$$
\newcommand{\uex}{{u_{\small\mbox{e}}}}
\newcommand{\half}{\frac{1}{2}}
\newcommand{\halfi}{{1/2}}
\newcommand{\xpoint}{\boldsymbol{x}}
\newcommand{\normalvec}{\boldsymbol{n}}
\newcommand{\Oof}[1]{\mathcal{O}(#1)}
\newcommand{\Ix}{\mathcal{I}_x}
\newcommand{\Iy}{\mathcal{I}_y}
\newcommand{\It}{\mathcal{I}_t}
\newcommand{\setb}[1]{#1^0} % set begin
\newcommand{\sete}[1]{#1^{-1}} % set end
\newcommand{\setl}[1]{#1^-}
\newcommand{\setr}[1]{#1^+}
\newcommand{\seti}[1]{#1^i}
\newcommand{\Real}{\mathbb{R}}
$$
The benefits of scaling
- It is difficult to figure out all the physical parameters of a case
- And it is not necessary because of a powerful: scaling
Introduce new \( x \), \( t \), and \( u \) without dimension:
$$ \bar x = \frac{x}{L},\quad \bar t = \frac{c}{L}t,\quad
\bar u = \frac{u}{a}
$$
Insert this in the PDE (with \( f=0 \)) and dropping bars
$$ u_{tt} = u_{xx}$$
Initial condition: set \( a=1 \), \( L=1 \), and
\( x_0\in [0,1] \) in (20).
In the code: set a=c=L=1, x0=0.8, and there is no need to calculate with
wavelengths and frequencies to estimate \( c \)!
Just one challenge: determine the period of the waves and an
appropriate end time (see the text for details).
