Analytical Solution for the two-dimensional wave equation, first step

Our first step is to make the ansatz $$\begin{equation*} u(x,y,t) = F(x,y) G(t), \end{equation*}$$ resulting in the equation $$\begin{equation*} FG_{tt}= c^2(F_{xx}G+F_{yy}G), \end{equation*}$$ or $$\begin{equation*} \frac{G_{tt}}{c^2G} = \frac{1}{F}(F_{xx}+F_{yy}) = -\nu^2. \end{equation*}$$

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