Tsunami Runup Height Estimator

Estimate how high a tsunami runs up a beach from its deep-water wave height, the shelf depth and the beach slope.

🌊 Tsunami Runup Height Estimator
Offshore wave height (H1)
m
0.110
Offshore water depth (d1)
m
5006000
Nearshore water depth (d2)
m
5500
Beach slope angle (β)
°
0.520
Estimated runup height
—
Nearshore wave height
—
Shoaling factor
—
Runup / nearshore height
—

🌊 What is Tsunami Runup Height?

Tsunami runup height is the maximum vertical elevation above the still sea level that tsunami water reaches on land. It is the number that field survey teams record after an event, usually by finding the highest water marks, debris lines or damaged vegetation. A tsunami that is only half a metre high in the deep ocean can produce runup of several metres on the coast, because the wave slows down, grows taller and then climbs the beach. This estimator links those two numbers with two classic results from coastal physics.

The first step is wave shoaling. A tsunami is a shallow-water wave, so its speed depends on water depth. As the wave moves from the deep ocean onto the continental shelf it slows, its energy is squeezed into a shorter wavelength, and its height grows. For waves moving over a gradually changing seabed, Green's law gives the new height from the ratio of the depths. The second step is the climb up the beach, which this tool models with the Synolakis (1987) relation for a non-breaking solitary wave on a plane sloping beach.

People use this kind of estimate for teaching, for quick sanity checks of survey numbers, for understanding why some coasts are hit harder than others, and for exploring how sensitive runup is to wave height, shelf depth and beach slope. Students in oceanography and coastal engineering often meet Green's law and the Synolakis formula in the same lecture, and this page lets you vary every input and watch the result and chart respond immediately.

It is important to be clear about what the result is not. It is an idealised estimate on a straight beach with a single wave, not a hazard forecast. Real tsunamis arrive as trains of long waves, are shaped by bays, reefs, harbours and river mouths, and flood inland according to ground elevation and roughness. Tide level, wave period and local bathymetry all change the answer. Always rely on official tsunami warning centres and local hazard maps for safety decisions, and treat this calculator as a learning and estimation tool only.

📐 Formula

H2 = H1 × (d1 ÷ d2)1/4
R ÷ d2 = 2.831 × √(cot β) × (H2 ÷ d2)5/4
H1 = tsunami wave height in deep water (m), d1 = offshore water depth (m)
H2 = shoaled wave height at the nearshore depth (m), d2 = nearshore water depth (m), d2 < d1
β = beach slope angle (degrees), cot β = 1 ÷ tan β
R = maximum runup height above still water (m)
Non-breaking limit (Synolakis 1987): H2 ÷ d2 < 0.8183 × (tan β)10/9
Wave speed: c = √(g × d), with g = 9.81 m/s²
Example: H1 = 0.5 m, d1 = 4000 m, d2 = 100 m, β = 4° → H2 = 1.26 m, R = 4.51 m.

Validity limits. Green's law assumes a long wave over a slowly varying seabed with no energy loss, and works best while H2 ÷ d2 stays small (well below about 0.3). The Synolakis formula was derived for a solitary wave on a plane beach and verified against laboratory tank experiments for waves that do not break before the shoreline. When the breaking criterion above is exceeded, the formula tends to overestimate runup. Both results are estimates, not forecasts.

📖 How to Use This Calculator

Steps

1
Enter the offshore wave. Type the tsunami wave height in deep water and the offshore water depth where that height applies, for example from a DART buoy reading.
2
Choose the nearshore depth and beach slope. Type a shelf depth shallower than the offshore depth, usually 50 to 200 m, and the beach slope angle in degrees. A 1 in 15 slope is about 3.8 degrees.
3
Read the runup and regime note. The calculator shows the shoaled wave height, the estimated runup height, and whether the wave is in the non-breaking regime, and plots runup against beach slope with your slope marked.

💡 Example Calculations

Example 1 — Moderate Open-Ocean Tsunami on a Gentle Beach

H1 = 0.5 m at d1 = 4000 m, shoaled to d2 = 100 m, beach slope 4°

1
Shoaling: (4000 ÷ 100)1/4 = 400.25 = 2.5149, so H2 = 0.5 × 2.5149 = 1.26 m
2
H2 ÷ d2 = 1.2574 ÷ 100 = 0.01257, and cot 4° = 14.301, so √(cot β) = 3.7816
3
R ÷ d2 = 2.831 × 3.7816 × 0.012571.25 = 2.831 × 3.7816 × 0.004211 = 0.04508, so R = 0.04508 × 100 m
Runup R = 4.51 m, which is 3.59 times the nearshore wave height (non-breaking, limit 0.0426)
Try this example →

