$$
\newcommand{\uex}{{u_{\small\mbox{e}}}}
\newcommand{\half}{\frac{1}{2}}
\newcommand{\halfi}{{1/2}}
\newcommand{\xpoint}{\boldsymbol{x}}
\newcommand{\normalvec}{\boldsymbol{n}}
\newcommand{\Oof}[1]{\mathcal{O}(#1)}
\newcommand{\Ix}{\mathcal{I}_x}
\newcommand{\Iy}{\mathcal{I}_y}
\newcommand{\It}{\mathcal{I}_t}
\newcommand{\setb}[1]{#1^0} % set begin
\newcommand{\sete}[1]{#1^{-1}} % set end
\newcommand{\setl}[1]{#1^-}
\newcommand{\setr}[1]{#1^+}
\newcommand{\seti}[1]{#1^i}
\newcommand{\Real}{\mathbb{R}}
$$
Numerical dispersion relation in 2D (1)
$$
\tilde\omega = \frac{2}{\Delta t}\sin^{-1}\left(
\left( C_x^2\sin^2 p_x + C_y^2\sin^ p_y\right)^\half\right)
$$
For visualization, introduce \( \theta \):
$$ k_x = k\sin\theta,\quad k_y=k\cos\theta,
\quad p_x=\half kh\cos\theta,\quad p_y=\half kh\sin\theta$$
Also: \( \Delta x=\Delta y=h \). Then \( C_x=C_y=c\Delta t/h\equiv C \).
Now \( \tilde\omega \) depends on
- \( C \) reflecting the number cells a wave is displaced during a time step
- \( kh \) reflecting the number of cells per wave length in space
- \( \theta \) expressing the direction of the wave
