$$
\newcommand{\uex}{{u_{\small\mbox{e}}}}
\newcommand{\half}{\frac{1}{2}}
\newcommand{\halfi}{{1/2}}
\newcommand{\xpoint}{\boldsymbol{x}}
\newcommand{\normalvec}{\boldsymbol{n}}
\newcommand{\Oof}[1]{\mathcal{O}(#1)}
\newcommand{\Ix}{\mathcal{I}_x}
\newcommand{\Iy}{\mathcal{I}_y}
\newcommand{\It}{\mathcal{I}_t}
\newcommand{\setb}[1]{#1^0} % set begin
\newcommand{\sete}[1]{#1^{-1}} % set end
\newcommand{\setl}[1]{#1^-}
\newcommand{\setr}[1]{#1^+}
\newcommand{\seti}[1]{#1^i}
\newcommand{\Real}{\mathbb{R}}
$$
A more general model PDE with variable coefficients
$$
\begin{equation}
\varrho(x)\frac{\partial^2 u}{\partial t^2} =
\frac{\partial}{\partial x}\left( q(x)
\frac{\partial u}{\partial x}\right) + f(x,t)
\tag{31}
\end{equation}
$$
A natural scheme is
$$
\begin{equation}
[\varrho D_tD_t u = D_x\overline{q}^xD_x u + f]^n_i
\end{equation}
$$
Or
$$
\begin{equation}
[D_tD_t u = \varrho^{-1}D_x\overline{q}^xD_x u + f]^n_i
\end{equation}
$$
No need to average \( \varrho \), just sample at \( i \)
