Statistical Analysis of Slow-Drift Responses

The statistical properties of second-order wave-induced response processes are investigated theoretically. Emphasis is placed on the slow-drift components. The assumed forcing waves are irregular with continuous frequency spectra. A spectral analysis of the response of a general system is made. It is shown that the slow-drift components are closely connected to the complex analytical signal and the Hilbert envelope of the wave elevation. A simple mathematical expression exists for the slow-drift components, based on the complex wave signal and the second-order impulse response of the system. By use of this explicit formula, the theoretical probability functions of slow-drift responses are investigated. The analysis is based on the Kac-Siegert method. A similar approach has earlier been applied to study the sum of both the low-frequency and the high-frequency second-order responses. Final calculations of the probability density functions are in general very complicated, but it can be simplified by the use of a simple idealized model for the second-order transfer function. Probability density curves for a few simple cases are presented.