Statistical Properties of Envelope Field for Gaussian Sea Surface

The envelope process is a useful analytical tool which is often used to study wave groups. Most research on statistical properties of the envelope, and thus of wave groups, was focused on one dimensional records. However for the marine application, an appropriate concept should be two dimensional in space and variable in time. Although a generalization to higher dimensions was introduced by Adler (1978), little work was done to investigate its features. Since the envelope is not defined uniquely and its properties depend on a chosen version, we discuss the definition of the envelope field for a two dimensional random field evolving in time which serves as a model of irregular sea surface. Assuming Gaussian distribution of this field we derive sampling properties of the height of the envelope field as well as of its velocity. The latter is important as the velocity of the envelope is related to the rate at which energy is transported by propagating waves. We also study how statistical distributions of group waves differ from the corresponding ones for individual waves and how a choice of a version of the envelope affects its sampling distributions. Analyzing the latter problem helps in determination of the version which is appropriate in an application in hand. NOMENCLATURE Envelope field evaluated at a position and a time instant . Gravity acceleration. Rayleigh distributed random amplitudes. Unitary spectrum for the sea surface, is the time angular frequency and its sign indicates the direction of propagation, is the azimuth. ! " Physical spectrum for the sea surface, $#$% , and &