$$
\newcommand{\uex}{{u_{\small\mbox{e}}}}
\newcommand{\half}{\frac{1}{2}}
\newcommand{\halfi}{{1/2}}
\newcommand{\xpoint}{\boldsymbol{x}}
\newcommand{\normalvec}{\boldsymbol{n}}
\newcommand{\Oof}[1]{\mathcal{O}(#1)}
\newcommand{\Ix}{\mathcal{I}_x}
\newcommand{\Iy}{\mathcal{I}_y}
\newcommand{\It}{\mathcal{I}_t}
\newcommand{\setb}[1]{#1^0} % set begin
\newcommand{\sete}[1]{#1^{-1}} % set end
\newcommand{\setl}[1]{#1^-}
\newcommand{\setr}[1]{#1^+}
\newcommand{\seti}[1]{#1^i}
\newcommand{\Real}{\mathbb{R}}
$$
Input data in the problem
- Initial condition \( u(x,0)=I(x) \): initial string shape
- Initial condition \( u_t(x,0)=0 \): string starts from rest
- \( c=\sqrt{T/\varrho} \): velocity of waves on the string
- (\( T \) is the tension in the string, \( \varrho \) is density of the string)
- Two boundary conditions on \( u \): \( u=0 \) means fixed ends (no displacement)
Rule for number of initial and boundary conditions:
- \( u_{tt} \) in the PDE: two initial conditions, on \( u \) and \( u_t \)
- \( u_{t} \) (and no \( u_{tt} \)) in the PDE: one initial conditions, on \( u \)
- \( u_{xx} \) in the PDE: one boundary condition on \( u \) at each boundary point