Example 2 — Doubling the Offshore Wave Height

H1 = 1 m at d1 = 4000 m, shoaled to d2 = 100 m, beach slope 4°

1
Shoaling is unchanged at 2.5149, so H2 = 1 × 2.5149 = 2.51 m
2
H2 ÷ d2 = 0.02515, so (H2 ÷ d2)1.25 = 0.010015
3
R ÷ d2 = 2.831 × 3.7816 × 0.010015 = 0.10722, so R = 0.10722 × 100 m
Runup R = 10.72 m, about 2.38 times Example 1 because runup scales with H1.25 (non-breaking, limit 0.0426)
Try this example →

Example 3 — Steeper Coast and Shallower Ocean

H1 = 0.5 m at d1 = 3000 m, shoaled to d2 = 100 m, beach slope 8°

1
Shoaling: (3000 ÷ 100)1/4 = 300.25 = 2.3403, so H2 = 0.5 × 2.3403 = 1.17 m
2
H2 ÷ d2 = 0.01170, and cot 8° = 7.115, so √(cot β) = 2.6675
3
R ÷ d2 = 2.831 × 2.6675 × 0.003849 = 0.02906, so R = 0.02906 × 100 m
Runup R = 2.91 m, only 2.48 times the nearshore height because the steeper beach limits the climb (non-breaking, limit 0.0925)
Try this example →

Example 4 — Large Wave on a Very Gentle Slope (Breaking Regime)

H1 = 2 m at d1 = 5000 m, shoaled to d2 = 200 m, beach slope 2°

1
Shoaling: (5000 ÷ 200)1/4 = 250.25 = 2.2361, so H2 = 2 × 2.2361 = 4.47 m
2
H2 ÷ d2 = 0.02236, and cot 2° = 28.636, so √(cot β) = 5.3513
3
R ÷ d2 = 2.831 × 5.3513 × 0.008647 = 0.13099, so R = 0.13099 × 200 m
4
Breaking check: the limit 0.8183 × (tan 2°)10/9 = 0.0197 is below H2 ÷ d2 = 0.0224, so the wave is flagged as breaking
Formula value R = 26.20 m, shown with a breaking warning, so treat it as an upper-end overestimate
Try this example →

❓ Frequently Asked Questions

What is tsunami runup height?+
Runup height is the maximum vertical elevation above the still sea level that tsunami water reaches on land at the point of furthest inland penetration. It is measured from the sea level at the time of the event, so it depends on the tide. It is different from the wave height at the coast and from the flow depth of water on the ground.
How do you calculate tsunami wave height near the shore?+
Use Green's law, H2 = H1 x (d1 / d2)^(1/4), where H1 is the wave height in deep water of depth d1 and H2 is the height in shallower water of depth d2. For a 0.5 m wave moving from 4000 m to 100 m depth, H2 is 0.5 x 40^0.25, which is about 1.26 m.
What is Green's law for waves?+
Green's law describes how the height of a long wave changes as it moves into water of slowly varying depth, assuming no energy is lost to reflection or friction. Wave height grows in proportion to depth raised to the power minus one quarter, so a wave that moves into water 16 times shallower doubles in height.
What is the Synolakis runup formula?+
Synolakis (1987) gave R/d = 2.831 x sqrt(cot beta) x (H/d)^(5/4) for a non-breaking solitary wave of height H climbing a plane beach of slope angle beta, where d is the constant water depth in front of the beach. R is the maximum runup height above still water.
Why does runup exceed the offshore wave height?+
As a tsunami enters shallow water it slows down, its wavelength shortens and its height grows by shoaling. Then the wave climbs the beach and converts its remaining kinetic energy into elevation. A 1 m open-ocean tsunami can therefore produce runup of several metres to ten metres or more on a gently sloping coast.
Does a steeper beach produce higher or lower runup?+
In the Synolakis non-breaking formula, steeper beaches give lower runup because runup scales with the square root of cot beta. A gentle slope lets the wave climb further. However, very gentle slopes make the wave break earlier, and breaking waves behave differently, so the formula should only be used inside its non-breaking limit.
When does a tsunami wave break on the beach?+
Synolakis gave a breaking criterion of H/d = 0.8183 x (tan beta)^(10/9). If the nearshore wave height divided by depth is above this value for the beach slope, the wave is expected to break before reaching the shoreline and the non-breaking runup formula tends to overestimate runup.
How fast does a tsunami travel?+
A tsunami is a shallow-water wave, so its speed is c = sqrt(g x d). In 4000 m of water that is about 198 m/s, or roughly 713 km/h, similar to a jet airliner. In 100 m of water the speed drops to about 31 m/s, or 113 km/h, which is why the wave steepens and grows near the coast.
Can this calculator predict how far inland a tsunami will flood?+
No. It estimates the vertical runup on an idealised straight beach only. Horizontal inundation distance depends on ground elevation, roughness, buildings, river channels and the full wave train, which are all outside this model. Official tsunami hazard maps and warnings should always be used for planning and safety.
What offshore depth and nearshore depth should I enter?+
For the offshore depth use the open-ocean depth where the wave height is known, such as a DART buoy depth of about 4000 to 6000 m. For the nearshore depth choose a point at the edge of the continental shelf or slope, usually 50 to 200 m, where the wave is still small compared with the depth.
How accurate is this tsunami runup estimate?+
It is a first-order educational estimate. Field runup values vary a lot from place to place because of bay shape, reefs, tide and wave period, and real tsunamis are long wave trains rather than single solitary waves. Expect real values to differ from this estimate by a factor of two or more.
Tags: Tsunami Runup Height Calculator Tsunami Runup Estimator Green's Law Calculator Wave Shoaling Calculator Synolakis Runup Formula Tsunami Wave Height Calculator Geophysics Calculator Calculator Formula Free Online

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What is tsunami runup height?

