Simple-periodic and non-periodic Lamé functions and their application in the theory of conical waveguides

Simple-periodic and non-periodic Lamé functions and their

application in the theory of conical waveguides

Citation for published version (APA):

Jansen, J. K. M. (1976). Simple-periodic and non-periodic Lamé functions and their application in the theory of

conical waveguides. Stichting Mathematisch Centrum. https://doi.org/10.6100/IR137818

DOI:

10.6100/IR137818

Document status and date:

Published: 01/01/1976

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NON-PERIODIC LAME FUNCTIONS AND THEIR

NON-PERIODIC LAME FUNCTIONS AND THEIR

APPLICATION IN THE THEORY OF CONICAL WAVEGUIDES

PROEFSCHRIFT

TER

VERKRIJGING VAN VE GRAAV VAN VOCTOR IN VE

TECHNISCHE WETENSCHAPPEN AAN VE TECHNISCHE

HOGESCHOOL EINDHOVEN, OP GEZAG VAN VE RECTOR

MAGNIF1CUS, PROF. VR.

P.

VAN VER LEEVEN, VOOR EEN

COMMISSIE AANGEWEZEN VOOR HET COLLEGE VAN

VEKANEN IN HIT OPENBAAR TE VERDEVIGEN OP

VRIJVAG

g

OKTOBER

1916

TE 16.00 UUR.

doolt

JOZEF KAREL MARIA JANSEN

GEBOREN TE HORST ILl

1976

dooJt de pJtomotoJten

PROF.VR. G.W. VELTKAMP

en

CHAPTER O. INTROVUCTION ANV SUMMARY 1

o.

O. I

n:tJr.oduetOJ!.Y

JtemaJl.k..6

1 O. 1. SummaJtY 1

0.2. ComputaUonai Jtema.Jt..1u

2

O.

3.

Appendix

3

0.4.

Re6eJten~e& 5

CHAPTER 1. CONICAL COO1<VINATE SYSTEMS 10

1.0. In:tJr.odu~on 10

1. 1. The geneJtal ~on.tcal ~001r.!U.na;te ~Y.6.:tem 10

1.

2.

VeetoJt opeJuttOJt.6

..i.n

.:the

genVtal

~on.tcal ~oOJ!.d..i.nate

~y~.:tem 11

1. 3. TJUgonometJU~

60Jtm 06

.:the

.6pheJto-~ona.e.

.6Y.6.:tem

13

1.4. S-ingie-vafued

6un~0n6

-in

:the

.6pheJto-~onal .6y~.:tem

19

1.5. Re6eJten~e&

20

CHAPTER 2. THE SCALAR HELMHOLTZ EQUATION IN THE CONICAL COORDINATE

SYSTEM 22

2.0.

IntJtodu~on

22

2.1.

Sep~on

06

.:the

van-iabie

r

22

2. 2. The tJtan.6VeJt.6e dependen~e 23 2.3.

Sepall.ation

06

.:the

valUabie&

e

and

<p 27

2.4. Appendix

38

2.5. Re6eJtence.6 46

-CHAPTER 3. LAME EQUATIONS

3.0.

In:tJr.odu~on

3. 1.

Soiu:ti.on.6

06

.:the

<p

Lame.

equ.a;Uon

3.2. SoiuUon.6

06

.:the e

Lame

equ.a;Uon

3.3. The

-in6,inUe WcU.agonai

matJlice&

3.4. Appe.n.dix

3. 5 • Re6eJten~e&

47

47

48

52

60

63

67

4.0.

I~oduction 4. 1. Ca1..c.ui.ttti.on 06 ugenva1..uu

4.2.

Ca1..c.ui.ttti.on 06 e£genvec.to~ 4. 3. AppencU..x 4.4. Re6eJLenc.u

69

69

77

80

82

CHAPTER

5.

ELECTROMAGNETIC HEWS IN THE SPHERO-CONAL SYSTEM 83

5.0. I~oduc.tion 83

5. 1.

