The Atkinson Wilcox theorem in ellipsoidal geometry

Transcription

1 J. Math. Anal. Appl The Atkinson Wilcox theorem in ellipsoidal geometry George Dassios Division of Applied Mathematics, Department of Chemical Engineering, University of Patras and ICE/HT-FORTH, Patras, Greece Received 10 April 00 Submitted by T. Fokas Abstract The famous Atkinson Wilcox theorem claims that any scattered field, no matter what the boundary conditions on the surface of the scatterer are, can be expanded into a uniformly and absolutely convergent series in inverse powers of distance and that once the leading coefficient of the expansion is known the full series can be recovered up to the smallest sphere containing the scatterer in its interior. The leading coefficient of the series is nothing else but the scattering amplitude. This is a very useful theorem, which provides the exact analogue of the Sommerfeld radiation condition, but it has the disadvantage of recovering the scattered field only outside the sphere circumscribing the scatterer. This means that an elongated obstacle which has a very large, as it compares to its volume, circumscribing sphere leaves a lot of exterior space where the scattered field cannot be recovered from its scattering amplitude. In the present work the Atkinson Wilcox theorem has been extended to the ellipsoidal system where the theorem as well as the relative recovering algorithm holds true all the way down to the smallest circumscribing ellipsoid. Considering the anisotropic character of the ellipsoidal geometry it is obvious that an appropriately chosen ellipsoid can fit almost every smooth convex obstacle. Furthermore, such a result offers the best opportunity to develop a hybrid method based on the theory of infinite elements. Two orientations dependent differential operators are introduced in the recurrence scheme which, as the ellipsoid degenerates to a sphere, one of them vanishes, while the other reduces to the Beltrami operator. A reduction to spherical geometry is also included. 00 Elsevier Science USA. All rights reserved. address: dassios@iceht.forth.gr X/0/$ see front matter 00 Elsevier Science USA. All rights reserved. PII: S00-47X

2 G. Dassios / J. Math. Anal. Appl Introduction The form that a scattered wave assumes at a large distance from the scattering region has a long history. It all started more than 130 years ago when Lord Rayleigh [,7, using Maxwell s idea of multipole expansion [18,19, demanded that the leading asymptotic form of the scattered field, far away from the scatterer, should be the field of an equivalent point wave source located within the smallest sphere that includes the scatterer in its interior. In other words, the scattered field in the radiation zone, should look like the fundamental solution of the Helmholtz equation in three special dimensions. This explicit condition was translated into an asymptotic condition for the scattered field in the neighborhood of infinity by Sommerfeld [4 in 191 and this is accepted to be the standard radiation condition of scattering theory up to these days. Successful attempts to generalize this condition to week solutions [0,1,5,6,30, to electromagnetism [7,0,3, to elasticity [9,16,17, to thermoelasticity [10,17 and to inhomogeneous waves [6 can be found in the literature, but it is important to notice that any known today form of radiation condition involves, in some form or another, the classical radiation condition of Sommerfeld. A real breakthrough, in the direction of conditions and representations for the scattered field, was the 1949 paper of Atkinson [1, who managed to replace the asymptotic radiation condition at infinity by an exact representation of the scattered field as a uniformly and absolutely convergent series in inverse powers of the distance between the observation point and a conveniently chosen point within the scattering region. The Atkinson series forms the wave analogue of Maxwell s multipole expansion in potential theory. Consequently, it is not accidental that the leading term of the Atkinson series provides the radiation condition proposed by Rayleigh and of course that it also satisfies the Sommerfeld condition. Besides the importance of the Atkinson s contribution to scattering theory the amazing, at the first sight but very reasonable at a second thought, result came from Wilcox [8, seven years later, when he proved that once the leading coefficient of the Atkinson expansion is known the full series can be recovered through an iterative process that generates in succession all the coefficients of the series from the leading one. These results were extended to electromagnetism by Wilcox [9, to scalar scattering in two dimensions by Karp [14, to elasticity by the author [8 and to thermoelasticity by Cakoni and the author [5. The real difficulties in all these extensions do not come from the expansion of the field into the relative series, but from the derivation of the appropriate iterative schemes that recover the sequence of coefficients from the leading one. For the more complicated vector fields this task takes more than straightforward extensions of standard procedures, but the important thing is that it can be done. During the last decay the Atkinson Wilcox theorem has been rediscovered since it provides a very efficient way to construct hybrid methods for solving scattering problems. Indeed, the Atkinson Wilcox theorem offers an excellent

