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Linearized (Airy) Wave Theory

Consider small amplitude waves: (small free surface slope)
\begin{figure}
\begin{center}
\epsfig{file=lfig192.eps,height=1.9in,clip=}
\end{center}
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Assume amplitude small compared to wavelength, i.e., $\frac{A}{\lambda } < < 1$. Consequently: $\frac{\phi }{{\lambda
^2} \mathord{\left/ {\vphantom {{\lambda ^2} T}} \right.
\kern-\nulldelimiterspace} T},\frac{\eta }{\lambda } < < 1$, and we keep only linear terms in $\phi $,$\eta $:

\begin{displaymath}\mbox{For example:\ } \left. {\left( \right)} \right\vert _{y...
...x{\tiny {discard}}} + \ldots
\mbox{\ \small {Taylor series}}
\end{displaymath}

Finally:

\begin{displaymath}\begin{array}{l}
1) \nabla ^2\phi = 0, - h < y < 0 \left\{ ...
...ac{\partial \phi }{\partial y} = 0_ ; y = 0 \\
\end{array}
\end{displaymath}


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\begin{center}
\epsfig{file=lfig193.eps,height=2.in,clip=}
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Linear (Airy) Waves in constant finite depth $h$ {infinite depth}
   
Given
$\phi $:
  
Solution of 2D Periodic Plane Progressive Waves (using separation of variables)
Math ...(solve (1), (2), (3))
Answer:
\begin{align}% latex2html id marker 239
\phi & = \frac{gA}{\omega }\sin \left( {...
...cos \left( {kx -
\omega t} \right); \left\{ {same} \right\} \notag
\end{align}

where
$A$ is the wave amplitude $ = H/2$.
1.
At $t = 0$ (say), $\eta =A\cos kx \to $ periodic in $x$ with wavelength: $\lambda = 2\pi/k [L]$ figure
\begin{figure}
\begin{center}
\epsfig{file=lfig194.eps,height=1.in,clip=}
\end{center}
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2.
At $x = 0$ (say), $\eta = A\cos \omega t \to $ periodic in $t$ with period: $T = 2\pi/\omega [T]$ figure
\begin{figure}
\begin{center}
\epsfig{file=lfig195.eps,height=1in,clip=}
\end{center}
\end{figure}
3.
$\eta = A\cos \left[ {k\left( {x - \frac{\omega }{k}t}
\right)} \right]_ ;\frac{\omega }{k} = \left[ V \right]$
Following a point with velocity $\frac{\omega }{k}$, i.e. $x_p = \left( {\frac{\omega }{k}} \right)t + const.$ the phase of $\eta $ does not change: $\frac{\omega }{k} =
\frac{\lambda }{T} \equiv V_p $ phase velocity.


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