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608 CHAPTER 16 The Wave Equation

0to1as r varies from 0 to R. In this way, we can use the formula for Fourier-Bessel coefficients
on (0,1).Wehave
2 1
a 0k = ξα 0 (Rξ)J 0 ( j 0k ξ)dξ for k = 1,2,3,··· ,
[J 1 ( j 0k )] 2 0
2 1
a nk = ξα n (Rξ)J n ( j nk ξ)dξ for k = 1,2,3,··· and n = 1,2,··· ,
[J n+1 ( j nk )] 2 0
and
2 1
b nk = ξβ n (Rξ)J n ( j nk ξ)dξ.
[J n+1 ( j nk )] 2 0
For a given problem, first perform the integrations with respect to θ to obtain the functions α 0 (r),
α n (r) and β n (r), written as Fourier-Bessel series. Then integrate to obtain the coefficients a nk and
b nk in these expansions. These integrations require a computational software package.

SECTION 16.8 PROBLEMS

1. Approximate the vertical deflections of the center of 2. Use the general solution derived in this section to
a circular membrane of radius 2 for any time t > prove the plausible fact that the center of the mem-
0 by computing the first three nonzero terms in the brane remains undeflected for all time if the initial
solution for the case c = 2 and the initial displace- displacement is an odd function of θ (that is, f (r,θ) =
2
2
ment f (r,θ) = (4 − r )sin (θ). Assume zero initial − f (r,−θ). Hint: The only integer order Bessel func-
velocity. tion that is different from zero at r = 0is J 0 .

16.9 Vibrations in a Rectangular Membrane

Suppose an elastic membrane is attached to a rectangular frame that occupies the region 0 ≤ x ≤
L,0 ≤ y ≤ K. The membrane is given an initial displacement and released with a given initial
velocity. We want the displacement function z(x, y,t).
The initial-boundary value problem for z is
2
2
2
∂ z ∂ z ∂ z
= c 2 + for 0 < x < L,0 < y < K,t > 0,
∂t 2 ∂x 2 ∂y 2
z(x,0,t) = z(x, K,t) = 0for 0 < x < L,t > 0,
z(0, y,t) = z(L, y,t) = 0for 0 < y < K,t > 0,
z(x, y,0) = f (x, y) for 0 < x < L,0 < y < K,
and
∂z
(x, y,0) = g(x, y) for 0 < x < L,0 < y < K.
∂t
We will solve this for the case g(x, y) = 0, so the membrane is displaced and released from
rest. To attempt a separation of variables, substitute z(x, y,t) = X(x)Y(y)T (t) into the wave
equation to get
2

XY T = c (X YT + XY T )
or
T Y X
− = .
c T Y X
2
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October 14, 2010 15:23 THM/NEIL Page-608 27410_16_ch16_p563-610