Tsunami Wave Speed Calculator
Solve the shallow-water wave speed relation v = √(g · h) for any of its three unknowns, with step-by-step math, a live ocean cross-section schematic, and companion metrics including travel time across ocean basins, wavelength, period, and depth classification.
Speed depends only on depth
For a shallow-water wave the speed is v = √(g·h). The wave height, wavelength, and period do not appear — the medium's depth alone controls how fast the disturbance propagates. This is why a tsunami crosses the Pacific at the speed of a jet aircraft, roughly 800 km/h, while it is barely noticeable as a wave on the open ocean (amplitude typically only 0.1–1 m).
Slowing down and shoaling near shore
As a tsunami enters shallow water, h decreases, so v decreases as √h. But the wave energy flux must be conserved, so the wave height grows dramatically — a process called shoaling. A 0.5 m open-ocean wave can grow to 5–15 m at the shoreline. The same physics explains why waves "peak up" and break on the beach.
Travel time is predictable
Because the deep-ocean tsunami speed is well-known (roughly 200 m/s), the arrival time at distant coastlines can be predicted hours in advance using only the ocean bathymetry. This is how tsunami warning centres — PTWC in Hawaii, JMA in Japan, and others — can issue alerts within minutes of a large earthquake. The travel time from Japan to Hawaii is roughly 8 hours; from Chile to Japan, roughly 22 hours.
The shallow-water approximation
The formula v = √(g·h) is the shallow-water (or long-wave) limit, valid when the wavelength λ is much larger than the depth h — typically λ/h > 20. Tsunami wavelengths are 100–500 km and ocean depths are 1–5 km, so λ/h is 20–500, well within the shallow-water regime. For wind waves (λ ≈ 100 m) in deep water (h ≈ 4 km), λ/h << 20, and a different dispersion relation applies: v = √(gλ/2π).
How to Use This Calculator
The shallow-water wave speed relation links three quantities: the wave speed, the water depth, and gravity. Pick which one you want the tool to solve, then fill in the others.
1. Choose what to solve for
Pick v, h, or g — the tool rearranges the relation automatically for whichever you choose.
2. Use realistic presets
Wave-speed presets span 20 m/s (near-shore) to 300 m/s (trench). Depth presets span 50 m (shelf) to 8 km (deep basin). Gravity presets include Moon, Mars, Earth, and Jupiter for planetary science applications.
3. Read the result and steps
The result card shows the solved quantity with engineering notation. The step-by-step panel walks through the multiplication g·h, the square root, and the conversion to km/h.
4. Check the ocean cross-section
The diagram shows a deep-ocean column with the tsunami crest at the surface, the propagation direction arrow, the wavelength dimension, and the water depth h — all updating live.
5. Switch to Advanced for depth
Advanced mode adds a chart of v vs. gravity, a 1% sensitivity check, and companion metrics including the wave speed in km/h and knots, the travel time to cross 1000 km and the Pacific, the approximate wavelength for a 1-hour period, and a depth classification.
6. Cross-check with the reference table
The reference section lists typical tsunami speeds for different ocean depths and the corresponding travel times across well-known ocean basins.
What This Tool Offers
Built around the shallow-water wave speed relation used throughout tsunami science, oceanography, and coastal engineering.
Three-way solving
Solve for wave speed, water depth, or gravity — whichever the problem gives you.
Live ocean cross-section
A deep-ocean column with the tsunami crest, propagation direction, wavelength dimension, and water depth, all updating live.
Dual comparison charts
v vs. water depth for three planetary gravities, plus v vs. gravity at three different depths.
Companion tsunami metrics
Advanced mode reports the speed in km/h and knots, the time to cross 1000 km and the whole Pacific, the wavelength for a 1-hour period, and a depth classification.
Built-in sensitivity check
See how much v moves for a 1% change in depth or gravity. The √ dependence means the sensitivity is 0.5% per 1% change in either input.
Reference tables
Typical tsunami speeds for depths from the shelf to the trench, and travel times across well-known ocean basins like Tokyo→Hawaii and Chile→Japan.
Reference Tables
Typical tsunami speeds and travel times for common ocean depths and trans-oceanic routes.
Tsunami wave speed vs. depth (Earth g = 9.80665 m/s²)
Speed v = √(g·h) for various ocean depths. Wavelength assumed 200 km for the period column.
| Depth h (m) | Speed v (m/s) | Speed (km/h) | Speed (knots) | Period for λ=200 km |
|---|
Travel Times Across Ocean Basins
Deep-ocean tsunami travel times computed with an average depth of 4,000 m (v ≈ 198 m/s). Actual times vary with the bathymetry along the path.
| Route | Distance (km) | Approx. travel time (h) | Notes |
|---|---|---|---|
| Near-field (epicentre to coast, 100 km) | 100 | 0.14 | Warning time is minutes |
| Regional (500 km) | 500 | 0.70 | Cross-basin alert |
| Tokyo → Hawaii | 6,200 | 8.7 | Classic 1960 Chile event track |
| Chile → Japan | 17,000 | 23.8 | Long trans-Pacific |
| Sumatra → India | 2,000 | 2.8 | 2004 Indian Ocean event |
| Alaska → California | 3,500 | 4.9 | 1964 Good Friday earthquake |
| Pacific-wide (average path) | 10,000 | 14.0 | Sum of many paths |
Variable Reference
| Symbol | Meaning | Typical range | Units |
|---|---|---|---|
| v | Wave speed | 10 – 300 | m/s |
| h | Water depth | 10 – 11,000 | m |
| g | Gravity | 1.6 – 25 | m/s² |
| λ | Wavelength | 100 – 500 | km |
| T | Period = λ / v | 10 – 60 | min |
What Does the Shallow-Water Wave Speed Measure?
