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602 CHAPTER 16 The Wave Equation

as a sum of a forward and a backward wave. Graph the cos(x) for −π/2 ≤ x ≤ π/2
15. f (x) =
initial position function and then graph the solution at 0 for |x| >π/2
selected times, showing the wave as a superposition of a
wave moving to the right and a wave moving to the left. 1 − x 2 for |x|≤ 1
16. f (x) =
0 for |x| > 1

2
sin(2x) for −π ≤ x ≤ π x − x − 2for −1 ≤ x ≤ 2
13. f (x) = 17. f (x) =
0 for |x| >π 0 for x < −1and for x > 2

3 2
1 −|x| for −1 ≤ x ≤ 1 x − x − 4x + 4for −2 ≤ x ≤ 2
14. f (x) = 18. f (x) =
0 for |x| > 1 0 for |x| > 2
16.7 Vibrations in a Circular Membrane I

Imagine an elastic membrane of radius R fastened onto a circular frame (such as a drumhead).
The membrane is set in motion from a given initial position and with a given initial velocity. In
polar coordinates, the membrane occupies the disk r ≤ R. Assume that the particle of membrane
at (r,θ) vibrates vertical to the x, y-plane and let the displacement of this particle at time t be
z(r,θ,t).
The wave equation in polar coordinates is
2
2
2
∂ z ∂ z 1 ∂z 1 ∂ z
= c 2 + + .
2
∂t 2 ∂r 2 r ∂r r ∂θ 2
We will assume axial symmetry, which means that the motion is independent of θ. Then z =
z(r,t) and the wave equation is
2
2
∂ z 2 ∂ z 1 ∂z
= c + .
∂t 2 ∂r 2 r ∂r
The initial position is z(r,0) = f (r) and the initial velocity is (∂z/∂t)(r,0) = g(r).
Attempt a solution z(r,t) = F(r)T(t). A routine calculation leads to
1 λ

F + F + F = 0 and T + λT = 0.
r c 2
2
If λ = ω > 0, this equation for F is a zero-order Bessel equation with solutions (bounded on the
disk r < R) that are multiples of
ω

r .
J 0
c
The equation for T is
2

T + ω T = 0
with solutions of the form

T (t) = a cos(ωt) + b sin(ωt).
For each positive number ω, we now have a function
ω ω

z ω (r,t) = a ω J 0 r cos(ωt) + b ω J 0 r sin(ωt).
c c
that satisfies the wave equation.

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October 14, 2010 15:23 THM/NEIL Page-602 27410_16_ch16_p563-610