ER..ectJr.omagne:U.c. 6-i..eR..d -i..n

:the bLteJL-i..oJL 06

a.

c.one

w-i..:th

eR..Up:ti.c.a1.. C!l.o,u-,6eetion 83

5.2.

ce.tu.,6-i..~c.a.:ti.o-n06 :the

modu

88

5.3. Re6eJLwc.u 89

CHAPTER 6. COMPUTATION OF THE ASSOCIATEV LEGENVRE FUNCTIONS OF THE

FIRST KINV 90

6.0. I~oduetion 90

6.1. Compu:ta.:ti.onai.. a6pect6 06 the thJc.ee.-teJLm Jtec.uMenc.e

Jte.R..a.:ti.On-6 90 6.2. AppencU..x 93 6.3. Re.6eJLenc.u 95 SAMENVATTING CURRICULUM VITAE

96

98

CHAPTER 0

INTROVUCTI0N ANV SUMMARY

The present work is a result of antenna research carried out in the preced-ing years by the Numerical Mathematics and Service group of the Department of Mathematics and the Theoretical Electrical Engineering group of the

De-partment of Electrical Engineering at the Technological University Eindhoven. The first research object concerned the investigation of a corrugated coni-cal-horn antenna with circular cross-section and large flare angle, the so-called "scalar feed" [lJ, [2J, [3J.

This was followed by a contract-research program of the European Space

Re-search and Technology Centre (ESTEC) for the investigation of the propaga-tion and radiapropaga-tion properties of an elliptical waveguide with anisotropic boundary conditions [4J, [5J. In the ESTEC report [4J we suggested some to-pics for further research work, and one of these is the investigation of a corrugated conical-hom feed with elliptical cross-section and large flare angle. This feed illuminates a parabolic satellite reflector which has an elliptical aperture and which is used for telecommunication purposes, e.g., for western Europe (see figure 0.1) or for time zones in the U.S.A. This problem, however, has so far appeared to be too difficult.

We investigate an easier problem, namely, the electromagnetic field inside a conical hom with an elliptical cross-section and an arbitrary flare an-gle, bounded by a perfectly conducting rather than an anisotropic surface. The mathematical results of this work and the expertise gained by it will be used as tools for further investigations of horns with anisotropic boun-dary conditions.

O.1. Sumnwty

The first problem in the investigation of the electromagnetic field inside a conical-hom feed with elliptical cross-section is to select a suitable coordinate system, with the following properties:

(1) the boundary of the cone must be a coordinate surface; (2) the scalar Helmholtz equation must be separable;

(3) the parametric representation of the coordinate system must be chosen so that the solutions of the separated equations are easy to find.

A coordinate system that satisfies these conditions is the eleventh coordi-nate system of Eisenhart, viz., the sphero-conal system parametrically re-presented by trigonometric functions as described for the first time by Kraus [6J in 1955. Separating the Helmholtz equation, we obtain three equa-tions:

(1) for the r dependence; the differential equation of the spherical Bessel functions;

(2) for the <p dependence: the Lame differential equation with periodic boun-dary conditions;

(3) for the

e

dependence: the Lame differential equation with non-periodic boundary conditions.

up to now there is virtually nothing known about the analytical solutions of the Lame differential equation with non-periodic boundary conditions. In this work, however, we show that they are connected with the periodic solu-tions of the Lame equation.

We observe the same phenomenon in the case of the solutions of the Mathieu equation by separating the Helmholtz equation in the elliptic-cylinder coordinates. Between the solutions of the separated equations of the scalar Helmholtz equation we have now found a relationship in four systems, viz., the cylindrical polar, the spherical polar, the elliptic-cylinder and the sphero-conal coordinate systems. Figure 0.2 displays an overview of these solutions, and i t is easy to see that these solutions transform into one another by the corresponding transition of the coordinate systems. As in the spherical polar coordinate system, the electromagnetic field inside a horn can be expressed in terms of two independent scalar Debye potentials. And in the same way as described in the spherical polar coordinate system we give a mode classification of the electromagnetic field.

0.2.

Compu.:ta;Uona1.!LeJIlaJr.lu,

We have developed a set of procedures in ALGOL 60 for calculating the perio-dic and non-perioperio-dic solutions of the Lame equations. These procedures, and directions for use, are obtainable from the author on request.

We have calculated the first forty dominant modes of the electromagnetic field inside a horn with an eccentricity of 0.9 and a flare angle of 60°. The results of the calculated periodic solutions were compared with the

nume-rical results of the finite-difference method with h2 extrapolation applied to the Lame differential equation with periodic boundary conditions. The

calculations a.greed to 10 decimals. The results of the computed non-periodic solutions were compared with those of a fifth-order Runge-Kutta method. These calculations agreed toll decimals. Both of the calculations mentioned above were performed in double-length arithmetic to guarantee high accuracy. All calculations were performed on the digital Burroughs computer B6700 of the Computer Centre of the Technological University Eindhoven.

