In this problem we consider the general casethat is, with u dependenceof the vibrating

Chapter 14, Problem 18

(choose chapter or problem)

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

\(a^{2}\left(\frac{\partial^{2} u}{\partial r^{2}}+\frac{1}{r} \frac{\partial u}{\partial r}+\frac{1}{r^{2}} \frac{\partial^{2} u}{\partial \theta^{2}}\right)=\frac{\partial^{2} u}{\partial t^{2}}\),  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,  \(0<\theta<2 \pi\)

\(\left.\frac{\partial u}{\partial t}\right|_{t=0}=g(r, \theta)\),  0<r<c, \(0<\theta<2 \pi\)

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

\(T^{\prime \prime}+a^{2} \lambda T=0, \quad \Theta^{\prime \prime}+\nu \Theta=0\)

\(r^{2} R^{\prime \prime}+r R^{\prime}+\left(\lambda r^{2}-\nu\right) R=0\)

(b) Let \(\lambda=\alpha^{2}\)  and \(\nu=\beta^{2}\) and solve the separated equations in part (a).

(c) Determine the eigenvalues and eigenfunctions of the problem.

(d) Use the superposition principle to determine a multiple series solution. Do not attempt to evaluate the coefficients.