Directional Spectra Models for Design Use
Authors: Borgman L.E. 
Published: October 2008

Abstract
The directional spectrum for ocean waves has not been widely used in design computations, although it is generally acknowledged that the directional distribution is a major and significant feature of waves in the ocean. This lack of use is largely due to the absence of reliably measured directional data for waves and the resulting unavailability of "design" versions of severe wave directional spectra.
The first problem, measuring the directional spectrum can be treated quite satisfactorily with techniques based on Fourier series analysis of cross-spectrum data from an array of wave recorders. The co- and quadrature-spectral densities are represented in terms of the direction and length of the line between the wave recorders, and the Fourier series coefficients for the directional spectrum. The Fourier coefficients are then extracted by solving a system of n equations in n unknowns.
Complete formulas are given for a four-gauge array.
The method is computationally checked against a set of computer-simulated wave records having a known theoretical directional spectrum and is shown to have satisfactory precision.
The interpretation of the Fourier series representation in terms of the implicit smoothing function or filter unavoidably involved in such representations is developed in some detail. A four or five gauge array is suggested as the minimum size for adequate resolution in the Fourier series.
The smoothing involved in working with only a finite number of the Fourier coefficients can be avoided if a parameterized version of the directional spectrum is introduced. Two possibilities, the circular normal1 and the wrapped-around normal2 are considered in some detail. In addition, various circular versions of the mean, standard deviation, and skewness are developed and used to predict the parameters in the model equations.
The various parameterized models for the directional spectra can be used in setting up "design" versions of severe wave conditions.

Introduction
The intuitive basis for the concept of the directional spectral density, p(f,?), is illustrated in Fig. 1. A single wave can be represented as a point in the frequency-angle plane with the value of (amplitude)2/2 written alongside. This latter value is the mean square sea surface elevation for a cosine wave. The representation can easily be extended to the case where the sea surface is the sum of a finite number of waves, each with its own direction of travel and amplitude. This is shown in (b) of the figure. The limit to the continuum involves the introduction of a function, called the directional spectral density, which has the property that(Mathematical Equation)(Available in full paper)
where the summation is over the dfxd? infinitesimal rectangle. Another interpretation is that p(f, ?) dfd? is the variance of the sea surface elevations for-a confused sea obtained by adding together only the waves with frequency and direction of travel in the dfxd? rectangle centered at (f, ?). Thus p(f, ?) is a function giving the allocation of the total variance among the various frequencies and directions. Since wave energy per unit sea surface area is proportional to the variance, p(f, ?) can also be considered as an allocation of wave energy.