The paper provides a detailed analysis for the second-order diffraction of monochromatic waves. For the second-order potential on the free surface, the paper proposed a forward prediction method for computing the integration on the free surface. By this method we only need to run the infinity integration on the free surface directly for a few points; a one-step quadrature can then be applied successively outward from the body for potentials at other points. For wave diffraction from a body of revolution with a vertical axis, the paper derives a new integral equation, which can cancel the leading singularity in the derivative of ring Green’s functions automatically. To obtain accurate results, different approaches are also used to deal with singularities in the ring Green’s functions in the integration on both the body surface and free surface. The method has been implemented for bodies of revolution with vertical axes, but the theory is also available for arbitrary bodies. A numerical examination is made to validate the numerical code by comparing the secondorder forces and moments on uniform and truncated cylinders and second-order diffraction potentials on the free surface with some published results. The comparison shows that the present results are in good agreement with those published. The method is also used to compute the second-order wave elevation around uniform and truncated cylinders. Ó 1998 Elsevier Science Ltd. All rights reserved.

Within the frame of potential theory and the assumption of weak nonlinearity of wave motion, a numerical method is developed for the third-order triple-frequency wave loads on fixed axisymmetric bodies in monochromatic incident waves. Applying the numerical code, numerical computations were carried out for surge and heave forces and pitch moments on a uniform cylinder, truncated cylinders and a hemisphere. Examinations were made of the contribution to third-order forces and moments from potentials at each wave order, and the relation of third-order forces and moments to wave number and drafts of cylinders.

A new set of equations of motion for wave propagation in water with varying depth is derived in this study. The equations expressed by the velocity potentials and the wave sur-face elevations include first-order non-linearity of waves and have the same dispersion characteristic to the extended Boussinesq equations. Compared to the extended Boussinesq equations, the equations have only two unknown scalars and do not contain spatial deriva-tives with an order higher than 2. The wave equations are solved by a finite element method. Fourth-order predictor-corrector method is applied in the time integration and a damping layer is applied at the open boundary for absorbing the outgoing waves. The model is applied to several examples of wave propagation in variable water depth. The computational results are compared with experimental data and other numerical results available in litera-ture. The comparison demonstrates that the new form of the equations is capable of calcu-lating wave transformation from relative deep water to shallow water.