Analytical Solution for the two-dimensional wave equation, final steps
The final step is to determine the coefficients \( B_{mn} \) and \( D_{mn} \) from the Fourier coefficients.
The equations for these are determined by the initial conditions \( u(x,y,0) = f(x,y) \) and
\( \partial u/\partial t|_{t=0}= g(x,y) \).
The final expressions are
$$
\begin{equation*}
B_{mn} = \frac{2}{L}\int_0^L\int_0^L dxdy f(x,y) \sin(\frac{m\pi x}{L})\sin(\frac{n\pi y}{L}),
\end{equation*}
$$
and
$$
\begin{equation*}
D_{mn} = \frac{2}{L}\int_0^L\int_0^L dxdy g(x,y) \sin(\frac{m\pi x}{L})\sin(\frac{n\pi y}{L}).
\end{equation*}
$$
Inserting the particular functional forms of \( f(x,y) \) and \( g(x,y) \) one obtains the final closed-form expressions.