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 $\rho^2= \nu^2-\kappa^2$.

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