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 $G^F$, the higher-order derivatives are analytically expressed by a combination of $G^F$ 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.

超奇异积分方程通常用于具有薄板或壳子结构的海洋结构的频域波衍射/辐射分析。他们的数值解需要具有奇点的自由表面格林函数的高阶导数，因此相应的评估非常具有挑战性。为了避免相关的数值困难，本文将为格林函数的自由表面部分和朗肯源部分的高阶导数提出替代公式。对于自由表面项$G^F$，高阶导数由 $G^F$本身及其一阶水平径向导数。此外，我们得出一个渐近表示，使我们能够精确处理该表示中的可移动奇点。通过与常规直接分化比较证明了所提出制剂的优越性。对于兰金源项，提出了由均匀偶极子分布在平板上引起的速度的解析表达式（涉及兰金源项的二阶导数），这与基于常数面板方法的数值实现直接相关。作为说明性示例，计算水下圆形不透水和多孔板的线性流体动力系数以进行验证。所提出的公式简单易行，可在超奇异积分方程中实现。