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Tsunami Wave Speed (Shallow Water Wave)

Calculates the propagation speed of a tsunami wave in open ocean, which behaves as a shallow-water wave relative to its very long wavelength.

Written by the Vorzep Team

GeologySeismologyTsunamis

Tsunami Wave Speed CalculatorShallow Water Wave

v = √(g · h)
v = wave speed (m/s)  ·  g = gravity (9.8 m/s²)  ·  h = water depth (m)
⟹ Solve⟷v, g, h
m/s²
m
m/s
Please fix the errors above.
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Presets:
Wave Speed—
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g: —h: —v: —
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v = √(g · h)  ·  Valid for shallow water waves (depth << wavelength)
v = sqrt(g*h)
Tsunami Wave Speed (Shallow Water Wave)

Variables

SymbolQuantityUnit
vWave speedm/s
gGravitational accelerationm/s2
hOcean depthm

What it means

In shallow water, tsunami waves travel at a speed determined by the water depth: v = √(g h), where g is gravitational acceleration and h is the water depth. In the deep ocean (h ≈ 4 km), speed is about 200 m/s (720 km/h); in shallow coastal waters, it slows down. This formula is used for tsunami propagation models to forecast arrival times and to issue warnings. It also applies to other long‑period shallow water waves. Understanding this relation is essential for tsunami hazard mitigation and for coastal engineering to design defences.

Worked example

Tsunami Wave Speed – Two Detailed Examples

Real‑World
Scenario: An oceanographer is modelling tsunami propagation across the Pacific Ocean. The average ocean depth is 4000 m. Using the shallow‑water wave speed formula v = √(g × h), they compute v = √(9.81 × 4000) = √39240 ≈ 198.1 m/s ≈ 713 km/h. This high speed explains why tsunamis can cross the ocean in hours, making early warning systems crucial for coastal communities.
ParameterValue
Ocean Depth (m)4000
1v = √(9.81 × 4000) = √39240 = 198.1 m/s
2Convert to km/h: 198.1 × 3.6 = 713.1 km/h
Result 713 km/h ✓ Tsunami speed in deep ocean
Scenario: As a tsunami approaches the coast, the water depth decreases to 1000 m. The wave speed reduces to v = √(9.81 × 1000) = √9810 ≈ 99.0 m/s ≈ 356 km/h. This slowing causes the wave to increase in height (shoaling), leading to the destructive run‑up at the shore. The oceanographer uses this to predict arrival times and hazard levels for coastal evacuation planning.
ParameterValue
Depth1000
1v = √(9.81 × 1000) = 99.0 m/s
2≈ 356.6 km/h
Result 356.6 km/h ✓ Slower coastal speed
Insight: Tsunami speed depends only on water depth in shallow water. As the wave enters shallower water, it slows and grows in height, which is why tsunamis become destructive near the coast.

Common mistakes

  • Tsunami wave speed: v = √(g·h) – for shallow‑water (long‑period) waves.
  • g: Gravitational acceleration (9.81 m/s²).
  • h: Water depth in metres – v in m/s.
  • Assumes: Long wavelength compared to depth – valid for deep oceans (h > 50 m).
  • Non‑linear effects: Near coastlines, wave shoaling and breaking occur – this is the deep‑water speed.

Applications

Tsunami wave speed in shallow water, v = √(g·h), where g is gravity and h is water depth, determines how fast a tsunami travels across the ocean. This formula shows that tsunami speed depends solely on depth; in deep oceans, tsunamis can travel at jet speeds (over 700 km/h). Oceanographers and warning centre scientists use this to estimate arrival times at coastlines, allowing timely evacuations. By knowing the bathymetry along the path, travel times can be calculated. This formula is essential for tsunami early warning systems and for understanding tsunami propagation. It is also used in coastal engineering to model wave run‑up and to design protective structures.

  • Tsunami travel‑time estimation for early warning
  • Coastal hazard assessment and evacuation planning
  • Ocean modelling of tsunami propagation and inundation
  • Design of coastal defences and breakwaters
  • Education on tsunami dynamics and physics