Analytical Solution for the two-dimensional wave equation, boundary conditions
The boundary conditions require that
F(x,y) = H(x)Q(y) are zero at the boundaries, meaning that
H(0)=H(L)=Q(0)=Q(L)=0 . This yields the solutions
\begin{equation*}
H_m(x) = \sin(\frac{m\pi x}{L}) \hspace{1cm} Q_n(y) = \sin(\frac{n\pi y}{L}),
\end{equation*}
or
\begin{equation*}
F_{mn}(x,y) = \sin(\frac{m\pi x}{L})\sin(\frac{n\pi y}{L}).
\end{equation*}
With
\rho^2= \nu^2-\kappa^2 and
\lambda = c\nu we have an eigenspectrum
\lambda=c\sqrt{\kappa^2+\rho^2}
or
\lambda_{mn}= c\pi/L\sqrt{m^2+n^2} .