Recursive computation of oblate spheroidal harmonics of the second kind and their first-, second-, and third-order derivatives

2012 ◽  
Vol 87 (4) ◽  
pp. 303-309 ◽  
Author(s):  
Toshio Fukushima
Keyword(s):  
Third Order ◽  
Spheroidal Harmonics ◽  
Oblate Spheroidal ◽  
Recursive Computation ◽  
Oblate Spheroidal Harmonics

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 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.

Probe size and 5th-order aberration

1987 ◽  
Vol 45 ◽  
pp. 134-135 ◽  
Author(s):  
Zhifeng Shao
Keyword(s):  
Electron Probe ◽  
Spherical Aberration ◽  
Wave Length ◽  
Cylindrical Symmetry ◽  
Electron Wave ◽  
Third Order ◽  
The Third ◽  
Probe Radius ◽  
Small Probe ◽  
Probe Size

A small electron probe has many applications in many fields and in the case of the STEM, the probe size essentially determines the ultimate resolution. However, there are many difficulties in obtaining a very small probe.Spherical aberration is one of them and all existing probe forming systems have non-zero spherical aberration. The ultimate probe radius is given byδ = 0.43Csl/4ƛ3/4where ƛ is the electron wave length and it is apparent that δ decreases only slowly with decreasing Cs. Scherzer pointed out that the third order aberration coefficient always has the same sign regardless of the field distribution, provided only that the fields have cylindrical symmetry, are independent of time and no space charge is present. To overcome this problem, he proposed a corrector consisting of octupoles and quadrupoles.

Children’s Recall of Approximations to English

1973 ◽  
Vol 16 (2) ◽  
pp. 201-212 ◽  
Author(s):  
Elizabeth Carrow ◽  
Michael Mauldin
Keyword(s):  
Language Development ◽  
Fourth Order ◽  
Third Order ◽  
Memory Processes ◽  
The Third ◽  
General Index ◽  
General Linguistic

As a general index of language development, the recall of first through fourth order approximations to English was examined in four, five, six, and seven year olds and adults. Data suggested that recall improved with age, and increases in approximation to English were accompanied by increases in recall for six and seven year olds and adults. Recall improved for four and five year olds through the third order but declined at the fourth. The latter finding was attributed to deficits in semantic structures and memory processes in four and five year olds. The former finding was interpreted as an index of the development of general linguistic processes.

The influence of third-order interactions on the density profile of associating hard spheres

1997 ◽  
Vol 91 (4) ◽  
pp. 761-767 ◽  
Author(s):  
D. HENDERSON ◽  
S. SOKOŁOWSKI ◽  
R. ZAGORSKI ◽  
A. TROKHYMCHUK
Keyword(s):  
Density Profile ◽  
Hard Spheres ◽  
Third Order
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