Abstract
The modified versions of the linear theoretical model of Longuet-Higgins (1983) are derived in this work and also compared with the laboratory experiments carried out in MARINTEK. The main feature of modifications is to replace the mean frequency in the formulation with the peak frequency of the wave spectrum. These two alternative forms of joint distributions are checked in three typical random sea states characterized by the initial wave steepness. In order to further explore the properties of these models, the associated marginal distributions of wave heights and wave periods are also researched with the observed statistics and some encouraging results are obtained.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
Boccotti, P., 2000. Wave mechanics for ocean engineering, Elsevier Oceanography Series 64, Elsevier, Amsterdam, 153–181.
Boukhanovsky, A. V. and Guedes Soares, C., 2009. Modelling of multipeaked directional wave spectra, Appl. Ocean Res., 31(2): 132–141.
Bretschneider, C. L., 1959. Wave Variability and Wave Spectra for Wind Generated Gravity Waves, U. S. Army Corps of Engineers, Beach Erosion Board, Technical Memorandum, 118, 10–40.
Cavanié, A., Arhan, M. and Ezraty, R., 1976. A statistical relationship between individual heights and periods of storm waves, in: Behavior of Offshore Structures (BOSS), 76, 354–363.
Chakrabarti, S. K. and Cooley, R. P., 1977. Statistical distribution of periods and heights of ocean waves, J. Geophys. Res., 82(9): 1363–1368.
Cherneva, Z. and Guedes Soares, C., 2011. Evolution of wave properties during propagation in a ship towing tank and an offshore basin, Ocean Eng., 38(17–18): 2254–2261.
Cherneva, Z., Guedes Soares, C. and Petrova, P., 2011. Distribution wave height maxima in storm sea states, J. Offshore Mech. Arct. Eng., 133(4): 041601.
Cherneva, Z., Tayfun, M. A. and Guedes Soares, C., 2009. Statistics of nonlinear waves generated in an offshore wave basin, J. Geophys. Res., 114(C08005).
Cummings, W. E., Bales, S. L. and Gentile, D. M., 1981. Hindcasting waves for engineering applications, Proc. Int. Symp. on Hydrody. in Ocean Eng., Trondheim, 1, 70–89.
Davidan, I. N., Lopatukhin, L. I. and Rozhkov, W. A., 1985. Vetrovoye Volneniye v Mirovom Okeane (Wind Waves in the Ocean), Gidrometeoizdat, Leningrad, 256–270. (in Russian)
Fedele, F., Cherneva, Z., Tayfun, M. A. and Guedes Soares, C., 2010. Nonlinear Schrödinger invariants and wave statistics, Phys. Fluids, 22(3): 036601.
Goda, Y., 1978. The observed joint distribution of periods and heights of sea waves, Proc. 16th Coastal Eng. Conf., Hamburg, Germany, Part I, 227–246.
Goda, Y., 2000. Random Seas and Design of Maritime Structures, Advanced Series on Ocean Engineering 15, World Scientific, 280–292.
Guedes Soares, C., 1984. Representation of double-peaked sea wave spectra, Ocean Eng., 11(2): 185–207.
Guedes Soares, C., 1991. On the occurrence of double peaked wave spectra, Ocean Eng., 18(1–2): 167–171.
Guedes Soares, C. and Henriques, A. C., 1998. Fitting a double-peak spectral model to measured wave spectra, In: Proc. 17th Int. Conf. Offshore Mechanics and Arctic Eng. (OMAE), New York, OMAE, 98–1491.
Guedes Soares, C. and Nolasco, M. C., 1992. Spectral modeling of sea states with multiple wave systems, J. Offshore Mech. Arct. Eng., 114(4): 278–284.
Guedes Soares, C., 2003. Probabilistic models of waves in the coastal zone, in: Advances in Coastal Modeling, C. Lakhan (Ed.), Elsevier Sciences B. V., Chapter 6, 159–187.
Guedes Soares, C., Cherneva, Z. and Antão, E., 2003. Characteristics of abnormal waves in North Sea storm sea states, Appl. Ocean Res., 25(6): 337–344.
Guedes Soares, C., Cherneva, Z. and Antão, E., 2004. Abnormal waves during the hurricane Camille, J. Geophys. Res., 109, C08008.
Guedes Soares, C., Cherneva, Z., Petrova, P. and Antão, E., 2011. Large waves in sea states, in: Marine Technology and Engineering, Guedes Soares et al. (Eds), Taylor & Francis Group, London, 79–95.
Haver, S., 1987. On the joint distribution of heights and periods of sea waves, Ocean Eng., 14(5): 359–376.
Lindgren, G. and Rychlik, I., 1982. Wave characteristics distributions for Gaussian waves–wave length, amplitude and steepness, Ocean Eng., 9(5): 411–432.
Longuet-Higgins, M. S., 1952. On the statistical distribution of heights of sea waves, J. Marine Res., 11(3): 245–265.
Longuet-Higgins, M. S., 1975. On the joint distribution of the periods and amplitudes of sea waves, J. Geophys. Res., 80(18): 2688–2694.
Longuet-Higgins, M. S., 1983. On the joint distribution of wave periods and amplitudes in a random wave field, Proc. R. Soc. Lond., 389(1797): 241–258.
Lucas, C., Boukhanovsky, A. and Guedes Soares, C., 2011. Modeling the climatic variability of directional wave spectra, Ocean Eng., 38(11–12): 1283–1290.
