Wave Equation in two Dimensions
If we insert this condition in Eq.
(23) we arrive at a
special formula for the first time step
$$
\begin{equation}
\tag{24}
u_{i,1}=u_{i,0}+\frac{\Delta t^2}{2\Delta x^2}\left(u_{i+1,0}-2u_{i,0}+u_{i-1,0}\right).
\end{equation}
$$
We need seemingly two different equations, one for the first time step
given by Eq.
(24) and one for all other time-steps
given by Eq.
(23). However, it suffices to use
Eq.
(23) for all times as long as we
provide \( u(i,-1) \) using
$$
\begin{equation*}
u_{i,-1}=u_{i,0}+\frac{\Delta t^2}{2\Delta x^2}\left(u_{i+1,0}-2u_{i,0}+u_{i-1,0}\right),
\end{equation*}
$$
in our setup of the initial conditions.