Tsunami Physics: From Seafloor Rupture to Coastal Run-Up

Q: Why is a tsunami classed as a shallow-water wave even in a 4 km-deep ocean?

A wave is shallow-water when its wavelength greatly exceeds the water depth. A tsunami's wavelength is typically 100-200 km, while even the deepest ocean trenches are only about 11 km — a depth-to-wavelength ratio far below the 1/20 threshold. That is why tsunamis obey c = sqrt(g*h), the same shallow-water speed formula that governs a wave in a puddle, rather than the deep-water formula used for ordinary wind swell.

Q: Why does a tsunami grow taller as it approaches the coast?

Wave speed c = sqrt(g*h) drops sharply as depth h decreases, from roughly 200 m/s over a 4,000 m abyssal plain to about 10 m/s on a shallow shelf. Because energy flux must be conserved as the wave slows and its wavelength compresses, the amplitude rises to compensate - a shoaling effect described by Green's Law, H2/H1 = (h1/h2)^(1/4), which can turn a barely-noticeable open-ocean swell into a multi-metre coastal wave.

Q: What actually triggers a tsunami?

About 80% of tsunamis come from submarine earthquakes on thrust faults, where sudden vertical seafloor displacement lifts or drops the entire water column above the rupture - the 2004 Indian Ocean earthquake displaced the seafloor by roughly 5 metres over a 1,200 km rupture zone. Submarine landslides, volcanic caldera collapses, and (in principle) large meteorite impacts can also generate tsunamis by abruptly displacing a large volume of water.

How a tsunami is born

About 80% of tsunamis come from submarine earthquakes on thrust faults, where the seafloor suddenly rises or drops over an area of thousands of square kilometres — the 2004 Indian Ocean earthquake (M9.1) displaced the seafloor roughly 5 metres over a 1,200 km rupture zone, and that displacement transfers straight into the water column sitting above it. Submarine landslides can be even more locally violent: the 1958 Lituya Bay rockfall in Alaska threw a wave that ran up 524 metres on the opposite hillside. Volcanic caldera collapses (the 2022 Hunga Tonga eruption) and, in principle, large meteorite impacts round out the list of triggers.

Why a 4 km-deep ocean still counts as "shallow water"

A wave is classified as a shallow-water wave whenever its wavelength λ is much greater than the water depth h — specifically h < λ/20. A tsunami's wavelength is typically 100-200 km, dwarfing even the deepest ocean trenches, so despite crossing genuinely deep water it behaves according to the shallow-water speed formula rather than the deep-water formula that governs ordinary wind swell.

c = √(g·h)           g = 9.81 m/s²

Pacific average depth 4,000 m:  c = √(9.81 × 4000) ≈ 198 m/s ≈ 713 km/h
Mariana Trench, 11,000 m:       c = √(9.81 × 11000) ≈ 1,039 km/h

Tsunami:  h/λ ≈ 4 km / 200 km = 0.02  ≪ 1/20  → shallow-water regime
Wind swell: h/λ ≈ 40 ≫ 1               → deep-water regime, different physics

live demo · a leapfrog-integrated shallow-water wave● LIVE

Shoaling: how a 1 m swell becomes a 4.5 m wall

As the seafloor rises toward shore, c = √(gh) falls fast — and because energy flux is conserved, the wave amplitude must rise to compensate. This is Green's Law, the shoaling relation that turns an imperceptible open-ocean swell into a destructive coastal wave. The 2004 Indian Ocean tsunami measured only about 60 cm in the deep ocean, tracked by satellite altimeters, yet reached run-ups above 30 m on Sumatra; the 2011 Tōhoku tsunami hit 40.1 m at Miyako, amplified further where V-shaped bays funnel wave energy inward.

H₂/H₁ = (h₁/h₂)^(1/4)          Green's Law

Ocean (h₁=4000 m, H₁=1 m) → shelf (h₂=10 m):
H₂ = 1 × (4000/10)^0.25 = 1 × 4.47 ≈ 4.5 m

Before the wave arrives, the sea often recedes dramatically — a leading depression wave drawing water offshore, sometimes hundreds of metres, minutes before the crest strikes. Recognising that drawback is one of the few pieces of natural, self-generated warning coastal residents get.

Detection: DART buoys and the 15-second relay

NOAA's DART (Deep-ocean Assessment and Reporting of Tsunamis) network — 39 buoys across the Pacific, Atlantic and Indian Oceans — anchors a bottom pressure recorder near 5,000 m depth that can detect a 1-centimetre tsunami and relay it via acoustic modem to a surface buoy and then satellite to warning centres in about 15 seconds. Combined with seismic detection, which pins down an earthquake's magnitude and thrust-fault mechanism within 3-5 minutes, this can give distant coastlines anywhere from 5 to 60 minutes of warning.

Frequently asked questions

Why is a tsunami classed as a shallow-water wave even in a 4 km-deep ocean?

A wave is shallow-water when its wavelength greatly exceeds the water depth. A tsunami's wavelength is typically 100-200 km, while even the deepest ocean trenches are only about 11 km — a depth-to-wavelength ratio far below the 1/20 threshold. That is why tsunamis obey c = √(g·h), the same shallow-water speed formula that governs a wave in a puddle, rather than the deep-water formula used for ordinary wind swell.

Why does a tsunami grow taller as it approaches the coast?

Wave speed c = √(g·h) drops sharply as depth h decreases, from roughly 200 m/s over a 4,000 m abyssal plain to about 10 m/s on a shallow shelf. Because energy flux must be conserved as the wave slows and its wavelength compresses, the amplitude rises to compensate — a shoaling effect described by Green's Law, H₂/H₁ = (h₁/h₂)^(1/4), which can turn a barely-noticeable open-ocean swell into a multi-metre coastal wave.

What actually triggers a tsunami?

About 80% of tsunamis come from submarine earthquakes on thrust faults, where sudden vertical seafloor displacement lifts or drops the entire water column above the rupture — the 2004 Indian Ocean earthquake displaced the seafloor by roughly 5 metres over a 1,200 km rupture zone. Submarine landslides, volcanic caldera collapses, and (in principle) large meteorite impacts can also generate tsunamis by abruptly displacing a large volume of water.

Try it live

Everything above runs in your browser — open Tsunami, pick a bathymetry preset, and click the ocean to trigger a seafloor uplift. Watch the wave race across deep water at hundreds of metres per second and pile up as it reaches the coast. Nothing is installed, nothing is uploaded.

▶ Open Tsunami simulation