Hong Life CalculatorsUnderstanding the tsunami and shallow-water wave speed calculator — concept and principle
A tsunami crosses the open ocean at the speed of a jet airliner and slows to the speed of a car near the coast. This calculator gives the speed of a shallow-water wave for a given water depth and the time it would take to travel a chosen distance. It is used in teaching, in rough tsunami arrival estimates and for any long wave such as tides or seiches.
How it is calculated
c = √(g × h)
- g = 9.81 m/s², the acceleration due to gravity.
- h is the water depth in metres.
- c comes out in m/s; multiplying by 3.6 converts it to km/h.
Travel time = distance (km) ÷ speed (km/h), in hours.
The principle
A wave counts as "shallow-water" when its wavelength is much longer than the depth, typically more than about 20 times. Tsunami wavelengths are often 100 km or more, so even a 4,000 m deep ocean is shallow for them. In that regime the whole water column moves back and forth with the wave, and the restoring force is gravity acting on the raised water surface. Balancing that pressure gradient against the inertia of the column gives c² = gh: deeper water means a larger mass that can be pushed along faster. Wave period does not matter, so all such waves travel together without spreading out.
Worked example
- Inputs: depth h = 4,000 m, distance 1,000 km.
- g × h = 9.81 × 4,000 = 39,240 m²/s².
- c = √39,240 = 198.09 m/s.
- In km/h: 198.09 × 3.6 = 713.13 km/h (713.1271976).
- Travel time: 1,000 ÷ 713.13 = 1.40 hours (1.402274382), about 84 minutes.
| Water depth (m) | Speed (km/h) |
|---|---|
| 1,000 | 356.5635988 |
| 2,000 | 504.2570773 |
| 3,000 | 617.5862693 |
| 4,000 | 713.1271976 |
| 5,000 | 797.3004453 |
| 6,000 | 873.3988779 |
| 8,000 | 1,008.514155 |
How to read the result
Average ocean depth is about 3,700–4,000 m, so tsunamis typically cross ocean basins at 700–800 km/h. As the wave reaches shallower water it slows: at 100 m depth c is about 113 km/h, and at 10 m about 36 km/h. Because the energy flux is nearly conserved, slowing causes the wave to grow taller, which is why a wave barely noticeable at sea can flood a coast.
Common mistakes
- Using a single depth for a whole route; real travel times come from integrating over the changing seafloor depth.
- Applying the formula to ordinary wind waves in deep water, which follow a different relation.
- Forgetting the unit change: c is in m/s until multiplied by 3.6.
Frequently asked questions
Why can a tsunami pass under ships unnoticed?
In deep water its height may be under a metre spread over a wavelength of more than 100 km, so the slope is too gentle to feel.
Why does the sea sometimes retreat before a tsunami?
If the trough of the wave arrives first, water is drawn away from the shore. It is a natural warning sign to move to high ground immediately.
Does the formula apply in lakes and harbours?
Yes, for long waves such as seiches the same √(gh) speed applies, as long as the wavelength is much larger than the depth.
Sources: R. G. Dean and R. A. Dalrymple, Water Wave Mechanics for Engineers and Scientists, World Scientific, 1991 · NOAA National Weather Service, Tsunami education resources