0.3.

Appendtx

In this section a new representation of the elliptic coordinates is intro-duced, and this contains the polar =ordinates as a special case by taking the focal distance (2h) zero. At the same time, the e,quations obtained from the Helmholtz equation on separation tend to the corresponding equations of the polar coordinate system.

The coordinates of the elliptic system denoted by r,rp are related to the Cartesian coordinates x,y by means of

x =

1h

2 + r2 cos(rp), y = r sin (rp) with h > 0 and 0 ~ rp < 2'l1, r ~ O.

First of all we observe that for h

=

0 the polar coordinate system is ob-tained. The coordinate curves are determined by the following two equations:

2 x 2 2 h =s (Ill)

1 •

These equations represent an ellipse and a hyperbola, respectively, with foci (h,O) and (-h,O).

The eccentricity of the ellipse is given by

If h

=

0, and consequently e

=

0, the ellipse becomes a circle and the equa~

tion of the hyperbola degenerates into

(_ _x _ _ ---L-) (__x _ +---L-) 0 ,

cos (rp) sin (rp) =s (rp) sin (rp)

Now we verify whether the coordinate curves are mutually perpendicular at each point in the plane. For that purpose we determine the tangent vectors to the parameter curves

~

and

~:

:

~-

_{aQ) - [}_(,,{2 + r2)Sin(Q)]

r cos(cp)

.

ax ax

It follows ~ndeedthat (a;'~)

=

o.

The scale factors of this coordinate system are

h

r :=

I

~I

dr =

0, ~(cp)

=

~(cp + 2n) ,

Again, if h = 0 these scale factors are identical with those of the polar coordinate system.

Now we shall investigate the separation of the Helmholtz equation.

We shall "Suppose that the function u = u(r,cp) satisfying the Helmholtz equa-tion

can be factored as

u (r, cp)

=

R(r) 4l(cp) •

Then we obtain the following two second-order differential equations:

d241 *2 2 2 2

--- + (k h sin (cp) + v )~

dcp2

in which v2 is the separation constant. Again, if h

=

0 we obtain the well-known differential equations of the polar coordinate system.

We can divide the cp solutions into four classes and we can expand these functions into trigonometric Fourier series [7;21J, [8;187J:

*

ce 2n(cp;k h)

*

ce 2n+1(cp;k h) 00 \ (2n) l. A 2R, cos (2R,cp) R,=O 00 \ (2n+1) l. A 2H1 cos((2R, + 1)cp) , R,=O

co *

L

B (2n+ 1) . «2R, se 2n+1(cp;k h) 2R,+1 s~n + 1)cp)

,

R,.=O co *

L

B (2n+2) . «2R, se 2n+2(cp;k h) 2R,+2 s~n + 2)cp)

.

R,=O

The corresponding r solutions are [7; 158]:

*

ce

2n(r;k h)

*

ce

_2n+ 1(r;k h)

*

ce 2n(0;k h) A(2n)

o

ce 2n+1

(0;k*h)~2

2 + r 00 I;' (2n+1) * L. (2R, + 1)A_U+₁ J₂R.+1 (k r), R,=O

*

se_2n+ 1(r;k h)

*

se 2n+1to;k h) ~k*hB(2n+1) 1 I;' (2n+1) * L. _{B2R.+1 J 2R.+1 (k r)} R,=O

*

Se 2n+2(r;k h) 0.4.

Re.6eJLenc.u

se2n+2(0;k*h)~2

+ r2 \k *2h2rB~2n+2) 00 I;' (2n+2) * L. (2R, + 2)B U +2 J 2R,+2 (k r). R,=O

[1J Jansen, J.K.M., M.E.J. Jeuken and C.W. Lambrechtse, The scalar feed. Eindhoven, Tech. Univ. Eindhoven, 1969. THE-Report 70-E-12. [2J Jansen, J.K.M., M.E.J. Jeuken and C.W. Lambrechtse, The scalar feed.

Arch. Elekt. Ubertr. 26 (1972),22-30.

