
1 dT 1 d R 1 dR 1 d 2 2
. ...(vii)
2
c T dt 2 R dr 2 Rr dr r 2 d 2
where 2 is any constant. We separate the variable in equation (vii) and write
1 d 2 2
d 2
thus we get

2
d R 1 dR 2 2 R 0 ...(viii)
dr 2 r dr r 2

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