Wave Group Velocity at Any Depth

Energy travels at the group speed, not the crest speed. The crests of a train of water waves move at the speed $c$, the wavelength divided by the period, but the energy travels at the group velocity $c_g$. That is the speed that carries wave energy to a coast, and it sets how a wave changes height as the water shoals. The two are linked by a factor $n$, so that $c_g = n\,c$, with $n = \tfrac{1}{2}\left(1 + \frac{2kd}{\sinh 2kd}\right)$ and $k = 2\pi/L$. In deep water $n$ is one half and the crests run through the group at twice its speed; in shallow water $n$ is one and the whole group moves with the crests.

Reading the terms. The wavenumber $k$ times the still-water depth $d$ compares the depth with the wavelength $L$. When $2kd$ is large, $\sinh 2kd$ is huge, the fraction in the brackets vanishes and $c_g$ tends to $c/2$; when it is small, the fraction tends to one and $c_g$ tends to $c$. The page needs both the crest speed and the wavelength at the site's depth, and the dispersion-relation page gives the wavelength. Dean and Dalrymple (1991, chapter 3) present the result from linear wave theory.

Checks by another route. The checks find $c_g = d\omega/dk$ by central difference of $\omega^2 = gk\tanh(kd)$, with $k$ found by bisection from the period, so neither the $n$ factor nor $\sinh$ appears in that working. A 10 s wave in 10 m of water has $k = 0.068032$ per metre, a wavelength of 92.35582 m and a crest speed of 9.2355817 m/s, and the page returns a group velocity of 8.068 m/s. An 8 s wave in 20 m gives 7.406 m/s, and a 6 s wave in 3 m gives 4.574 m/s.

The two limits. In 1,000 m of water a 10 s wave has a crest speed of 15.607768 m/s and a group velocity of 7.8039 m/s, matching the deep-water value $gT/4\pi = 7.80388$ m/s. In 1 m of water the shallow limit is $c_g = c = \sqrt{gd} = 3.13156$ m/s, and for a 60 s wave there (a wavelength of 187.858 m and a crest speed of 3.130973 m/s) the derivative gives 3.12981 m/s, within 0.06 % of $\sqrt{gd}$. The deep-water page gives the first limit from the period alone.