On the distributions of wave periods, wavelengths, and amplitudes in a random wave field

Abstract

On the basis of a narrow‐band Gaussian process, two theoretical probability density functions (PDFs) are derived by applying an approximate Hilbert transform to the process represented by a general form. The first PDF is for the joint distribution of the wave period τ and amplitude ρ of sea waves. It has the same merits as the PDF previously derived by Longuet‐Higgins [1983] , in being asymmetric in τ and depending only on the spectral width parameter ν, but gains an advantage that it predicts an exactly Rayleigh distribution of ρ and so facilitates its handling in theory and application in practice. In virtue of the advantage, a relatively simple conditional PDF of τ assuming ρ is derived, which may be used to predict an arbitrarily defined characteristic conditional wave period assuming the wave height from the average wave period. The second PDF is for the joint distribution of wavelengths and amplitudes in a unidirectional Gaussian wave field with narrow spectrum. It has the same form as the first PDF but depends only on the parameter μ defined by μ 2 = m 0 m 4 / m 2 2 − 1, where m n is the n th moment of spectrum. From this joint PDF, a PDF for the distribution of wavelengths of sea waves is also derived. Numerical simulation and laboratory experiments of wind waves and their results are reported. It is shown that the PDFs derived in this paper give fairly good fits to the simulated data and suitably filtered laboratory data.