Analytical Solution for the two-dimensional wave equation,
The lhs and rhs are independent of each other and we obtain two differential equations
$$
\begin{equation*}
F_{xx}+F_{yy}+F\nu^2=0,
\end{equation*}
$$
and
$$
\begin{equation*}
G_{tt} + Gc^2\nu^2 = G_{tt} + G\lambda^2 = 0,
\end{equation*}
$$
with \( \lambda = c\nu \).
We can in turn make the following ansatz for the \( x \) and \( y \) dependent part
$$
\begin{equation*}
F(x,y) = H(x)Q(y),
\end{equation*}
$$
which results in
$$
\begin{equation*}
\frac{1}{H}H_{xx} = -\frac{1}{Q}(Q_{yy}+Q\nu^2)= -\kappa^2.
\end{equation*}
$$