$$
\newcommand{\uex}{{u_{\small\mbox{e}}}}
\newcommand{\half}{\frac{1}{2}}
\newcommand{\halfi}{{1/2}}
\newcommand{\xpoint}{\boldsymbol{x}}
\newcommand{\normalvec}{\boldsymbol{n}}
\newcommand{\Oof}[1]{\mathcal{O}(#1)}
\newcommand{\Ix}{\mathcal{I}_x}
\newcommand{\Iy}{\mathcal{I}_y}
\newcommand{\It}{\mathcal{I}_t}
\newcommand{\setb}[1]{#1^0} % set begin
\newcommand{\sete}[1]{#1^{-1}} % set end
\newcommand{\setl}[1]{#1^-}
\newcommand{\setr}[1]{#1^+}
\newcommand{\seti}[1]{#1^i}
\newcommand{\Real}{\mathbb{R}}
$$
Discretization of variable-coefficient wave equation in operator notation
$$
\begin{equation}
\lbrack D_tD_t u = D_x\overline{q}^{x}D_x u + f\rbrack^{n}_i
\tag{29}
\end{equation}
$$
We clearly see the type of finite differences and averaging!
Write out and solve wrt \( u_i^{n+1} \):
$$
\begin{align}
u^{n+1}_i &= - u_i^{n-1} + 2u_i^n + \left(\frac{\Delta x}{\Delta t}\right)^2\times \nonumber\\
&\quad \left(
\half(q_{i} + q_{i+1})(u_{i+1}^n - u_{i}^n) -
\half(q_{i} + q_{i-1})(u_{i}^n - u_{i-1}^n)\right)
+ \nonumber\\
& \quad \Delta t^2 f^n_i
\tag{30}
\end{align}
$$
