Directional Wave Polyspectra Beyond Stationarity
Authors: Borgman Leon E. 
Published: October 2008

Abstract
The fast Fourier transform algorithm has dominated routine wave analysis for the past 30 years. However, the need to understand nonlinear, nonstationary, directional storm waves for conceptual design of ultra-large marine structures will require new modes of analysis. A variety of possible approaches are reviewed and illustrated with data.

Introduction
The tittle for this presentation was intentionally selected to be excessively ambitious. It was intended to motivate a broad review of current wave analysis procedures, and to provide suggestions for profitable new directions for study. The title encapsulates three aspects of confused seas in the ocean environment which are poorly understood. These are:the extent and importance of nonlinear features in wave hydrodynamics (denoted by the reference to polyspectra), the nature of departures from statistical stationarity in hoth time and space, andthe intrinsic involvement of directionality in the nonlinear nonst, ationary mix of waves which occur in the ocean.
The lack of current, understanding of these topics was a strong and broadly held concern of the participants in a recent, workshop sponsored by the Office of Naval Research, August 19 20, 1997, in Arlington, Virginia1, Forty scientists and engineers attended the workshop, including representatives from educational institutions, private industry, and government with expertise spanning a wide range of disciplines in ocean science and engineering. A conclusion of the workshop was that "confirmed theoretical and numerical models of the fully three-dimensional, nonlinear problem do not exist and the suitability of lower dimensional or order approximations is unknown. Furthermore, data analysis and statistical techniques must be extended beyond the typically linear, stationary approaches and then proven to be effective in providing useful results." If this is accepted as true, what then should be sound, practical approaches to go beyond linear, stationary and/or unidirectional analysis of confused seas in the deep ocean'? The following thoughts and suggestions by the author are intended to encourage a dialogue within the ocean engineering community as to what should be the best future directions for study. Analysis with the FFT Algorithm For about 30 years, l'Outine analysis of wave data has been dominated by the fast Fourier transform (FFT) algorithm. The algorithm is incredibly fast, easy to use, and highly appropriate within the context of lineal' directional mixt. 1lres of waves. If the water level elevation time history is denoted by [ ?(n?t); n = 0,1,2, ..., N - 1], the FFT representation or Fourier series
(mathematical equation)(available in full paper)
both ?(n?t) and A(m?f) are formally periodic in time, with period N ?t, and in frequency, with period N ? f, If stationarity is imposed on this periodicity, to extend the formulas formally outside the data interval of (0, N ?t), the computational structure for spectral estimation becomes remarkably simple and convenient. However, there is a dark side to this. For if the representation is both periodic and stationary, then the Fourier coefficient Am = Um - iVm is uncorrelated with any other coefficient Am' with 0 &lt;m, m' &lt; N /2.