Comparison With Observation At Sea Of Period Or Height Dependent Sea State Parameters From A Theoretical Model

ABSTRACT A theoretical joint probability density for individual wave heights and wave periods, originally developed to describe storm conditions, is here compared with some 2,000 routine wave recordings in the Bay of Biscay. From the joint probability density, all other mean sea-state parameters (HT1/3, TH1/3, T1/3,) can be computed using H1/3, T and (Symbol available in full paper), the spectrum width parameter. The systematic discrepancy existing between theory and observation can, if necessary, be corrected empirically. INTRODUCTION The paper published by Cartwright and Longuet Higgins is widely used to predict the height of sea waves and to compute the significant wave height, h1/3, or similar variables h1/N, as well as the expectancy of the maximum, E(hmax), of a given number of waves, starting from mo respectively, the total energy and the width parameter of the spectrum. To describe a sea state, it is necessary to have at least a characteristic period as well as a characteristic height. In the "zero-up-crossing" wave analysis, the following periods usually appear. T1/ 3 = mean of the highest third of the zero-upcrossing periods; TH1/3 = mean of the periods connected to the waves used to compute H1/ 3; T = mean of the zero-up-crossing periods. In the same way, T1/N and TH1/N can be defined. Empirical relationships between these characteristic periods, based on observation at sea can be found in the literature.8 The theoretical model developed at C.N.E.X.O. is based on the theory of Gaussian noise as established by Rice7 and leads to an explicit formula for the joint probability density of wave heights and periods; this density is fixed, given three parameters a characteristic height, a characteristic period. Then, it is possible, by appropriate integrations, to relate the different average heights to the associated average periods that describe a given sea state. Although the narrow band spectrum hypothesis is not always satisfied in reality, computed values of mean quantities from observations at sea remain quite close to their theoretical equivalents; the discrepancy, being systematic, can be corrected, if necessary, for the theoretical model. THE THEORETICAL MODEL The model is developed, using as a starting point the joint probability density for a Gaussian noise signal to have a value?1, at a maximum and a second derivative, with respect to time ?3, in the notation of Cartwright and Longuet-Higgins.(Mathematical equation available in full paper) To each positive maximum we have assigned a sinusoidal wave having for amplitude ?1 and period T given by(Mathematical equation available in full paper)