For a stationary two-dimensional random field evolving in time, we derive the intensity distributions of appropriately defined velocities of crossing contours.The results are based on a generalization of the Rice formula.The theory can be applied to practical problems where evolving random fields are considered to be adequate models. We study dynamical aspects of deep sea waves by applying the derived results to Gaussian fields modeling irregular sea surfaces.In doing so, we obtain distributions of velocities for the sea surface as well as for the envelope field based on this surface.Examples of wave and wave group velocities are computed numerically and illustrated graphically.
In this paper, we are interested in analyzing the dynamics of the sea by studying the distributions of different notions of velocity. In order to accomplish this goal, we proceed in two steps. Firstly, we identify different motions of the surface through appropriately defined velocities. Secondly, we derive the distributions of the defined velocities and we compute their densities given the spectrum of the underlying field. Distributions of the velocities can be studied at various regions of the surface such as points of local extremes, level crossing contours or regions with large curvature and so on. These distributions are different even if the same notion of velocity is considered. This leads to studies of the distribution of a random field given that another field (describing for example, level crossing contours) takes a fixed value. For computation of such a distribution we utilize a generalized Rice formula.
In this section we present various notions of velocities that can be defined for the random sea surface. All these velocities reflect different aspects of the kinematic features of the sea and therefore can be used accordingly to the problem at hand.