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Directional Wave Spectrum & JONSWAP Parameter Calculator

Build JONSWAP spectral parameters from Hs, Tp and gamma, with cosine-power directional spreading.

Last reviewed by the myCalculator Editorial Team

Input Parameters

m
s

γ = 1.0 gives Pierson-Moskowitz; 3.3 is the standard JONSWAP mean value.

Low s means a broad confused wind sea; high s means narrow long-crested swell.

°

Calculation Results

Significant Wave Height (Hs)
3 m
Peak Period (Tp)
10 s
Spectrum Type
Standard JONSWAP — fetch-limited developing sea
Directional Spreading
Moderate — typical wind sea (±24.4°)
Peak Enhancement Factor (γ)
3.3
Spreading Exponent (s)
10
Mean Wave Direction
225° (SW) — coming from
Peak Frequency (fp)
0.1 Hz
Phillips Constant (α)
0.00299
Peak Spectral Density S(fp)
17.48 m²/Hz
Spectral Moment m₀
0.5625 m²
Zero-Crossing Period (Tz)
7.81 s
Mean Period (T1)
8.33 s
Deep-Water Wavelength at Tp
156.1 m
Summary: A JONSWAP spectrum with Hs 3 m, Tp 10 s and gamma 3.3 describes a Standard JONSWAP sea. Directional spreading exponent s = 10 gives a ±24.4° spread about 225° (SW) — moderate.
Step-by-Step Calculation Solution
1.JONSWAP: S(f) = alpha g² (2pi)^-4 f^-5 exp(-1.25 (f/fp)^-4) x gamma^r
2.where r = exp(-(f - fp)² / (2 sigma² fp²)), sigma = 0.07 for f ≤ fp and 0.09 for f > fp
3.fp = 1/Tp = 1/10 = 0.1 Hz
4.m₀ = (Hs/4)² = (3/4)² = 0.5625 m²
5.alpha = 5.061 (Hs²/Tp⁴)(1 - 0.287 ln gamma) = 0.00299
6.Peak density S(fp) = 17.48 m²/Hz
7.Directional spread D(theta) = cos^2s(theta/2), s = 10
8.Spreading width sigma_theta = sqrt(2/(s+1)) = 24.4°
9.gamma = 1.0 reduces JONSWAP exactly to the Pierson-Moskowitz spectrum.

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Formula & How Directional Wave Spectrum & JONSWAP Parameter Calculator Works

Mathematical Formula
S(f) = αg²(2π)⁻⁴f⁻⁵ exp(-1.25(f/fp)⁻⁴) γ^r | D(θ) = cos^2s(θ/2)

The JONSWAP spectrum was derived from the Joint North Sea Wave Project and describes fetch-limited seas that have not fully developed. Its peak enhancement factor gamma sharpens the Pierson-Moskowitz form around the peak frequency — at gamma = 1 the two are identical. Real seas are also directional, so a spreading function distributes energy about the mean direction. The cosine-power model with exponent s controls that spread: a low s describes a short-crested confused wind sea, while a high s describes long-crested groundswell arriving from a distant storm.

How to Calculate Directional Wave Spectrum & JONSWAP Parameter Calculator (Step-by-Step)
  1. Enter Significant Wave Height Hs (m) (m): Input your target parameter value.
  2. Enter Peak Period Tp (s) (s): Input your target parameter value.
  3. Enter Peak Enhancement Factor γ: Input your target parameter value.
  4. Enter Directional Spreading Exponent s: Input your target parameter value.
  5. Enter Mean Wave Direction (° from) (°): Input your target parameter value.
  6. Apply Formula: Calculate using S(f) = αg²(2π)⁻⁴f⁻⁵ exp(-1.25(f/fp)⁻⁴) γ^r | D(θ) = cos^2s(θ/2).
  7. Review Results: View exact computed answers, metrics, and worked mathematical proofs.
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Frequently Asked Questions (FAQs)

What does the JONSWAP gamma parameter mean?

Gamma sharpens the spectral peak. A value of 1.0 reduces JONSWAP exactly to Pierson-Moskowitz (fully developed sea), while 3.3 is the North Sea mean and higher values describe young, strongly fetch-limited seas.

What is directional spreading?

Real waves arrive across a range of directions, not a single heading. The cos^2s model sets that width: s below 5 is a confused short-crested sea, while s above 30 is long-crested groundswell.

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