[3J Jansen, J.K.M. and M.E.J. Jeuken, Surface waves in the corrugated co-nical horn. Electronics Lett. ~ (1972), 342-344.

[4J Jansen, J.K.M. and M.E.J. Jeuken, Propagation and radiation properties of elliptical waveguide with anisotropic boundary. Eindhoven, Tech. Univ. Eindhoven, 1973. ESTEC-contract no. 1657/72 HP. [5J Jansen, J.K.M. and M.E.J. Jeuken, Circularly polarised horn antenna

with an asymmetrical pattern. Budapest, Proc. fifth COlI. Microwave Comm., 3 (1974), 179-188.

[6] Kraus, L., Diffraction by a plane angular sector (doctoral disserta-tion). New York, New York University, 1955.

[7] McLachlan, N.W., Theory and application of Mathieu functions. Oxford, Clarendon Press, 1951.

[8] Meixner, J. and F. W. Schafke, Mathieusche Funktionen und Spharoid Funktionen. Berlin etc., Springer,..Verlag, 1954.

0.1.

j

7

j

j

j

j

j

j

j

j

j

j

j

j

j

j

j

j

j

j

j

j

j

j

j

j

j

j

j

j

*2 Solutions of the separated Helmholtz equation ~u + k u = O.

Elliptic-cylinder coordinate system (r,~,z)

00 00 \ (2m) - \ (2m) [L A₂R. J2R- (k c r)J[L A₂R- cos(2R-~)], m = 0,1,2, ••• R-=O n R-=O 00 00 [g (r)

!

(2R- _{+ 1) A2R-+l}(2m+l)J_2R-+l(kc r)][

!

A₂(2m+l)R-+l cos«2R-+l)~)], R-=O n R-=O 0,1,2, • •• e-Ynz m = 00 00 [g (r)

!

UB(2m) r)][

!

(2m) . 1,2, ••• U JU (kc B2R- s~n(U~)], m = R-=1 n R-=1 \ (2m+l) \ (2m+l) [L B₂R-+l J U+1(kc r)][ L B2R-+l sin«U+l)~)], m=0,1,2, ••• R-=O n R-=O where g(r) =

~2

+ r2/r.

Cylindrical polar coordinate system (r,~,z)

J 2m(kc r)cos(2m~), m = 0,1,2, ••• n J 2m+1(kc r)cos«2m+l)~), m = 0,1,2, ••• n J 2m c(k r)sin(2m~), m = 1,2, ••• n J 2m+1(kc r)sin«2m+l)~), m = 0,1,2, ••• n

Solutions of the separated Helmholtz equation

~u

+ k*2 u =

o.

Sphero-conal coordinate system (r,6,~)

""

~ (2m) 2R. ~ (2m) ]

[ l . T(2R.)A

2t P\I (cos(6»][ l. A2t c.os(2R.~) , m=O,1,2, •••

t=O n t=O ~ (2m+l) ] • [ l . B 2H1 sin(2Hl)~) , m = 0,1, ••• t=O where f(61k)

A-

k 2 cos2(6) sin(6)

Spherical polar coordinate system (r,6,~)

2m

P\I (cos (6» cos(2m~), m = 0, 1 ,2, •••

n

2m·H

P\I (cos (6» cos ( (2m + 1)~), m = 0, 1 , 2 , •••

n 2m P\I (cos(6»sin(2m~), m = 1,2, ••• n p2m+l(cos(6»sin«2m+

1)~),

m = 0,1,2, ••• \I n

CHAPTER 7

CONICAL COORDINATE SYSTEMS

1.0. I~duct£on

Of the eleven coordinate systems of Eisenhart [7;656J, [8;94J in which the scalar wave equation is separable, we shall need the sphero-aonaZ system.

However, we shall first study conical coordinate systems in general.

1. 1. The. ge.neJWi. c.ofLi.c.a1. c.oOJr.cUnate. .61j.6t:.e.m

This system is based on a family of concentric spheres and an orthogonal net of curves on the unit sphere. The conical coordinates, denoted by

r,e,~, are related to the familiar Cartesian coordinates x,y,z by x

=

r~(e

,11»,

or

x

=

rfl(e,~), y

=

rf2(e,~), Z

=

rf3(e,~) ,

where f

i (i = 1,2,3) are the Cartesian components of the unit vector

of

de-fined in a certain domain D of the e,~ plane to be specified later on. We have

and

1, or

f~

+

f~

+ f; 1 ,

The tangent vectors to the parameter curves at the point (r,e,~) are given by

~

=

fax =

r

21

1£

=

ar -' ae ae ' a~

The length of these tangent vectors have the nature of scale factors and we define them as:

h :=

I~I

r ar

Points for which heh~ 0 are singular points of the parametric representa-tion. In the vicinity of these points there is no one-to-one mapping on the Cartesian coordinates.

