Diffraction-radiation of regular waves by offshore structures and flows around ships advancing in calm water or in regular waves are commonly analyzed via potential-flow methods based on the Green functions that satisfy the corresponding free-surface boundary conditions. This realistic, practical, and widely-used approach requires evaluation of free-surface flows due to arbitrary (notably constant, linear or quadratic) distributions of singularities (sources and dipoles) over (flat or curved) panels of various shapes (notably rectangles and triangles). Indeed, reliable, efficient and practical methods to evaluate the flows due to arbitrary singularity-distributions over hull-surface panels is a crucial core-element of the Green-function method in ship and offshore hydrodynamics. This core-issue is the object of the study. Free-surface flows due to singularity-distributions over panels (and related ‘influence coefficients’) are ordinarily evaluated via a two-step procedure that involves evaluation of a Green function $G$ and its gradient $\nabla G$ (a Fourier integration) and subsequent integration of $G$ and $\nabla G$ over a panel (a space integration). This common approach involves notorious analytical and numerical complexities related to the complicated singularities of the Green functions in ship and offshore hydrodynamics. An alternative general approach, applicable to generic dispersion relations and arbitrary distributions of singularities, is expounded. This alternative approach is based on a Fourier–Kochin representation of free-surface effects, in which the space integration over hull-panels is performed first and the Fourier integration is performed subsequently. Thus, the Green function and its gradient are not evaluated in this approach. Indeed, this usual first step is bypassed, and the flow due to a singularity distribution is evaluated directly. A major advantage of the Fourier–Kochin method is that the panel-integration is a trivial task as it merely involves integration of an exponential–trigonometric function. A crucial element of the approach expounded in the study is a general analytical decomposition of free-surface effects into waves and a local flow. The waves in this fundamental flow decomposition are expressed as single integrals along the dispersion curves defined by the dispersion relation in the Fourier plane, and the local flow is given by a double Fourier integral that has a smooth integrand dominant within a compact region of the Fourier plane. This analytical flow representation does not involve approximations, i.e. is mathematically exact, as is verified via numerical applications for two main classes of flows in ship and offshore hydrodynamics: ships advancing in calm water, and diffraction-radiation of regular waves by offshore structures. These applications also demonstrate that the general approach expounded in the study provides a practical, remarkably simple, basis that is well suited for accurate and efficient evaluation of flows due to arbitrary singularity-distributions. Indeed, the approach yields a smooth wave and local-flow decomposition that avoids the complexities related to the evaluation and subsequent hull-panel integration of the singular wave and local-flow components in the classical Green functions of ship and offshore hydrodynamics. The general approach expounded in the study is applicable to other classes of flows, notably wave diffraction-radiation by ships advancing through waves in deep water and by offshore structures in finite water-depth, and these important particular applications will be considered in sequels to the study.

通常通过基于满足相应自由表面边界条件的格林函数的势流方法，通过势流方法来分析海上结构和绕在平静水中或规则波中前进的船舶周围流向的规则波的衍射辐射。这种现实，实用和广泛使用的方法需要评估自由表面流动，这是由于各种形状（尤其是矩形和矩形）的（平面或弯曲）面板上的奇异点（源和偶极子）的任意（尤其是恒定，线性或二次）分布所致。三角形）。确实，可靠，高效，实用的方法来评估船体表面面板上任意的奇异性分布所引起的流量，是船舶和海上流体动力学中格林函数方法的关键核心要素。该核心问题是研究的对象。$G$ 及其梯度 $\nabla G$ （傅里叶积分）和随后的积分 $G$ 和 $\nabla G$在面板上（空间集成）。这种通用方法涉及到臭名昭著的分析和数值复杂性，这些复杂性与船舶和海上流体动力学中格林函数的复杂性有关。阐述了适用于一般色散关系和奇点的任意分布的另一种通用方法。这种替代方法基于自由表面效应的傅里叶-科钦表示，其中首先执行对船体面板的空间积分，然后执行傅里叶积分。因此，在此方法中未评估Green函数及其梯度。实际上，绕过了这个通常的第一步，并直接评估了由于奇异性分布引起的流量。傅里叶-科钦方法的一个主要优点是面板集成是一项微不足道的任务，因为它仅涉及指数-三角函数的集成。该研究中阐述的方法的关键要素是将自由表面效应分解为波浪和局部流动的一般分析。该基本流动分解中的波表示为沿着由傅立叶平面中的弥散关系定义的弥散曲线的单个积分，局部流动由双重傅里叶积分给出，该双重傅里叶积分在傅里叶的紧致区域内具有光滑的被积体主导飞机。这种分析性流量表示形式不涉及近似值，即数学上是精确的，正如船舶和海上流体力学中两大类流量的数值应用所证实的那样：船舶在平静的水中前进，海上结构发出规则波的衍射辐射。这些应用还表明，研究中阐述的一般方法提供了一种实用的，非常简单的基础，该基础非常适合由于任意奇异分布而对流量进行准确，高效的评估。确实，该方法产生了平稳的波浪和局部流分解，避免了与评估和随后的船舶和海上流体动力学经典格林函数中的奇异波浪和局部流分量的船体面板集成相关的复杂性。研究中阐述的一般方法适用于其他类型的流动，特别是通过深水波前进的船舶和有限水深的近海结构进行波衍射辐射，