Read the gravity–capillary dispersion relation

Take a horizontal fluid layer of uniform depth h, density ρ, and surface tension σ, with an impermeable flat bottom. For small-amplitude waves in an inviscid incompressible fluid with irrotational motion, a velocity potential obeys Laplace’s equation below the surface. The kinematic boundary condition makes the surface move with the fluid. The linear dynamic condition balances pressure with gravity and the surface-tension curvature contribution. Solving the depth dependence and matching these conditions gives the gravity–capillary dispersion relation.

The idealization neglects viscosity, background current, and the inertia of the medium above the surface, whose pressure is treated as uniform. Wavenumber k is positive here and refers to a horizontal spatial period. Small wave slope and sufficiently small displacement relative to the relevant geometric scales justify linearizing the moving boundary. These assumptions differ from compressible sound: incompressibility is compatible with surface propagation because the restoring dynamics involve the moving boundary rather than compression of the fluid bulk.

Deep water means kh is much larger than one, so the hyperbolic tangent is close to one. Gravity dominance additionally requires σk²/(ρg) much smaller than one. Under both conditions, angular frequency approaches the square root of gk, phase speed is the square root of g/k, and group speed is half of phase speed. Saying only that a wave is in deep water does not establish gravity dominance; sufficiently short deep-water ripples can instead be dominated by surface tension.

For deep-water capillary dominance, the same ratio is much larger than one. Angular frequency then scales as k to the three-halves power, and group speed is three-halves of phase speed. The crossover wavenumber is the square root of ρg/σ, where the two restoring terms are equal. Shallow water is another independent limit, kh much smaller than one. There the leading relation becomes ghk²+(σh/ρ)k⁴. It is nondispersive only when the capillary term is also negligible.

In deep water with both restoring terms retained, differentiating the square-root dispersion gives group velocity. Dividing by phase velocity yields a ratio that interpolates from one half in the gravity limit to three halves in the capillary limit. At the crossover, group and phase speeds coincide even though the overall dispersion is not linear. Equality at one wavenumber does not make all packets nondispersive; nearby components can still accumulate different group delays and distort the envelope.

Use the spectral lab to inspect how shorter spatial scales enter a surface-like profile, while keeping its limits explicit. At mode count twelve, predict whether raising coefficient decay from zero to three suppresses fine ripples, then compare with mode count one. These settings change only a prescribed finite sum of sin(nx−t)/nᵖ. Its modes all have the same temporal frequency, so it does not obey the gravity–capillary dispersion relation, model depth, or predict the relative motion of a real surface-wave envelope and its crests.