A model is suggested that is capable of estimating the kinematics to third order of approximation of unidirectional and multidirectional random waves in deep water and finite depth. The model is applied in deep water to a surface spectrum which involves a mean JONSWAP spectrum and a directional spreading function in an independent frequency form and in finite depth to the surface spectrum which is obtained from a re-examined T M A model alread derived by the present authors. from that proposed by Bows et al. following the self similarity shape hypothesis and using an extensive set of field data. The Tayfun theory is extended to obtain the time histories of the surface elevation and to control and regulate the value of the skewness and kurtosis coefficients, and it uses the non-linear Fenton theory both to specify to third order of approximation the time history of the surface elevation and to construct the time histories of the horizontal and vertical velocities and accelerations of the unidirectional random waves.
As mentioned out by Zhang et al. (1992), the precise knowledge of the kinematics of surface waves is very important for many engineering applications which range from the estimate of the wave load on offshore structure components to the evaluation of the sediment transport in shallow water regions. At present, the kinematics of regular waves can be accurately predicted by higher order non-linear theories (Dean, 1965; Monkmeyer, 1970; Schwartz, 1974; Cokelet, 1977; Chaplin, 1980; Fenton, 1985, Pierson, 1993) whereas the kinematics of random waves, even in unidirectional conditions, has not yet reached sufficient levels of knowledge, in spite of the significant contributions provided by many authors (Wheeler, 1970; Rodenbush and Forristall, 1986; Gudmestad, 1990; Kim et aI., 1990; Zhang et aI., 1991, 1992, Donelan et aI., 1992).