Massel, S. R., 1996. Ocean Surface Waves: Their Physics and Prediction, Advanced Series on Ocean Engineering 11, World Scientific, 172–180.
Mori, N., 2003. Effects of wave breaking on wave statistics for deep-water random wave train, Ocean Eng., 30(2): 205–220.
Mori, N. and Janssen, P. A. E. M., 2006. On kurtosis and occurrence probability of freak waves, J. Phys. Oceanogr., 36(7): 1471–1483.
Naess, A., 1985. On the distribution of crest-to-trough wave heights, Ocean Eng., 12(3): 221–234.
Nair, N. U., Muraleedharan, G. and Kurup, P., 2003. Erlang Distribution Model for Ocean Wave Periods, J. Ind. Geophys. Union, 7(2): 59–70.
Onorato, M., Cavaleri, L., Fouques, S., Gramstad, O., Janssen, P. A. E. M., Monbaliu, J., Osborne, A. R., Serio, M., Stansberg, C.T., Toffoli, A. and Trulsen, K., 2009. Statistical properties of mechanically generated surface gravity waves: a laboratory experiment in three-dimensional wave basin, J. Fluid Mech., 627, 235–257.
Onorato, M., Osborne, A., Serio, M., Cavaleri, L., Brandini, C. and Stansberg, C., 2006. Extreme waves, modulational instability and second order theory: Wave flume experiments on irregular waves, Eur. J. Mech. B, 25(5): 583–601.
Petrova, P., Cherneva, Z. and Guedes Soares, C., 2006. Distribution of crest heights in sea states with abnormal waves, Appl. Ocean Res., 28(4): 235–245.
Petrova, P., Cherneva, Z. and Guedes Soares, C., 2007. On the adequacy of second-order models to predict abnormal waves, Ocean Eng., 34(7): 956–961.
Petrova, P. and Guedes Soares, C., 2008. Maximum wave crest and height statistics of irregular and abnormal waves in an offshore basin, Appl. Ocean Res., 30(2): 144–152.
Rice, S. O., 1944. Mathematical analysis of random noise, Bell Syst. Tech. J., 23(3): 282–332.
Rice, S. O., 1945. Mathematical analysis of random noise, Bell Syst. Tech. J., 24(1): 46–156.
Rodriguez, G. and Guedes Soares, C., 1999a. The bivariate distribution of wave heights and periods in mixed sea states, J. Offshore Mech. Arct. Eng., 121(2): 102–108.
Rodriguez, G. R. and Guedes Soares, C., 1999b. Uncertainty in the estimation of the slope of the high frequency tail of wave spectra, Appl. Ocean Res., 21(4): 207–213.
Rodriguez, G. R., Guedes Soares, C. and Machado, U., 1999a. Uncertainty of the sea state parameters resulting from the methods of spectral estimation, Ocean Eng., 26(10): 991–1002.
Rodriguez, G. R., Guedes Soares, C. and Ocampos, F. J., 1999b. Experimental evidence of the transition between power law models in the high frequency range of the gravity wave spectrum, Coast. Eng., 38(4): 249–259.
Shemer, L. and Sergeeva, A., 2009. An experimental study of spatial evolution of statistical parameters in a unidirectional narrow-banded random wave field, J. lGeophys. Res., 114(C1): C01015.
Shemer, L., Sergeeva, A. and Liberzon, D., 2010. Effect of the initial spectrum on the spatial evolution of statistics of unidirectional nonlinear random waves, J. Geophys. Res., 115(C12): C12039.
Srokosz, M., 1988. A note on the joint distribution of wave height and period during the growth phase of a storm, Ocean Eng., 15(4): 379–387.
Stansell, P., Wolfram, J. and Linfoot, B., 2004. Improved joint probability distribution for ocean wave heights and periods, J. Fluid Mech., 503, 273–297.
Tayfun, M. A., 1980. Narrow-band nonlinear sea waves, J. Geophys. Res., 85(C3): 1548–1552.
Tayfun, M. A., 1990. Distribution of large wave heights, J. Waterw. Port. Coast. Ocean Eng., ASCE, 116(6): 107–118.
Tayfun, M. A. and Fedele, F., 2007. Wave-height distributions and nonlinear effects, Ocean Eng., 34(11–12): 1631–1649.
Xu, D., Li, X., Zhang, L. Z., Xu, N. and Lu, H. M., 2004. On the distributions of wave periods, wave lengths and amplitudes in a random wave field, J. Geophys. Res., 109(C5): C05016.
Zhang, H. D., Cherneva, Z., Guedes Soares, C. and Onorato, M., 2013a. Comparison of distributions of wave heights from nonlinear Schrödinger equation simulations and laboratory experiments, in: Proc. 32nd Int. Conf. Offshore Mechanics and Arctic Eng. (OMAE), Nantes, France, OMAE2013-11633.
Zhang, H. D., Cherneva, Z. and Guedes Soares, C., 2013b. Joint distributions of wave height and period in laboratory generated nonlinear sea states, Ocean Eng., 74, 72–80.
Author information
Authors and Affiliations
Corresponding author
Additional information
The project was financially supported by the European Union (Grant No. 234175) and the Portuguese Foundation for Science and Technology (Grant No. SFRH/BD/98983/2013).
Rights and permissions
About this article
Cite this article
Zhang, H.D., Guedes Soares, C. Modified joint distribution of wave heights and periods. China Ocean Eng 30, 359–374 (2016). https://doi.org/10.1007/s13344-016-0024-8
Received:
Revised:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s13344-016-0024-8