$$
\newcommand{\uex}{{u_{\small\mbox{e}}}}
\newcommand{\half}{\frac{1}{2}}
\newcommand{\halfi}{{1/2}}
\newcommand{\xpoint}{\boldsymbol{x}}
\newcommand{\normalvec}{\boldsymbol{n}}
\newcommand{\Oof}[1]{\mathcal{O}(#1)}
\newcommand{\Ix}{\mathcal{I}_x}
\newcommand{\Iy}{\mathcal{I}_y}
\newcommand{\It}{\mathcal{I}_t}
\newcommand{\setb}[1]{#1^0} % set begin
\newcommand{\sete}[1]{#1^{-1}} % set end
\newcommand{\setl}[1]{#1^-}
\newcommand{\setr}[1]{#1^+}
\newcommand{\seti}[1]{#1^i}
\newcommand{\Real}{\mathbb{R}}
$$
Why \( C\leq 1 \) is a stability criterion
Assume \( C>1 \). Then
$$
\underbrace{\sin\left(\frac{\tilde\omega\Delta t}{2}\right)}{>1} = C\sin\left(\frac{k\Delta x}{2}\right)
$$
- \( |\sin x| >1 \) implies complex \( x \)
- Here: complex \( \tilde\omega = \tilde\omega_r \pm i\tilde\omega_i \)
- One \( \tilde\omega_i < 0 \) gives \( \exp(i\cdot i\tilde\omega_i) =
\exp (\tilde\omega_i) \) and exponential growth
