Ocean Wave Speed Calculator
Find the wavelength, phase speed and group speed of an ocean wave from its period and the water depth, and see how the wave slows in shallow water.
🌊 What is Ocean Wave Speed?
Ocean wave speed is the rate at which a surface gravity wave advances across the sea. For a regular wave train there are really two speeds. The phase speed c is how fast a single crest moves, equal to the wavelength divided by the period. The group speed cg is how fast a packet of waves, and the energy it carries, travels. Both depend on the wave period and on the water depth, and the link between them is the linear dispersion relation of Airy wave theory.
The key idea is dispersion: waves of different periods travel at different speeds. In deep water, long period waves outrun short period ones, which is why a distant storm sends long swell ahead of the choppier waves. In shallow water the speed stops depending on period and depends only on depth, which is why every wave slows as it approaches a beach. Coastal engineers use these speeds to design harbors and breakwaters, surfers and forecasters use them to time the arrival of swell, and oceanographers use them to track storm waves across whole basins.
A common misconception is that water travels along with the wave. It does not. Water particles move in nearly closed orbits, and only the wave form and its energy propagate. Another is that waves travel at sqrt(gh) everywhere. That formula holds only for very long waves in shallow water, such as a tsunami in the open ocean. A 10 second wind swell in deep water moves at about 15.6 m/s, not at 99 m/s, even in 1000 m of water. The classification by the ratio h/L of depth to wavelength makes the distinction exact: deep water above 0.5, shallow water below 0.05, and an intermediate regime in between where neither limit works.
This calculator solves the full dispersion relation numerically, so it is valid in every regime. It also shows the deep-water and shallow-water limits next to your answer, and draws two charts so you can see how far your wave is from each limit. For the shallow-water speed alone, see the tsunami wave speed calculator.
📐 Formula
The energy flux of a wave per unit crest length is E cg, where E is the wave energy density. That is why the group speed is called the energy transport speed: it sets how fast wave energy arrives at a coast.
📖 How to Use This Calculator
Steps
💡 Example Calculations
Example 1 — 10 s Swell in Deep Ocean
T = 10 s, h = 1000 m
Example 2 — 8 s Wave at 10 m Depth
T = 8 s, h = 10 m
Example 3 — Near-Shore 6 s Wave at 3 m
T = 6 s, h = 3 m
Example 4 — Long Period 14 s Swell at 50 m
T = 14 s, h = 50 m
❓ Frequently Asked Questions
🔗 Related Calculators
How do you calculate the speed of an ocean wave?
Solve the dispersion relation omega^2 = g k tanh(kh) for the wavenumber k, where omega = 2 pi / T and h is the water depth. Then wavelength L = 2 pi / k and phase speed c = L / T. In deep water this reduces to c = gT/(2 pi), about 1.56 T m/s.
What is the difference between phase speed and group speed?
Phase speed c is how fast an individual crest moves. Group speed cg is how fast a packet of waves, and its energy, moves. They are related by cg = n c, where n = 0.5 (1 + 2kh / sinh 2kh). In deep water n is 0.5, in shallow water n is 1.
How fast does a 10 second swell travel in deep water?
A 10 s wave in deep water has a wavelength of 156.08 m and a phase speed of 15.61 m/s (about 56 km/h or 30 knots). Its group speed is half of that, 7.80 m/s, so the energy of the swell crosses an ocean at roughly 28 km/h.
When is water considered deep for waves?
Water is classed as deep when the depth is more than half the wavelength (h/L greater than 0.5). At that depth tanh(kh) is within about 0.3 percent of 1, so the bottom has no practical effect on the wave speed.
When is water considered shallow for waves?
Water is classed as shallow when the depth is less than one twentieth of the wavelength (h/L below 0.05). Then tanh(kh) is approximately kh and the speed becomes c = sqrt(g h), independent of the period.
Why do waves slow down in shallow water?
As the depth falls below about half a wavelength, the bottom interacts with the orbital motion of the water. Since the period is fixed, the wavelength and the speed both decrease according to the dispersion relation. This slowing is what makes waves bend toward the shore (refraction).
Why does the period of a wave stay constant as it approaches the shore?
The frequency is set by the source, such as the wind or an earthquake, and a stable wave train cannot gain or lose crests in steady conditions. So the period is conserved while the wavelength and speed change with depth.
Is the speed of a tsunami the same as an ordinary wave?
A tsunami has a wavelength of tens to hundreds of kilometers, far greater than the ocean depth, so it is a shallow-water wave in the open ocean and travels at sqrt(g h), about 200 m/s in 4000 m of water. Wind waves are much shorter and follow the deep-water speed.
Why is there no simple formula for wavelength at any depth?
The dispersion relation is implicit because the wavenumber k appears both inside and outside the tanh term. It must be solved numerically, here by Newton iteration. Explicit approximations such as Eckart and Hunt exist for hand calculation, with errors of about 5 percent and 0.1 percent.
What units does the calculator use?
Wave period is in seconds, water depth in meters. Wavelength is in meters, speeds in meters per second, wavenumber in radians per meter. Gravity is taken as 9.80665 m/s squared.
How accurate is linear wave theory?
Airy theory is accurate for small amplitude waves, where wave height is small relative to both wavelength and depth. For steep waves or waves near breaking, nonlinear corrections raise the speed by a few percent.