In fluid dynamics, dispersion of water waves generally refers to frequency dispersion, which means that waves of different wavelengths travel at different phase speeds. Water waves, in this context, are waves propagating on the water surface, with gravity and surface tension as the restoring forces. As a result, water with a free surface is generally considered to be a dispersive medium. For a certain water depth, surface gravity waves – i.e. waves occurring at the air–water interface and gravity as the only force restoring it to flatness – propagate faster with increasing wavelength. On the other hand, for a given (fixed) wavelength, gravity waves in deeper water have a larger phase speed than in shallower water. In contrast with the behavior of gravity waves, capillary waves (i.e. only forced by surface tension) propagate faster for shorter wavelengths. Besides frequency dispersion, water waves also exhibit amplitude dispersion. This is a nonlinear effect, by which waves of larger amplitude have a different phase speed from small-amplitude waves. Frequency dispersion for surface gravity waves This section is about frequency dispersion for waves on a fluid layer forced by gravity, and according to linear theory. For surface tension effects on frequency dispersion, see surface tension effects in Airy wave theory and capillary wave. Wave propagation and dispersion The simplest propagating wave of unchanging form is a sine wave. A sine wave with water surface elevation η( x, t ) is given by: where a is the amplitude (in metres) and θ = θ( x, t ) is the phase function (in radians), depending on the horizontal position ( x , in metres) and time ( t , in seconds): with and where: λ is the wavelength (in metres), T is the period (in seconds), k is the wavenumber (in radians per metre) and ω is the angular frequency (in radians per second). Characteristic phases of a water wave are: the upward zero-crossing at θ = 0, the wave crest at θ = ½ π, the downward zero-crossing at θ = π and the wave trough at θ = 1½ π. A certain phase repeats itself after an integer m multiple of 2π: sin(θ) = sin(θ+m•2π). Essential for water waves, and other wave phenomena in physics, is that free propagating waves of non-zero amplitude only exist when the angular frequency ω and wavenumber k (or equivalently the wavelength λ and period T ) satisfy a