Advances in Mathematical Physics

Volume 2015 (2015), Article ID 572458, 12 pages

http://dx.doi.org/10.1155/2015/572458

## Connection Formulae between Ellipsoidal and Spherical Harmonics with Applications to Fluid Dynamics and Electromagnetic Scattering

^{1}Department of Chemical Engineering, University of Patras, 26504 Patras, Greece^{2}Institute of Chemical Engineering Sciences, Stadiou Street, P.O. Box 1414, Platani, 26504 Patras, Greece

Received 26 July 2014; Accepted 30 October 2014

Academic Editor: George Dassios

Copyright © 2015 Michael Doschoris and Panayiotis Vafeas. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

The environment of the ellipsoidal system, significantly more complex than the spherical one, provides the necessary settings for tackling boundary value problems in anisotropic space. However, the theory of Lamé functions and ellipsoidal harmonics affiliated with the ellipsoidal system is rather complicated. A turning point would reside in the existence of expressions interlacing these two different systems. Still, there is no simple way, if at all, to bridge the gap. The present paper addresses this issue. We provide explicit formulas of specific ellipsoidal harmonics expressed in terms of their counterparts in the classical spherical system. These expressions are then put into practice in the framework of physical applications.

#### 1. Introduction

The ellipsoidal coordinate system, by its very nature, is demanding concealing numerous difficulties. The main reason can be associated with the acquisition of solutions for miscellaneous operators. Even in the case of the Laplacian, deriving the corresponding eigensolutions is a nontrivial task. The French engineer and mathematician Gabriel Lamé in the mid nineteenth century, following an ingenious argument, separated variables for the Laplace operator arriving at the functions, which carry nowadays his name. Taking the product of Lamé functions leads to the ellipsoidal harmonics.

But the complications regarding the particular system do not end here. In contrast to the theory of spherical harmonics, only ellipsoidal harmonics of low order have been computed in closed form [1, 2]. Why? First of all, a recursive technique in order to generate Lamé functions does not exist. Although we know that Lamé functions are connected by three-term recurrence relations [3], to the authors knowledge no procedure calculating the corresponding coefficients has been proposed so far. This particular impediment forces us to undergo an involved algorithm from which the Lamé functions are determined. This essentially two-step operation requires the computation of the roots of polynomial functions allowing nontrivial solutions for the initial linear homogeneous systems. We note that the previously indicated algorithm can be applied analytically only for Lamé functions up to the seventh degree [2]. Higher-order terms demand computational implementations [4] introducing numerical instability, which in the sequence is transferred to the calculation of the corresponding Lamé function.

The aforementioned hurdles could in theory be avoided on the assumption that the functions of Lamé and corresponding ellipsoidal harmonics would be able to be expressed in terms of Legendre functions and spherical harmonics, respectively. Although, in principle, the possibility exists, no general formulae connecting these functions are available. The absence of such relations is justified bearing in mind that ellipsoidal harmonics are not reducible in a straightforward and unique way to the corresponding spherical harmonics. Nonetheless, distinct ellipsoidal harmonics can be represented in terms of a finite set of spherical harmonics of degree equal or less in view of the initial ellipsoidal harmonic.

The present communication aims towards this direction. The section following is devoted to a brief introduction of the peculiarities of the ellipsoidal system, Lamé functions, and ellipsoidal harmonics (missing details can be found in [1]). We then continue elaborating explicit relations connecting ellipsoidal harmonics with the spherical ones. Finally, Section 3, introducing two real-world problems, showcases the efficiency of the derived formulas.

#### 2. Mathematical Background

##### 2.1. The Confocal Ellipsoidal Coordinate System

The ellipsoidal coordinates of each point are provided that . The coordinate , which determines a family of confocal ellipsoids, is comparable to the radial variable in spherical coordinates. On the other hand, the coordinates and specify a family of confocal hyperboloids of one and two sheets, respectively, and correlate to the angular variables and .

In the ellipsoidal coordinate system each direction is unique providing a particular perception of anisotropic space. Since any direction exhibits its own character, the ellipsoidal coordinate system requires the introduction of a reference ellipsoid establishing the variations in angular dependence, a direct analogy to the unit sphere. This reference ellipsoid defined by is not one of a kind but must incorporate the physical reality at hand. The constants are the squares of the semifocal distances and with fixed parameters determining the reference semiaxes.

A decisive aspect when concerned with boundary value problems in ellipsoidal coordinates resides in the spectral decomposition of the Laplacian. Separating variables leads to three identical ordinary differential equations, known as Lamé’s equation. The corresponding solutions are the so-called Lamé functions of the first kind , where designates the set of nonnegative integers and , as well as the matching second kind functions .

Analytically, the first, second, and third degree Lamé functions of the first kind are presented below. The variable represents either variable , , or . Therein, where the constants and , as well as and , , are given as where represents the Kronecker delta. The above constants satisfy the following relations: respectively.

The corresponding Lamé functions of the second kind are for every and , where is an elliptic integral.

In view of problems where the boundary consists of a triaxial confocal ellipsoid, the product of two Lamé functions belonging to the same class defines the surface ellipsoidal harmonics ; that is, whereas designate the interior ellipsoidal harmonics.

On the other hand, the exterior ellipsoidal harmonics are specified as where

##### 2.2. Connecting Ellipsoidal and Spherical Harmonics

We already mentioned in the introduction the paucity of general formulas associating ellipsoidal harmonics with the same degree or less spherical harmonics. Another way to comprehend this is the following. As the triaxial ellipsoid deteriorates to a sphere, the ellipsoidal harmonics reduce to the so-called spheroconal harmonics which are a form of spherical harmonics. The spheroconal system, which incorporates the radial coordinate of the spherical system with the coordinates of the ellipsoidal system that specify orientation over any ellipsoidal surface , is established on the same parameters , , thus preserving its ellipsoidal characteristic. Nevertheless, although it seems that a general framework cannot be established, it is possible to represent distinct ellipsoidal harmonics with reference to finite terms of spherical harmonics (see Figure 1 for an illustration).