$$
\newcommand{\uex}{{u_{\small\mbox{e}}}}
\newcommand{\half}{\frac{1}{2}}
\newcommand{\halfi}{{1/2}}
\newcommand{\xpoint}{\boldsymbol{x}}
\newcommand{\normalvec}{\boldsymbol{n}}
\newcommand{\Oof}[1]{\mathcal{O}(#1)}
\newcommand{\Ix}{\mathcal{I}_x}
\newcommand{\Iy}{\mathcal{I}_y}
\newcommand{\It}{\mathcal{I}_t}
\newcommand{\setb}[1]{#1^0} % set begin
\newcommand{\sete}[1]{#1^{-1}} % set end
\newcommand{\setl}[1]{#1^-}
\newcommand{\setr}[1]{#1^+}
\newcommand{\seti}[1]{#1^i}
\newcommand{\Real}{\mathbb{R}}
$$
Numerical wave propagation (3)
Taking the square root:
$$
\sin\left(\frac{\tilde\omega\Delta t}{2}\right)
= C\sin\left(\frac{k\Delta x}{2}\right)
$$
- Exact \( \omega \) is real
- Look for a real solution \( \tilde\omega \) of
- Then the sine functions are in \( [-1,1] \)
- Lef-hand side in \( [-1,1] \) requires \( C\leq 1 \)
Stability criterion
$$
C = \frac{c\Delta t}{\Delta x} \leq 1
$$
