Methodology: Deep Water Wave Physics & Swell Propagation

This entry documents the core wave physics equations that power our swell propagation and arrival time predictions. These are not approximations. They are the fundamental relationships governing ocean wave behavior.

The Dispersion Relation

The fundamental relationship between wave frequency (ω), wavenumber (k), and water depth (d):

ω² = gk · tanh(kd)

Where:
  ω = angular frequency (rad/s)
  g = 9.81 m/s² (gravitational acceleration)
  k = wavenumber (2π/wavelength)
  d = water depth (m)

This equation determines how waves of different periods travel at different speeds. This is the basis for swell dispersion and arrival time calculations.

Deep Water Wavelength

L₀ = gT² / 2π ≈ 1.56T² (meters)

For a 12-second swell: L₀ = 1.56 × 144 = 225m

Group Velocity (Energy Propagation)

Swell energy travels at the group velocity, not the phase velocity. This is critical for arrival time predictions:

Deep water: Cg₀ = gT / 4π ≈ 0.78T (m/s)

For a 12-second swell: Cg = 0.78 × 12 = 9.4 m/s = 34 km/h

At arbitrary depth, group velocity includes a depth correction factor:

Cg = (C/2) × [1 + 2kd / sinh(2kd)]

Wave Energy Density

E = ρgHs² / 16

Where:
  ρ = 1025 kg/m³ (seawater density)
  Hs = significant wave height (m)

Energy Flux (Power per Unit Crest)

P = E × Cg (watts per meter of wave crest)

Shoaling Coefficient

Describes wave height change due to depth change (assuming no energy loss):

Ks = √(Cg₁ / Cg₂)

Refraction (Snell's Law for Waves)

Kr = √(cos(θ₁) / cos(θ₂))

Where angles are relative to bottom contours.

Bottom Friction Dissipation

Based on Nielsen (1992) formulation for orbital velocity at the seabed:

Orbital velocity: ub = πH / [T × sinh(kd)]

Energy remaining: E_final = E_initial × exp(-αL)

Breaking Wave Limit (McCowan)

Hb ≈ 0.78d

Waves break when height exceeds ~78% of local depth.

References

• Holthuijsen, L.H. (2007). Waves in Oceanic and Coastal Waters. Cambridge University Press.
• Dean, R.G. & Dalrymple, R.A. (1991). Water Wave Mechanics for Engineers and Scientists.
• Nielsen, P. (1992). Coastal Bottom Boundary Layers and Sediment Transport.
• SWAN Technical Documentation (TU Delft).