Introduction
A forecast or a wave buoy gives a sea state as two numbers: the significant wave height and a period. Behind them is a spectrum and a statistical distribution of individual waves, and many decisions depend on the waves the two numbers do not show: the largest wave in the next three hours, how often a given height will be exceeded, the zero-crossing period that sets the number of wave encounters. This calculator builds the JONSWAP or Pierson–Moskowitz spectrum of DNV-RP-C205 from Hs and Tp (or Tz), integrates its moments, and applies the Rayleigh wave-height distribution.
It is for masters and superintendents reading weather routing and forecast data, marine operations planners checking lifting and towing limits, and students of ocean waves.
Formulas
Pierson–Moskowitz: S_PM(ω) = (5/16) Hs² ωp⁴ ω⁻⁵ exp[−(5/4)(ω/ωp)⁻⁴]
JONSWAP: S_J(ω) = (1 − 0.287 ln γ) S_PM(ω) γ^exp[−(ω − ωp)²/(2σ²ωp²)], σ = 0.07 / 0.09
DNV γ rule: γ = 5 (Tp/√Hs ≤ 3.6), exp(5.75 − 1.15 Tp/√Hs) (3.6–5), 1 (≥ 5)
Moments: m_n = ∫ ωⁿ S(ω) dω
Periods: Tz = 2π √(m0/m2), T1 = 2π m0/m1, Te = 2π m₋₁/m0
Rayleigh heights: H_rms = Hs/√2, P(H > h) = exp[−(h/H_rms)²]
Highest 1/n: H1/n = H_rms [√(ln n) + (n√π/2) erfc √(ln n)]
Maximum of N waves: H_mp = H_rms √(ln N), E[H_max] ≈ H_rms [√(ln N) + 0.5772/(2√(ln N))], N = duration/TzAssumptions and limitations
- One stationary, single-peaked sea state in deep water.
- The DNV γ rule is for wind seas; swell needs its own spectrum.
- Rayleigh statistics are for a narrow-band sea and slightly overestimate the largest waves of real storms.
- Shallow water, currents and crossing seas are not modelled.
Worked example
Hs = 4 m and Tp = 9 s, a three-hour sea state, and a wave height of interest of 7 m.
- Tp/√Hs = 4.5, so the DNV rule gives γ = exp(5.75 − 1.15 × 4.5) = 1.78.
- Integrating the spectrum gives Tz = 6.66 s, T1 = 7.20 s and Te = 7.91 s.
- H_rms = 2.83 m; the highest tenth average 5.09 m and the highest hundredth 6.67 m.
- In 3 hours there are 1622 waves; the most probable maximum is 7.69 m and the expected maximum 7.99 m.
- One wave in about 460 exceeds 7 m, and the chance that at least one does in the 3 hours is 97 %.
FAQ
Why is the maximum wave so much larger than Hs?
Hs is roughly the mean of the highest third. Among a thousand or more waves the largest is typically 1.8–2 times Hs.
Which period do I have, Tp or Tz?
Buoys and hindcasts usually give Tp, or Tm02, which equals Tz; ship observations and older reports give a mean period close to Tz. Choose the matching input; the other periods are derived from the spectrum.
What does γ do?
It sharpens the spectral peak. A young, growing wind sea has γ around 3.3; a fully developed sea has γ = 1, the Pierson–Moskowitz spectrum.
Are these numbers conservative?
Mostly. Real storm seas are slightly broad-banded and their largest waves are about 5–10 % lower than the Rayleigh values.
Can I use this for swell?
Only roughly. Swell is narrow-banded with a long period; the DNV rule gives γ = 1 for it, which is too broad.
Related calculators
- Wave properties and encounter period — length, celerity and encounter period of a single wave.
- Beaufort force and sea state — the sea that goes with a wind.
- STAwave-1 added wave resistance — the speed loss in head seas.
- Roll period — synchronous and parametric roll in the waves of this sea state.