A Finite-Depth Unified Theory for the Linear and Second-Order Problems of Slender Ships

In this paper an extension of the unified slender-body theory is introduced to solve the finite-depth seakeeping problem of a slender ship. The far-and near-field behaviors of the velocity potential in finite depth are introduced, and a new kernel of the integral equation is derived for the heave and pitch motions of a slender ship at zero speed. The kernel of the integral equation is expressed in a series form which makes the integral equation easy to solve. Based on the present theoretical background, computations were carried out, and the hydrodynamic coefficients and motion RAOs were obtained. The computational results are compared with WAMIT, and a nice agreement is shown. The present method is extended to the computation of the second-order mean drift forces and moment in infinite and finite depth. Motion RAOs of sway, roll and yaw are obtained using strip theory, and the drift forces using the far-field momentum equations. The results show favorable agreement with WAMIT. Using the drift forces, the wave drift damping matrix is obtained for infinite depth. Aranha's formula is applied, and the damping coefficients are compared with the existing data. The present study shows that unified theory is an efficient and accurate design tool for slender ships.