Page 393 - DMTH504_DIFFERENTIAL_AND_INTEGRAL_EQUATION

Differential and Integral Equation

Notes Similarly using other boundary condition, we get
n
A 2 0 and 2 n = 1, 2, 3, ...
b
Now (vii) becomes

m x n y
y
x
u x , ,t A cos t B sin t x sin sin
mn mn mn mn mn ...(viii)
a b
2 m 2 n 2
where mn .
a 2 b 2
Since the wave equation is linear and homogeneous, therefore sums of any number of different
solution will still be a solution.

Thus instead of (viii) an appropriate solution of u x , ,t is
y

m x n y
y
u x , ,t A mn cos mn t B mn sin mn t sin sin ...(ix)
m 1 n 1 a b

where 2 2 mn 2 m 2 n 2
a 2 b 2
Now using the initial conditions (ii), we have

m x m y
u x , ,0 A mn sin sin f , x y .
y
m 1 n 1 a b
)
This series is called the double Fourier series of ( ,f x y therefore.
2 2 a b m x n y
A mn . f , x y sin sin dxdy
a b x 0 y 0 x b

u m x n y
and C mn mn sin sin g , x y . ...(x)
B
t t 0 m 1 n 1 a b
Therefore,

2 2 a b m x n y
c mn mn , , g , x y sin sin dxdy
B
a b x 0 y 0 x b
4 a b m x n y
or B mn , g , x y sin sin dxdy ...(xi)
abc mn x 0 y 0 x b

Hence the solution of two dimensional wave equation is given by (ix) with the coefficients (x)
and (xi) satisfying all the conditions (i) and (ii).

23.1.3 The Vibrations of a Circular Membrane

In the case of the circular membrane we naturally have recourse to polar co-ordinates with the
origin at the centre. In this case the equation of motion obtained in Cartesian co-ordinates must

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