Ultimate Image Singularities in Oblate Spheroidal Coordinates with Applications in Hydrodynamics

This study exploits the Touvia Miloh oblate spheroid theorem with a special focus on hydrodynamical applications. The theorem provides explicit relations that express the oblate spheroidal harmonics, given in terms of the fundamental solutions of the Laplace equation. Here, the theorem is employed to transform the underlying Green’s function into the relevant coordinate system and, consequently, to formulate the diffraction potential. The case considered refers to the axisymmetric placement of the spheroid, namely, symmetrical axis perpendicular to the free surface. The mathematical formulations have been implemented numerically providing exceptionally accurate computations, which manifests the consistency and robustness of the relevant formulas.

Gravity field due to a homogeneous oblate spheroid: Simple solution form and numerical calculations

Gravity field due to a homogeneous oblate spheroid: Simple solution form and numerical calculations We present a simple derivation of the interior and exterior gravitational potentials due to oblate spheroid and also its gravity field components by using the fundamental solution of the Laplace equation in oblate spheroidal coordinates. Application of the method of separation of variables provides an expression for the potential in terms of oblate spheroidal harmonics of degree n = 0, 2. This solution is more concise and suitable for the numerical calculations in comparison with infinite series in spherical harmonics. Also presented are the computations in the form of potential isolines inside and outside the spheroid, as well as for the gravity field components. These reveal some interesting properties of the gravity field of this fundamental geophysical body useful for the applied gravimetry.

II.—Oblate spheroidal harmonics and their applications

The harmonic functions appropriate to the oblate spheroid, which are of the form P n (ζ), q n (ζ), or P n (ιζ), Q n (ιζ), when the large letters denote the usual Legendre functions, have received but little attention. Yet they provide, as we shall show in this memoir, a very elegant analysis of a variety of physical problems. We propose to exhibit a series of illustrations of their use, together with a large body of analysis whose applications extend very far, and lead to elegant solutions, in an analytical form, of problems which are in many cases new. In other cases—for example, the classical problems of electrified circular discs under influence—geometrical methods which lead to serious limitations have alone been effective hitherto. The analysis by spheroidal harmonics is shown to be intimately associated with that by other methods, such as the Fourier-Bessel integral method, and important theorems of analysis are involved. We may begin with a brief summary of the more important expressions already known for these functions. If a potential function ϕ satisfies ∇ 2 ϕ = 0 and a transformation to cylindrical coordinates ( z, ρ, ω ) is made, ∂ 2 ϕ / ∂ ρ 2 + 1/ ρ ∂ ϕ / ∂ ρ ∂ 2 ϕ / ∂ z 2 + 1/ ρ 2 ∂ 2 ϕ / ∂ ω 2 = 0, where ρ is distance from the axis.