$$
\newcommand{\uex}{{u_{\small\mbox{e}}}}
\newcommand{\half}{\frac{1}{2}}
\newcommand{\halfi}{{1/2}}
\newcommand{\xpoint}{\boldsymbol{x}}
\newcommand{\normalvec}{\boldsymbol{n}}
\newcommand{\Oof}[1]{\mathcal{O}(#1)}
\newcommand{\Ix}{\mathcal{I}_x}
\newcommand{\Iy}{\mathcal{I}_y}
\newcommand{\It}{\mathcal{I}_t}
\newcommand{\setb}[1]{#1^0} % set begin
\newcommand{\sete}[1]{#1^{-1}} % set end
\newcommand{\setl}[1]{#1^-}
\newcommand{\setr}[1]{#1^+}
\newcommand{\seti}[1]{#1^i}
\newcommand{\Real}{\mathbb{R}}
$$
Discrete wave components in 2D
$$
\lbrack D_tD_t u = c^2(D_xD_x u + D_yD_y u)\rbrack^n_{q,r}
$$
This equation admits a Fourier component
$$
u^n_{q,r} = e^{i(k_x q\Delta x + k_y r\Delta y -
\tilde\omega n\Delta t)}
$$
Inserting the expression and using formulas from the 1D analysis:
$$
\sin^2\left(\frac{\tilde\omega\Delta t}{2}\right)
= C_x^2\sin^2 p_x
+ C_y^2\sin^2 p_y
$$
where
$$ C_x = \frac{c^2\Delta t^2}{\Delta x^2},\quad
C_y = \frac{c^2\Delta t^2}{\Delta y^2}, \quad
p_x = \frac{k_x\Delta x}{2},\quad
p_y = \frac{k_y\Delta y}{2}
$$
