Wave Energy Density

A wave in the sea holds energy in two forms, and linear theory says they are exactly equal. Half of it is potential — water lifted into the crests and removed from the troughs against gravity — and half is kinetic, in the orbital motion of the water beneath. Adding them and averaging over a wavelength gives $E = \rho g H^{2}/8$, joules per square metre of sea surface, integrated through the whole water column.

The equipartition is not a coincidence. Every linear oscillator does it, from a pendulum to an LC circuit, and it is one of the tells that the wave really is behaving linearly. As soon as a wave becomes nonlinear — steepening as it shoals, growing a peaked crest — the two halves stop being equal, and the imbalance is a measure of how far outside Airy theory you have wandered.

The factor of eight is the thing to get right, and it is where most errors live. This formula is written for wave HEIGHT $H$, trough to crest. Written for amplitude $a = H/2$, the identical energy is $\rho g a^{2}/2$. Both forms appear constantly, sometimes on facing pages of the same book, and using one where the other was meant is a factor of four. Physics texts tend to write amplitude; coastal engineering texts almost always write height, because height is what a buoy reports and what a person can see. Check which one you have before you square anything.

The square is the other half of the story. Energy goes as $H^{2}$, so a sea twice as high carries four times the energy, and — running it the other way — halving the wave height behind a breakwater means removing three-quarters of the incoming energy. That is why armour units get large so fast, and it is why a transmission coefficient $K_t$ quoted for height must never be read as an energy fraction: the energy fraction is $K_t^{2}$, and a structure that "transmits 30%" of the height transmits only 9% of the energy, which makes it look far better than a careless reader will credit, or far worse, depending which way the confusion runs.

Now the honesty about $H$, and it is the most important paragraph on this page. Significant wave height $H_s$ is the mean height of the highest one-third of the waves in a record. It is not the height of any particular wave, and it is roughly what a trained observer reports when asked to eyeball a sea — which is why it was defined that way in the first place. In a Rayleigh-distributed sea, the mean height is about $0.63\,H_s$, the highest one wave in ten is about $1.27\,H_s$, one in a hundred about $1.67\,H_s$, and the largest wave in a three-hour storm record runs near $1.8$ to $2.0\,H_s$. A structure designed to $H_s$ is designed to be overtopped by hundreds of waves in a single storm.

That has a direct consequence for this equation. Substituting $H_s$ here gives the energy of a notional wave of that height, which is not the mean energy density of the sea state. The mean energy uses the root-mean-square height, $H_{rms} \approx H_s/\sqrt{2}$, so the true mean energy density of the sea is about half what a naive substitution of $H_s$ returns. Both numbers are useful and they answer different questions; the mistake is not knowing which one you have computed.

Finally, the density. It is an input on this page and always will be. Open ocean sits near 1025 kg/m³, fresh water at 1000, and the difference is 2.5% in every energy and power figure. The Great Lakes generate a wave climate that has sunk large ships, and they are fresh; an estuary is somewhere in between and moves with the tide. Baking 1025 into the arithmetic would be a small error made silently, which is the worst kind.