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16.8 Vibrations in a Circular Membrane II 605

0 j 1 j 2 j y = J (x)
0
3
FIGURE 16.24 Third normal mode.

SECTION 16.7 PROBLEMS

1. Let c = R = 1, f (r) = 1 − r,and g(r) = 0. Approxi- 2. Repeat Problem 1 with f (r) = 1 −r .
2
mate the coefficients a 1 through a 5 in the solution for 3. Repeat Problem 1 with f (r) = sin(πr).
the motion of the membrane, and graph the fifth partial
sum for a selection of different times.

16.8 Vibrations in a Circular Membrane II

We will expand the discussion of vibrations in a circular elastic membrane to include a
dependence of the displacement function on θ. Now the problem is
2
2
2
∂ z ∂ z 1 ∂z 1 ∂ z
= c 2 + +
∂t 2 ∂r 2 r ∂r r ∂θ 2
2
∂z
z(r,θ,0) = f (r,θ), (r,θ,0) = 0for 0 ≤r < R,−π ≤ θ ≤ π,t > 0.
∂t
Thus, we assume that the membrane is released from rest with the initial displacement function
f (r,θ).
In cylindrical coordinates, θ can be replaced by θ +2nπ for any integer n, so we also impose
the periodicity conditions
∂z ∂z
z(r,−π,t) = z(r,π,t) and (r,−π,t) = (r,π,t).
∂θ ∂θ
To separate the variables, set
z(r,θ,t) = F(r) (θ)T (t).
Upon substituting this into the wave equation we obtain
T F + (1/r)F 1

= + =−λ,
c T F r
2
2
for some separation constant λ. The reason for this is that the left side depends only on t and the
right side only on r and θ, and these variables are independent. Then
2
T + c T = 0

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October 14, 2010 15:23 THM/NEIL Page-605 27410_16_ch16_p563-610