3 830 G. Dassios / J. Math. Anal. Appl opportunity to use analytic results for the purpose of avoiding discreetizing an unbounded domain, such as the domain exterior to the scatterer, where the scattered field lives [11 13,15. An elegant idea based on the Atkinson Wilcox theorem was suggested by Burnett [ and by Burnett and Holford [3,4 who proposed the following infinite element method. The discreetization of the unbounded domain is replaced by the discreetization of the surface of a spheroid including the scatterer in its interior. Then the spheroidal patches are used as the base elements for the corresponding infinite elements, which are generated from each surface patch. An infinite element is the subset of the exterior domain which is laterally restricted by the spheroidal coordinate curves that spring out from any point of the boundary of a given surface patch. This is an excellent way to keep the discreetization procedure to a minimum and to take care of the infinite space through the analytic information that the Atkinson Wilcox theorem provides within each infinite element, where the scattered wave propagates with a known and recoverable fashion. Spheroidal geometry offers a significant improvement to the spherical geometry, since it allows for two instead of one-indepent variables, but it still restricts ourselves to rotational symmetry. In order to have a genuine three-dimensional technique for solving scattering problems we should move up to the triaxial ellipsoidal geometry, which provides the three-dimensional anisotropic analogue of the one-dimensional isotropic spherical case. In that sense, an extension of Atkinson Wilcox theorem to ellipsoidal geometry is the best one can expect for fitting solid scatterers of any shape. This extension forms the purpose of the present work. In Section the scalar scattering problem in terms of the ellipsoidal coordinate system is postulated and the Atkinson Wilcox theorem is stated and proved. Section 3 develops the analytic algorithm that provides the recurrence formulae necessary to express all the coefficients of the expansion in terms of the leading one that, as it is shown, coincides with the scattering amplitude. The interesting result here is that up to the sixth coefficient the recurrence formulae involve all previous coefficients, but for the determination of coefficients of higher order only the last five coefficients are needed. Hence, from a recurrence relation of second order, for the sphere, we move up to a recurrence relation of sixth order for the ellipsoid. This property reflects the much more complicated way that the scattered field is expressed in ellipsoidal geometry, which in turn takes care of all the freedom the ellipsoidal system allows as it compares to the spherical case. Two-second order angular differential operators are involved in the recurrence scheme, which correspond to the ellipsoidal analogue of Beltrami s operator in spherical coordinates. Finally, the reduction of the theory to spherical geometry is demonstrated in Section 4.

4 G. Dassios / J. Math. Anal. Appl The expansion theorem Let us assume that a scalar incident field u i is disturbed by an obstacle, which occupies the closure of the bounded open domain V. The boundary S = V of the obstacle is considered to be C 1 -smooth. As a result of the interaction between the incident field u i and the obstacle, a scattered field u is generated, which lives in the exterior open domain V = V S c and satisfies Sommerfeld s radiation condition at infinity. If the time dependence is introduced via the spectral component exp{ iωt} of angular frequency ω, then the spatial form of the above scattering problem is postulated in mathematical terms as follows [7,9. Find the scattered field u, which solves the Helmholtz s equation + k ur = 0 in V 1 satisfies one of the boundary conditions ur = u i r on S D or ur = n n ui r on S N or ur + ikνur = n n ui r ikνu i r on S R where /n denotes outward normal differentiation on S. Furthermore, the scattered field satisfies the asymptotic condition 1 ur ikur = O r r, r 3 uniformly over directions. The wave number k is connected to the angular frequency ω via the dispersion relation ω = c k 4 where c is the phase velocity of the medium occupying the region of propagation V and ν is a dimensionless constant known as Robin s constant [9. Given that u represents excess pressure field, the Dirichlet condition D characterizes the obstacle as soft, the Neumann condition N as hard and the Robin condition R as resistive. If in addition, the obstacle is capable of sustaining vibrations in its interior, caused by the incident field, then the obstacle is characterized as penetrable. In this case, an interior field u is also generated in V, which satisfies equation + η k u r = 0 in V 5

