commit eba73f69e728b6abd663a48b0d1cf5c93975f973
parent 15ad16c5a256b66892ea3ea60afe002b097a492e
Author: Ivan Gankevich <igankevich@ya.ru>
Date: Thu, 18 Jul 2019 22:32:16 +0300
Model.
Diffstat:
1 file changed, 20 insertions(+), 6 deletions(-)
diff --git a/main.tex b/main.tex
@@ -84,20 +84,34 @@ crest propagation on their length and period. As a result
\includegraphics[width=\textwidth]{graphics/01-gerstner.png}
\includegraphics[width=\textwidth]{graphics/02-gerstner.png}
\caption{Analytic solutions: progressive Gerstner wave (top),
- extremely high or standing wave (bottom).\label{fig-gerstner}}
+ a wave with critical height~--- a standing wave
+ (bottom).\label{fig-gerstner}}
\end{figure}
-\(r_W=1.134\lambda_Wh_W/4\pi\text{ }\left[\text{m}\right]\)
-
+Gerstner wave (fig.~\ref{fig-gerstner}) is a cycloid, fluid parcel trajectory
+radius \(r_W=1.134\lambda_Wh_W/4\pi\text{ }\left[\text{m}\right]\) of which is fixed
+relative to \(z_W\)~--- flat wavy surface level, hence \(z\)-coordinates of the
+crest and trough are the same. Here \(\lambda_W\) is the wave length,
+\(h_W\)~--- relative wave height defined on the interval \([0..1]\) with
+\(h_W=1\) being the maximum wave height for which the crest does not
+break (fig.~\ref{fig-gerstner}). Vertical displacement of a fluid parcel
+is given by
\begin{equation*}
\zeta_Z = r_W \cos x_W \exp\left(-2\pi z_W / \lambda_W\right)
- \qquad \left[\text{m}\right]
+ \qquad \left[\text{m}\right].
\end{equation*}
-
+Horizontal displacement of the same fluid parcel with respect to its initial
+position for progressive wave is given by analogous equation, but with
+a shift by one fourth of the phase:
\begin{equation*}
\zeta_X = -r_W \sin x_W \exp\left(-2\pi z_W / \lambda_W\right)
- \qquad \left[\text{m}\right]
+ \qquad \left[\text{m}\right].
\end{equation*}
+Critical wave height of Gerstner waves (fig.~\ref{fig-gerstner}) gives the correct
+ratio of wave height to wave length, but 60 degree slope limit for standing
+wave with steepness \(\approx{}1/4\) as well as 30 degree slope limit for
+progressive (traveling) wave with steepness \(\approx{}1/7\) are not correctly
+captured by the model.
\begin{figure}
\centering
