next up previous
Next: Characteristics of a Plane Up: Dispersion Relationship Previous: Dispersion Relationship

Solution of the Dispersion Relationship


\begin{displaymath}\omega ^{2} = gk \tanh kh
\end{displaymath}

Property of $\tanh kh$:

\begin{displaymath}\tanh kh = \frac{\sinh kh}{\cosh kh} = \frac{1 - e^{ - 2kh}}{...
...all { short waves or deep water}}) \\
\end{array}} \right.
\end{displaymath}

1.
Deep water or short waves, $kh >> 1 (h > \sim \lambda/2)$
\begin{align}& \omega ^{2} = gk,\ \lambda = \frac{g}{2\pi }T^2 (\lambda
\mbox{(...
...frac{2\pi }{g}V_p^2 \leftarrow \mbox{\ frequency dispersion} \notag
\end{align}

2.
Shallow water or long waves, $kh << 1,\ ( h /\lambda <
\sim 1 / 20 \mbox{in practice})$
\begin{align}& \omega ^2 \cong gk \cdot kh \to \omega = \sqrt {gh} \ k ;\ \lambd...
...{T} = \sqrt {gh}
\leftarrow \mbox{no frequency dispersion } \notag
\end{align}

3.
Intermediate depth or wavelength. Need to solve

\begin{displaymath}\omega ^{2} = gk \tanh kh
\end{displaymath}

given $\omega , h$ for $k$ (given $k, h$ for $\omega $ - easy!)
(a)
Use tables or graphs (e.g. JNN fig. 6.3)

\begin{displaymath}\omega ^{2 \quad } = gk \tanh kh = gk_{\infty }, \mbox{\ then...
... }{\lambda _\infty } =
\frac{V_p }{V_{p \infty} } = \tanh kh
\end{displaymath}

(b)
Use numerical approximation (hand calculator)
i.
calculate $C = \omega ^{2}h/g$.
ii.




Keyword Search