توزیع دقیق امواج آشکار در دریاهای نامنظم

We discuss the long-run distributions of several characteristics for the apparent waves in a Gaussian sea. Three types of one-dimensional wave records are considered: 1) the seaway in time at a fixed position; 2) the instantaneous profile along a horizontal line; 3) the encountered seaway. Exact integral forms of the joint long run distributions are derived for the apparent periods, lengths, and heights. Results of numerical approximations of these distributions are presented in examples. For the computations we considered, as the input spectra, empirical estimates of the frequency spectra as well as JONSWAP type spectra. Effective algorithms are discussed and utilized in the form of a comprehensive computer package of numerical routines.

All recorded wave data exhibit essential irregularity of the sea surface. This irregularity led to the nowadays commonly accepted conviction that any rigorous approach to the seaway has to involve statistical description. The sea surface is then considered as a random two-dimensional field evolving in time arising as a limit of sums of regular sinusoidal wave trains with random phases. The directional spectrum or spectral density—an important characteristic for this model—represents half of the squared amplitude of the sinusoidal component in a given direction and with a particular frequency. The spectrum itself, however, does not specify a random field completely, and some further distributional assumptions are needed. The precise distributional character of the seaway revealed in the field data is rather complicated but Gaussian distributions to a great extent constitute adequate approximations of the empirical ones. In fact, a Gaussian random surface can be obtained as the first order approximation of the solutions to differential equations based on the hydrodynamic theory of deep-water waves. Since the theory of Gaussian random processes is well understood, the assumption about the Gaussian nature of a sea surface equips us with a whole range of powerful theoretical tools which can be applied to statistical analysis. For further discussion of this assumption see also such fundamental works as St Denis and Pierson (1953) and Lewis (1988). In this paper, following numerous other works in the field, we assume the same Gaussian framework as in both the quoted references. Despite this approximate Gaussian character, there is no doubt that real wave records also exhibit some non-Gaussian features; an important example is the vertical asymmetry characterised by average larger crest heights than trough depths. A physically meaningful approach should involve higher than first order approximations to the Stokes equation. This would lead to the non-linear theory of random sea surfaces which constitutes a research area in its own right. Deriving long run distributions for non-linear, non-Gaussian waves is a theoretically advanced topic which would go beyond the scope of this presentation. Nevertheless, one can circumvent the problem of asymmetry between sizes of crests and troughs by assuming that the surface elevation, although itself not Gaussian, is a smooth transformation of a Gaussian surface. In the literature, there are known parametric forms of either transformations or their inverses (see Ochi and Ahn, 1994) as well as non-parametric methods of their estimation (see Rychlik et al., 1997). Clearly, once we have given the inverse transformation, the approach demonstrated in this work can be applied to the transformed wave records, assuming that they are correctly modeled by a Gaussian process. In Section 4.1, we apply the inverse transformation method to the wave record with evident vertical asymmetry in order to diminish the influence of this non-Gaussian trait.The wave surface is clearly a two-dimensional phenomenon and its study should naturally deal with two-dimensional objects. However theoretical studies of random surfaces still face major difficulties. For example, the evolution in time of the two-dimensional contours of fixed levels—natural objects of study—is highly erratic, often resulting in their merging, splitting or disappearing, and thus very difficult to deal with statistically. Much simpler, yet still meaningful, are studies of one-dimensional records. They can be extracted from a photograph of the sea surface as, for example, the instantaneous profile along a line in some fixed horizontal direction on the sea, or can be obtained directly as a record taken in time at a fixed position in space as by means of a wave pole or distance meters. The encountered sea, another important one-dimensional record, can be collected at a moving point as by means of a ship-borne wave recorder. Fig. 1 illustrates the irregularity of a one-dimensional record which could be collected by one of the three methods mentioned above. In fact, these are fragments of real-life data on the sea elevation collected in time at a fixed point of the North Sea. To analyze these data we need natural and operational definitions of the individual wave, its period, height, and possibly some other meaningful characteristics. For random seas, however, the individual wave can be defined in various, not necessarily, equivalent ways. In Fig. 1, we present two examples of possible definitions: the zero down-crossing method and the maximum-to-maximum method. In this