In the same way we can calculate the area of the sea surface consisting of an arbitrary number of intersecting regular waves. Under natural conditions, wave profiles are constantly changing with time in random fashion. Owing to the complex energy GSK2118436 transfer from
the atmosphere to the ocean and vice versa, the resulting surface waves are multidirectional. Information about a time series of surface displacements at a given point is usually available from a wave recorder or from numerical simulation. For the purpose of this paper we use the simulation approach and assume that a confused sea is the summation of many independent harmonics travelling in various directions. These harmonics are superimposed with a random phase φ, which is uniformly distributed on (–π, π). Thus we have ( Massel & Brinkman 1998) equation(80) ζ(x, y, t)=∑m=1M1∑n=1N1amn cos[km xcosθn+km ysinθn−ωmt+φmn],in
which the deterministic amplitudes amn are prescribed by the following formula: equation(81) amn2=2S1(ω,θ)ΔωmΔωn,where S1(ω, θ) is the input frequency-directional Stem Cell Compound Library spectrum, Δωm denotes the band-width of the mth frequency, and Δωn is the band-width of the nth wave angle. The wave numbers km are given by the dispersion relation equation(82) ωm2=gkmtanh(kmh)and M1 and N1 are the respective numbers of frequencies and directions used in the simulation. We represent the input frequency-directional spectrum S1(ω, θ) in the form of the product of the frequency spectrum S1(ω) and the directional spreading D(θ), in which the JONSWAP frequency spectrum ( eq. (12)) is used, and for the directional spreading function D(θ) we adapt formula (20) with parameter s = 1. To simulate the sea surface, a time series of M 1 = 155 frequencies non-uniformly distributed in the frequency band 0.5 ωp < ω < 6ωp and N 1 = 180 directions (Δθ = 2°) were used. When the surface displacement ζ=ζ(x, y, t)ζ=ζ(x, y, t) is known, the area of random sea surface over the plain rectangle a × b is given by eq. (79). Let us assume that an area of 1 km × 1 km is covered by surface
waves induced by a wind of velocity changing from U = 2m/s to U = 25m/s and fetch X = 100 km . The relationship between the relative increase in area δ and wind Cell press speed U is shown in Figure 8. In a very severe storm, when U = 25 m s−1 and significant wave height Hs = 4.57 m, the increase δ approaches the value of δ = 0.77%. This paper examines some geometrical features of ocean surface waves, which are of special importance in air-sea interaction and incipient wave breaking. In particular, the paper demonstrates the influence of directional spreading on the statistics of sea surface slopes. Theoretical analysis and comparison with the available experimental data show that unimodal directional spreading is unable to reproduce properly the observed ratio of the cross-wind/up-wind mean square slopes.