k2 : Wave groups can only be discerned in case of a narrow-banded signal, with the wave-number difference k1 − k2 small compared to the mean wave number ½ (k1 + k2). Multi-component wave patterns The effect of frequency dispersion is that the waves travel as a function of wavelength, so that spatial and temporal phase properties of the propagating wave are constantly changing. For example, under the action of gravity, water waves with a longer wavelength travel faster than those with a shorter wavelength. While two superimposed sinusoidal waves, called a bichromatic wave, have an envelope which travels unchanged, three or more sinusoidal wave components result in a changing pattern of the waves and their envelope. A sea state – that is: real waves on the sea or ocean – can be described as a superposition of many sinusoidal waves with different wavelengths, amplitudes, initial phases and propagation directions. Each of these components travels with its own phase velocity, in accordance with the dispersion relation. The statistics of such a surface can be described by its power spectrum. Dispersion relation In the table below, the dispersion relation ω2 = [Ω(k)]2 between angular frequency ω = 2π / T and wave number k = 2π / λ is given, as well as the phase and group speeds. Deep water corresponds with water depths larger than half the wavelength, which is the common situation in the ocean. In deep water, longer period waves propagate faster and transport their energy faster. The deep-water group velocity is half the phase velocity. In shallow water, for wavelengths larger than twenty times the water depth, as found quite often near the coast, the group velocity is equal to the phase velocity. History The full linear dispersion relation was first found by Pierre-Simon Laplace, although there were some errors in his solution for the linear wave problem. The complete theory for linear water waves, including dispersion, was derived by George Biddell Airy and published in about 1840. A similar equation was also found by Philip Kelland at around the same time (but making some mistakes in his derivation of the wave theory). The shallow water (with small h / λ) limit, ω2 = gh k2, was derived by Joseph Louis Lagrange. Surface tension effects In case of gravity–capillary waves, where surface tension affects the waves, the dispersion relation becomes: with σ the surface tension (in N/m).