5 83 G. Dassios / J. Math. Anal. Appl and the transmission conditions u r = u i r + ur, 6 n u r = β u i r + ur. 7 n The constant η in 5 stands for the relative index of refraction while the constant β in 7 denotes the ratio of the mass densities in V and V whenever both media in V and in V are lossless. For anyone of the above scattering problems the scattered field u satisfies the integral representation ur = 1 [ ur e ik r r 4π nr r r eik r r r r nr ur dsr 8 S over the surface S, foreveryr in V. Suppose now that the triaxial ellipsoid x 1 α1 + x α + x 3 α3 = 1 9 with 0 <α 3 α α 1 < +, is the best externally fitting ellipsoid to the surface S of the obstacle, in the sense that it minimizes the volume of the region bounded by the ellipsoid 9 and the surface S. Given the ellipsoid 9 an ellipsoidal coordinate system ρ,µ,ν [9,19can be introduced with focal ellipse where x 1 h + x h = 1, x 3 = h 1 = α α 3, h = α 1 α 3, h 3 = α 1 α 11 are the squares of the semifocal distances. The family of confocal ellipsoids that is generated by the focal ellipse 10 is given by x1 ρ + x ρ h + x 3 3 ρ h = 1 1 where ρ [h, +. The focal ellipse 10 corresponds to ρ = h, the ellipsoid 9 is obtained when ρ = α 1 while as ρ + the ellipsoid 1 degenerates to a sphere. Hence, the ellipsoidal coordinate ρ can be thought of as the radial coordinate, while the other two ellipsoidal coordinates µ [h 3,h and ν [ h 3,h 3 can be thought of as the angular coordinates. From each point on the focal ellipse 9 a curve, defined by the intersection of the surfaces µ = constant and ν = constant, crosses vertically the plane of the focal ellipse.

6 G. Dassios / J. Math. Anal. Appl The ellipsoidal coordinates ρ,µ,ν are connected to the Cartesian coordinates x 1,x,x 3 via the expressions x 1 = h 1 h 3 ρµν, h ρ<+, x = h 1 1 h 3 ρ h 3 µ h 3 h 3 ν, h 3 µ h, 13 x 3 = h 1 1 h ρ h h µ h ν, h 3 ν h 3, where besides the family of confocal ellipsoids which corresponds to the ρ-coordinate, the µ-coordinate represents a family of confocal hyperboloids of one sheet and the ν-coordinate represents a family of confocal hyperboloids of two sheets. We are now in a position to state and prove the basic expansion theorem. Theorem. Let u be a classical solution of Helmholtz s equation 1 in V,which satisfies the Sommerfeld radiation condition 3. Letρ 0 α 1, +, so that the ellipsoidal surface S ρ0 given by x 1 ρ0 + x ρ0 + x 3 h 3 ρ0 = 1 14 h lies entirely within the open domain V,whereρ,µ,ν denotes the ellipsoidal coordinates introduced in 13.Thenu has an expansion of the form uρ, µ, ν = eikρ ρ F n µ, ν ρ n 15 which converges absolutely and uniformly for every ρ ρ 0. The expansion 15 can be differentiated with respect to the variables ρ, µ and ν any number of times and all the resulting series converge also absolutely and uniformly for ρ ρ 0. Proof. Since the ellipsoid ρ = α 1, as it is given by 9, circumscribes the surface of the scatterer S the distance between the two surfaces ρ = α 1 and S vanishes. On the other hand, the family of confocal ellipsoids 1 is the coordinate family of ellipsoidal surfaces which, as ρ travels the interval h, +, fills up the complement of the focal ellipse passing exactly once from each point in space. In other words, for ρ 1 ρ the ellipsoids ρ = ρ 1 and ρ = ρ have an empty intersection. If fact, for ρ 1 <ρ the ρ = ρ 1 ellipsoid lies entirely within the ellipsoid ρ = ρ. Consequently the distance between any two members of the family 1 corresponding to the values ρ 1 and ρ, with ρ 1 ρ,isalwayspositive. In view of Green s second identity the surface S in the representation 8 can be deformed to the surface S α1, as it is given by 1. That is ur = 1 [ ur 4π nr S α1 e ik r r r r eik r r r r nr ur dsr 16