paper we consider mostly zero down-crossing waves. Namely, the apparent individual wave at fixed time or position is defined as the part of the record between two consecutive down-crossings of the zero seaway level (the latter often more descriptively referred to as the still water level). For individual waves one can consider various natural characteristics, among them apparent periods and apparent heights (amplitudes). The pictorial definitions of these two characteristics are given in Fig. 1.Having precisely defined characteristics of interest, one can extract their frequency (empirical) distributions from a typical sufficiently long record. For example, measurements of the apparent period and height of waves could be taken over a sufficiently long observation time to form an empirical two-dimensional distribution. This distribution will represent some aspects of a given sea surface. Clearly, because of the irregularity of the sea, empirical frequencies will vary from record to record, however if the sea is in “steady” condition, which corresponds mathematically to the assumption that the observed random field is stationary and ergodic, their variability for sufficiently large records will be insignificant. Such limiting distributions (limiting with respect to observation time, for records measured in time, increasing without bound) are termed the long-run distributions. What makes these distributions particularly attractive, and what is our main subject of study, is the possibility of their exact computation from the mathematical form of a random seaway. No field data are needed, given an estimated or assumed directional spectrum, although they can be used, e.g., for cross-validation of the model considered. The roots of our approach to computing long-run distributions can be tracked back to Rice, 1944 and Rice, 1945 whose celebrated formula for the expected value of the number of crossings in a given interval was derived in connection with studies of noise in electrical circuits. Since then further development, more directly related to analysis of random sea surfaces, has occured. Fundamental work in this area includes: the pioneering work of Longuet-Higgins (1952) later extended in several papers (see Longuet-Higgins, 1975 and Longuet-Higgins, 1983, and references therein) where, for example, the long-run distributions of the apparent periods and lengths for narrow band Gaussian waves are given in an analytical form and compared to the empirical distributions obtained from the field data, and Lindgren, 1972a and Lindgren, 1972b introducing a rigorous treatment of the longrun distributions through so called Slepian models, allowing computation of the exact long-run marginal distributions of the apparent wave lengths or periods and also the joint densities of the lengths and amplitudes for the maximum-to-maximum method. For a more complete account of the developments in this area see Section 3.2. Despite all this progress, a general approach which would be applicable to arbitrary apparent wave characteristics and their joint multidimensional distributions was still lacking. To quote Longuet-Higgins (1975): “The joint distribution of wave heights and periods is still more difficult, in the most general case”. In this respect, our work addresses this issue by obtaining a method which is general in the following two important aspects. First, our derivations and, in consequence, numerical algorithms do not depend on the form of the spectrum which is used to model the sea surface. Thus spectra can be theoretical ones, like those from the JONSWAP family, or they can be estimated from empirical records. Moreover, our results can be applied to arbitrary one-dimensional records given that they are Gaussian and smooth enough. The relative position of a ship's bow with respect to the sea surface is an important example for which long-run distributions could be of interest in the analysis of such important events as wet deck and slamming. Second, in principle, multivariate long-run distributions in integral form can be derived for a large class of apparent wave characteristics. The only limitation in practical applications is the effectiveness of numerical algorithms used for computations of the integrals involved. This is an issue here as the generality of the method is achieved at the cost of the relative complexity of the integral formulas. However, accounting for constant improvement in computer performance, one will be able to use the proposed method to study effectively more and more complex multidimensional distributions. To initiate the unified approach to numerical problems in computation of the involved integrals, we implement numerical routines in the form of a computer package of FORTRAN subroutines accessible from the MATLAB© software. This package is available from the authors on request. To conclude this section we mention two possible extensions of the methods presented. Since the second order and thus non-linear approximations of the sea surface are not Gaussian, it is of interest to extend studies of the long-run distributions beyond the Gaussian class of processes and relate them to analysis of the sea surface. Another non-trivial and important issue is to consider the long-run distribution for the wave contours in two-dimensional records. We plan to investigate these research topics in some forthcoming work.