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 functional relationship: the frequency dispersion relation
The dispersion relation has two solutions: ω = +Ω(k) and ω = −Ω(k), corresponding to waves travelling in the positive or negative x–direction. The dispersion relation will in general depend on several other parameters in addition to the wavenumber k. For gravity waves, according to linear theory, these are the acceleration by gravity g and the water depth h. The dispersion relation for these waves is:
an implicit equation with tanh denoting the hyperbolic tangent function.
An initial wave phase θ = θ propagates as a function of space and time. Its subsequent position is given by:
This shows that the phase moves with the velocity:
which is called the phase velocity.