Applied Ocean Research 31 (2009) 65–73 Contents lists available at ScienceDirect Applied Ocean Research journal homepage: www.elsevier.com/locate/apor Exact asymmetric slope distributions in stochastic Gauss–Lagrange ocean waves Georg Lindgren ∗ Mathematical Statistics, Lund University, Box 118, SE-22100 Lund, Sweden article info Article history: Received 10 February 2009 Received in revised form 5 May 2009 Accepted 14 June 2009 Available online 30 June 2009 Keywords: Wave steepness Front–back asymmetry Fourier snapshot abstract The stochastic Lagrange wave model is a realistic alternative to the Gaussian linear wave model, which has been successfully used in ocean engineering for more than half a century. This paper presents exact slope distributions and other characteristic distributions at level crossings for symmetric and asymmetric Lagrange space and time waves. These distributions are given as expectations in a multivariate normal distribution, and they have to be evaluated by simulation or numerical integration. Interesting characteristic variables are: slopes obtained by asynchronous sampling in space or time, slopes in space or time, and horizontal particle velocity, when waves are observed when the water level crosses a predetermined level. © 2009 Elsevier Ltd. All rights reserved. 1. Introduction In the safety analysis of marine vessels and structures the statistical asymmetry properties of irregular ocean waves have drawn considerable interest. The linear Gaussian wave model does not provide any solution, and descriptions based on nonlinear wave equations are hard to calibrate and generalize. However, more or less continuously sampled data from offshore platforms are available, giving empirical distributions of various asymmetry characteristics. An example is Stansell et al. [1], who analyzed 5 Hz sampled sea surface data from a North Sea platform at 130 m water depth, giving front–back steepness distributions for various individual wave heights, showing considerable front–back asymmetry. Neither the linear nor the second order random wave theories, based on Gaussian elementary waves, allow such asymmetry. The alternative stochastic Lagrange model is a promising model, that agrees with observations and allows theoretical analysis of many of its statistical properties, some of them to be shown in this paper. A thorough study of irregular, stochastic, Lagrange models was made by Gjøsund [2], and the theory was further developed by Socquet-Juglard et al. [3] and Fouques [4]. Also Woltering and Daemrich [5] studied empirical properties of the stochastic model, based on the previous studies of regular waves. Systematic theoretical studies of the statistical properties of Lagrange ocean has just begun. In a series of recent papers, [6–10], Lindgren and Åberg have derived expressions for the exact ∗ Tel.: +46 462228547; fax: +46 462224623. E-mail address: georg@maths.lth.se. URL: http://www.maths.lth.se/matstat/staff/georg/. statistical distributions of many individual wave characteristics, including steepness/slope, both for the space formulation (with time frozen), and the time formulation (at fixed location). In this paper we extend and systematize the theoretical results concerning asymmetry properties of different first order Lagrange model. Emphasis is on front–back asymmetry. We restrict the analysis to unidirectional 2D waves, with one time and one space co-ordinate. The main message in the presentation is that the statistical correlations between the vertical and horizontal water particle movements, uniquely determine the exact skewness and asymmetry distributions in the first order stochastic Lagrange model. For engineering purposes there are many variables that are of interest, related to upcrossing or downcrossing in space or time. Quantities to be dealt with in this paper are slopes in asynchronous sampling (AS, AT), space slopes in space waves (SS), time slopes (TT), space slopes (ST), and horizontal particle speed (VT), when waves are observed at level crossings in space or time, respectively. All these are quantities important for calculation of wave impact, for example on offshore structures: the TT-case shows how fast the water level will rise, once it has reached a high level, case ST deals with the geometry of the wave when it reaches the high level, and case VT deals with forces that a wave can exert on a marine or offshore structure. The original results for slope distributions at crossings in Sec- tions 5 and 6 were derived in [6,9,10]. The asynchronous distribu- tions in Section 4 are new. 2. Stochastic Lagrange models 2.1. The Gauss–Lagrange model A stochastic Lagrange wave is a stochastic version of the Miche waves, the depth dependent modification of the Gerstner waves; 0141-1187/$ – see front matter © 2009 Elsevier Ltd. All rights reserved. doi:10.1016/j.apor.2009.06.002