Wave dispersion · JONSWAP spectrum · Ship heave · Sea state · Swell propagation
Model ocean surface waves from deep-water dispersion to shallow-water breaking. See how swell travels thousands of km, how waves interact with a ship hull, and calculate sea state from the JONSWAP wave spectrum.
The dispersion relation ω² = gk·tanh(kd) connects wave frequency to wavenumber and depth. In deep water (d ≫ λ/2): phase speed c = √(g/k), group speed cg = c/2. In shallow water (d ≪ λ/20): c = √(gd), cg = c. Longer waves travel faster — this is why swell from distant storms arrives as long, smooth waves before the short choppy waves from nearby wind.
The JONSWAP (Joint North Sea Wave Project) spectrum S(f) = αg²/(2πf)&sup5;·exp(−5/4·(f_p/f)⁴)·γ^(exp) describes the energy distribution of wind-generated seas. The peak frequency f_p determines the dominant wave period. The significant wave height Hₛ ≈ 4√(m₀) where m₀ is the zeroth spectral moment.
A ship in waves behaves as a spring-mass-damper: z̈ + 2ζω_n·ż + ω_n²·z = F_wave/m. Resonance occurs when the ship's natural heave period T_ship ≈ T_wave, leading to large motions even in moderate seas. For a 100m ship, the natural period ≈ 8–12s. Naval architects tune this away from typical peak wave periods.
Use the presets to set sea state from Calm to Storm. Adjust significant wave height Hₛ, peak period, water depth, and ship length. Watch the animated wave profile and ship heave. The spectrum chart (lower half) shows the JONSWAP energy distribution. Check the Resonance indicator — it turns red when the ship period is close to the peak wave period.