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\begin{document}

water wave dispesion relations. goal: finding \( w(k) \) for \( k=\frac{2\pi }{\lambda } \). \( v_{ph}=\frac{w}{k} \). \( v_{gp}=\frac{\partial w}{\partial k_{i}} \). Variables:
\( w,k,g,h,\rho _{water},\gamma _{surface} \). The number of \( \Pi  \)'s is 6-3=3.\( \Pi _{1}=\frac{gk}{w^{2}} \), \( \Pi _{2}=\frac{\gamma k^{3}}{\rho w^{2}} \), \( \Pi _{3}=kh \). Then \( \Pi ^{-1}_{1}=f_{2}(\Pi _{2},\Pi _{3}) \) so \( w^{2}=gkf_{2}(\frac{\gamma k^{3}}{\rho w^{2}},hk) \).

\begin{itemize}

\item specialize to surface waves: hk\(>\)\(>\)1, then \( w^{2}=gkf_{3}(\frac{\gamma k^{3}}{\rho w^{2}}) \). Solution =
sum of short and long surface wave solutions = \( w^{2}=gk+\gamma k^{3}/\rho  \)

\item specialize to long surface waves, then \( \gamma \rightarrow 0 \). then \( w^{2}=C_{1}gk \). (surface gravity
waves \(>\)10cm) Calculation yields \( C_{1}=1 \).

\item specialize to short surface waves (gravity drops out) \( \frac{\rho w^{2}}{\gamma k^{3}}=C_{2} \) Calculation
yields \( C_{2}=1 \), so \( w^{2}=\frac{C_{2}\gamma k^{3}}{\rho } \)

\item specialize to shallow water waves: \( kh<<1 \) and long waves, so \( h>>(\frac{\gamma }{\rho g})^{1/2} \) then \( w^{2}=gkf_{5}(hk) \).
By calculation, you get \( w^{2}=ghk^{2} \)

\end{itemize}

Examples:

\begin{itemize}

\item deep water waves

\item ship wakes

\item shallow water waves

\end{itemize}

\end{document}
