Second-order wave diffraction in two dimensions

A method for constructing the second-order diffraction potential associated with two-dimensional bodies is described. By introducing analytic functions such as the square of the first-order complex velocity and taking their real or imaginary parts, harmonic functions are generated which are products of first-order quantities. A particular solution for the second-order potential which satisfies the inhomogeneous free surface condition is formed from a combination of these harmonic functions. This particular solution is found to contain a vortex-like term which dominates the behaviour of the second-order potential at large depths and high frequencies. High frequency approximations to the second-order horizontal and vertical forces are developed and compared with computations of the full second-order forces on a semi-circular cylinder.