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Gravity Waves in Shallow Water

Consider the so-called shallow water limit,
\begin{displaymath}
k\,d\ll 1,
\end{displaymath} (1141)

in which the depth, $d$, of the water is much less than the wavelength, $\lambda=2\pi/k$, of the wave. In this limit, the gravity wave dispersion relation (1127) reduces to
\begin{displaymath}
\omega= (g\,d)^{1/2}\,k,
\end{displaymath} (1142)

since $\tanh(x)\rightarrow x$ as $x\rightarrow 0$. It follows that the phase velocities and group velocities of gravity waves in shallow water all take the fixed value
\begin{displaymath}
v_p = v_g = (g\,d)^{1/2},
\end{displaymath} (1143)

irrespective of wave number. We conclude that--unlike deep water waves--shallow water gravity waves are non-dispersive in nature: i.e., waves pulses and plane waves all propagate at the same speed. Note, also, that the velocity (1143) increases with increasing water depth.

For a plane wave of wave number ${\bf k}=k\,{\bf e}_x$, in the limit $k\,d\ll 1$, Equation (1125) yields

\begin{displaymath}
\phi(x,z,t) = A\,[1+k^2\,(z+d)^2/2]\,\cos(\omega\,t-k\,x).
\end{displaymath} (1144)

Hence, Equations (1113) and (1133) give [cf., Equations (1151)-(1154)]
$\displaystyle \xi_x(x,z,t)$ $\textstyle =$ $\displaystyle -a\,(k\,d)^{-1}\,\cos(\omega\,t-k\,x),$ (1145)
$\displaystyle \xi_z(x,z,t)$ $\textstyle =$ $\displaystyle a\,(1+z/d)\,\sin(\omega\,t-k\,x),$ (1146)
$\displaystyle v_x(x,z,t)$ $\textstyle =$ $\displaystyle a\,\omega\,(k\,d)^{-1}\,\sin(\omega\,t-k\,x)$ (1147)
$\displaystyle v_z(x,z,t)$ $\textstyle =$ $\displaystyle a\,\omega\,(1+z/d)\,\cos(\omega\,t-k\,x).$ (1148)

Here, $a$ is again the amplitude of the vertical oscillation at the water's surface. According to the above expressions, the passage of a shallow water gravity wave causes a water particle located a depth $h$ below the surface to execute an elliptical orbit, of horizontal radius $a/(k\,d)$, and vertical radius $a\,(1-h/d)$, about its equilibrium position. Note that the orbit is greatly elongated in the horizontal direction. Furthermore, its vertical radius decreases linearly with increasing depth such that it becomes zero at the bottom (i.e., at $h=d$). As before, whenever the particle's vertical displacement attains a maximum value the particle is moving horizontally in the same direction as the wave, and vice versa.


next up previous
Next: Energy of Gravity Waves Up: Waves in Incompressible Fluids Previous: Gravity Waves in Deep
Richard Fitzpatrick 2012-04-27