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Analytical Solution for the two-dimensional wave equation, separation of variables

Since the lhs and rhs are again independent of each other, we can separate the latter equation into two independent equations, one for x and one for y , namely \begin{equation*} H_{xx} + \kappa^2H = 0, \end{equation*} and \begin{equation*} Q_{yy} + \rho^2Q = 0, \end{equation*} with \rho^2= \nu^2-\kappa^2 .