Unsteady three-dimensional surface theory with the free surface effect: Supercritical case

The unsteady linearized theory in the supercritical case is solved by the integral equation method. The Kernel originating in the potential of the Havelock source is expressed as a Fourier series with respect to the azimuth angle, the coefficients of which are simple integrals. It is shown that convergence disappears on the free-surface downstream, behind the integration point. An asymptotic approximation of the remainder is given. This provides a way of finding the singularity. The integral equation yielding the local lift coefficient is solved by a collocation method similar to that of Multhopp. Numerical results of horizontal and vertical vibrating plates moving horizontally are given.