Second order solution of circular disc impact problem

The second order solution of the disc impact problem is derived for the initial stage of the disc motion. The disc is of circular shape and is originally floating on the free surface, then it suddenly starts to penetrate the water with a constant entry velocity. The boundary value problem for the second order velocity potential is of mixed type with a Dirichlet condition being imposed on the free surface and a Neumann boundary condition on the disc surface. Both conditions are given in terms of the first order solution derived in a previous paper. The second order problem is complicated by singular behavior of the velocity potential on the free surface. For this reason the solution is presented as the sum of three different contributions. The first two terms are introduced to match the singular behavior of the velocity potential on the free surface whereas the third one, which is regular at the disc edge, provides the complete satisfaction of the boundary conditions both over the free surface and on the body.