Progressive Cross Waves due to the Horizontal Oscillations of a Vertical Cylinder in Water. Evolution Equations

The paper deals with the theoretical analysis of progressive cross waves excited due to the horizontal oscillations of a vertical, surface-piercing circular cylinder in water of constant depth. Although cross waves are a phenomenon well known in laboratory wave tanks, it seems that they have not been observed around horizontally oscillating structures in fluid up to now. Such observations have recently been carried out by the authors on various models of offshore gravity platforms subjected to earthquake-like horizontal excitation in a water tank. The theoretical analysis of the problem is based on a method developed by Becker and Miles (1992) for the radial cross waves due to the motion of an axisymmetric cylindrical wavemaker. Whitham's average-Lagrangian approach is applied. It is shown that the energy transfer to the cross wave is described by the functional which is quadratic, both in the forced basic wave and in the cross wave. Therefore, the solution to second-order problems is necessary for the derivation of the evolution equations. The evolution of the cross wave is found to be described by two complex nonlinear partial differential equations with coefficients depending on a slow radial variable both in linear and nonlinear terms. The evolution equations are coupled through the nonlinear terms and through the boundary conditions as well.