Page 629 - Advanced_Engineering_Mathematics o'neil

16.9 Vibrations in a Rectangular Membrane 609

The left side depends only on y and t and the right only on x, and these variables are independent,
so both sides must be constant:
T Y X
− = =−λ.
2
c T Y X
Then
T Y
X + λX = 0 and + λ = .

c T Y
2
The equation for T and Y has one side dependent only on t and the other side only on y.
Therefore, for some constant μ,
T Y
+ λ = =−μ.
c T Y
2
Then
2
Y + μY = 0 and T + c (λ + μ)T = 0.

Separation of variables has introduced two separation constants. From the boundary conditions,
X(0) = X(L) = Y(0) = Y(K) = 0.
We have solved these problems for X and Y before, obtaining eigenvalues and eigenfunctions
2
n π 2 nπx
λ n = , X n (x) = sin
L 2 L
and
2
m π 2 mπy
μ m = ,Y m (x) = sin
L 2 K
with n and m varying independently over the positive integers. The problem for T becomes
2 2 2 2
n π m π
2
T + c + T = 0.

L 2 K 2
With zero initial velocity we have T (0) = 0. Therefore, T (t) must be a constant multiple of

cos(α nm πct), where

n 2 m 2
α nm = + .
L 2 K 2
For each positive integer n and m, we now have functions
nπx mπy

z nm (x, y,t) = a nm sin sin cos(α nm πct).
L K
that satisfy the wave equation and the boundary conditions, as well as the condition of zero initial
velocity. To satisfy z(x, y,0) = f (x, y), attempt a superposition, which is now a double sum:
∞ ∞

z(x, y,t) = z nm (x, y,t).
n=1 m=1
We must choose the coefficients so that
∞ ∞

nπx mπy
z(x, y,0) = a nm sin sin = f (x, y).
L K
n=1 m=1
If we think of y as fixed for the moment, then f (x, y) = h y (x) is a function of x.Now

∞ ∞

mπy nπx
f (x, y) = h y (x) = a nm sin sin
K L
n=1 m=1
is the Fourier sine expansion in x of f (x, y) on [0, L]. Therefore, the coefficient of sin(nπx/L),
which is the entire sum in square brackets, is the Fourier sine coefficient of this function. For a
given n,

Copyright 2010 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s).
Editorial review has deemed that any suppressed content does not materially affect the overall learning experience. Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it.

October 14, 2010 15:23 THM/NEIL Page-609 27410_16_ch16_p563-610