$$
\newcommand{\uex}{{u_{\small\mbox{e}}}}
\newcommand{\half}{\frac{1}{2}}
\newcommand{\halfi}{{1/2}}
\newcommand{\xpoint}{\boldsymbol{x}}
\newcommand{\normalvec}{\boldsymbol{n}}
\newcommand{\Oof}[1]{\mathcal{O}(#1)}
\newcommand{\Ix}{\mathcal{I}_x}
\newcommand{\Iy}{\mathcal{I}_y}
\newcommand{\It}{\mathcal{I}_t}
\newcommand{\setb}[1]{#1^0} % set begin
\newcommand{\sete}[1]{#1^{-1}} % set end
\newcommand{\setl}[1]{#1^-}
\newcommand{\setr}[1]{#1^+}
\newcommand{\seti}[1]{#1^i}
\newcommand{\Real}{\mathbb{R}}
$$
A slightly generalized model problem
Add source term \( f \) and nonzero initial condition \( u_t(x,0) \):
$$
\begin{align}
u_{tt} &= c^2 u_{xx} + f(x,t),
\tag{11}\\
u(x,0) &= I(x), \quad &x\in [0,L]
\tag{12}\\
u_t(x,0) &= V(x), \quad &x\in [0,L]
\tag{13}\\
u(0,t) & = 0, \quad & t>0,
\tag{14}\\
u(L,t) & = 0, \quad &t>0
\tag{15}
\end{align}
$$
