# Water Wave Calculator - Dispersion, Wave Power, Tsunami Speed, Kelvin Wake

> Water waves: phase and group speed from wavelength or period and depth, orbit animation, wave power, tsunami travel time, shoaling, breaking, wind seas.

- URL: https://qilingtools.com/en/physics/water-waves
- Category: Physics Tools (https://qilingtools.com/en/physics)
- Tags: [Physics](https://qilingtools.com/en/tags/physics), [Education](https://qilingtools.com/en/tags/education)
- Last updated: 2026-10-29
- Price: Free, no sign-up
- Privacy: Runs locally in the browser; files are never uploaded

Built on linear (Airy) wave theory, the tool solves the dispersion relation ω² = gk·tanh(kh) from a wavelength or period and the water depth, gives phase speed, group speed and the depth regime, and animates the surface and the circular or elliptical paths of water particles to scale, with optional surface tension for capillary waves. The other tabs compute wave energy and power per metre of crest, tsunami speed, arrival time and growth near the coast by Green's law, shoaling of wind waves to their breaking depth, fully developed seas from the Pierson–Moskowitz spectrum next to a Beaufort table, and the 19.47° Kelvin ship wake.

## How to use

1. On the dispersion tab enter a wavelength or period, the depth and an amplitude, and watch the animation and speeds
2. On the energy tab pick a sea state or enter significant wave height and period to get the power per metre
3. On the tsunami tab set ocean depth, distance and near-shore depth to see travel time and amplitude growth
4. On the wind tab enter wind speed and fetch and compare with the Beaufort table; on the wake tab change ship speed

## Features

- Full dispersion relation with optional surface tension and automatic deep, intermediate or shallow regime
- To-scale animation of the surface and particle orbits: circles in deep water, ellipses in shallow
- Phase and group speed vs period, against the deep gT/2π and shallow √(gh) limits
- Wave energy, wave power in kW/m and total power per km of coast
- Tsunami speed and arrival time, Green's law amplification, linear shoaling and the 0.78 breaking criterion
- Pierson–Moskowitz spectrum, JONSWAP fetch-limited growth, Beaufort table and a Kelvin wake diagram

## FAQ

### What is the difference between deep and shallow water waves?

It depends on depth relative to wavelength. When the water is deeper than half a wavelength the bottom barely feels the wave: these are deep-water waves, whose speed c = gT/2π depends only on period, so long-period swell outruns short waves (dispersion). When the depth is less than a twentieth of the wavelength the wave is a shallow-water wave with c = √(gh), the same for all wavelengths. In between you need the full tanh(kh) relation.

### Why is a tsunami barely noticeable at sea but devastating at the coast?

A tsunami's wavelength is hundreds of kilometres, so even a 4000 m deep ocean is shallow for it. It travels at √(gh) ≈ 200 m/s, about 700 km/h, yet its amplitude may be only tens of centimetres and ships hardly notice it. Approaching land, the water gets shallower and the wave slows while its energy flux stays nearly constant, so the height grows as h^(−1/4) by Green's law and the water behind keeps piling in, producing run-up of metres to tens of metres.

### Does the water travel with the wave?

Not in linear theory. The wave shape moves at the phase speed, but each water particle traces a closed circle or ellipse in place. In deep water the circles shrink with depth as e^{kz}, down to 4% at half a wavelength; in shallow water the orbits flatten into ellipses and become a back-and-forth motion near the bed. The red dot in the animation is one particle. Real large waves add a slow Stokes drift, which is not modelled here.

### How much power do ocean waves carry?

The energy density of an irregular sea is E = ρgHs²/16 and the power is energy times group speed. In deep water P ≈ 0.49·Hs²·Te kW/m: a 2 m, 8 s sea carries about 16 kW per metre of crest, and a North Atlantic winter sea of 5 m at 12 s about 150 kW/m. Wave energy converters can capture only part of that and must survive extreme storm loads.

### Why is the Kelvin wake always about 19.5°?

Only waves whose phase speed component along the track matches the ship's speed stay steady relative to it, and in deep water their energy travels at half the phase speed, so it lags behind. Adding up all these waves confines the wave energy to a wedge of half-angle arcsin(1/3) ≈ 19.47°, whatever the speed. Speed only sets the transverse wavelength λ = 2πU²/g; in shallow water or at very high Froude numbers the angle changes.

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