Significant Wave Height Calculator
Find the significant wave height from a wave spectrum or a measured wave record, and see how likely a much larger wave is.
🌊 What is Significant Wave Height?
Significant wave height, written Hs or Hm0, is the standard single number used to describe how rough the sea is. A real sea surface is made of many waves of different heights and periods, so no one height describes it. Hs was originally defined by Walter Munk and Harald Sverdrup in the 1940s as the mean height of the highest one third of waves, written H1/3. That definition was chosen because it agrees closely with the wave height a trained observer estimates by eye, who tends to notice the larger waves and ignore the small ones. Today Hs is usually computed from instruments, as four times the standard deviation of the sea-surface elevation, or equivalently four times the square root of the zeroth moment m0 of the wave spectrum.
Hs is the input to almost every practical wave calculation. Marine forecasters report it, ship routing and offshore operations set weather limits in terms of it, and engineers use it to design breakwaters, platforms and cables. Once you know Hs, the Rayleigh distribution gives the whole statistics of individual waves: the root-mean-square height, the mean height, the average of the highest tenth, and the probability that any single wave exceeds a given height. With the number of waves in a storm, the Longuet-Higgins result also gives the expected height of the largest wave, which is what really threatens a structure or a small vessel.
A common misunderstanding is that Hs is the typical or maximum wave. It is neither. The average wave is only about 63 percent of Hs, about one wave in seven is higher than Hs, and over a few thousand waves the largest is typically 1.5 to 2 times Hs. Another is that the two definitions are interchangeable in every sea. In a perfectly Rayleigh sea H1/3 is within 0.1 percent of 4 sigma, but real seas with swell, steep waves or shallow-water effects can differ by 5 to 10 percent.
This calculator covers both routes. Mode A starts from the elevation standard deviation or the spectral moment, as you would get from a wave buoy, a spectral model or a pressure gauge. Mode B starts from a handful of individual wave heights, as you might read from a staff gauge, a photo sequence or a short time series, and tests how close your record is to the Rayleigh prediction. To translate the period into wavelength and speed, use the ocean wave speed calculator.
📐 Formula
The expected maximum uses the leading term of the Longuet-Higgins formula. The full expression adds a small positive correction of gamma / (2 ln N) in relative terms (gamma = 0.5772, Euler's constant), roughly 4 percent for a few thousand waves. The calculator reports the leading term, which is the form most often quoted in engineering notes.
📖 How to Use This Calculator
Steps
💡 Example Calculations
Example 1 — A 2 m Sea (Mode A)
σ = 0.5 m, 3 hours, Tz = 7 s
Example 2 — A 5 m Storm from m0 (Mode A)
m0 = 1.5625 m², 6 hours, Tz = 10 s
Example 3 — A Record of 10 Waves (Mode B)
Heights 1.2, 2.4, 1.8, 3.1, 0.9, 2.2, 1.5, 2.8, 1.1, 1.9 m
Example 4 — A Short Record of 7 Waves (Mode B)
Heights 0.8, 1.4, 2.0, 1.1, 1.7, 0.6, 1.3 m, rows 8 to 10 blank
❓ Frequently Asked Questions
🔗 Related Calculators
How do you calculate significant wave height?
From a wave spectrum, Hs = 4 sqrt(m0), where m0 is the area under the variance density spectrum. Equivalently Hs = 4 sigma, with sigma the standard deviation of the sea-surface elevation. From a record, Hs is approximated by H1/3, the mean of the highest third of waves.
What does significant wave height mean?
It is a single number that summarizes sea state. Historically it was defined as the mean height of the highest one third of waves, and it matches what an experienced observer reports as the typical height of the larger waves.
What is the relation between Hs and Hrms?
For a Rayleigh distributed sea, Hrms = Hs / sqrt(2), which is about 0.707 Hs. Equivalently H1/3 = 1.416 Hrms, which is a useful check on measured wave records.
How do you find the maximum wave height from Hs?
Use Longuet-Higgins: the expected maximum in N waves is about (Hs/2) sqrt(ln N). For a 3 hour record of 7 s waves, N is about 1543 and Hmax is about 1.35 Hs.
What is H1/10?
H1/10 is the mean of the highest 10 percent of waves. For a Rayleigh sea it is about 1.27 Hs, so a sea with Hs = 2 m has H1/10 near 2.55 m.
Is significant wave height the same as H1/3?
Nearly. H1/3 is the original observational definition and Hs = 4 sigma is the spectral definition. In a Rayleigh sea H1/3 is 1.001 times Hs, so they agree to within about 0.1 percent. In real seas, they can differ by 5 to 10 percent.
How many waves are in a storm?
Divide the duration in seconds by the mean zero-crossing period. A 6 hour storm with 10 s waves contains 21,600 / 10 = 2160 waves.
Why is the largest wave bigger than Hs?
Hs is an average over the highest third. Individual waves follow a Rayleigh distribution with a long tail, so about 1 wave in 7 exceeds Hs (P = e^-2 = 13.5 percent) and over thousands of waves the largest is typically 1.5 to 2 times Hs.
How is the mean wave height related to Hs?
For the Rayleigh distribution, Hmean = (sqrt(pi)/2) Hrms = 0.886 Hrms, which equals about 0.627 Hs. So the average wave is only about 63 percent of the significant wave height.
How many waves do I need to estimate Hs from a record?
At least 100 for a reasonable estimate, and standard wave buoy records use about 1200 waves (20 minutes). With only 10 waves, the observed H1/3 can differ from the true Hs by 10 to 20 percent.
What units does the calculator use?
Heights and standard deviation are in meters, m0 is in square meters, duration is in hours and the mean zero-crossing period is in seconds.