Cycle Amplitude Distributions for Gaussian Processes - Exact and Approximative Results

Georg Lindgren and K. Bertram Broberg

Centre for Mathematical Sciences
Mathematical Statistics
Lund Institute of Technology,
Lund University,

ISSN 1403-9338
Wave cycles, i.e. pairs of local maxima and minima, play an important role in many engineering fields. Many cycle definitions are used for specific purposes, such as crest-trough cycles in wave studies in ocean engineering and rainflow cycles for fatigue life predicition in mechanical engineering. The simplest cycle, that of a pair of local maximum and the following local minimum is also of interest as a basis for the study of more complicated cycles. This paper presents and illustrates modern computational tools for the analysis of different cycle distributions for stationary Gaussian processes with general spectrum. It is shown that numerically exact but slow methods will produce distributions in almost complete agreement with simulated data, but also that approximate and quick methods work well in most cases.
Of special interest is the relation between the cycle average and the cycle amplitude for the simple maximum-minimum cycle and its implication for the corresponding amplitude. It is observed that for a Gaussian process with rectangular box spectrum, these quantities are almost independent and that the amplitude is not far from a Rayleigh distribution. It will also be shown that
had there been a Gaussian process where exact independence hold then the amplitude would have had an exact Rayleigh distribution. Unfortunately no such Gaussian process exists.
Key words:
crest-trough waves, max-min waves, rainflow cycles, random fatigue, spectral density, stationary Gaussian process, wave period distribution