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\begin{document}

properties of deep water waves

\begin{itemize}

\item \( v_{ph}=\frac{w}{k} \) for gravity waves: \( \frac{w}{k}=(\frac{g}{k})^{1/2} \) short waves aren't fast.

\item for capillary waves: \( \frac{w}{k}=(\frac{\gamma k}{\rho }) \) really short waves are fast. this means there
is a minimum at \( v_{ph}=23cm/s \)at \( \lambda \simeq 1.7cm \). Impacts at less than \( v_{phmin} \)do not excite waves.

\item \( v_{gp}=\frac{\partial w}{\partial k} \)for gravity waves, \( v_{gp}=\frac{1}{2}v_{ph}=\frac{1}{2}(\frac{g}{k})^{1/2} \)

\item for capillary waves, \( v_{gp}=\frac{3}{2}v_{ph}=\frac{3}{2}(\frac{\gamma k}{\rho })^{1/2} \). There is a minimum \( v_{gpmin}=17.7cm/s \) and \( \lambda =4.4cm \).

\item surfing is best on the first day.

\end{itemize}

\end{document}