Second-order oscillatory forces on a body in waves

When a small amplitude, water-wave train is incident upon a fixed body, a second-order analysis predicts that the body experiences a steady force and a force at twice the frequency of the incident wave. The double-frequency force is comprised of integrals of products of linear quantities over the surface of the body and the mean waterline and a term due to the second-order potential. An application of Green's theorem to the first-order potential and its horizontal derivative shows that the integral of the first order terms over the body is related in a simple way to the waterline integral and the far-field representation of the linear, diffraction potential. A minor modification of the analysis yields the farfield formulae for the drift force.