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 = θ0 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. Phase velocity A sinusoidal wave, of small surface-elevation amplitude and with a constant wavelength, propagates with the phase velocity, also called celerity or phase speed. While the phase velocity is a vector and has an associated direction, celerity or phase speed refer only to the magnitude of the phase velocity. According to linear theory for waves forced by gravity, the phase speed depends on the wavelength and the water depth. For a fixed water depth, long waves (with large wavelength) propagate faster than shorter waves. In the left figure, it can be seen that shallow water waves, with wavelengths λ much larger than the water depth h, travel with the phase velocity with g the acceleration by gravity and cp the phase speed. Since this shallow-water phase speed is independent of the wavelength, shallow water waves do not have frequency dispersion. Using another normalization for the same frequency dispersion relation, the figure on the right shows that for a fixed wavelength λ the phase speed cp increases with increasing water depth. Until, in deep water with water depth h larger than half the wavelength λ (so for h/λ > 0.5), the phase velocity cp is independent of the water depth: with T the wave period (the reciprocal of the frequency f, T=1/f ). So in deep water the phase speed increases with the wavelength, and with the period. Since the phase speed satisfies cp = λ/T = λf, wavelength and period (or frequency) are related. For instance in deep water: The dispersion characteristics for intermediate depth are given below. Group velocity Interference of two sinusoidal waves with slightly different wavelengths, but the same amplitude and propagation direction, results in a beat pattern,