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Gravity Waves in Deep Water

Consider the so-called deep water limit,

$\displaystyle k\,d\gg 1,$ (11.22)

in which the depth, $ d$ , of the water greatly exceeds the wavelength, $ \lambda=2\pi/k$ , of the wave. In this limit, the gravity wave dispersion relation (11.21) reduces to

$\displaystyle \omega = (g\,k)^{1/2},$ (11.23)

because $ \tanh(x)\rightarrow 1$ as $ x\rightarrow \infty$ . It follows that the phase velocity of gravity waves in deep water is

$\displaystyle v_p = \frac{\omega}{k} = \left(\frac{g}{k}\right)^{1/2}.$ (11.24)

Note that this velocity is proportional to the square root of the wavelength. Hence, deep-water gravity waves with long wavelengths propagate faster than those with short wavelengths. The phase velocity, $ v_p=\omega/k$ , is defined as the propagation velocity of a plane wave with the definite wave number, $ k$ [and a frequency given by the dispersion relation (11.23)] (Fitzpatrick 2013). Such a wave has an infinite spatial extent. A more realistic wave of finite spatial extent, with an approximate wave number $ k$ , can be formed as a linear superposition of plane waves having a range of different wave numbers centered on $ k$ . Such a construct is known as a wave pulse (Fitzpatrick 2013). As is well known, wave pulses propagate at the group velocity (Fitzpatrick 2013),

$\displaystyle v_g = \frac{d\omega}{dk}.$ (11.25)

For the case of gravity waves in deep water, the dispersion relation (11.23) yields

$\displaystyle v_g = \frac{1}{2}\left(\frac{g}{k}\right)^{1/2} = \frac{1}{2}\,v_p.$ (11.26)

In other words, the group velocity of such waves is half their phase velocity.

Let $ \xi$ $ ({\bf r},t)$ be the displacement of a particle of water, found at position $ {\bf r}$ and time $ t$ , due to the passage of a deep water gravity wave. It follows that

$\displaystyle \frac{\partial\mbox{\boldmath$\xi$}}{\partial t} = {\bf v},$ (11.27)

where $ {\bf v}({\bf r},t)$ is the perturbed velocity. For a plane wave of wave number $ {\bf k}=k\,{\bf e}_x$ , in the limit $ k\,d\gg 1$ , Equation (11.19) yields

$\displaystyle \phi(x,z,t) = A\,{\rm e}^{\,k\,z}\,\cos(\omega\,t-k\,x).$ (11.28)

Hence, [cf., Equations (11.45)-(11.48)]

$\displaystyle \xi_x(x,z,t)$ $\displaystyle = -a\,{\rm e}^{\,k\,z}\,\cos(\omega\,t-k\,x),$ (11.29)
$\displaystyle \xi_z(x,z,t)$ $\displaystyle = a\,{\rm e}^{\,k\,z}\,\sin(\omega\,t-k\,x),$ (11.30)
$\displaystyle v_x(x,z,t)$ $\displaystyle = a\,\omega\,{\rm e}^{\,k\,z}\,\sin(\omega\,t-k\,x),$ (11.31)
$\displaystyle v_z(x,z,t)$ $\displaystyle = a\,\omega\,{\rm e}^{\,k\,z}\,\cos(\omega\,t-k\,x),$ (11.32)


$\displaystyle p_1 =\rho\,g\,\xi_z,$ (11.33)

where use has been made of Equations (11.7), (11.9), and (11.27). Here, $ a$ is the amplitude of the vertical oscillation at the water's surface. According to Equations (11.29)-(11.32), the passage of the wave causes a water particle located a depth $ h$ below the surface to execute a circular orbit of radius $ a\,{\rm e}^{-\,k\,h}$ about its equilibrium position. The radius of the orbit decreases exponentially with increasing depth. Furthermore, 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. (See Figure 11.1.)

Figure: Motion of water particles associated with a deep water gravity wave propagating in the $ x$ -direction.
\epsfysize =2.25in

Finally, if we define $ h(x,z,t) = \xi_z(x,z,t)-z$ as the equilibrium depth of the water particle found at a given point and time then Equations (11.3) and (11.33) yield

$\displaystyle p(x,z,t) = p_0 + \rho\,g\,h(x,z,t).$ (11.34)

In other words, the pressure at this point and time is the same as the unperturbed pressure calculated at the equilibrium depth of the water particle.

next up previous
Next: Gravity Waves in Shallow Up: Waves in Incompressible Fluids Previous: Gravity Waves
Richard Fitzpatrick 2016-03-31