The theoretical foundations of the boundary integral method are considered for inviscid monochromatic internal waves, and an analytical approach is presented for the solution of the boundary integral equation for oscillating bodies of simple shape: an elliptic cylinder in two dimensions, and a spheroid in three dimensions. The method combines the coordinate stretching introduced by Bryan and Hurley in the frequency range of evanescent waves, with analytic continuation to the range of propagating waves by Lighthill's radiation condition. Not only are the waves obtained for arbitrary oscillations of the body, with application to radial pulsations and rigid vibrations, but also the distribution of singularities equivalent to the body, allowing later inclusion of viscosity in the theory. Both a direct representation of the body as a Kirchhoff–Helmholtz integral involving single and double layers together, and an indirect representation involving a single layer alone, are considered. The indirect representation is seen to require a certain degree of symmetry of the body with respect to the horizontal and the vertical. As the surface of the body is approached the single- and double-layer potentials exhibit the same discontinuities as in classical potential theory.

中文翻译：

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分层流体中振动体的边界积分

考虑了非粘性单色内波边界积分法的理论基础，提出了求解简单形状的振动体边界积分方程的解析方法：二维椭圆柱体和三维椭球体. 该方法将 Bryan 和 Hurley 在渐逝波频率范围内引入的坐标拉伸与 Lighthill 辐射条件下对传播波范围的解析延拓相结合。不仅是针对物体的任意振动获得的波，适用于径向脉动和刚性振动，而且是等效于物体的奇点分布，允许稍后在理论中包含粘度。将身体直接表示为涉及单层和双层的基尔霍夫-亥姆霍兹积分，以及仅涉及单层的间接表示。间接表示被认为需要身体相对于水平和垂直具有一定程度的对称性。当接近物体表面时，单层和双层电位表现出与经典电位理论中相同的不连续性。