For a water–air interface (with and ) the waves can be approximated as pure capillary waves – dominated by surface-tension effects – for wavelengths less than . For wavelengths above the waves are to good approximation pure surface gravity waves with very little surface-tension effects. Interfacial waves For two homogeneous layers of fluids, of mean thickness h below the interface and h′ above – under the action of gravity and bounded above and below by horizontal rigid walls – the dispersion relationship ω2 = Ω2(k) for gravity waves is provided by: where again ρ and ρ′ are the densities below and above the interface, while coth is the hyperbolic cotangent function. For the case ρ′ is zero this reduces to the dispersion relation of surface gravity waves on water of finite depth h. When the depth of the two fluid layers becomes very large (h→∞, h′→∞), the hyperbolic cotangents in the above formula approaches the value of one. Then: Nonlinear effects Shallow water Amplitude dispersion effects appear for instance in the solitary wave: a single hump of water traveling with constant velocity in shallow water with a horizontal bed. Note that solitary waves are near-solitons, but not exactly – after the interaction of two (colliding or overtaking) solitary waves, they have changed a bit in amplitude and an oscillatory residual is left behind. The single soliton solution of the Korteweg–de Vries equation, of wave height H in water depth h far away from the wave crest, travels with the velocity: So for this nonlinear gravity wave it is the total water depth under the wave crest that determines the speed, with higher waves traveling faster than lower waves. Note that solitary wave solutions only exist for positive values of H, solitary gravity waves of depression do not exist. Deep water The linear dispersion relation – unaffected by wave amplitude – is for nonlinear waves also correct at the second order of the perturbation theory expansion, with the orders in terms of the wave steepness (where a is wave amplitude). To the third order, and for deep water, the dispersion relation is so This implies that large waves travel faster than small ones of the same frequency. This is only noticeable when the wave steepness is large. Waves on a mean current: Doppler shift Water waves on a mean flow (so a wave in a moving medium) experience a Doppler shift.