Wave Power from the Energy Flux (Energy Times Group Velocity)

A wave train carries power, and the useful measure is the power crossing a line one metre long drawn parallel to the crests, in watts per metre of crest. It is the energy lying on each square metre of sea surface, $E$, times the speed at which that energy travels, $c_g$. The crests move at the phase speed, but the energy moves at the group velocity, which is half the phase speed in deep water and equal to it in shallow water. So the group velocity, not the crest speed, is the one to multiply by. This is a result of linear wave theory, treated in the chapters on linear waves in Dean and Dalrymple's 1991 text on water wave mechanics, and not a fitted rule.

Unlike the familiar deep-water power formula, which has the deep-water group velocity folded into it, this form holds at any depth. It is the one to use for a buoy in 30 metres of water, where a long swell already feels the bottom. Feed it the energy of the sea itself, $E = \rho g H_s^{2}/16$ for a sea of significant height $H_s$, and the group velocity at the energy period. For a single regular wave of height $H$ the energy is $\rho g H^{2}/8$.

Take a regular wave of height 1 m in seawater (1 025 kg/m³, $g = 9.80665$ m/s²). Then $E = 1\,256.477$ J/m². A 10 s wave in deep water has $c_g = gT/4\pi = 7.803884$ m/s, and the product is 9 805.40 W/m, or 9.8054 kW/m. A second case is a sea of significant height 2 m and energy period 8 s: $E = 2\,512.954$ J/m², $c_g = 6.243107$ m/s, and $P = 15.68864$ kW/m. The deep-water formula $\rho g^{2} H_s^{2} T_e/(64\pi)$ gives 15.68864 as well, and the 0.49 $H_s^{2} T_e$ rule gives 15.6896.

The common deep-water rule of thumb is 0.5 kW per m³ s times $H_s^{2}$ times $T_e$, a rounded coefficient. For $H_s = 3$ m and $T_e = 10$ s it gives 45 kW/m. The exact coefficient, $\rho g^{2}/64\pi$, is 0.4903 in seawater, so the product of the energy 5 654.147 J/m² and 7.803884 m/s is 44.124 kW/m, 2 % under the rounded rule. Backwards, 15.68864 kW/m at 6.243107 m/s corresponds to 2 512.95 J/m².