$$
\newcommand{\uex}{{u_{\small\mbox{e}}}}
\newcommand{\half}{\frac{1}{2}}
\newcommand{\halfi}{{1/2}}
\newcommand{\xpoint}{\boldsymbol{x}}
\newcommand{\normalvec}{\boldsymbol{n}}
\newcommand{\Oof}[1]{\mathcal{O}(#1)}
\newcommand{\Ix}{\mathcal{I}_x}
\newcommand{\Iy}{\mathcal{I}_y}
\newcommand{\It}{\mathcal{I}_t}
\newcommand{\setb}[1]{#1^0} % set begin
\newcommand{\sete}[1]{#1^{-1}} % set end
\newcommand{\setl}[1]{#1^-}
\newcommand{\setr}[1]{#1^+}
\newcommand{\seti}[1]{#1^i}
\newcommand{\Real}{\mathbb{R}}
$$
Representation of waves as sum of sine/cosine waves
Build \( I(x) \) of wave components \( e^{ikx} = \cos kx + i\sin kx \):
$$
I(x) \approx \sum_{k\in K} b_k e^{ikx}
$$
- \( k \) is the frequency of a component (\( \lambda = 2\pi/k \) corresponding wave length)
- \( K \) is some set of all \( k \) needed to approximate \( I(x) \) well
- \( b_k \) must be computed (Fourier coefficients)
Since \( u(x,t)=\half I(x-ct) + \half I(x+ct) \):
$$
u(x,t) = \half \sum_{k\in K} b_k e^{ik(x - ct)}
+ \half \sum_{k\in K} b_k e^{ik(x + ct)}
$$
Our interest: one component \( e^{i(kx -\omega t)} \), \( \omega = kc \)
