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*}