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\begin{document}


\begin{center}
Vibrating membrane
\end{center}

\noindent Consider%
\begin{align*}
\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) \quad 0\leq
x\leq L\quad 0\leq y\leq H \\
u(x,0) &= \alpha (x,y) \\
\frac{\partial u}{\partial t}(x,0) &= \beta (x,y) \\
u(0,y,t) &= u(L,y,t)=u(x,0,t)=u(x,H,t)=0
\end{align*}%
\noindent We can use two approaches

\begin{description}
\item[(a)] $u(x,y,t)=h(t)\phi (x,y)$

\item[(b)] $u(x,y,t)=f(x)g(y)h(t)$
\end{description}

\bigskip

\noindent We shall use the second approach. Putting this into the wave
equation we get%
\begin{align*}
fg\frac{d^{2}h}{dt^{2}} &= hc^{2}\left( g\frac{d^{2}f}{dx^{2}}+f\frac{d^{2}g%
}{dy^{2}}\right) \\
\frac{1}{c^{2}}\frac{1}{h}\frac{d^{2}h}{dt^{2}} &= \frac{1}{f}\frac{d^{2}f}{%
dx^{2}}+\frac{1}{g}\frac{d^{2}g}{dy^{2}}=-\lambda
\end{align*}%
\noindent So%
\begin{equation*}
\frac{d^{2}h}{dt^{2}}+\lambda c^{2}h=0
\end{equation*}

\noindent Now%
\begin{equation*}
\frac{1}{f}\frac{d^{2}f}{dx^{2}}=-\lambda -\frac{1}{g}\frac{d^{2}g}{dy^{2}}%
=-\mu
\end{equation*}

\noindent or%
\begin{align*}
\frac{d^{2}f}{dx^{2}}+\mu f &= 0\qquad f(0)=f(L)=0 \\
\frac{d^{2}g}{dy^{2}}+\left( \lambda -\mu \right) g &= 0\qquad g(0)=g(H)=0
\end{align*}

\noindent To satisfy the boundary conditions we need to choose%
\begin{align*}
\mu _{n} &= \left( \frac{n\pi }{L}\right) ^{2}\qquad f_{n}(x)=\sin \frac{%
n\pi }{L}x \\
\lambda _{nm}-\mu _{m} &= \left( \frac{m\pi }{H}\right) ^{2}\qquad
g_{nm}(y)=\sin \frac{m\pi }{H}y \\
\text{So \ \ }\lambda _{nm} &= \mu _{m}+\left( \frac{m\pi }{H}\right) ^{2}=%
\frac{\left( n^{2}+m^{2}\right) \pi ^{2}}{H^{2}}
\end{align*}

\noindent with eigenfunctions%
\begin{equation*}
u_{nm}(x)=\sin \frac{n\pi }{L}x\sin \frac{m\pi }{H}y
\end{equation*}

Therefore%
\begin{align*}
u(x,y,t) &= \tsum\limits_{m=1}^{\infty }\tsum\limits_{n=1}^{\infty}
A_{nm}\sin \frac{n\pi }{L}x\sin \frac{m\pi }{H}y\cos (c\sqrt{\lambda _{nm}}t)
\\
& +\tsum\limits_{m=1}^{\infty }\tsum\limits_{n=1}^{\infty }B_{nm}\sin \frac{%
n\pi }{L}x\sin \frac{m\pi }{H}y\sin (c\sqrt{\lambda _{nm}}t)
\end{align*}

\noindent with%
\begin{align*}
A_{nm} &= \frac{2}{L}\frac{2}{H}\tint\limits_{0}^{L}\tint\limits_{0}^{H}%
\alpha (x,y)\sin \frac{n\pi }{L}x\sin \frac{m\pi }{H}ydydx \\
c\sqrt{\lambda _{nm}}B_{nm} &= \frac{2}{L}\frac{2}{H}\tint\limits_{0}^{L}%
\tint\limits_{0}^{H}\beta (x,y)\sin \frac{n\pi }{L}x\sin \frac{m\pi }{H}ydydx
\end{align*}

\end{document}
