Hong Life CalculatorsUnderstanding the ocean wave calculator — concept and principle
The period of an ocean wave, the time between two crests passing a fixed point, is easy to measure and is reported in every surf and marine forecast. From the period alone you can work out how long and how fast the wave is in deep water, and with the depth you can find its actual wavelength anywhere. Surfers, sailors, coastal engineers and students use these relations.
How it is calculated
- Deep-water wavelength: L₀ = gT² ÷ 2π ≈ 1.56 T² (m)
- Deep-water speed: c₀ = gT ÷ 2π ≈ 1.56 T (m/s)
- Wavelength at depth h: L = (gT² ÷ 2π) × tanh(2πh ÷ L), solved by repeating the calculation 50 times starting from L₀
- Deep water? yes if h > L/2
T is the period in seconds, h the depth in metres and g = 9.81 m/s².
The principle
Linear (Airy) wave theory gives the dispersion relation ω² = gk tanh(kh), where ω = 2π/T and k = 2π/L. In deep water kh is large, tanh(kh) is almost 1, and the relation simplifies to L₀ = gT²/2π: longer-period waves are longer and faster, which is why swell from distant storms arrives ahead of shorter waves. When the depth is less than about half the wavelength, the water motion reaches the bottom, tanh(kh) drops below 1 and the wave shortens and slows. Since L appears on both sides, the calculator iterates until the value settles.
Worked example
- Inputs: period T = 10 s, depth h = 100 m.
- L₀ = 9.81 × 10² ÷ 2π = 981 ÷ 6.2832 = 156.13 m (156.1309992).
- c₀ = 9.81 × 10 ÷ 2π = 15.61 m/s (15.61309992), about 56 km/h.
- At 100 m: 2πh/L ≈ 4.03, tanh ≈ 0.99936, so L = 156.13 × 0.99936 = 156.03 m (156.0317579).
- Check: L/2 = 78.0 m, and 100 m > 78.0 m, so yes (h > L/2), deep water.
| Wave period (s) | Deep-water wavelength (m) |
|---|---|
| 2.5 | 9.758187448 |
| 5 | 39.03274979 |
| 7.5 | 87.82368703 |
| 10 | 156.1309992 |
| 12.5 | 243.9546862 |
| 15 | 351.2947481 |
| 20 | 624.5239967 |
How to read the result
A 10 s swell is a typical long-period groundswell; wind chop is often 3–6 s, with deep-water lengths of only 14–56 m. When the "deep water?" answer is no, the wave feels the bottom, slows and steepens, and eventually breaks. Energy in deep water travels at the group speed, half of c₀, so a swell's energy arrives at about 7.8 m/s for a 10 s period.
Common mistakes
- Using the deep-water formula in shallow surf zones; there the wave is much shorter.
- Confusing wave period with the time between sets of waves.
- Reading c₀ as how fast energy arrives; the group speed is half as large in deep water.
Frequently asked questions
Why is "1.56 T²" so often quoted?
g/2π = 9.81/6.2832 ≈ 1.56, so in metres the deep-water wavelength is about 1.56 times the period squared.
When does a wave count as shallow-water?
When the depth is less than about 1/20 of the wavelength; then the speed becomes √(gh).
Why do long-period waves surf better?
They carry more energy per unit height and start feeling the bottom farther offshore, so they build into more powerful breakers.
Sources: R. G. Dean and R. A. Dalrymple, Water Wave Mechanics for Engineers and Scientists, World Scientific, 1991 · U.S. Army Corps of Engineers, Coastal Engineering Manual (EM 1110-2-1100), Part II