| name | ocean-wave-loads |
| description | Ocean wave load analysis — linear wave theory, Morison equation, diffraction (large structures), irregular sea state (JONSWAP, Pierson-Moskowitz), response spectrum, fatigue, API RP 2A. |
| metadata | {"priority":7,"promptSignals":{"phrases":["ocean wave","wave load","Morison equation","sea state","JONSWAP spectrum","offshore wave","irregular waves"],"minScore":3}} |
Ocean Wave Loads — Complete Skill
Linear Wave Theory (Airy Waves)
Surface elevation:
η(x,t) = H/2 × cos(kx - ωt)
H = wave height; k = wave number = 2π/λ; ω = angular frequency = 2π/T; λ = wavelength; T = period
Dispersion relation (finite depth):
ω² = g k tanh(kd) [d = water depth]
Deep water (kd > π): ω² ≈ gk → λ = gT²/2π = 1.56 T² [m; T in sec]
Shallow water (kd < π/10): c = √(gd)
Water particle velocities (finite depth):
Horizontal: u = ω H/2 × cosh(k(z+d))/sinh(kd) × sin(kx-ωt)
Vertical: w = ω H/2 × sinh(k(z+d))/sinh(kd) × cos(kx-ωt)
Deep water simplification:
u ≈ ω H/2 × e^(kz) × sin(kx-ωt) [z measured up from surface; z<0]
Amplitude decays exponentially; negligible below z ≈ -λ/2
Morison Equation (Small Diameter Structures)
Valid when D/λ < 0.2 (diffraction negligible)
Force per unit length:
F/L = C_M × ρ × (πD²/4) × u̇ + C_D × ρ/2 × D × u|u|
C_M = inertia coefficient (1.5–2.0; isolated cylinder → C_M = 2.0 for long slender)
C_D = drag coefficient (0.6–1.2; depends on Re, KC number; cylindrical: C_D ≈ 1.0)
u = horizontal particle velocity; u̇ = horizontal particle acceleration
Keulegan-Carpenter number: KC = U_max T / D
KC < 3: inertia dominant; KC > 15: drag dominant; KC ≈ 5–15: mixed
Maximum force (monopile, regular wave):
F_max = C_M × ρ π D²/4 × A_w + C_D × ρ/2 × D × (A_w)²... [phase-dependent; compute numerically]
Diffraction (Large Structures, D/λ > 0.2)
Linear diffraction theory required for large diameter structures (SPAR platforms, caissons, TLPs)
Incident wave + scattered wave → total velocity potential
First-order wave force: F = ρ ω² × V × C_a × η (approximate; from Froude-Krylov + diffraction)
Full solution: boundary element method (BEM) — panel codes: WAMIT, ANSYS AQWA, OrcaFlex
Irregular Sea State Representation
Real ocean: superposition of many frequencies with random phases
Wave spectrum S(ω): energy distribution over frequency
Significant wave height: H_s = 4√(m₀) where m₀ = ∫S(ω)dω (zeroth moment)
Mean period: T_m = 2π√(m₀/m₂)
JONSWAP Spectrum (Fetch-Limited)
S(ω) = α g² ω⁻⁵ exp(-1.25(ω_p/ω)⁴) × γ^exp(-(ω-ω_p)²/(2σ²ω_p²))
γ = peak enhancement factor (1.0 = PM; typical North Sea: 3.3; range 1–7)
ω_p = peak frequency = 2π/T_p; σ = 0.07 (ω ≤ ω_p); 0.09 (ω > ω_p)
α = Phillips constant
Pierson-Moskowitz (PM, Fully Developed Sea)
S(ω) = α g² ω⁻⁵ exp(-β(g/(U_10 ω))⁴)
α = 0.0081; β = 0.74; U_10 = wind speed at 10m above surface
Structural Response in Irregular Seas
Response Amplitude Operator (RAO)
RAO(ω) = H_response(ω) / H_wave(ω) (transfer function)
σ_response² = ∫ RAO(ω)² × S(ω) dω (mean square response)
σ_response = standard deviation of response
Significant response: ξ_s = 4σ_response (3-hr storm)
Most Probable Maximum (MPM) Response
In storm duration D (3 hrs standard):
N_cycles = D / T_m
ξ_MPM = σ_response × √(2 ln N_cycles) [for Rayleigh distributed peaks]
Fatigue from Waves (Long-Term)
Use scatter diagram (H_s vs. T_p sea state frequency tables):
For each sea state: σ_stress from RAO × S(ω); compute fatigue damage from S-N + Miner's rule
D_i = Σ n_i / N_i (DNV-RP-C203 S-N curves)
Annual damage: D_annual = ΣD_i over all sea states
Life = 1/D_annual
API RP 2A Design Requirements (Offshore Structures)
Design wave:
Return period: 100-year wave (L1 structure); 50-year (L2)
Design H_max = H_1/3 × 1.86 (for Rayleigh statistics, P_exceed = 0.01%)
Load combination (API RP 2A-WSD):
Dead + Live + Wave + Current + Wind → combine factored loads
100-year return period wave (Gulf of Mexico example):
H_s ≈ 12–15 m (100-year); H_max ≈ 24 m; T ≈ 14–16 s
Base shear and overturning moment:
Compute from integrated Morison force over all members at 100-year wave conditions
Current Loading
Current profile: logarithmic (wind-driven) + tidal
Combined current + wave: vector sum velocities
Modified Morison with (u_wave + u_current): F/L = C_M ρπD²/4 u̇_wave + C_D ρD/2 (u_wave + u_current)|u_wave + u_current|
Output
Provide: H_s, T_p (design sea state), H_max and T_max (extreme wave), Morison force profile [kN/m vs. depth], total base shear F [kN] and overturning moment M [kN·m], RAO peak [m/m], σ_response [m], ξ_MPM [m], fatigue annual damage, C_M and C_D selection.
