Ocean Wave Calculator (Wavelength, Speed & Power)
Find the wavelength, speed, group speed and power of ocean waves from the wave period and water depth with the linear dispersion relation — deep, intermediate or shallow water — for surf and swell forecasts.
How it works
Dispersion relation ω² = g k tanh(k d), ω = 2π/T, solved for k · L = 2π/k, c = L/T · deep water L₀ = gT²/2π ≈ 1.56 T² m · group speed c_g = n c, n = ½ [1 + 2kd / sinh(2kd)] · energy E = ρ g H_s² / 16, power P = E c_g (ρ = 1,025 kg/m³)
Ocean waves of a given period travel at different speeds depending on the depth. In deep water (depth more than half the wavelength) the speed depends only on the period, c = gT/2π, and the energy travels at half the wave speed — which is why swell from a distant storm arrives as a group. As waves enter shallower water they slow down, shorten and steepen until they break; in very shallow water every wave travels at √(g d), the same law that sets tsunami speed. Linear (Airy) wave theory is accurate for small, non-breaking waves and is the basis of surf and swell forecasts.
Worked example
A 10-second swell in 20 m of water: the deep-water wavelength would be 156.1 m, but solving the dispersion relation at 20 m gives a wavelength of 121.2 m — intermediate depth. The crests move at 43.6 km/h (12.12 m/s) and the energy at 33.4 km/h (9.27 m/s). With a significant height of 1.5 m the swell carries about 13.1 kW per metre of crest.
Sources: NOAA — Ocean waves (wave speed and period) · Wikipedia — Dispersion (water waves)
Frequently asked questions
How fast do ocean waves travel?
In deep water about 1.56 × period m/s: a 10 s swell moves at about 15.6 m/s (56 km/h), and the swell group at half that speed. In shallow water waves slow to √(g × depth).
What is the difference between phase speed and group speed?
Phase speed is how fast individual crests move; group speed is how fast the wave energy — and so the swell itself — travels. In deep water the group moves at half the crest speed.
When is water "deep" for a wave?
When the depth is more than half the wavelength; the wave then does not feel the bottom. Below one-twentieth of the wavelength it is shallow-water, where speed depends only on depth.
How much power is in ocean waves?
Deep-water wave power is about 0.49 × H_s² × T kW per metre of crest (H_s in metres, T in seconds). Open-ocean coasts often average 20–60 kW/m.
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This calculator is for educational purposes. Double-check important results before acting on them.