Tsunami Physics: From Seafloor to Shore

In the open ocean, a tsunami is barely noticeable — 1 metre high, travelling at 800 km/h, with wavelengths of 200 km. Hours later, as it enters shallow water, it slows to 50 km/h and piles up to 30 metres. The physics of this transformation is governed by a single elegant formula.

1. Generation Mechanisms

Tsunamis are generated by any sudden large-scale displacement of a water body:

Submarine earthquakes (80% of tsunamis): Thrust faults raise or lower the seafloor over areas of thousands of km². Only earthquakes with M ≥ 7.5 on thrust faults generate significant tsunamis. The vertical seafloor displacement is directly transferred to the water column — the entire column moves upward or downward as a unit. 2004 Indian Ocean (M9.1): vertical fault displacement ~5 m over the entire ~1,200 km rupture zone.
Submarine landslides: Can generate more focused, locally catastrophic waves. The 1958 Lituya Bay megatsunami (Alaska) — triggered by a rockfall — created a wave that ran up 524 metres up the opposite hillside. The 1929 Grand Banks landslide generated a damaging trans-Atlantic tsunami.
Volcanic collapses: Caldera collapses or flank failures. The 2022 Hunga Tonga-Hunga Ha'apai eruption generated a tsunami through a combination of the atmospheric pressure wave and direct caldera collapse.
Meteorite impacts: No confirmed historical examples, but modelling shows an asteroid 200 m in diameter impacting the ocean would generate devastating coastal waves globally.

2. Shallow-Water Wave Theory

A wave is "shallow-water" when its wavelength λ ≫ water depth h (specifically, h < λ/20). For tsunamis:

Shallow-water wave speed: c = \sqrt(g \cdot h) g = 9.81 m/s² h = water depth (m) Average Pacific depth = 4,000 m: c = \sqrt(9.81 \times 4,000) = \sqrt39,240 \approx 198 m/s \approx 713 km/h Deep ocean trench (Mariana, h = 11,000 m): c = \sqrt(9.81 \times 11,000) \approx 1,039 km/h Why tsunamis are "shallow-water" waves: Ocean depth \approx 4 km Tsunami wavelength \approx 200 km h/λ \approx 4/200 = 0.02 << 1/20 ✓ (firmly in shallow-water regime) Compare to wind waves: Ocean swell depth \approx 4 km, wavelength \approx 100 m h/λ \approx 40 >> 1 (deep-water regime) c = \sqrt(gλ/2π) — very different physics

3. Ocean Propagation

Because c = √(gh), tsunamis slow down in shallower water and speed up over deeper water. This creates refraction — the wavefront bends to follow depth contours, much like light bending in optics.

Key features of open-ocean propagation:

Low amplitude in open ocean: The 2004 Indian Ocean tsunami had a wave height of only 60 cm in the deep ocean mid-way to Africa. Satellites (TOPEX/Poseidon, Jason-1) measured the wave as it passed.
Long wavelength: 100-500 km wavelength means there's no single "crash" — the sea level rises over periods of minutes, not seconds.
Energy conservation: In the 2004 tsunami, wave energy was detected on the Pacific coast of North America, 20,000 km from the source.
Wave splitting from ridges: Seamount chains and mid-ocean ridges reflect and refract tsunami waves, creating complex interference patterns at coastlines.

4. Shoaling & Amplification

As the tsunami approaches shore and depth decreases, it slows dramatically. Energy is conserved, so decreasing wave speed must be compensated by increasing amplitude — wave shoaling:

Green's law for shoaling amplification: H₂/H₁ = (h₁/h₂)^(1/4) H = wave height, h = water depth From ocean (h₁=4,000m, H₁=1m) to nearshore (h₂=10m): H₂ = 1 \times (4000/10)^(1/4) = 1 \times (400)^0.25 = 1 \times 4.47 \approx 4.5 m Beyond this, nonlinear effects and friction further modify run-up. Real amplification can be much greater due to coastal geometry: V-shaped bays concentrate wave energy 2011 Tōhoku: deep bays amplified waves to 40+ m in some locations

The "drawback" effect: Before a tsunami arrives, the sea often recedes dramatically — a leading depression wave drawing water offshore. In 2004, this visible drawback (the sea retreating hundreds of metres) occurred minutes before the wave struck, giving crucial but brief warning to those who recognised the sign. Many people walked toward the exposed seafloor in curiosity rather than running to high ground.

5. Run-Up & Inundation

Run-up height R (measured above sea level at the furthest inland reach) is what determines destruction. It is not simply the wave height — it depends strongly on coastal bathymetry, topography, and wavelength.

Maximum run-up (nonlinear shallow-water, sloped beach): R / H \approx 2.831 \times (H/λ)^(-1/2) \times tan(slope)^(-1/2) Notable historical run-ups: 2004 Indian Ocean: max 30+ m (Sumatra), average 5-8 m 2011 Tōhoku: max 40.1 m (Miyako, Japan), average 10-15 m 1958 Lituya Bay: 524 m (megatsunami from rockfall) 1883 Krakatoa: 35 m (volcanic caldera collapse) Inundation modelling uses: • Non-linear shallow-water equations (NSWW/MOST model) • Depth-averaged, finite-difference solvers • Validated with tide gauge and field survey data • GEBCO bathymetry (1-arc-minute resolution globally)

6. Detection & Early Warning

The Pacific Tsunami Warning Center (PTWC) and NOAA's DART (Deep-ocean Assessment and Reporting of Tsunamis) buoy network provide the primary warning infrastructure:

Seismic detection: Earthquake magnitude and focal mechanism are determined within 3-5 minutes. A magnitude threshold (M7.5+, thrust fault, submarine) triggers a tsunami alert. Can provide 5-60 minutes of warning for distant coasts.
DART buoys: 39 buoys in the Pacific, Atlantic, and Indian Oceans. A bottom pressure recorder (BPR) at ~5,000 m depth measures sea-surface height with 0.01 mm precision. Detects a 1-cm tsunami in the open ocean. Data sent via acoustic modem to surface buoy → satellite → warning centres in ~15 seconds.
Coastal tide gauges: Confirm tsunami arrival and provide real-time amplitude at specific locations. More than 1,000 gauges globally.
GPS coastal geodesy: Seafloor GPS arrays (especially in Japan) measure seafloor deformation in real-time, improving source modelling and early height estimates.

7. Mitigation Engineering

Seawall barriers: Japan built 400 km of concrete seawalls up to 12.5 m high after the 2011 disaster. The 2011 tsunami overtopped many existing 6-8 m walls. Some new walls would withstand a similar event; others would still be overtopped by a 40 m run-up.
Vertical evacuation structures: Reinforced concrete towers 20+ m tall designed to shelter people who cannot reach high ground in time. Used in Japan, Indonesia, and the US Pacific coast.
Coastal parks and forests: Dense vegetation (mangroves, Casuarina trees) reduces tsunami velocity and debris transport. Modelling shows 30-metre-wide, 100-metre-deep forests reduce flow speed by 50% and force by 75%.
Land use planning: Setback zones prohibiting residential construction in inundation areas. Most effective long-term mitigation strategy.
Probabilistic Tsunami Hazard Analysis (PTHA): Analogous to seismic hazard analysis — computing probability of exceeding a given run-up height at any coastal location over a 50- or 100-year interval. Required for siting nuclear power plants, LNG terminals, and critical infrastructure.