Chapter 13.2, Problem 17E

In this problem we consider the general case—that is, with θ dependence—of the vibrating circular membrane of radius c:

$\begin{array}{l}{a}^2\left(\frac{{\partial }^{2}u}{\partial r^2}+\frac{1}{r}\frac{\partial u}{\partial r}+\frac{1}{r^2}\frac{{\partial }^{2}u}{\partial {\theta }^2}\right)=\frac{{\partial }^{2}u}{\partial t^2},0<r<c,t>0\\ u(c,\theta ,t)=0,0<\theta <2\pi ,t>0\\ u(r,\theta ,0)=f(r,\theta ),0<r<c,0<\theta <2\pi \\ {\frac{\partial u}{\partial t}|}_{t=0}=g(r,\theta ),0<r<c,0<\theta <2\pi .\end{array}$

1. (a) Assume that $u=R(r)\Theta (\theta )T(t)$ and that the separation constants are –λ and –v. Show that the separated differential equations are

   $\begin{array}{l}{T}^{″}+a^2\lambda T=0,{\Theta }^{″}+v\Theta =0\\ r^2{R}^{″}+r{R}^{\prime }+(\lambda r^2-v)R=0.\end{array}$

2. (b) Let λ = α² and v = β² and solve the separated equations.
3. (c) Determine the eigenvalues and eigenfunctions of the problem.
4. (d) Use the superposition principle to determine a multiple series solution. Do not attempt to evaluate the coefficients.