The set of orthogonal unit vectors in the r, e and <p directions, which vary from point to point, are defined as

e

=

dK

=

f, e

=

1-

~

= .!...

d~

e

=.!...

~

=

1-

~

-r dr - -6 he de h* de ' -<p h<p d<p h* d<p ,

e

<p

where

Thus, we can write each vector at the point (r,e,<p) in a unique way as:

1.2. Ve.etOJL opeJta.tOJL6

..[n.

:the.

ge.n.eJr.al c.on...i.c.al c.ooJLcii..n.a.te.

.6

Y.6te.m

In this section we shall deduce some vector identities in the general or-thogonal curvilinear coordinates [3;298J.

Let

Then

div F

=

div(F e + (Fee

e + Fe»

- r-r - <p-<p

For convenience we shall define

divF and

div F

:=

div F e

=.!...

L(r2F )

r- r-r 2 dr r

r

The operators with index r are the radial operators and those with index t are the transversal ones, i.e. transversal in relation to r.

Similarly we have for the curl of a vector curl F

with

curl F := (curl _F)e~e + (curl F) e

~ -Ql~ 1 dF d i d dF = ~--=:.

-

- ( h F )}e +,:...{h e ~r(heFe) - ~}~m h dQl dr Ql Ql

-e

a dO T Ql and Let now then

In the same way as before we define

and

With

a

the scalar Laplace operator we have

divrgradrg +

~

divtgradtg

This can be written as

with

and

In differential geometry the scalar transverse Laplace operator lit is known as the Beltrami operator or the second differentiator of Beltrami [1;225].

1.3. TJLi..gonome.:t.JLi..c. 60/tm

06

.the t.pheJtO-c.onlLf. -!l!!t..tem

The sphero-conal system is usually described mathematically in the algebra-ic form and/or in the elliptalgebra-ic-functional form [7;659], [8; 105].

In 1955 however, Kraus described the system with the help of trigonometric. functions. This paraI)letric representation is very important to the present work and therefore we shall investigate the trigonometric form [4], [5],

[6].

The coordinates of the sphere-conal system, denoted by r,e,~, are related to the Cartesian coordinates by

x

=

r cos(~)sin(e) , y where

o

< k < 1, 0 < k' < 1, k2 + k,2 1 , and r ?: 0, D := {(I),~)

I

0 ~ e ~ 'II, 0 ~ ~ < 2'11} •

z

y

I I I I 1 I I I I /~ I / . / I I I I Figure 1.1. / / /

-- - { - - - I \ I \ I

J

\e

I /' I / , I / I I / I \ \ /I \ \ 1/ : I / ... / ''( I I I I I \ I \ \ \ \ \ \ \ \ \

First of all we Observe that if k

=

1, and consequently k'

=

0, this coor-dinate system reduces to the spherical polar coorcoor-dinate system [8;99J. Now we verify whether the coordinate curves are mutually perpendicular at each point in space. For that purpose we determine the tangent vectors to

ax

a;K

~ [3

J

the parameter curves

r; ,

as

and alp ;298 , cos(lp)sin (6)

r cos (cp) cos (a)

-r sin(cp)sin(a)

~=

r cos(CP)/l - k2cOs2(a)

-rk,2cos (a)sin(cp) cos (cp)/A _ k,2 sin2(cp) It follows that, indeed,

(2a

2.a)

= 0 ~.

Cla ' Clcp

We also find that the vector product ~x Clx

Clr

ae

is a posi tive multiple of the vector

~

and hence r, a, cp form, in this or-der, a right-handed system of coordinates. The scale factors of this coor-dinate system are

h :=

I~

1

,

r Clr h a :=

I

~:J

=

r h_cp :=

I~I

_Clcp

=

r 2 2 2 . 2 k' cos (cp) + k s~n (a) 1 _k2cos 2 (a) k,2cos2(cp) + k2sin2 (a)

1 - k,2 sin2(cp)

We observe again that if k

=

1 these scale factors are identical with those of the spherical polar coordinate system.

