$$
\newcommand{\uex}{{u_{\small\mbox{e}}}}
\newcommand{\half}{\frac{1}{2}}
\newcommand{\halfi}{{1/2}}
\newcommand{\xpoint}{\boldsymbol{x}}
\newcommand{\normalvec}{\boldsymbol{n}}
\newcommand{\Oof}[1]{\mathcal{O}(#1)}
\newcommand{\Ix}{\mathcal{I}_x}
\newcommand{\Iy}{\mathcal{I}_y}
\newcommand{\It}{\mathcal{I}_t}
\newcommand{\setb}[1]{#1^0} % set begin
\newcommand{\sete}[1]{#1^{-1}} % set end
\newcommand{\setl}[1]{#1^-}
\newcommand{\setr}[1]{#1^+}
\newcommand{\seti}[1]{#1^i}
\newcommand{\Real}{\mathbb{R}}
$$
Boundary and initial conditions
We need one boundary condition at each point on \( \partial\Omega \):
- \( u \) is prescribed (\( u=0 \) or known incoming wave)
- \( \partial u/\partial n = \normalvec\cdot\nabla u \) prescribed
(\( =0 \): reflecting boundary)
- open boundary (radiation) condition: \( u_t + \boldsymbol{c}\cdot\nabla u =0 \)
(let waves travel undisturbed out of the domain)
PDEs with second-order time derivative need two initial conditions:
- \( u=I \),
- \( u_t = V \).
