TABLE OF CONTENTS In this Section we will apply linear system theory outlined in Section 1 above, to find the spectra and statistics of the output (vessel responses). In the time domain, of course, we already have the necessary tools; see Sections 2.5 and 3.4 above for radiation and wave exciting forces, respectively. The wave exciting forces depend on the time history of the wave elevation, which can be calculated from the wave spectrum using Eq. (4.125), for example. However if statistics of the responses are of primary interest, and only wave-induced forces are applied, it is much more efficient to do the computations in the frequency domain; these computations amount to first finding the response spectra and then finding the moments of the response spectra, as will be shown below.
The relationship between the spectrum of the ship motions and that of the incident waves, under the assumption of linearity and time-invariance, was derived in Section 1 above:
where S ff and S xixi correspond to input (wave) and output (motion) spectra. H( ?) is the frequency response function or "Response Amplitude Operator" (RAO), which is just the amplitude of the response per unit wave amplitude in the frequency domain. Note that Eq. (5.187) provides a means to compute the spectrum of any process that is linearly proportional to the wave height, including motions, velocities, accelerations, and forces computed using the linear theory described above.
Eq. (5.187) can be written in the...