7 834 G. Dassios / J. Math. Anal. Appl which holds true for every point r with ellipsoidal coordinates ρ,µ,ν and ρ>α 1. Furthermore, the distance between S α1 and S ρ0 with ρ 0 >α 1 is positive and let it be denoted by δ>0. Introduce now the vector R = r r 17 for which R = r r 18 and R = r r r r 19 where the cup on the top of a vector indicates unit length. Taking into consideration that the unit outward normal on the ellipsoid S α1 is given by ˆρ = α 1 α α 3 α 1 µ α 1 ν r 3 ˆx i ˆx i i=1 1 = h 1 h h 3 α1 α µ 1 ν [h 1 α α 3 µ ν ˆx 1 + α 1 h α 3 µ h 3 h 3 ν ˆx + α 1 α h 3 h µ h ν ˆx 3 and that the normal differentiation on S α1 is expressed as nr =ˆρ r = α i α α 3 α 1 µ α 1 ν the representation 16 assumes the form where uρ, µ, ν = 1 4π S α1 eikr R ˆρ r r = [ uα 1,µ,ν ˆρ r r 1 ikr R 3 e ikr α α 3 α 1 µ α 1 ν 0 ρ 1 ρ uρ,µ,ν ρ =α 1 dsr [ 3 α 1 α α 3 x i x i α1 µ α1 ν α 1. 3 i=1 i

8 G. Dassios / J. Math. Anal. Appl Long but tedious calculation lead to the expression [ R = r r = ρ + µ + ν + α1 + µ + ν h h 3 1/ ρ + µ + ν h h 3 α 1 + µ + ν h h3 cos γ 4 where the positive square root is chosen and cos γ = ˆr ˆr 1 = h 1 h h 3 ρ + µ + ν h h 3 α1 + µ + ν h h 3 [ h 1 α 1ρµνµ ν + h α h 3 ν h 3 ν + h 3 α 3 ρ h 3 ρ h µ h 3 µ h 3 h h h h µ ν µ ν. 5 Since the point of integration r varies on the ellipsoid ρ = α 1 while the point of observation r lies on or outside the ellipsoid ρ = ρ 0 it follows that R = r r δ>0 6 where δ is the distance between the surfaces S α1 and S ρ0. In fact, if we extend the function Rρ, as it is given by 4, into the complex plane the inequality 6 still holds true for all complex ρ with ρ ρ 0. Indeed, for ρ = ρ 1 + iρ 7 the components of the vector r become complex and we obtain Rρ = r r r r = r r + r Rer r 8 where denotes complex conjugation. If ρ is a root of R,then r r + r = Rer r Re r + r 9 or r r = Re r + Imr Re r 30 which implies that Imr = 0 31 and consequently the roots of Rρ have to be real.

9 836 G. Dassios / J. Math. Anal. Appl Consider now the compact set Kn,δ= { z C } Re z [ n, n, Im z [ δ,δ 3 for any fixed n N and positive δ. If there is a sequence of complex numbers r m Kn,δ such that lim r m r =0 33 m for r on the ellipsoid ρ = α 1, then by compactness, there exists a subsequence r mi converging to r 0 and r 0 r =0. But this is a contradiction of 31. If r m forms a sequence which does not belong to the compact set Kn,δ for any n N, then there exists a subsequence r mi such that lim r m i = 34 i which again contradicts the possibility of lim r m i r =0. 35 i Therefore, 6 holds for any ρ C with ρ ρ 0. This implies that the square root in 4 and consequently the expression f 1 ρ = eikr ρ 36 R is an analytic function of the variable α 1 /ρ for ρ>α 1. Therefore, the series A n f 1 ρ = ρ n 37 n=1 converges absolutely and uniformly for ρ ρ 0 >α 1 where the coefficients A n in 37 involve the variables µ, ν, µ and ν. A similar argument, also based on the fact that the expression r r with r living on S α1 and r not entering the interior of S ρ0 is always greater or equal to δ, implies that the function f ρ =ˆρ r r 1 ikr R 3 e ikr ρ 38 is analytic in the variable α 1 /ρ for ρ>α 1. Hence, the series B n f ρ = ρ n 39 n=1 converges absolutely and uniformly for ρ ρ 0 >α 1 with the coefficients B n depending on µ, ν, µ and ν. Multiplying the series 37 by the expression α α 3 α1 µ α1 ν ρ uρ,µ,ν ρ =α 1