The shallow-water wave speed is the propagation speed of a long-wavelength disturbance through a body of water whose depth is much smaller than the wavelength. This is precisely the regime in which tsunamis travel: an open-ocean tsunami has a wavelength of 100–500 km, while the ocean depth is only 1–5 km, so the ratio of depth to wavelength is about 1:50, well within the shallow-water limit. The remarkable result is that the speed depends only on gravity and depth — not on the wave height. That is why a tsunami can be imperceptible on the open ocean (amplitude often less than 0.5 m) and yet move across the entire Pacific basin at jet-aircraft speed.
Shallow-Water Wave Speed Formula
The defining relationship is:
Rearranged to solve for depth or gravity:
The corresponding wavelength and period are related by:
And the shallow-water validity condition is:
Shallow-Water Wave Variables
| Symbol | Meaning | Notes |
|---|---|---|
| v | Wave speed | Propagation speed of the disturbance |
| g | Gravitational acceleration | 9.80665 m/s² on Earth's surface |
| h | Water depth | The parameter that controls speed in the shallow-water regime |
| λ | Wavelength | Distance between successive crests; typically 100–500 km for tsunamis |
| T | Period | Time between successive crests; typically 10–60 min for tsunamis |
How Each Parameter Moves the Wave Speed
Wave speed is proportional to the square root of both gravity and depth. A 1% increase in either raises the speed by 0.5%. This square-root dependence has important implications: to double the speed, you need to quadruple the depth. That is why the deep ocean (h ≈ 4 km) and the continental shelf (h ≈ 200 m) produce such different wave speeds — 198 m/s versus 44 m/s, a factor of about 4.5. When the tsunami reaches the shelf, it slows dramatically and its energy is squeezed into a much shallower column, producing the coastal amplification that makes tsunamis so destructive.
Shallow-Water Wave Speed Calculation Examples
g · h = 9.80665 × 4,000 = 39,226.6 m²/s²
v = √39,226.6 = 198.06 m/s
In km/h: 198.06 × 3.6 = 713.0 km/h
In knots: 198.06 × 1.94384 = 385.0 kt
h = v² / g = 100² / 9.80665 = 10,000 / 9.80665 = 1,019.7 m
Roughly the depth of a continental shelf edge.
g = v² / h = 220² / 6,000 = 48,400 / 6,000 = 8.067 m/s²
Which would apply to a fluid ocean at a location with reduced effective gravity.
Frequently Asked Questions
Because in the shallow-water (long-wave) limit, the wave is a pure translation of the water column — every water particle moves horizontally, and the wave travels as a long, low-amplitude disturbance. In this regime, the linearised shallow-water equations reduce to ∂²η/∂t² = g·h·∂²η/∂x², whose wave speed is √(g·h), independent of amplitude. The formula remains accurate for tsunamis even at coastal amplitudes of 10 m because the ocean is still much deeper than the wave amplitude, and the ratio of amplitude to depth stays small until the wave is right at the shore.
In the deep ocean (4 km depth), a tsunami travels at roughly 200 m/s = 720 km/h = 390 knots. This is comparable to a cruising commercial jet. In the deepest parts of the Pacific, the speed can reach 220–250 m/s (800–900 km/h). As the wave enters shallow water, it slows dramatically — at 100 m depth it moves at only 31 m/s (113 km/h), and at 10 m depth at 10 m/s (36 km/h). This slowing is what causes the wave to pile up and become a destructive wall of water at the coastline.
Wind waves have wavelengths of 50–200 m and periods of 5–15 s. In deep water (h ≈ 4 km), the ratio λ/h is much less than 20, so the shallow-water approximation does not apply. Instead, wind waves in deep water obey the deep-water dispersion relation v = √(gλ/2π), which depends on wavelength rather than depth. As wind waves approach shore, λ/h eventually falls below 20 and they too transition to shallow-water behaviour — this is exactly why all waves, regardless of origin, slow down and steepen as they approach a beach.
Yes, if the body has a liquid ocean. On Saturn's moon Titan, the gravity is 1.35 m/s² and the lakes and seas of liquid methane are hundreds of metres deep — the same physics applies, though at slower speeds (v = √(1.35 × 300) ≈ 20 m/s for a 300 m deep sea). On Jupiter's moon Europa, the ice shell hides a global ocean that may be 100 km deep; the gravity is 1.31 m/s², giving a hypothetical wave speed of √(1.31 × 100,000) ≈ 362 m/s. The formula is genuinely universal for any shallow-water long-wave disturbance.
In a narrow channel or a lake, the same formula applies with the local depth. Seiches — standing oscillations in enclosed basins like Lake Geneva or the Adriatic Sea — have periods that depend on the basin length and the shallow-water wave speed. A seiche with a 30-minute period in a 30-km-long lake corresponds to a wave speed of about 33 m/s, consistent with a mean depth of about 110 m. The same physics also governs tides in estuaries, river bores, and hydraulic jumps.
For a flat-bottomed, uniform ocean, it is essentially exact. For a real ocean with varying bathymetry, the speed at each point is determined by the local depth, and the total travel time is computed by integrating 1/v along the ray path. Tsunami warning centres use this integration over gridded bathymetry to forecast arrival times with typical accuracies of a few minutes across the Pacific — impressive given path lengths of 10,000 km. Non-linear effects, dispersion, and wave refraction by seafloor features introduce smaller corrections that are handled by more complete numerical models.
Variables
| Symbol | Quantity | Unit |
|---|---|---|
| v | Wave speed | m/s |
| g | Gravitational acceleration | m/s2 |
| h | Ocean depth | m |
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| Parameter | Value |
|---|---|
| Ocean Depth (m) | 4000 |
| Parameter | Value |
|---|---|
| Depth | 1000 |
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
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