In deep water (depth greater than about half a wavelength) the dispersion relation ω² = gk links angular frequency ω = 2π/T to wavenumber k = 2π/λ, independent of wave height. Solving gives λ = gT²/(2π) and phase speed c = λ/T = gT/(2π) — longer-period swell always travels faster and farther, which is why distant storm swell arrives as long, fast, gentle waves while local wind chop is short and slow.
Group speed, the speed energy actually travels at, is cg = c/2 in deep water — individual crests appear to overtake the group and vanish at its front edge. Wave steepness H/λ is independent of the dispersion relation but sets a physical limit: linear wave theory breaks down and waves start to break near H/λ ≈ 1/7 (~0.143).
The small circles beneath the surface show water-particle orbits from linear wave theory — circular at the surface, shrinking exponentially with depth (radius ∝ ekz), so a swimmer feels almost nothing a wavelength down. Wind direction blends in a shorter, choppier secondary wave riding on top of the swell; chop is strongest when the wind blows the same way the swell travels and weakest when it blows against it.
- Try a long period (14–16 s) at low height: fast, gentle ocean swell.
- Try a short period (3–4 s) at high height: steep, slow local wind waves — watch the breaking warning.