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\begin{equation}
  C_g = \partial \omega / \partial k
\end{equation}
The wave energy propagates with the packet at the group velocity. 

These two velocities nealy coincide when $k, l$ are large. However, if $ \sqrt{k^2 + l^2} < f / \sqrt{gH} $, the difference becomes large. The inverse number of the criterion measure, $\sqrt{gH} / f$, is called the {\it deformation radius}, i.e. for the wave with larger wavelength than the deformation radius, the group velocity doesn't coincide with the phase velocity.

%======================================================================
\section{Exercises}

\begin{figure}[b]
  \begin{center}
    \includegraphics[width=10cm,keepaspectratio,clip]{fig/CA2_01.epsf}
    \caption{Result of Example 1.}
    \label{fig:CA2_01}
  \end{center}
\end{figure}

Though the shallow-water equations are horizontally 2-dimensional, the plane wave can be treated 1-dimensionally by steering the x-axis parpendicular to crest lines of the wave, without loss of generality.
It means $ \partial / \partial y \equiv 0$, so that the shallow-water equations reduce to
\begin{eqnarray}
  u_t =& f v - g h_x &\, \mbox{$x$-component of the equation of motion} \\
  v_t =& - f u       &\, \mbox{$y$-component of the equation of motion} \\
  h_t =& - H  u_x    &\, \mbox{the continuity equation} 
\end{eqnarray}

Perform the program named as "C8" in GFD Menu to study time-integration of these governintypedstream@NSURLRequest NSObject iNSURL c NSString+-http://www.cospar2004.org/images/index_15.gif d<NSHTTPURLRequestParameters NSDictionary 
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