File:Membrane exampleA.gif

English: Graphical representation of the solution to the wave equation for 2D membrane bounded by a rectangular region given by:

${\displaystyle \nabla ^{2}u={\frac {1}{c^{2}}}u_{tt}\ (0\leq x\leq \pi ),\ (0\leq y\leq \pi ),\ (t\geq 0)}$

where c = 1 subject to the boundary conditions:

${\displaystyle {\begin{aligned}u(0,y,t)=0,&\ u(\pi ,y,t)=0\\u(x,0,t)=0,&\ u(x,\pi ,t)=0\end{aligned}}}$

with the initial displacement of the membrane given by:

${\displaystyle {\begin{aligned}u(x,y,0)&=x(\pi -x)y(\pi -y)\\u_{t}(x,y,0)&=0\end{aligned}}}$

The solution is:

${\displaystyle u(x,y,t)=\sum _{n=1}^{\infty }\sum _{m=1}^{\infty }C_{mn}\sin[(2n-1)x]\sin[(2m-1)y]\cos(\omega _{mn}t)}$

where:

${\displaystyle \omega _{mn}={\sqrt {m^{2}+n^{2}}}}$

${\displaystyle {\begin{aligned}C_{mn}&={\frac {4}{\pi ^{2}}}\int _{0}^{\pi }\sin(mx)\int _{0}^{\pi }x(x-\pi )y(y-\pi )\sin(ny)\ dy\ dx\\&={\frac {4\times 4\left(-1+(-1)^{m}\right)\left(-1+(-1)^{n}\right)}{\pi ^{2}m^{3}n^{3}}};\ n,m=1,2,\ldots \\&={\frac {64}{\pi ^{2}(2m-1)^{3}(2n-1)^{3}}};\ n,m=1,2,\ldots \end{aligned}}}$