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 $$\begin{equation*} u(x,y,t) = \sum_{mn=1}^{\infty} u_{mn}(x,y,t) = \sum_{mn=1}^{\infty}F_{mn}(x,y)G_{mn}(t). \end{equation*}$$

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