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

$$v_p = (g\,d)^{1/2}\left[\frac{\tanh(k\,d)}{k\,d}\right]^{1/2}.$$ (1149)

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

$$\frac{v_g}{v_p} =\frac{1}{2}\left[1+\frac{2\,k\,d}{\sinh(2\,k\,d)}\right].$$ (1150)

Note, 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

$$\xi_x(x,z,t) = -a\,\frac{\cosh[k\,(z+d)]}{\sinh(k\,d)}\,\cos(\omega\,t-k\,x),$$ (1151)

$$\xi_z(x,z,t) = a\,\frac{\sinh[k\,(z+d)]}{\sinh(k\,d)}\,\sin(\omega\,t-k\,x),$$ (1152)

$$v_x(x,z,t) = a\,\omega\,\frac{\cosh[k\,(z+d)]}{\sinh(k\,d)}\,\sin(\omega\,t-k\,x),$$ (1153)

$$v_z(x,z,t) = a\,\omega\,\frac{\sinh[k\,(z+d)]}{\sinh(k\,d)}\,\cos(\omega\,t-k\,x).$$ (1154)

Now, the mean kinetic energy per unit surface area associated with a gravity wave is defined

$$K = \langle\int_{-d}^\zeta \frac{1}{2}\,\rho\,v^{\,2}\,dz\rangle,$$ (1155)

where

$$\zeta(x,t) = a\,\sin(\omega\,t-k\,x)$$ (1156)

is the vertical displacement at the surface, and

$$\langle \cdots\rangle = \int_0^{2\pi} (\cdots)\,\frac{d(k\,x)}{2\pi}$$ (1157)

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 (1153) and (1154) that, to second order in $a$,

$$K = \frac{1}{4}\,\rho\,a^2\,\omega^2\int_{-d}^0 \frac{\cosh[... ...\frac{1}{4}\,\rho\,g\,a^2\,\frac{\omega^2}{g\,k\,\tanh(k\,d)}.$$ (1158)

Making use of the general dispersion relation (1127), we obtain

$$K = \frac{1}{4}\,\rho\,g\,a^2.$$ (1159)

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

$$U = \langle\int_{-d}^\zeta \rho\,g\,z\,dz\rangle + \frac{1}{2}\,\rho\,g\,d^{\,2},$$ (1160)

which yields

$$U = \langle \frac{1}{2}\,\rho\,g\,(\zeta^{\,2}-d^{\,2})\rang... ...,g\,d^{\,2} = \frac{1}{2}\,\rho\,g\,\langle\zeta^{\,2}\rangle,$$ (1161)

or

$$U = \frac{1}{4}\,\rho\,g\,a^{\,2}.$$ (1162)

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

$$E = K+ U = \frac{1}{2}\,\rho\,g\,a^{\,2}.$$ (1163)

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