خانواده فرآیندهای تصادفی غیر- خطیِ باند- باریک برای مکانیک امواج دریا

A bi-parametric family of non-linear stochastic processes is introduced, to investigate the properties of second-order random processes with a narrow-band spectrum in the mechanics of the sea waves. In particular, the expressions of the probability density function and of the probabilities of exceedance of the absolute maximum and absolute minimum are obtained for this stochastic family. The analytical results are particularized for some processes of basic interest in the mechanics of the sea waves: the free surface displacement, and the fluctuating wave pressure beneath the sea surface.

The effects of non-linearity for random (wind-generated) sea waves were firstly investigated by Longuet-Higgins [1]. He achieved the first three terms of the Gram–Charlier series for the probability density function of the normalized free surface displacement, which is correct for any shape of the energy spectrum. Later Tayfun [2] obtained the probability density function and the probability of exceedance of the crest (absolute maximum) for the free surface displacement in an undisturbed wave field. The probability of exceedance of the trough (absolute minimum) was then derived by Tung and Huang [3]. The recent book of Boccotti [4] deeply develops the linear theory of random sea waves, and the effects of finite bandwidth. As for the non-linearity effects, it is emphasized that the probability of exceedance of the absolute minimum of the fluctuating wave pressure beneath the sea surface usually is markedly greater than the probability of exceedance of the absolute maximum, especially if the waves are subject to reflection. These conclusions are based on two recent small-scale field experiments and have some important consequences in the design of submerged floating tunnels and vertical breakwaters. In this paper a new theoretical approach is proposed to investigate the effects of non-linearity for the mechanics of the sea waves. In particular a bi-parametric family of non-linear stochastic processes is introduced, which includes the processes ‘free surface displacement’ and ‘fluctuating wave pressure’, both for waves in an undisturbed field (progressive waves) and for waves interacting with structures. Some statistical properties of the stochastic family are derived. Firstly the characteristic function (by using the Laplace transform) and the probability density function (by inverse-Fourier transforming the characteristic function) are obtained Moreover both the distributions of the absolute maximum and of the absolute minimum are achieved. All these properties depend upon two parameters α 1 and α 2 of the family. Finally, some applications are considered: the process ‘free surface displacement’ and the process ‘fluctuating wave pressure’, both for progressive waves and for waves in front of a vertical wall. The expressions of the parameters α 1 and α 2 , which enable us to quickly predict the effects of non-linearity, are obtained for the above-mentioned processes. The new approach is valid for most of the second-order processes in the mechanics of the sea waves, except for special cases relating to the interaction of strongly non-linear waves with structures (as the fluctuating wave pressure in front of a vertical wall near the seabed on deep water)

The properties of the family (1) of stochastic processes have been investigated. For this purpose the analytical expressions of the probability density function and of both the distributions of the absolute maximum and of the absolute minimum have been obtained. It is proven that all these properties depend upon two deterministic parameters named α 1 and α 2 . For zero mean processes we have α 1 =− α 2 , and the family has only one degree of freedom. We have shown that if both α 1 and α 2 approach zero, the non-linearity vanishes. As a consequence the probability density function tends to be Gaussian (according to the theory of wind-generated waves of Longuet-Higgins [1] and Phillips [6]) and both the probabilities of exceedance of the absolute maximum and of the absolute minimum tend to the Rayleigh distribution (according to Longuet-Higgins [7]). We have obtained also that for α 1 > 0 each realization of a process belonging to the family is a sequence of waves which have the crest amplitude (absolute maximum) greater than the trough amplitude (absolute minimum, in absolute value); for α 1 < 0 each wave has the trough amplitude greater than the crest amplitude. Finally, in the applications we have obtained the expressions of the parameters α 1 and α 2 for some sea wave processes, as functions of ε,ky and kd . In particular we have obtained that the surface waves have the crest greater than the trough, and furthermore, for a fixed kd , the effects of non-linearity at a vertical wall are greater than the non-linear effects in an undisturbed field. For the fluctuating wave pressure (at a point beneath the sea surface) α 1 is generally less than zero; in this case the trough of the fluctuating wave pressure are greater than the crest. Furthermore, as for the surface waves, the effects of non-linearity for the fluctuating wave pressure on a vertical wall are greater than in an undisturbed field. These theoretical conclusions agree well with the results of two small-scale field experiments by Boccotti [4], as we can see by comparing the data and the analytical predictions for the probabilities of exceedance of the wave crest P(ξ high >ξ) and of the wave trough P(ξ low >ξ) . Fig. 8 shows the comparison for both the free surface displacements (left panel) and the fluctuating wave pressure at a fixed depth beneath the sea surface (right panel), in an undisturbed field on deep water (we have re-examined the original source data of the small-scale field experiment described by Boccotti et al. [8]; see also Sections 10.9 and 10.10 of Boccotti [4]). Fig. 9 shows the comparison for the fluctuating wave pressure on a vertical wall (data by Boccotti [4]; see his Fig. 13.3