Energy of Gravity Waves

It is easily demonstrated, from the analysis contained in the previous sections, that a gravity wave of arbitrary wavenumber $k$ , propagating horizontally through water of depth $d$ , has a phase velocity

$$v_p = (g\,d)^{1/2}\left[\frac{\tanh(k\,d)}{k\,d}\right]^{1/2}.$$ (11.43)

Moreover, the ratio of the group to the phase velocity is

$$\frac{v_g}{v_p} =\frac{1}{2}\left[1+\frac{2\,k\,d}{\sinh(2\,k\,d)}\right].$$ (11.44)

It follows that neither the phase velocity nor the group velocity of a gravity wave can ever exceed the critical value $(g\,d)^{1/2}$ . It is also easily demonstrated that the displacement and velocity fields associated with a plane gravity wave of wavenumber $k\,{\bf e}_x$ , angular frequency $\omega$ , and surface amplitude $a$ , are

$$\xi_x(x,z,t) = -a\,\frac{\cosh[k\,(z+d)]}{\sinh(k\,d)}\,\cos(\omega\,t-k\,x),$$ (11.45)

$$\xi_z(x,z,t) =a\,\frac{\sinh[k\,(z+d)]}{\sinh(k\,d)}\,\sin(\omega\,t-k\,x),$$ (11.46)

$$v_x(x,z,t) =a\,\omega\,\frac{\cosh[k\,(z+d)]}{\sinh(k\,d)}\,\sin(\omega\,t-k\,x),$$ (11.47)

$$v_z(x,z,t) =a\,\omega\,\frac{\sinh[k\,(z+d)]}{\sinh(k\,d)}\,\cos(\omega\,t-k\,x).$$ (11.48)

The mean kinetic energy per unit surface area associated with a gravity wave is defined

$$K = \langle\int_{-d}^\zeta \frac{1}{2}\,\rho\,v^{\,2}\,dz\rangle,$$ (11.49)

where

$$\zeta(x,t) = a\,\sin(\omega\,t-k\,x)$$ (11.50)

is the vertical displacement at the surface, and

$$\langle \cdots\rangle = \int_0^{2\pi} (\cdots)\,\frac{d(k\,x)}{2\pi}$$ (11.51)

is an average over a wavelength. Given that $\langle \cos^2(\omega\,t-k\,x)\rangle =\langle \sin^2(\omega\,t-k\,x)\rangle=1/2$ , it follows from Equations (11.47) and (11.48) that, to second order in $a$ ,

$$K = \frac{1}{4}\,\rho\,a^{\,2}\,\omega^{\,2}\int_{-d}^0 \frac{\co... ...,d)}\,dz=\frac{1}{4}\,\rho\,g\,a^{\,2}\,\frac{\omega^{\,2}}{g\,k\,\tanh(k\,d)}.$$ (11.52)

Making use of the general dispersion relation (11.21), we obtain

$$K = \frac{1}{4}\,\rho\,g\,a^{\,2}.$$ (11.53)

The mean potential energy perturbation per unit surface area associated with a gravity wave is defined

$$U = \langle\int_{-d}^\zeta \rho\,g\,z\,dz\rangle + \frac{1}{2}\,\rho\,g\,d^{\,2},$$ (11.54)

which yields

$$U = \langle \frac{1}{2}\,\rho\,g\,(\zeta^{\,2}-d^{\,2})\rangle + \frac{1}{2}\,\rho\,g\,d^{\,2} = \frac{1}{2}\,\rho\,g\,\langle\zeta^{\,2}\rangle,$$ (11.55)

or

$$U = \frac{1}{4}\,\rho\,g\,a^{\,2}.$$ (11.56)

In other words, the mean potential energy per unit surface area of a gravity wave is equal to its mean kinetic energy per unit surface area.

Finally, the mean total energy per unit surface area associated with a gravity wave is

$$E = K+ U = \frac{1}{2}\,\rho\,g\,a^{\,2}.$$ (11.57)

This energy depends on the wave amplitude at the surface, but is independent of the wavelength, or the water depth.