10 G. Dassios / J. Math. Anal. Appl and the series 39 by the expression uα 1,µ,ν, adding them together and integrating the resulting expressions over S α1 we arrive at uρ, µ, νe ikρ = n=1 F n µ, ν ρ n 40 from which the expansion 15 follows immediately. Hence, the proof of the theorem is completed. 3. The recurrence scheme As it was mentioned in the introduction, the practical importance of the Atkinson Wilcox expansion theorem is connected to the possibility of recovering all the coefficients of the expansion from the first one. In fact, as ρ the ellipsoidal variable ρ is reducedto the sphericalvariable r and the expansion 15 provides the asymptotic form ur = F 0 µ, ν eikr r where F 0 µ, ν = 1 4π S [ + O 1 r u i nr r + ur 41 + ikˆr ˆn u i r + ur e ikˆr r dsr 4 is the scattering amplitude [9 in the direction ˆr specified by the ellipsoidal variables µ and ν. In order to obtain the recurrence relation for the coefficients F n of the expansion 15, we need to apply the ellipsoidal form of the Helmholtz operator on 15 and claim uniform convergence for ρ ρ 0 in order to perform term-byterm differentiation. By virtue of 13 the Laplace s operator in ellipsoidal coordinates assumes the form = ρ h 3 ρ h ρ µ ρ ν ρ + ρρ h 3 + ρρ h ρ µ ρ ν ρ 1 + µ ρ µ ν M + 1 ν ρ ν µ N 43 where

11 838 G. Dassios / J. Math. Anal. Appl and M = µ h 3 h µ µ = µ h 3 µ h N = h 3 ν h ν ν µ h 3 h µ µ µ + [ µ µ h 3 h 3 ν h ν ν + µ µ h µ 44 = ν h 3 ν h ν + [ ν ν h 3 + ν ν h ν. 45 Inserting 15 into Eq. 1, using and performing term-by-term differentiation we arrive at ρ h 3 ρ h [ ρ µ ρ ν eikρ ikρ 1 n n F n µ, ν ρ n+3 + ρρ h 3 ρ h ρ µ ρ ν eikρ µ ρ µ ν eikρ 1 ν ρ ν µ eikρ ikρ 1 n F nµ, ν MF n µ, ν ρ n+1 NF n µ, ν ρ n+1 ρ n+ + k e ikρ F n µ, ν ρ n+1 = If the nonvanishing exponential exp{ikρ} is eliminated from 46 and the whole expression is multiplied by ρ µ ρ ν the following form is obtained G n ρ F nµ, ν ρ n+3 + ν ρ µ ν + ρ µ ρ ν k MF n µ, ν ρ n+1 + µ ρ ν µ NF n µ, ν ρ n+1 F n µ, ν ρ n+1 = 0 47 where G n ρ = ρ h 3 ρ h [ ikρ 1 n n + [ ρ ρ h 3 + ρ ρ h ikρ 1 n = k ρ 6 iknρ 5 + [ k h + h 3 + nn + 1 ρ 4 + ikn + 1 h + 3 h ρ 3 [ k h h 3 + n + 1 h + h 3 ρ ikn+ 1h h 3 ρ + n + 1n + h h 3. 48

12 G. Dassios / J. Math. Anal. Appl Every term in 47 involves a series in inverse powers of ρ multiplied by a polynomial in ρ. Therefore, we can rearrange terms and write 47 as [ N M µ ν F 0 + k h µ + h 3 ν F 0 ikf 1 ρ [ + ik h + h 3 F0 + N M µ ν F 1 + k h µ + h 3 ν F 1 + F 1 4ikF + [ n n 1h h 3 F n 3 ikn 1h h 3 F n n=1 + ν M µ N µ ν F n 1 n h + h 3 Fn 1 + µ ν h h 3 k F n 1 + ikn + 1 h + h 3 Fn + N M µ ν F n+1 + h µ + h 3 ν k F n+1 + n + 1n + F n+1 ikn+ F n+ ρ n = Equating to zero the coefficient of ρ, the constant term and the coefficients of all powers of 1/ρ in 49 we obtain the recurrence formulae that we are seeking. In fact, all coefficients involve algebraic relations between the semifocal distances h, h 3, the wave number k and the angular ellipsoidal coordinates µ, ν,aswell as the two second order differential operators D 1 = 1 µ ν M + 1 µ N, 50 ν D = ν µ ν M µ µ ν N 51 where M and N are given in 44 and 45, respectively. The operators D 1 and D involve differentiation in µ and ν alone and they play, for the ellipsoidal geometry, the role that Beltrami s operator plays for the spherical geometry [8. They represent the angular part of Laplace s operator. Note that the function d µ, ν = h µ + h 3 ν = ρ r 5 entering all the coefficients in 49, expresses the difference between the square of the ellipsoidal distance ρ and the square ofthe Euclideandistance r, as a function of the direction specified by µ and ν.

