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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} .