The determination of the joint probability density of wave heights and periods in the short-term statistics of sea waves of any band-width is of great importance in ...
COASTAL ENGINEERING ELSEVIER
Coastal Engineering 22 (1994) 201-215
On the theory of the joint probability of heights and periods of sea waves Constantine D. Memos Department of Civil Engineering, National Technical University of Athens, Athens, Greece
(Received 24 July 1992; accepted after revision 24 June 1993)
Abstract The determination of the joint probability density of wave heights and periods in the short-term statistics of sea waves of any band-width is of great importance in modelling coastal wave processes, environmental loading on floating vessels, etc. This research is based on known probability relations as well as on simple theoretical considerations. Assuming Gaussian waves of any band-width, a set of differential equations has been formulated giving the desired probability density function of wave heights and periods. Non-symmetrical wave heights about the mean can be accommodated, while some simplifying approximations are discussed. Numerical results are presented in a companion paper appearing in this issue.
1. Introduction Many coastal engineering problems, ranging from processes such as shoaling and wave breaking to vessel motions, involve statistics of the sea surface associated with both wave height and wave period. It is, therefore, quite clear that in a random sea better knowledge of the joint probability of height and period would improve substantially the available methods dealing with relevant coastal processes. The theoretical joint distribution proved to be an intractable problem for the broad-band sea, whilst some progress has been made for the narrow-band spectrum. The main investigations of this problem resulted in the following four approximations: Longuet-Higgins (1975), Cavani6 et al. (1976), Lindgren and Rychlik (1982) and Longuet-Higgins (1983). In the first paper Longuet-Higgins developed previous results by Rice ( 1944, 1945) for noise in electrical circuits and produced a joint density function for narrow spectra. He assumed statistically independent phases of the constituent individual waves, thus producing a linear Gaussian model. Comparison of his result with visual observations of wave properties in a narrow banded 0378-3839/94/$07.00 © 1994 Elsevier Science B.V. All rights reserved SSDI0378-3839 (93) E0025E
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C.D. Memos / Coastal Engineering 22 (1994) 201-215
spectrum published by Bretschneider (1959) showed good agreement. However, the underlying assumption of narrow-banded frequency spectrum renders the result not good for unfiltered records of surface elevation Goda (1978). Also, the expression for the probability density function is symmetrical about a non-dimensionalized characteristic period. This implies in effect that the correlation between wave height and period is zero, a deduction not supported by real life data. In fact, there appears to be a strong tendency for individual waves of small heights to have short periods, and so the joint probability density function of wave heights and periods deviates from the symmetrical shape indicated by the above result. A reformulation of this theory was provided by Longuet-Higgins (1983) by expressing differently the zero-upcrossing local wave period in order to get rid of the symmetry in the resulting joint density. His new formulation, based on envelope theory, can simulate the characteristics of observed waves, in such a way that short-period waves tend to have smaller heights. The distribution agrees well with wave data taken in the North Atlantic (Chakrabarti and Cooley, 1977) and in the Sea of Japan (Goda, 1978). Nevertheless, there are still shortcomings in the theory, as for example the overall correlation coefficient between wave heights and periods is slightly negative, which causes the significant period to be shorter than the mean period, in contrast with the observed characteristics of sea waves (Goda, 1985). Both results of Longuet-Higgins have been used by various researchers for applications or further elaboration. For example, Swift (1988) used both expressions to calculate cumulative probability of peak wave forces on cylindrical elements. Tayfun (1990) used the same technique to induce asymmetry as that of Longuet-Higgins and obtained an expression for the joint density function of large wave heights and periods. Cavani6 et al. (1976) define a wave period based upon the elevation and acceleration at the crest, by assuming a sine curve there. The joint density of elevation and acceleration for a narrow-band spectrum is given by Rice ( 1944, 1945) from which Cavani6 et al. derived the joint density of wave heights and periods. The results agreed rather well with analysis of 200 twenty-minute storm wave recordings in the North Sea. A practical problem with this formulation lies in estimating the value of the spectral width parameter which depends on the fourth moment of the variance spectrum. The values of this parameter are thus more descriptive of the high frequency cut-off than of the sea state as a whole. In fact, for the usual f - 5 high frequency tail, f = frequency, the fourth moment of the spectrum tends slowly to infinity as the integration is taken to higher frequencies. Comparisons of the distributions given by Cavani6 et al. and by Longuet-Higgins with observations and with numerically derived data are given by Shum and Melville (1984) and by Srokosz and Challenor (1987). They found in general good agreement in the region of high waves of a narrow spectrum. In the last of the previously mentioned papers Lindgren and Rychlik (1982) give approximations to the theoretical probability density within bounds, regardless of bandwidth, assuming a Gaussian sea surface. The solutions are based on a certain model process, and can be applied to bi-modal spectra as well, but are not closed form relationships. They are expressed in terms of wave height and period defined between a crest and the following trough. This definition, including negative maxima and positive minima of the elevation, is a serious drawback for engineering applications, since it allows a great deal of high frequency components to distort both the wave height and period probability structures. As with
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CavaniE's result, the existence of the fourth moment of the spectrum is required, which implies that the method can only be applied if a high frequency cut-off is specified. Also, in Lindgren and Rychlik's paper (1982) a constraint is implicitly assumed, namely that within every length of record equal to the maximum period, as defined above, falls either none or one minimum. In other words, the ratio of the maximum to the minimum wave period in any record is less than 2. The numerical evaluation of the result is based on knowledge of the entire spectrum rather than on a spectral width parameter. Therefore, a considerable amount of computer time is needed for carrying out the required numerical integrations. In an older work, Rice and Beer (1965) used a different approach to establish the distribution of wave heights without employing the above mentioned constraint on the wave period. However, their procedure did not include the necessary computational aspects. In the following, an analytical formulation for the joint density of the general broad-band problem is presented under justifiable assumptions. This is based on probability relations and simple theoretical considerations. Assuming Gaussian waves of any band-width, a set of differential equations has been formulated describing the probability density function of wave heights and periods. Some simplifying approximations are also given. A novel feature appears to be the capability of the developed model to cope with non-symmetrical wave heights about the mean. Numerical results are presented in a companion paper (Memos and Tzanis, 1994) showing good agreement with real life data.
2. Theoretical development The usual assumption is used that the sea surface elevation x above the mean level at a fixed point is well represented by a Gaussian process x ( t ) at times t for which the environmental conditions are well established, so that the process be regarded as statistically stationary. No restriction is made regarding the narrowness of the variance spectrum of the underlying process. The definition of the crest-to-trough wave height H and the period T between successive up- (or down-) crossings of the free surface with the mean level is given in Fig. 1. In this figure the ergodic process x ( t ) at a fixed point in the ocean is represented. It should be noted that in engineering applications it is quite customary to deal with the wave height H rather than with crest elevations ~, since the former is the physical quantity more closely related to the loading of the coastal structures. The wave height H is formally defined as H = s ~, -s~2
s~2
