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Next: Solution of the Dispersion Up: Water Waves Previous: Linearized (Airy) Wave Theory

Dispersion Relationship

So far, any $\omega ,k$ combination is allowed. Must satisfy FSBC, substitute $\phi $ into equation (3):

\begin{displaymath}- \omega ^2\cosh kh + gk \sinh kh = 0,
\end{displaymath}

which gives

\begin{displaymath}\underbrace{\omega ^2 = gk \tanh kh ; \left\{ {\omega ^2 = gk}
\right\}}_{\mbox{ Dispersion Relationship}}
\end{displaymath}

Given $k$ (and $h$) $ \to \omega $.
Given
$\omega $ (and $h$), ...
From (6):

\begin{displaymath}C \equiv \frac{\omega ^2h}{g} = \left( {kh} \right)\tanh
\left( {kh} \right)
\end{displaymath}


\begin{displaymath}\frac{C}{kh} = \tanh kh \to k \left( \mbox{only} 1 \right)
\end{displaymath}

Dispersion Relationship (6) uniquely relates $\omega $ and $k$, given $h$.

\begin{displaymath}\omega = \omega (k;h) \mbox{\ or\ } k = k(\omega;h)
\end{displaymath}

In general, $k \uparrow $ as $\omega \uparrow $ or $\lambda \uparrow $ as $T \uparrow $.

\begin{displaymath}\frac{\lambda }{T} = V_p = \frac{\omega }{k} = \sqrt
{\frac{g}{k}\tanh kh}; \left\{ {V_p = \sqrt {\frac{g}{k}} }
\right\}
\end{displaymath}

$V_{p} \uparrow $ as $T \uparrow $ or $\lambda \uparrow $. For fixed $k$ (or $\lambda )$, $V_{p} \uparrow $ as $h \uparrow $ ( from shallow water)

\begin{displaymath}V_{p} = V_{p}(k) \mbox{\ or\ } V_{p}(\omega ) \to \mbox{\
frequency dispersion }
\end{displaymath}



 

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