Abstract
A joint theoretical probability density for individual wave heights and periods, originally developed to describe storm conditions, is compared with about 2,000 routine wave recordings in the Bay of Biscay. From this joint probability density, all other mean sea-state parameters (HT 1/3, - TH 1/3, T 1/3 . . .) can be computed using H 1/3, T, and, the spectrum width parameter. The systematic discrepancy existing between theory and observation can be corrected empirically if necessary.
Introduction
Cartwright and Longuet-Higgins predicted the height of sea waves and computed the significant wave height, h 1/3, or similar variables, h 1/n, as well as the expectancy of the maximum, E(h max), of a given number of waves. They started with mo and, the total energy and the width parameter of the spectrum, respectively.To describe a sea state, a characteristic period and characteristic height are necessary. In the "zero-up-crossing" wave analysis the following periods usually appear: (1) T 1/3 is the mean of the periods usually appear:T 1/3 is the mean of the highest third of the zero-up-crossing periods,TH 1/3 is the mean of the periods connected to the waves used to compute H 1/31 andThis the mean of the zero-up-crossing periods.In the same way, T 1/n and TH 1/n also can be defined.
Wiegel found empirical relationships between these characteristic periods, based on observation at sea. Our theoretical model is based on the theory of Gaussian noise as established by Rice and leads to an explicit formula for the joint probability density of wave heights and periods. probability density of wave heights and periods. This density is fixed when given these parametersa characteristic height, a characteristic period, and epsilon. Then, by appropriate integrations, we can 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, computed values of mean quantities from observations at sea remain close to their theoretical equivalents. Any systematic discrepancy can be corrected if necessary.
THE THEORETICAL MODEL
A model was developed using as a starting point, the joint probability density for a Gaussian noise signal with a maximum value, 1, and a second derivative with respect to time, 3, as
........................................(1)
We have assigned to each positive maximum a sinusoidal wave with amplitude 1 and period T given by
SPEJ
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ON JOINT DISTRIBUTION OF WAVE HEIGHTS AND DIRECTIONS
