$$
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$$
Numerical wave propagation (2)
The complete scheme,
$$
\begin{equation*}
\lbrack D_tD_t e^{ikx}e^{-i\tilde\omega t} = c^2D_xD_x e^{ikx}e^{-i\tilde\omega t}\rbrack^n_q
\end{equation*}
$$
leads to an equation for \( \tilde\omega \):
$$
\begin{equation}
\sin^2\left(\frac{\tilde\omega\Delta t}{2}\right)
= C^2\sin^2\left(\frac{k\Delta x}{2}\right),
\tag{49}
\end{equation}
$$
where \( C = \frac{c\Delta t}{\Delta x} \) is the Courant number