Analytical Solution for the two-dimensional wave equation
We develop here the closed-form solution for the \( 2+1 \) dimensional wave equation with the following boundary and initial conditions
$$
\begin{equation*}
\begin{array}{cc} c^2(u_{xx}+u_{yy}) = u_{tt}& x,y\in(0,L), t>0 \\
u(x,y,0) = f(x,y) & x,y\in (0,L) \\
u(0,0,t)=u(L,L,t)=0 & t > 0\\
\partial u/\partial t|_{t=0}= g(x,y) & x,y\in (0,L)\\
\end{array} .
\end{equation*}
$$