$$
\newcommand{\uex}{{u_{\small\mbox{e}}}}
\newcommand{\half}{\frac{1}{2}}
\newcommand{\halfi}{{1/2}}
\newcommand{\xpoint}{\boldsymbol{x}}
\newcommand{\normalvec}{\boldsymbol{n}}
\newcommand{\Oof}[1]{\mathcal{O}(#1)}
\newcommand{\Ix}{\mathcal{I}_x}
\newcommand{\Iy}{\mathcal{I}_y}
\newcommand{\It}{\mathcal{I}_t}
\newcommand{\setb}[1]{#1^0} % set begin
\newcommand{\sete}[1]{#1^{-1}} % set end
\newcommand{\setl}[1]{#1^-}
\newcommand{\setr}[1]{#1^+}
\newcommand{\seti}[1]{#1^i}
\newcommand{\Real}{\mathbb{R}}
$$
Properties of the solution of the wave equation
$$
\begin{equation*} \frac{\partial^2 u}{\partial t^2} =
c^2 \frac{\partial^2 u}{\partial x^2}
\end{equation*}
$$
Solutions:
$$
u(x,t) = g_R(x-ct) + g_L(x+ct)
$$
If \( u(x,0)=I(x) \) and \( u_t(x,0)=0 \):
$$
u(x,t) = \half I(x-ct) + \half I(x+ct)
$$
Two waves: one traveling to the right and one to the left
