Analytical Solution for the two-dimensional wave equation, separation of variables and solutions
The solution for \( G \) is
$$
\begin{equation*}
G_{mn}(t) = B_{mn}\cos(\lambda_{mn} t)+D_{mn}\sin(\lambda_{mn} t),
\end{equation*}
$$
with the general solution of the form
$$
\begin{equation*}
u(x,y,t) = \sum_{mn=1}^{\infty} u_{mn}(x,y,t) = \sum_{mn=1}^{\infty}F_{mn}(x,y)G_{mn}(t).
\end{equation*}
$$