Frequency of maxima of non-narrow banded stochastic processes

In many engineering problems one deals with quantities that must be considered to be of a stochastic or random nature. This is true for the natural environment such as wind, waves, and earthquakes, which are the driving mechanism behind the loadings on a wide variety of land-based and offshore structures. From a design perspective it is important to determine the expected highest value of a stochastic process, and structural fatigue life. The relevant procedures are reasonably well established for processes that have narrow band spectra, but it is much less clear how to deal with non-narrow band cases. In this paper it is shown that the extremes of a Gaussian, non-narrow band process are asymptotically equal to the extremes calculated according to the narrow band formula. Also demonstrated is that fatigue estimates may, with good accuracy, be based on the narrow band formula unless the bandwidth becomes extremely large. These statements are illustrated by examples of a process with (1) a low pass box spectrum, and (2) a Pierson-Moskowitz wave amplitude spectrum. It is also shown that the bandwidth parameter may in some cases be counter-intuitive, as the Pierson-Moskowitz spectrum has a larger bandwidth parameter than the box spectrum, even when both cover the same frequency range.