Introduction

Ellipsoidal geometries in fields governed by anisotropic potentials have many important applications in science and modern technology. To list a few, reference is made to electrostatics, electromagnetics, acoustic scattering, brain imagining, tumor growth simulation and apparently, hydrodynamics. In relevant applications the solution of the field equation is described by ellipsoidal harmonics, first obtained by Gabriel Lamé in 1837. Lamé in his celebrated paper on temperature distribution in an ellipsoid[1] has managed to separate the Laplace equation in a coordinate system consisting of second degree system is known today as the ellipsoidal fact, Lamé himself proposed the classification of the solutions of the obtained ordinary differential equation in four classes having a particular structure and he produced the first few solutions in each of these functions generate the ellipsoidal harmonics.

It is worth mentioning that this particular analysis by Lamé, gave rise to what we refer today as the theory of curvilinear coordinates, a theory also developed by Lamé[2]. Many famous mathematicians produced excellent results on Lamé functions and ellipsoidal harmonics during the whole of 19th century. A fairly complete collection of these results as well as historical notes can be found in the recently published book of Dassios[3].

All these efforts resulted in a well-defined theoretical structure for dealing with boundary value problems in ellipsoidal geometry, which, unfortunately, cannot be very effective without the use of computational techniques. Some attempts to produce numerical solutions of Lamé functions, and therefore of ellipsoidal harmonics, appear in the literature in the early 1960’s. In that context, reference is made to the books of Arscott and Khabaza[4] and Arscott[5].

In hydrodynamics, the quest for analytical solutions for hydrodynamic boundary value problems in the Laplace domain involving ellipsoidal geometries is indeed a great challenge. To the authors’ best knowledge, the only attempts in that respect were those due to Havelock[6] and Miloh[7]. Havelock[6]considered the wave resistance (only) problem of ellipsoidal forms and provided rather simplified formulae for its calculation. Havelock’s[6] theory didn’t employ the actual ellipsoidal harmonics, i.e.Lamé functions, whilst no numerical calculations were performed. Miloh[7] employed the Lagally theorem[8-10]to analyze the general maneuvering of ellipsoids. The study indeed relied on the ellipsoidal harmonics but,again no numerical implementation was performed.

The analytical approach to the solution for the hydrodynamic diffraction problem by immersed ellipsoids involves two major difficulties. The first is associated with the derivation of closed forms that satisfy the conditions of the governing boundary value problem and the second is the numerical calculation of the ellipsoidal harmonics of both kinds. In fact, the latter must be obtained for arbitrary large degrees and orders to allow achieving convergent results. However,nearly all the existing studies which provide formulas for calculating the Lamé functions stop at degree n = 3 [3]. Only recently Dassios and Satrazemi[11]provided formulas up to degree =7n . Even these formulas however require numerical solution.

In the present contribution we have undertaken both tasks, i.e., the two major difficulties mentioned in the previous paragraph. In particular, we developed a robust and efficient algorithm that was accordingly implemented into a computer code that is able to calculate the Lamé functions of both kinds and for arbitrary large degrees and orders. In addition, the concerned methodology can properly calculate the orthogonality constants by employing the general orthogonality relation satisfied by the Lamé functions of the first kind. The mathematics behind the numerical algorithm and the computer code is that of the general theory of Lamé functions that can be found in the book of Dassios[3].

As far as the analytic solutions of the hydrodynamic problems considered at present are concerned,namely the attraction force and the added mass for an ellipsoid moving steadily in parallel and close to a rigid wall as well as the ship crossing encounter problem, they were obtained by applying the method of ultimate image singularities of tri-axial ellipsoids which were rigorously derived by Miloh[12].

1. A steadily translating tri-axial ellipsoid close to a rigid wall

1.1 The ellipsoidal coordinate system

Ellipsoidal coordinates (,,)λ μ ν are defined relative to the basic ellipsoid

where

The three surfaces λ=constant (ellipsoids),μ=constant(hyperboloids of one sheet), ν=constant (hyperboloids of two sheets) form a triply orthogonal coordinate system in space. For ξ=λ,μ or ν, these surfaces satisfy the equation

The ellipsoids λ =constant are confocal with that for λ=a1. The limiting member of this family, given by λ=h2, is the area of the z-plane bounded by the“focal ellipse”