Deep Water Wave Celerity
Calculate the celerity (phase velocity) of an ocean wave in deep water, c = g·T ÷ (2π), from the period T (s) and gravity g. The result, in m/s, is the speed at which the wave crest propagates. In deep water, longer-period waves travel faster — a phenomenon called dispersion, which makes long-period swell reach the coast before the short waves generated by the same storm. The celerity is half the group velocity (at which energy travels) in deep water. It is a base concept of wave hydrodynamics. Enter the wave period.
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Deep water celerity
Celerity (or phase speed) is the speed at which the crest of a wave travels across the sea surface. In deep water, c = g·T ÷ (2π), which depends only on the period T. The crucial fact is that longer-period waves travel faster — a property called dispersion. That has a fascinating and very practical consequence: when an ocean storm generates a spectrum of waves of many periods, they separate as they travel (the long ones in front, the short ones behind), and what reaches a distant beach first is the long-period swell, smooth and regular, days ahead of the short waves. Surfers and forecasters use this to predict the arrival of a swell event: by measuring the period of the swell that begins to arrive, one knows where the bigger waves sit in the train (still longer periods travelled faster and have already gone by, while shorter ones are yet to come). Celerity also explains refraction: as a wave moves into shallower water, the part over the shallower bottom slows down and the wave front bends. An important detail of wave physics: in deep water the celerity (crest speed) is twice the group velocity (the speed at which energy travels) — which is why individual crests 'run' through a wave group. Enter the wave period.
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Wave Group Velocity
Calculate the group velocity of an ocean wave in deep water, c_g = g·T ÷ (4π), from the period T (s). The result, in m/s, is the speed at which the wave energy (and the 'envelope' of a wave group) propagates — exactly half the celerity (phase velocity) in deep water. This difference explains a curious phenomenon: within a wave group, individual crests appear at the rear, advance through the group (faster than it) and disappear at the front. The group velocity is what matters for energy transport and predicting swell arrival at the coast. Enter the wave period.
Ocean Wavelength
Calculate the wavelength of an ocean wave in deep water, L = g·T² ÷ (2π), from the wave period T (s) and gravity g (9.81 m/s²). The result, in meters, is the distance between two successive crests — in deep water, it depends only on the period. Long-period waves (swell from distant storms) have much larger wavelengths than local wind waves. The wavelength sets the depth at which the wave 'feels' the bottom (about L/2), starts to refract and shoal until it breaks. It is a fundamental parameter of linear wave theory and coastal engineering. Enter the wave period.
Shallow Water Wave Celerity
Calculate the celerity of a wave in shallow water, c = √(g·h), from the water depth h (m) and gravity g. The result, in m/s, is the propagation speed when the depth is much smaller than the wavelength — a situation in which the wave 'feels' the bottom and its speed depends only on depth, no longer on the period. This is why waves refract as they approach the coast (the deeper part travels faster) and why tsunamis travel at hundreds of km/h in the deep ocean and slow down (piling up energy) as they reach the coast. Enter the water depth.
Wave Period
Calculate the period of a wave, T = 1 ÷ f, from the frequency f (Hz). The result, in seconds, is the time between two successive crests passing a fixed point — one of the most important properties of an ocean wave. The period determines the wavelength and celerity (in deep water), the depth at which the wave interacts with the bottom, and classifies the sea state: local wind waves have short periods (3-8 s), while swell from distant storms has long periods (10-20 s), travels faster and penetrates deeper. The period is measured by buoys and used in wave forecasting. Enter the wave frequency.
Wave Power
Estimate the power flux of an ocean wave per meter of wave front, P ≈ 0.5 × H² × T, from the wave height H (m) and the period T (s), in deep water (seawater). The result, in kW/m, is the power available per meter of wave front width — a key indicator of wave energy potential for converters (WECs). Coasts exposed to ocean swells (European Atlantic, Pacific) reach 30-70 kW/m, a significant renewable resource. The power grows with the square of the height and linearly with the period. Enter the wave height and period.
Wave Steepness
Calculate the steepness of a wave, s = H ÷ L, dividing the wave height H by the wavelength L. The dimensionless result is the relative steepness of the wave — the taller it is for its length, the steeper. Steepness has a physical limit: deep-water waves break when s exceeds about 1/7 (0.143), as the crest becomes unstable (120° angle). Young storm waves are steep; swell that travels long distances is gentle (low steepness). Steepness governs wave stability, breaking and vessel comfort. Enter the wave height and wavelength.
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