Higher-order derivatives of the Green function in hyper-singular integral equations

Hyper-singular integral equations are often applied in the frequency-domain wave diffraction/radiation analyses of marine structures with thin plates or shell sub-structures. Their numerical solutions require the higher-order derivatives of the free-surface Green function featuring hyper-singularity, and hence the corresponding evaluation is very challenging. To circumvent the associated numerical difficulties, this paper will propose alternative formulations for the higher-order derivatives of both free-surface and Rankine-source parts of the Green function. For the free-surface term GF, the higher-order derivatives are analytically expressed by a combination of GF itself and its first-order horizontal radial derivative. Further, we derive an asymptotic representation, enabling us to deal exactly with a removable singularity in this representation. The superiority of the proposed formulation is demonstrated by comparing with a conventional direct differentiation. For the Rankine-source term, analytical expressions for the velocities induced by a uniform dipole distribution over a flat panel (involving second derivatives of the Rankine source term) are presented, which is directly relevant to numerical implementation based on constant panel methods. As illustrative examples, linear hydrodynamic coefficients of submerged circular impermeable and perforated plates are calculated for verification purposes. The proposed formulas are simple and easy to implement in the hyper-singular integral equations.