HOW FAST ARE THE TWO-DIMENSIONAL GAUSSIAN WAVES? Anastassia Baxevani 1 , Krzysztof Podg´ orski 2 , Igor Rychlik 1 1 Center for Mathematical Sciences, University of Lund Lund, Sweden 2 Department of Mathematical Sciences, IUPUI Indianapolis, Indiana, USA ABSTRACT For a stationary two-dimensional random field evolving in time, we derive the intensity distributions of appropri- ately 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 evolv- ing random fields are considered to be adequate models. We study dynamical aspects of deep sea waves by ap- plying the derived results to Gaussian fields modeling irregular sea surfaces. In doing so, we obtain distribu- tions 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. Key words: directional spectrum, Gaussian sea, Rice formula, velocities, level crossing contours, wave groups. INTRODUCTION 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 spec- trum of the underlying field. Distributions of the velocities can be studied at various regions of the surface such as points of local ex- tremes, 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. Notation and Assumptions The sea surface elevation at a position p =(x, y) and time t is represented by a homogeneous real Gaussian field W (p,t) . Sta- tistical properties of the field W (p,t) are uniquely identified by its spectrum S(λ) . For example in the case of discrete spectrum, the Gaussian field becomes a sum of cosine functions with random amplitudes and phases W (p,t)=  λ∈Λ  2S(λ)∆(λ)Rj cos(λ1x + λ2y + λ3t + j ), (1) where Λ is any subset of R 3 such that −Λ ∩ Λ= ∅ and −Λ ∪ Λ= R 3 −{0} for example, Λ =  (λ1,λ2,λ3) ∈ R 3 : λ3 ≥ 0  , ∆(λ) are infinitesimally small increments,  Rj  is a se- quence of independent Rayleigh random variables having density fR(r)= re − r 2 2 ,r ≥ 0 and  j  is a sequence of independent uni- formly distributed random variables in [0, 2π] also independent of  Rj  . In this case the covariance function can be written as the Fourier integral R(p,t)=  R 3 exp(i · (λ1x + λ2y + λ3t))S(λ)dλ. (2) The Hilbert transform of the process W is defined as ˆ W (p,t)=  λ∈Λ  2S(λ)∆(λ)Rj sin(λ1x + λ2y + λ3t + j ) (3) and the real envelope process is given by E(p,t)=  W (p,t) 2 + ˆ W (p,t) 2 . (4) We shall write the first and second order derivatives of W with respect to x , y and t as Wu = ∂W ∂u , Wuv = ∂ 2 W ∂u∂v , u, v = x, y, t. The spectral moments λ ijk , if they are finite, are defined as λ ijk =2  Λ λ i 1 λ j 2 λ k 3 S(λ)dλ . Some of the most often used co- variances of the field, its Hilbert transform and their derivatives Proceedings of The Twelfth (2002) International Offshore and Polar Engineering Conference Kitakyushu, Japan, May 26–31, 2002 Copyright © 2002 by The International Society of Offshore and Polar Engineers ISBN 1-880653-58-3 (Set); ISSN 1098-6189 (Set) 18