scholarly journals Ultimate Image Singularities in Oblate Spheroidal Coordinates with Applications in Hydrodynamics

2020 ◽  
Vol 8 (1) ◽  
pp. 32 ◽  
Author(s):  
Ioannis K. Chatjigeorgiou ◽  
Eva Loukogeorgaki ◽  
Eirini Anastasiou ◽  
Nikos Mantadakis
Keyword(s):  
Free Surface ◽  
Coordinate System ◽  
Oblate Spheroid ◽  
Fundamental Solutions ◽  
Special Focus ◽  
Spheroidal Harmonics ◽  
Oblate Spheroidal ◽  
Oblate Spheroidal Harmonics ◽  
Spheroidal Coordinates ◽  
Accurate Computations

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.

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

2011 ◽  
Vol 41 (4) ◽  
pp. 307-327 ◽  
Author(s):  
Milan Hvoždara ◽  
Igor Kohút
Keyword(s):  
Gravity Field ◽  
Separation Of Variables ◽  
Oblate Spheroid ◽  
Numerical Calculations ◽  
Simple Solution ◽  
Simple Derivation ◽  
Spheroidal Harmonics ◽  
Oblate Spheroidal ◽  
Solution Form ◽  
Oblate Spheroidal Harmonics

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.

scholarly journals II.—Oblate spheroidal harmonics and their applications

1924 ◽  
Vol 224 (616-625) ◽  
pp. 49-93 ◽  
Keyword(s):  
Harmonic Functions ◽  
Integral Method ◽  
Large Body ◽  
Oblate Spheroid ◽  
Analytical Form ◽  
Legendre Functions ◽  
Spheroidal Harmonics ◽  
Oblate Spheroidal ◽  
Physical Problems ◽  
Oblate Spheroidal Harmonics

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.

Blind Robust 3-D Mesh Watermarking Based on Oblate Spheroidal Harmonics

2009 ◽  
Vol 11 (1) ◽  
pp. 23-38 ◽  
Author(s):  
J.M. Konstantinides ◽  
A. Mademlis ◽  
P. Daras ◽  
P.A. Mitkas ◽  
M.G. Strintzis
Keyword(s):  
Spheroidal Harmonics ◽  
Oblate Spheroidal ◽  
Oblate Spheroidal Harmonics

scholarly journals Eigenfunction Expansions for the Stokes Flow Operators in the Inverted Oblate Coordinate System

2016 ◽  
Vol 2016 ◽  
pp. 1-6
Author(s):  
Maria Hadjinicolaou ◽  
Eleftherios Protopapas
Keyword(s):  
Coordinate System ◽  
Fluid Flows ◽  
Scalar Function ◽  
Order Partial Differential Equation ◽  
Coordinate Systems ◽  
Prolate Spheroidal ◽  
Spherical System ◽  
Oblate Spheroidal ◽  
Generalized Eigenfunctions ◽  
Spheroidal Coordinates

When studying axisymmetric particle fluid flows, a scalar function,ψ, is usually employed, which is called a stream function. It serves as a velocity potential and it can be used for the derivation of significant hydrodynamic quantities. The governing equation is a fourth-order partial differential equation; namely,E4ψ=0, whereE2is the Stokes irrotational operator andE4=E2∘E2is the Stokes bistream operator. As it is already known,E2ψ=0in some axisymmetric coordinate systems, such as the cylindrical, spherical, and spheroidal ones, separates variables, while in the inverted prolate spheroidal coordinate system, this equation acceptsR-separable solutions, as it was shown recently by the authors. Notably, the kernel space of the operatorE4does not decompose in a similar way, since it accepts separable solutions in cylindrical and spherical system of coordinates, whileE4ψ=0semiseparates variables in the spheroidal coordinate systems and itR-semiseparates variables in the inverted prolate spheroidal coordinates. In addition to these results, we show in the present work that in the inverted oblate spheroidal coordinates, the equationE′2ψ=0alsoR-separates variables and we derive the eigenfunctions of the Stokes operator in this particular coordinate system. Furthermore, we demonstrate that the equationE′4ψ=0  R-semiseparates variables. Since the generalized eigenfunctions ofE′2cannot be obtained in a closed form, we present a methodology through which we can derive the complete set of the generalized eigenfunctions ofE′2in the modified inverted oblate spheroidal coordinate system.

Torsional oscillation of a homogeneous elastic spheroid

1962 ◽  
Vol 52 (3) ◽  
pp. 469-484 ◽  
Author(s):  
Tatsuo Usami ◽  
Yasuo Satô
Keyword(s):  
Fundamental Mode ◽  
Free Oscillation ◽  
Torsional Oscillation ◽  
Numerical Calculations ◽  
Higher Harmonics ◽  
Oblate Spheroidal ◽  
The Earth ◽  
Spectral Peaks ◽  
The Difference ◽  
Spheroidal Coordinates

abstract There are several causes for the observations of splitting of the spectral peaks determined from the free oscillation of the earth. In this paper, the splitting due to the ellipticity is studied assuming a homogeneous earth described by oblate spheroidal coordinates. Ellipticity causes the iTn mode to split into (n + 1) modes, while the earth's rotation causes it to split into (2n + 1) modes. 1/297.0 is adopted as the ellipticity of the earth. Numerical calculations are carried out for the fundamental mode (n = 2, 3, 4) and for the first higher harmonics (n = 1). The difference between the extreme frequencies for each value of n is 0.7% (n = 2), 0.5% (n = 3), and 0.4% (n = 4).

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