Runup height is the maximum vertical elevation above the still sea level that tsunami water reaches on land at the point of furthest inland penetration. It is measured from the sea level at the time of the event, so it depends on the tide. It is different from the wave height at the coast and from the flow depth of water on the ground.

How do you calculate tsunami wave height near the shore?

Use Green's law, H2 = H1 x (d1 / d2)^(1/4), where H1 is the wave height in deep water of depth d1 and H2 is the height in shallower water of depth d2. For a 0.5 m wave moving from 4000 m to 100 m depth, H2 is 0.5 x 40^0.25, which is about 1.26 m.

What is Green's law for waves?

Green's law describes how the height of a long wave changes as it moves into water of slowly varying depth, assuming no energy is lost to reflection or friction. Wave height grows in proportion to depth raised to the power minus one quarter, so a wave that moves into water 16 times shallower doubles in height.

What is the Synolakis runup formula?

Synolakis (1987) gave R/d = 2.831 x sqrt(cot beta) x (H/d)^(5/4) for a non-breaking solitary wave of height H climbing a plane beach of slope angle beta, where d is the constant water depth in front of the beach. R is the maximum runup height above still water.

Why does runup exceed the offshore wave height?

As a tsunami enters shallow water it slows down, its wavelength shortens and its height grows by shoaling. Then the wave climbs the beach and converts its remaining kinetic energy into elevation. A 1 m open-ocean tsunami can therefore produce runup of several metres to ten metres or more on a gently sloping coast.

Does a steeper beach produce higher or lower runup?

In the Synolakis non-breaking formula, steeper beaches give lower runup because runup scales with the square root of cot beta. A gentle slope lets the wave climb further. However, very gentle slopes make the wave break earlier, and breaking waves behave differently, so the formula should only be used inside its non-breaking limit.

When does a tsunami wave break on the beach?

Synolakis gave a breaking criterion of H/d = 0.8183 x (tan beta)^(10/9). If the nearshore wave height divided by depth is above this value for the beach slope, the wave is expected to break before reaching the shoreline and the non-breaking runup formula tends to overestimate runup.

How fast does a tsunami travel?

A tsunami is a shallow-water wave, so its speed is c = sqrt(g x d). In 4000 m of water that is about 198 m/s, or roughly 713 km/h, similar to a jet airliner. In 100 m of water the speed drops to about 31 m/s, or 113 km/h, which is why the wave steepens and grows near the coast.

Can this calculator predict how far inland a tsunami will flood?

No. It estimates the vertical runup on an idealised straight beach only. Horizontal inundation distance depends on ground elevation, roughness, buildings, river channels and the full wave train, which are all outside this model. Official tsunami hazard maps and warnings should always be used for planning and safety.

What offshore depth and nearshore depth should I enter?

For the offshore depth use the open-ocean depth where the wave height is known, such as a DART buoy depth of about 4000 to 6000 m. For the nearshore depth choose a point at the edge of the continental shelf or slope, usually 50 to 200 m, where the wave is still small compared with the depth.

How accurate is this tsunami runup estimate?

It is a first-order educational estimate. Field runup values vary a lot from place to place because of bay shape, reefs, tide and wave period, and real tsunamis are long wave trains rather than single solitary waves. Expect real values to differ from this estimate by a factor of two or more.

📌 Quick Tips

💡Tsunami waves in the open ocean are usually less than a metre high but hundreds of kilometres long, so they pass unnoticed by ships. The danger appears only as the wave shoals and slows in shallow water.
💡Green's law should be applied from deep water to a depth where the wave is still small compared with the water depth, typically 50 to 200 m on the continental shelf. Pushing it to 5 or 10 m depth makes H/d large and the result unreliable.
💡Runup is much more sensitive to wave height than the shoaling alone suggests. In the Synolakis relation runup scales with wave height to the power 1.25, so doubling the offshore height raises runup by about 2.4 times.
💡Flatter beaches give larger runup in this formula, but they also push the wave toward breaking. Check the regime note under the results: if it says breaking, treat the runup as an upper-end figure.
💡This tool is an educational estimate on an idealised plane beach. Real inundation depends on local bathymetry, tide, bays, rivers and structures, so never use it as a hazard forecast or evacuation guide.