Expressions have been derived, from the standpoint of Gaussian random theory, for the joint and marginal distributions of wave envelope amplitude and local wave period when the sampling frequency is equal to the local wave frequency. The new marginal p.d.f. for wave envelope amplitudes shows substantial non-zero probability density at very small amplitudes. The new joint and marginal distributions are found to compare favourably with data obtained from both high frequency measurements of real ocean waves in extreme storms and from measurements of numerical simulations of moderately broadbanded processes with Pierson–Moskowitz spectra. The new p.d.f. of wave envelope amplitudes is found to provide a better approximation to the p.d.f.s of the simulated zero down-crossing wave amplitude than either the traditional Rayleigh p.d.f., applicable for infinitesimal bandwidth; or the narrow bandwidth approximation given by M. S. Longuet-Higgins (Proc. R. Soc. Lond. A, 389: 241–258, 1983). The reason for this improvement is that our method takes into account that small waves are likely to have shorter periods than large waves, rather than assuming a constant wave period. There are, however, limitations to the approach adopted which assumes that the individual wave amplitudes can be obtained from the amplitude of the wave envelope. These limitations become more severe as the bandwidth increases. The results obtained apply not only to sea waves, but to any Gaussian linear random process.
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