91Ó°ÊÓ

The vertical displacement of a nonuniform membrane satisfies $$ \frac{\partial^{2} u}{\partial t^{2}}=c^{2}\left(\frac{\partial^{2} u}{\partial x^{2}}+\frac{\partial^{2} u}{\partial y^{2}}\right), $$ where \(c\) depends on \(x\) and \(y\). Suppose that \(u=0\) on the boundary of an irregularly shaped membrane. (a) Show that the time variable can be separated by assuming that $$ u(x, y, t)=\phi(x, y) h(t) . $$ Show that \(\phi(x, y)\) satisfies the eigenvalue problem \(\nabla^{2} \phi+\lambda \sigma(x, y) \phi=0 \quad\) with \(\phi=0\) on the boundary. What is \(\sigma(x, y)\) ? (b) If the eigenvalues are known (and \(\lambda>0\) ), determine the frequencies of vibration.

Solve Laplace's equation inside a circular cylinder subject to the boundary conditions (a) \(u(r, \theta, 0)=\alpha(r, \theta), \quad u(r, \theta, H)=0, \quad u(a, \theta, z)=0\) (b) \(u(r, \theta, 0)=\alpha(r) \sin 7 \theta, \quad u(r, \theta, H)=0, \quad u(a, \theta, z)=0\) (c) \(u(r, \theta, 0)=0, \quad u(r, \theta, H)=\beta(r) \cos 3 \theta, \quad \frac{\partial u}{\partial r}(a, \theta, z)=0\) (d) \(\frac{\partial u}{\partial z}(r, \theta, 0)=\alpha(r) \sin 3 \theta, \quad \frac{\partial u}{\partial z}(r, \theta, H)=0, \quad \frac{\partial u}{\partial r}(a, \theta, z)=0\) (e) \(\frac{\partial u}{\partial z}(r, \theta, 0)=\alpha(r, \theta), \quad \frac{\partial u}{\partial z}(r, \theta, H)=0, \quad \frac{\partial u}{\partial r}(a, \theta, z)=0\) Under what condition does a solution exist?

Consider a vibrating quarter-circular membrane, \(0

Consider Laplace's equation \(\nabla^{2} u=0\) in a three-dimensional region \(R\) (where \(u\) is the temperature). Suppose that the heat flux is given on the boundary (not necessarily a constant). (a) Explain physically why \(\oiint \nabla \boldsymbol{\nabla} \cdot \hat{\mathbf{n}} d S=0\). (b) Show this mathematically.

Solve for \(u(r, t)\) if it satisfies the circularly symmetric heat equation $$ \frac{\partial u}{\partial t}=k \frac{1}{r} \frac{\partial}{\partial r}\left(r \frac{\partial u}{\partial r}\right) $$ subject to the conditions $$ \begin{aligned} &u(a, t)=0 \\ &u(r, 0)=f(r) \end{aligned} $$ Briefly analyze the \(\lim _{t \rightarrow \infty}\).