A family of narrow-band non-linear stochastic processes for the mechanics of sea waves

A bi-parametric family of non-linear stochastic processes is introduced, to investigate the properties of second-order random processes with a narrow-band spectrum in the mechanics of the sea waves. In particular, the expressions of the probability density function and of the probabilities of exceedance of the absolute maximum and absolute minimum are obtained for this stochastic family. The analytical results are particularized for some processes of basic interest in the mechanics of the sea waves: the free surface displacement, and the fluctuating wave pressure beneath the sea surface.