Analytical calculation of the added mass and the wave damping due to an oscillating horizontal circular cylinder using the bipolar coordinates

Abstract

The added mass and the wave damping due to an oscillating horizontal circular cylinder are analytically calculated using the bipolar coordinate. In the bipolar coordinate, a point is uniquely defined by the intersection of two orthogonal circles. These circles can be used to represent the mean surface of partially- and fully submerged oscillating circular cylinder along with the mean position of the free surface. By expressing boundary conditions on the surface of the cylinder and the free surface in terms of the bipolar coordinate, the Laplace equation in terms of the velocity potential is analytically solved, and, thus, the added mass and the wave damping are analytically obtained. Both partially/fully submerged and heave/surge cases are considered. The analytical results are compared with existing numerical studies using the boundary element method. They agree with each other very well. When the oscillating frequency converges to zero or infinite, the added mass is analytically calculated with two methods using the simplified free surface boundary condition and the difference of the relative moving directions of two circular cylinders submerged in the infinite fluids. And, the negative added mass is explained by the kinetic energy and potential energy of the fluids.