Diffraction of nonlinear random waves by a vertical cylinder in deep water

Abstract

Abstract A solution, exact to second-order, is presented for the nonlinear diffraction of random waves by a fixed, surface-piercing vertical circular cylinder in deep water. The incident wave field is considered as a stationary random process, and the nonlinear diffraction problem is analyzed utilizing the Stokes perturbation expansion procedure combined with a Fourier-Stieltjes spectral representation of the stationary random wave kinematics. The second-order velocity potential is explicitly obtained by applying a modified form of Weber's Integral Theorem to invert the inhomogeneous second-order free-surface condition. Particular attention is directed towards the second-order diffraction forces on the cylinder. The spectral description of the second-order diffraction forces involves a complicated integral expression with highly oscillatory wave-wave interaction kernels and multiple convolutions of the linear wave spectrum. The present approach provides a complete spectral description of the second-order diffraction forces, and yields the spectral densities of the diffraction forces at the sum and difference frequencies. Numerical results are presented which illustrate the spectral content of the diffraction force due to an incident wave field represented by a superposition of waves described by band-limited white noise processes centered at different frequencies.