13 840 G. Dassios / J. Math. Anal. Appl In view of 49 we observe that F 1 is given in terms of F 0 via ikf 1 = [ D 1 + k d F F is given in terms of F 1 and F 0 via 4ikF = [ D 1 + k d + F 1 + ik h + h 3 F0. 54 F 3 is given in terms of F, F 1 and F 0 via 6ikF 3 = [ D 1 + k d + 6 F + 3ik h + h 3 F1 + [ D + k µ ν h h 3 h + h 3 F0. 55 F 4 is given in terms of F 3, F, F 1 and F 0 via 8ikF 4 = [ D 1 + k d + 1 F 3 + 5ik h + h 3 F + [ D + k µ ν h h 3 4 h + h 3 F1 ikh h 3 F 0 56 and for each n 5 the coefficient F n is expressed in terms of the five previous coefficients F n 1, F n, F n 3, F n 4 and F n 3 as follows ikn+ F n+ = [ D 1 + k d + n + 1n + F n+1 + ikn + 1 h + h 3 Fn + [ D + k µ ν h h 3 n h + h 3 Fn 1 ikn 1h h 3 F n + n 1n h h 3 F n Therefore, for the general coefficient, the recurrence relation is of the sixth order. 4. Reduction to the sphere As it is well known, the process of reducing results for the ellipsoid to corresponding results for the sphere, is not always easy. This is so, because the ultimate singularity set for the ellipsoidal system consists of the focal ellipse, which is a two-dimensional region, while the corresponding set for the spherical system consists of a single point. This is responsible for the many indeterminant forms that appear when the three semiaxes of the ellipsoid tend to a common value. In this section we will consider the case where the three parameters α 1, α and α 3 tend to a constant value α. Hence, the triaxial ellipsoid 9 will be reduced to a sphere of radius α. Consequently, the semifocal distances h 1, h and h 3 will vanish, the ellipsoidal distance ρ will be reduced to the spherical distance r, and the angular ellipsoidal coordinates µ and ν will also vanish.

14 G. Dassios / J. Math. Anal. Appl The eccentric angular variables θ and φ are connected to µ and ν via the relations cos θ = µν h h 3, µ h 3 h 3 ν 58 sin θ cos φ =, 59 h 1 h 3 h sin θ sin φ = µ h ν, 60 h 1 h where θ, φ are, either the eccentric angles that determine the point µ, ν on the ellipsoid ρ = constant, or the spherical angles that determine the point µ, ν on the sphere r = constant. Let D s 1 and Ds be the limiting values of the differential operators D 1 and D, respectively, as the ellipsoid reduces to a sphere. Then, in view of the above discussion and the vanishing of d, formulae 53 and 54 reduce to ikf s 1 = Ds 1 F s 0, 61 4ikF s = [ D s 1 + F s 1 6 while formulae 55, 56 and 57 are incorporated into iknf s n = [ D s 1 + nn 1 F s n 1 + Ds F s n 3 63 for every n 3, where by Fn s we denote the value of the coefficient F n in the limit as the ellipsoid tends to a sphere. Next we calculate D s 1 and Ds. Since no simple formulae exist to express θ and φ with respect to µ and ν, we are forced to work via the Cartesian system where x 1, x, x 3 are easily expressible in terms of ρ, µ, ν andalsointermsofr, θ, φ. In view of 13 and the chain rule we obtain ρ h 3 µ h 3 ν µ = ρν h h 3 = x 1 µ x 1 + h 1 h 3 µ h 3 ρ h µ h ν h 1 h h µ x 1 + µx µ h 3 x 3 x + µx 3 µ h x x 3 64 and through appropriate use of 64 as well as long calculations we obtain the following Cartesian form of the operator M

