91Ó°ÊÓ

In this problem we consider the general case- that is, with \(\theta\) dependence - of the vibrating circular membrane of radius \(c\) : $$ \begin{aligned} &a^{2}\left(\frac{\partial^{2} u}{\partial r^{2}}+\frac{1}{r} \frac{\partial u}{\partial r}+\frac{1}{r^{2}} \frac{\partial^{2} u}{\partial \theta^{2}}\right)=\frac{\partial^{2} u}{\partial t^{2}}, \quad 00 \\ &u(c, \theta, t)=0, \quad 0<\theta<2 \pi, t>0 \end{aligned} $$ \(u(r, \theta, 0)=f(r, \theta), \quad 0

Consider an idealized drum consisting of a thin membrane stretched over a circular frame of radius 1 . When such a drum is struck at its center, one hears a sound that is frequently described as a dull thud rather than a melodic tone. We can model a single drumbeat using the boundary-value problem solved in Example \(1 .\) (a) Find the solution \(u(r, t)\) given in (9) when \(c=1, f(r)=0\), and $$ g(r)= \begin{cases}-v_{0}, & 0 \leq r

In Problems \(21-24\), solve the given boundary-value problem. $$ \begin{aligned} &\frac{\partial^{2} u}{\partial r^{2}}+\frac{1}{r} \frac{\partial u}{\partial r}+\frac{\partial^{2} u}{\partial z^{2}}=0, \quad 0

In Problems \(21-24\), solve the given boundary-value problem. $$ \begin{aligned} &\frac{\partial^{2} u}{\partial r^{2}}+\frac{1}{r} \frac{\partial u}{\partial r}+\frac{\partial^{2} u}{\partial z^{2}}=0, \quad 0

In Problems \(21-24\), solve the given boundary-value problem. $$ \begin{aligned} &\frac{\partial^{2} u}{\partial r^{2}}+\frac{1}{r} \frac{\partial u}{\partial r}+\frac{\partial^{2} u}{\partial z^{2}}=0, \quad 00 \\ &u(1, z)=0, \quad z>0 \\ &u(r, 0)=100, \quad 0