Wave Equation in two Dimensions, discretizing
We discretize again time and position,
$$
\begin{equation*}
u_{xx}\approx \frac{u(x+\Delta x,t)-2u(x,t)+u(x-\Delta x,t)}{\Delta x^2},
\end{equation*}
$$
and
$$
\begin{equation*}
u_{tt}\approx \frac{u(x,t+\Delta t)-2u(x,t)+u(x,t-\Delta t)}{\Delta t^2},
\end{equation*}
$$
which we rewrite as
$$
\begin{equation*}
u_{xx}\approx \frac{u_{i+1,j}-2u_{i,j}+u_{i-1,j}}{\Delta x^2},
\end{equation*}
$$
and
$$
\begin{equation*}
u_{tt}\approx \frac{u_{i,j+1}-2u_{i,j}+u_{i,j-1}}{\Delta t^2},
\end{equation*}
$$
resulting in
$$
\begin{equation}
\tag{23}
u_{i,j+1}=2u_{i,j}-u_{i,j-1}+\frac{\Delta t^2}{\Delta x^2}\left(u_{i+1,j}-2u_{i,j}+u_{i-1,j}\right).
\end{equation}
$$