Experimental Determination Of The Quadratic Transfer Function Governing Slowly Oscillating Phenomena In Irregular Waves
Authors: Rolf Rolf 
Published: October 2008

ABSTRACT
This report discusses the functional series approach to the analysis of nonlinear responses of offshore structures as can be measured in model tests. Emphasis is laid on the second order slowdrift oscillations.
A short theoretical discussion of the problem is given. As a result of this discussion an equation for estimating the quadratic transfer function of a response that is proportional to the wave height squared is presented. The analysis involves computation of the cross-bispectrum between response and wave. It is shown that Fast Fourier Transform routines can be used in computing the crossbispectrum.
Examples on analysis of both simulated data and model test results are given.

INTRODUCTION
This report is part of a research program which centers on nonlinear wave forces on floating structures resulting from nonlinear interactions between contigous parts of the wave spectrum, see e.g. references (1), (2), (3) and (4). Traditionally these non-linearities in the wave forces have been neglected as "second order effects" and thus considered small compared to the forces having a linear relationship to the wave heights. This is a reasonable approximation having resulted in good tools for predicting wave loads and other wave induced responses for floating structures. There are, however, occasions where a linear theory can not adequately predict nor describe an observed wave response. The slow-drift oscillations of moored floating structures and the ship's rolling are examples of such phenomena.
When analyzing the recorded responses of moored large-volume structures to an irregular wave field, a distinct characteristic property presents itself. It turns out that the motion of the structures take place within two different frequency domains. One corresponds directly to the waves, while the other takes place in a frequency domain characterized by longer periods than those of the wave field itself. The long period motion is often denoted by "slow drift oscillations" when one is dealing with the horizontal surge motion. The same partition into two different frequency domains may also occur for other responses such as the pitch and heave motions.
The frequency domains of the slow-drift oscillations can a priori be determined. It turns out that the slowly oscillating motion takes place in a very narrow frequency interval centered around the natural frequency of oscillation. See e.g. refs. (2) and (7). However, in estimating the magnitude of these second order effects it seems that there does not exist manageable theoretical or numerical procedures to handle this problem. One will therefore have to rely on model tests to get good predictions of the second order slow drift oscillations.
Model tests in irregular waves serve as a physical simulation of the wave environment and the floating structure's response to this. If the experiment is carefully designed (see ref. (7)), valuable observations of the slow-drift response in the particular wave field can be made. It is desireable, however, to extract more knowledge from these tests to give a better understanding of the physical phenomena involved and thereby resulting in less costly procedures for reliable estimates of the slowd rift oscillations.