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

${\displaystyle {\mathrm{\nabla }}^{2}u=\frac{1}{c^2}{u}_{tt}(0\le x\le \pi ),(0\le y\le \pi ),(t\ge 0)}$

where c = 1 subject to the boundary conditions:

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

with the initial displacement of the membrane given by:

${\displaystyle \begin{array}{cc}u(x,y,0)& =x(\pi -x)y(\pi -y)\\ u_t(x,y,0)& =0\end{array}}$

The solution is:

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

where:

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

${\displaystyle \begin{array}{cc}{C}_{mn}& =\frac{4}{{\pi }^2}{\int }_0^{\pi }\sin (mx){\int }_0^{\pi }x(x-\pi )y(y-\pi )\sin (ny)dydx\\ & =\frac{4\times 4(-1+(-1{)}^m)(-1+(-1{)}^n)}{{\pi }^2{m}^3{n}^3};n,m=1,2,\dots \\ & =\frac{64}{{\pi }^2(2m-1{)}^3(2n-1{)}^3};n,m=1,2,\dots \end{array}}$