Coastal & Ocean Waves formula solvers

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D I L S W

Deep-Water Wave Celerity

c_0 = \frac{g\,T}{2\pi}

Coastal & Ocean WavesHow fast an individual crest travels in deep water, set by the period and nothing else. Long waves outrun short ones, which is why the swell from a distant storm arrives sorted — the longest periods first — and why a rising swell with a lengthening period is the classic warning that weather is coming.

Deep-Water Wave Power per Metre of Crest

P = \frac{\rho g^{2} H^{2} T}{64\pi}

Coastal & Ocean WavesThe rate at which a deep-water wave train delivers energy across a line one metre long drawn along the crest. It is the energy density carried at the group velocity, and it is how every marine-energy resource on earth is quoted — because a wave front has no width of its own, so the resource has to be stated per metre of it.

Deep-Water Wavelength from Period

L_0 = \frac{g\,T^{2}}{2\pi}

Coastal & Ocean WavesIn water deeper than half a wavelength, a wave's length is fixed by its period alone — nothing about the wind that made it, the depth beneath it or how big it is enters. This is the first number to compute about any sea state, because it is the length every other coastal criterion is measured against.

Depth-Limited Breaking Wave Height

H_b = \gamma_b\, d_b

Coastal & Ocean WavesA wave running up a beach breaks when its height reaches a fixed fraction of the water depth. The fraction is near 0.78 for a solitary wave on a flat bottom — McCowan's classical result — and it is why the surf zone has an outer edge you can see, and why the wave that reaches a structure is often limited by the water in front of it rather than by the storm offshore.

Iribarren Number (Surf-Similarity Parameter)

\xi = \frac{\tan\beta}{\sqrt{H / L_0}}

Coastal & Ocean WavesThe beach slope divided by the square root of the wave steepness. Ramón Iribarren and Casto Nogales published it in 1949, and it turns out to classify almost everything about a surf zone: whether a wave spills, plunges, collapses or surges, how much of it runs up, how much reflects, and how armour on a slope behaves.

Linear Wave Dispersion Relation

\omega^{2} = g\,k\,\tanh(k d), \qquad \omega = \frac{2\pi}{T},\quad k = \frac{2\pi}{L}

Coastal & Ocean WavesThe one relation the whole of linear wave theory rests on, valid at every depth. The deep-water and shallow-water formulas are both limits of this, and between those limits — which is where most coastal engineering happens — there is nothing else to use. It is transcendental in the wavenumber, so that direction is solved numerically here rather than pretended into a closed form.

Shallow-Water Wave Celerity

c = \sqrt{g\,d}

Coastal & Ocean WavesOnce the water is shallower than about a twentieth of the wavelength, the period drops out of the celerity entirely and every wave travels at the square root of g times the depth. This is why tsunamis cross oceans at jetliner speed, why a bore runs up an estuary at a fixed pace, and why waves bend to face the beach.

Shoaling Coefficient

K_s = \sqrt{\frac{c_{g0}}{c_g}}

Coastal & Ocean WavesAs a wave moves into shallower water it slows down, and because it keeps delivering the same energy per second, the height has to rise to compensate. The shoaling coefficient is the ratio H/H₀ that results, and it comes straight out of conserving energy flux between deep water and the point of interest.

Wave Energy Density

E = \frac{\rho g H^{2}}{8}

Coastal & Ocean WavesThe mean energy in the water column per square metre of sea surface, averaged over a wavelength. Half of it is potential — water lifted into the crests and taken out of the troughs — and half is kinetic, in the orbital motion. It goes as the SQUARE of the height, which is why a sea twice as high is four times the problem.

Wave Steepness

S = \frac{H}{L}

Coastal & Ocean WavesHeight divided by wavelength: the one dimensionless number that says whether a wave is a gentle swell or a breaking sea. It sets whether the wave is stable, how hard it hits, and — through the surf-similarity parameter — what kind of breaker it becomes when it reaches the beach.