Gravity Wave Dispersion — Shallow vs Deep Water

ω² = gk·tanh(kh) · deep: c_p=√(g/k) · shallow: c_p=√(gh) (non-dispersive)

c_p = m/s
c_g = m/s
kh =
Regime:
The dispersion relation ω²=gk·tanh(kh) governs all water waves. Deep water (kh≫1): ω≈√(gk), dispersive — each frequency travels at its own speed, spreading wave packets. Shallow water (kh≪1): ω≈k√(gh), non-dispersive — tsunamis behave this way. The group velocity c_g=dω/dk carries energy; phase velocity c_p=ω/k carries wave crests. The dispersion curve is shown below; click to set wavenumber.