01955nas a2200121 4500008004100000245009900041210006900140260001900209300000700228520153200235100002901767856003701796 2008 eng d00aOn some consequences of the canonical transformation in the Hamiltonian theory of water waves.0 asome consequences of the canonical transformation in the Hamilto bECMWFc11/2008 a433 aWe discuss some consequences of the canonical transformation in the Hamiltonian theory of water waves(Zakharov, 1968). Using Krasitskii's canonical transformation we derive general expressions for the second order wavenumber and frequency spectrum, and the skewness and the kurtosis of the sea surface. For deep-water waves, the second-order wavenumber spectrum and the skewness play an important role in understanding the so-called sea state bias as seen by a Radar Altimeter. According to the present approach, but in contrast with results obtained by Barrick andWeber (1977), in deep-water second-order effects on the wavenumber spectrum are relatively small. However, in shallow water where waves are more nonlinear, the second-order effects are relatively large and help to explain the formation of the observed second harmonics and infra-gravity waves in the coastal zone. Second-order effects on the directional frequency spectrum are as a rule more important, in particular it is shown how the Stokes frequency correction affects the shape of the frequency spectrum, and it is also discussed why in the context of second-order theory the mean square slope cannot be estimated from time series. The kurtosis of the wave field is a relevant parameter in the detection of extreme sea states. Here, it is argued that, in contrast perhaps to one's intuition, the kurtosis decreases while the waves approach the coast. This is related to the generation of the wave-induced current and the associated change in mean sea level.1 aJanssen, Peter, A. E. M. uhttps://www.ecmwf.int/node/10217