15 84 G. Dassios / J. Math. Anal. Appl M = µ h 3 h µ µ = µ h h 3 x1 + µ h x 1 + µ h 3 x3 + µ h x 3 µ h 3 h µ µ x x x1 x x 1 x + µ h 3 x1 x 3 + µ x x 3 x 1 x x x 3 + µ µ h µ h [ x1 3 µ 4 x1 + x µ h 3 x3 + µ h x3 65 where the ellipsoidal system enters explicitly only through the coordinate µ. Similarly we obtain ν = ρµ ρ h 3 µ h 3 ν h h 3 x 1 h 1 h 3 h 3 ν x = x 1 ν ρ h h µ ν h 1 h h ν x 1 + νx ν h 3 x 3 x + νx 3 ν h and the following Cartesian form of the operator N N = h 3 ν h h ν 3 ν ν h ν ν = ν h h 3 x1 + ν h x x 1 x + ν h 3 x3 + ν h x1 x x 3 x 1 x + ν h 3 x1 x 3 + ν x x 3 x 1 x 3 x x 3 + ν ν h ν h [ x1 x 3 ν 4 x1 + ν h 3 x x3 + ν h x3 x x

16 G. Dassios / J. Math. Anal. Appl where again the only non-cartesian variable that appears in 67 is the ellipsoidal coordinate ν. Next we substitute the expression 65 for M and the expression 67 for N into the form 50 and 51, perform the appropriate calculations and use 13 to express D 1 and D in the following Cartesian forms and D 1 = ρ x1 [ x 1 x 1 + x x 1 + x x 3 x x 3 [ D = h x 1 + x + x 1 x 1 x x1 [ + h 3 x 1 + x 3 + x 1 x 1 x 3 ρ µ + ν + ρ h 3 x x + ρ h x 3 x3 + x 3 + x 1 x + x 1 x 3 x x 3 x 1 x x 1 x 3 x 1 + h h 3 x 1 ρ x x 1 x x 3 + x 1 x + x 3 x 1 x + x 1 x 3 68 x 1 x The advantage of the Cartesian forms 68 and 69 is that in the limit, as the ellipsoid tends to the sphere, they do not lead to any indeterminant forms anymore. The indeterminacies have been eliminated through algebraic manipulations and the expressions 68 and 69 are continuous functions with respect to the limit α 1,α,α 3 α,α,α. 70 Consequently, the limit 70 implies that and D 1 D s 1 = x + x 3 D s 0. x 1 + x1 + x 3 x + x1 + x x3 + x x 3 + x 1 x 1 x 3 x x 3 x 3 [x 1 x + x 1 x 3 x 1 x + x + x 3 x In view of the operator x 1 + x + x 3 = r =r x 1 x x 3 r x

17 844 G. Dassios / J. Math. Anal. Appl the operator D s 1 is written as where D s 1 = r x1 x1 [ x 1 x 1 + r x x r r = r r r [ x + x 1 x x 1 x 1 x 1 + x 3 x 3 x 1 x 3 x 3 x 3 x 3 x 1 + x 1 x 1 + r x3 x3 x 3 + x x 3 x + x x + x 1 + x x x 3 x 3 x x x x 1 x x 3 x 3 = r r r x 1 + x + x 3 x 1 + x + x 3 x 1 x x 3 x 1 x x 3 = r r r r r = r r r r r r r = B 74 B = 1 sin θ sin + 1 θ θ sin θ φ 75 is the Beltrani operator. Hence, as the ellipsoid reduces to the sphere D 1 becomes Beltrami s operator and D vanishes. Then the recurrence scheme recovers the Wilcox relation [8 iknfn s = [ B + nn 1 Fn 1 s 76 which holds true for every n 1. Acknowledgment Thanks are due to professor Antonios Charalambopoulos for fruitful discussion during the preparation of the present manuscript. References [1 F.V. Atkinson, On Sommerfeld s radiation condition, Philos. Mag. Ser

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