The statistical properties of the second-order Froude-Krylov force on a cylinder (whether a vertical cylinder or a horizontal submerged cylinder), for narrow-band spectra, are investigated. For this purpose two families of stochastic processes are defined and for each family the probability density function and the probabilities of exceedance of the absolute maximum and of the absolute minimum are obtained. It is then proven that the abovementioned Froude-Krylov force processes belong to these stochastic families. The predictions for the Froude-Krylov force on a horizontal submerged cylinder agree with the results of a small-scale field experiment.
The amplitude of the wave force on a large structure may be obtained as the product of the Froude-Krylov force (which is defined as the force on the equivalent water volume) and the diffraction coefficient of the wave force (Sarpkaya and Isaacson, 1981). According to the linear theory of wind-generated waves (Longuet- Higgins, 1963; Phillips, 1967) the linear Froude-Kxylov force, whether on a vertical cylinder or on a horizontal submerged cylinder, represents a random Gaussian process of time. Therefore both the absolute maximum and the absolute minimum of the linear Froude-Krylov force have the same Rayleigh distribution, if the spectrum is very narrow (Longuet-Higgins, 1952). Boccotti (2000) has shown that, for large horizontal cylinders, the two random processes wave force on the solid cylinder, and Froude-Krylov wave force have nearly the same very narrow spectrum, the same nonlinearity effects, and the same statistical properties: equal distribution of the normalized crest-to-trough heights, distribution of the normalized absolute maximum and distribution of the normalized absolute minimum. This conclusion is based on the evidence of a small-scale field experiment which consisted in the real time comparison of the wave forces on a horizontal submerged cylinder and on an ideal equivalent water cylinder (see also Boccotti, 1996).
