Physics · Classical Mechanics · Oscillations & Waves
Tsunami Wave Speed Calculator
Calculates tsunami wave speed using the shallow-water wave approximation based on ocean depth.
Calculator
Formula
v is the tsunami wave speed (m/s), g is the gravitational acceleration (9.81 m/s²), and h is the ocean depth (m). This formula is valid when the wavelength of the wave is much greater than the water depth — the defining condition of a shallow-water wave.
Source: Lamb, H. (1932). Hydrodynamics (6th ed.). Cambridge University Press. Also referenced in NOAA Tsunami Preparedness documentation.
How it works
Tsunamis are long-period gravity waves generated by sudden seafloor displacement — typically caused by submarine earthquakes, landslides, or volcanic eruptions. Unlike wind-driven surface waves, a tsunami's wavelength (often 200–600 km) far exceeds the ocean depth (typically 1–11 km). This places tsunamis firmly in the shallow-water wave regime, where wave speed depends only on depth and gravity, not on wavelength or period.
The governing formula is derived from the linearized shallow-water wave equations: v = √(g · h), where v is the wave propagation speed in meters per second, g is gravitational acceleration (9.81 m/s²), and h is the local water depth in meters. As a tsunami moves from deep ocean into shallower coastal water, its speed decreases proportionally with the square root of depth. This deceleration causes the wave to compress and increase in height — a phenomenon known as shoaling — which explains why tsunamis that are barely noticeable at sea can become devastating walls of water onshore.
In practice, tsunami travel-time models (such as NOAA's MOST model) integrate this formula over varying bathymetric profiles across an entire ocean basin. However, the simple single-depth version remains a powerful tool for quick estimation, classroom instruction, and back-of-envelope emergency planning. For example, knowing the average Pacific Ocean depth of about 4,000 m, one can immediately estimate that a trans-Pacific tsunami travels at roughly 720 km/h — comparable to a commercial jetliner.
Worked example
Suppose a major earthquake strikes off the coast of Japan in water 4,000 m deep. How fast does the resulting tsunami travel, and how long would it take to reach Hawaii, approximately 6,200 km away?
Step 1 — Identify inputs: Ocean depth h = 4,000 m, gravitational acceleration g = 9.81 m/s².
Step 2 — Apply the formula:
v = √(9.81 × 4000) = √39,240 ≈ 198.1 m/s
Step 3 — Convert to km/h:
198.1 m/s × 3.6 = 713.0 km/h
Step 4 — Estimate travel time to Hawaii:
Time = Distance ÷ Speed = 6,200 ÷ 713 ≈ 8.7 hours
This matches real-world historical records closely: the 2011 Tōhoku tsunami reached Hawaii in approximately 7–8 hours, with actual variation due to the non-uniform depth of the Pacific basin. The simple formula provides a remarkably useful first estimate.
Limitations & notes
The shallow-water formula v = √(g·h) assumes a uniform, flat ocean bottom, which is never strictly true. Real ocean basins have ridges, trenches, and continental shelves that cause local variations in wave speed. More sophisticated models integrate the formula continuously over measured bathymetric data. Additionally, the formula describes the phase speed of the leading wave front and does not account for wave dispersion, frequency content, or the complex wave-interference patterns generated by extended rupture zones. Near the coast, nonlinear effects (wave breaking, bottom friction, and inundation dynamics) dominate, and the shallow-water approximation breaks down. The formula also assumes incompressible water and ignores Earth's rotation (Coriolis effect), which becomes relevant for very long travel distances. Finally, this calculator uses a standard gravitational acceleration of 9.81 m/s²; slight variations exist with latitude and elevation, though their effect on the result is negligible for most purposes.
Frequently asked questions
Why does tsunami speed depend on depth rather than wave height?
Tsunamis are shallow-water gravity waves, meaning their behavior is governed by the restoring force of gravity acting on the entire water column. In this regime, the wave speed depends only on water depth and gravitational acceleration. Wave height (amplitude) affects energy content but has negligible influence on propagation speed in the linear approximation.
How fast is a typical tsunami in the open ocean?
In the deep Pacific Ocean, where average depths are around 4,000–5,000 m, tsunamis typically travel between 700 and 800 km/h — roughly the cruising speed of a commercial aircraft. In shallower seas like the Mediterranean (average depth ~1,500 m), speeds drop to around 430 km/h.
What happens to tsunami speed as it approaches the shore?
As the ocean shallows toward the coast, the wave speed decreases in proportion to the square root of the depth. A wave traveling at 700 km/h in 4,000 m of water will slow to about 36 km/h in 10 m of water. This deceleration compresses the wave, causing its height to increase dramatically — the process called shoaling.
Is this formula the same as the one used by NOAA for tsunami warnings?
The same physical principle underlies NOAA's models, but operational systems like MOST (Method of Splitting Tsunamis) apply the formula iteratively across thousands of bathymetric grid cells to account for the real, non-uniform ocean floor. The simple single-depth version here is ideal for estimates and education, while operational forecasting requires full numerical integration.
Can I use this calculator for regular ocean wind waves?
No. Wind-generated surface waves are deep-water or intermediate-water waves, meaning their speed depends on wavelength and wave period, not water depth. The shallow-water approximation only applies when the water depth is less than about 1/20th of the wavelength. For typical wind waves with wavelengths of tens of meters, the deep-water formula c = √(gλ / 2π) is more appropriate.
Last updated: 2025-01-15 · Formula verified against primary sources.