$$
\newcommand{\uex}{{u_{\small\mbox{e}}}}
\newcommand{\half}{\frac{1}{2}}
\newcommand{\halfi}{{1/2}}
\newcommand{\xpoint}{\boldsymbol{x}}
\newcommand{\normalvec}{\boldsymbol{n}}
\newcommand{\Oof}[1]{\mathcal{O}(#1)}
\newcommand{\Ix}{\mathcal{I}_x}
\newcommand{\Iy}{\mathcal{I}_y}
\newcommand{\It}{\mathcal{I}_t}
\newcommand{\setb}[1]{#1^0} % set begin
\newcommand{\sete}[1]{#1^{-1}} % set end
\newcommand{\setl}[1]{#1^-}
\newcommand{\setr}[1]{#1^+}
\newcommand{\seti}[1]{#1^i}
\newcommand{\Real}{\mathbb{R}}
$$
Generalization: damping
Why do waves die out?
- Damping (non-elastic effects, air resistance)
- 2D/3D: conservation of energy makes an amplitude reduction by
\( 1/\sqrt{r} \) (2D) or \( 1/r \) (3D)
Simplest damping model (for physical behavior, see demo):
$$
\begin{equation}
\frac{\partial^2 u}{\partial t^2} + \color{red}{b\frac{\partial u}{\partial t}}
= c^2\frac{\partial^2 u}{\partial x^2} + f(x,t),
\tag{32}
\end{equation}
$$
\( b \geq 0 \): prescribed damping coefficient.
Discretization via centered differences to ensure \( \Oof{\Delta t^2} \) error:
$$
\begin{equation}
[D_tD_t u + bD_{2t}u = c^2D_xD_x u + f]^n_i
\tag{33}
\end{equation}
$$
Need special formula for \( u^1_i \) + special stencil (or ghost cells)
for Neumann conditions.
