Tsunami Speed & Travel Time Calculator: Arrival Time, Wavelength and Height at the Coast

How it works

c = √(g × h), g = 9.80665 m/s²; travel time = distance ÷ c (or Σ dᵢ ÷ √(g × hᵢ) along a path); λ = c × T; full theory: ω² = g·k·tanh(k·h), c = ω ÷ k; Green’s law: H₂ = H₁ × (h₁ ÷ h₂)^¼ × (b₁ ÷ b₂)^½

A tsunami’s wavelength is hundreds of kilometres, far longer than the ocean is deep. It therefore behaves as a shallow-water wave everywhere, and its speed depends only on the depth: c = √(g × h). Over a 4 km deep ocean it moves at about the speed of a jet airliner, yet it may be well under a metre high and pass under ships unnoticed. Travel time is distance divided by speed. For a real route where the depth changes, enter the path as segments (distance and depth per line) and the calculator adds up the time for each, the same way GIS travel-time maps are built. The wave period sets the wavelength (λ = c × T). The calculator also solves the full linear dispersion relation, ω² = g·k·tanh(k·h), as a check. For earthquake tsunamis with 10–60 minute periods the two speeds agree within a fraction of a percent. For short-period landslide waves, √(g·h) runs too fast. Near the coast the front of the wave slows down, the back catches up, and the energy is squeezed into less water, so the wave grows. Green’s law estimates this growth as the fourth root of the depth ratio, times the square root of any narrowing of a bay or channel. It is an idealised estimate. It stops working once the wave breaks (about when the height reaches 0.78 × the depth), and it does not give run-up on land, which coastal shape, resonance and the wave’s own momentum can make several times larger.

Worked example

With the default inputs: c = √(9.80665 × 4,000) = 198.06 m/s = 713 km/h (443 mph). Crossing 3,000 km takes 3,000,000 ÷ 198.06 = 15,147 s = 4.21 hours (4 h 12 min). A 20-minute (1,200 s) period gives λ = 198.06 × 1,200 = 237.7 km, so the depth is 1/59 of the wavelength, a shallow-water wave. Full wave theory gives 198 m/s (99.8% of √(g·h)). In 10 m of water the wave slows to 9.9 m/s (36 km/h), and Green’s law grows a 0.5 m deep-water wave to 0.5 × (4,000 ÷ 10)^¼ = 2.24 m.

Sources: NOAA: tsunami fast facts (speed vs depth) · Esri Learn ArcGIS: calculate travel time for a tsunami (segment method) · Calculator Academy: wave celerity and the dispersion relation

Frequently asked questions