Gravity Waves in Deep Water

Consider the so-called deep water limit,

$$k\,d\gg 1,$$ (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

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

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

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

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

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

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

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

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

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

$$\xi_x(x,z,t) = -a\,{\rm e}^{\,k\,z}\,\cos(\omega\,t-k\,x),$$ (11.29)

$$\xi_z(x,z,t) = a\,{\rm e}^{\,k\,z}\,\sin(\omega\,t-k\,x),$$ (11.30)

$$v_x(x,z,t) = a\,\omega\,{\rm e}^{\,k\,z}\,\sin(\omega\,t-k\,x),$$ (11.31)

$$v_z(x,z,t) = a\,\omega\,{\rm e}^{\,k\,z}\,\cos(\omega\,t-k\,x),$$ (11.32)

and

$$p_1 =\rho\,g\,\xi_z,$$ (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.

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

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