The coordinate surfaces are determined by the following equations

(1.1)

(1. 2)

(1.3)

x2 + k2

l

sin2(a) 1 - k2cos 2 (a) cos (a)2 '

2 Y

Equation (1.1) represents a sphere with centre at the origin. If 6 ~ ~/2

equation (1.2) represents a cone with the vertex at the origin. The cross-section of this cone with a plane z

=

Zo ~ 0 is an ellipse satisfying the equation 2 2 x Y --=-2---";;"-2-- + -;:"2----:::2:-"---:2:----zotan (6) zO(sec (6)/k - 1) 1 •

The major axis, lying in the y,z plane, is denoted by 2a, in which

a

=

and the minor axis, lying in the x,z plane,is denoted by 2b, in which

The eccentricity is e := k'

2

2

k cos (6) z

--...

-~

,

\:---b-\ ... - ~-+_..­ \ \ \ \ \ \ \ \ Figure 1.2. y

Equation (1.3) represents an elliptic cone with the vertex at the origin. The cross-section of this cone with a plane y = YO " 0 is an ellipse which satisfies the equation

2 2

-".2--2"...:z:.---,2,---+ -2-""x-2

-yO(csc (ql)/k' - 1) yocot (ql)

1, ql " O,'If •

The major axis, lying in the y,z plane, is denoted by 2a, in which

and the minor axis, lying in the x,y plane, is denoted by 2b, in which

The eccentricity is z

-Figure 1.3.

,

\ \ \ \ a ,

I

I

-)

... I

,

I ' I

,

' \

,

.... \ I .... ... \ I I ... \ I ... \ II

...\.)1

y

We observe that i f k

=

1 equation (1.3) degenerates into

(_x -.:i....-) (_-_x_ _ + -.:i....-)

=

0

cos (ql) sin (ql) cos(cp) sin (ql) , and this is the equation of a pair of planes.

Now we investigate the one-one correspondence between the Cartesian coordi-nate system and the sphere-conal system. For that purpose we consider the functional determinant det(2.£ 2.£

~)

= h h h

ar ' aa '

a~

r

6 ~ 2 r 2 2 2 . 2 k' cos (~) + k s~n (a)

If r

=

0 or (cos(~)

=

0 and sin(6} = O}, the functional determinant is zero

and we have locally no one-to-one mapping on the Cartesian coordinate sys-tem.

Each point (0,6,~) is mapped onto the origin of the Cartesian coordinate system. (cos(~)

a

0, ~ 6 0, _~

a

71, ~ 6 71, _~

=

0 and sin(a)

=

O} holds if:

71/2; this corresponds to the half-plane kIZ - ky

=

0, y~0, x

=

0;

371/2; corresponding to k'z + ky

=

0, Y S 0, x

=

0; 71/2; corresponding to k'z + ky

=

0, Y~ 0, x

=

0; 371/2; corresponding to k'z - ky

=

0, Y S 0, x

=

O.

y

(i) (ii)

We observe that if e

= a

the elliptic cone (1.2) degenerates into a sector of the y,z plane determined by the conditions x = 0, Iyl ~ (k'/k)z.

To each point inside this sector there exist two coordinate triples, viz.,

(r,O,~) and (r,O,n-~).

If ~

=

n/2 the elliptic cone (1.3) degenerates into a sector of the y,z plane determined by the conditions x

=

0, Izi ~ (k/k')y.

At each point inside this sector, however, the mapping from (r,ei~) to (x,y,z) is one-to-one.

We observe that if k

=

1 the sectors corresponding to e =

a

and e = n dege-nerate into the z axis (this is also true in the spherical polar coordinate system).

If

e

=

n/2 the elliptic cone (1.2) degenerates into the whole x,y plane, and if ~ =

a

the elliptic cone (1.3) degenerates into the whole x,z plane.

1.4.

Sbtgle-vai.ued

6unc:UOM -<-11.

.the "phvw-col1.al

"y".tem

It is convenient to enlarge the domain D of definition to -00 < e,~ < "".

First of all, we observe that in this extended domain the following rela-tions hold:

~(r,e,~) = ~(r,e,~+2n), periodicity relation.

~(r,e,~)

=

~(r,-e,n-~), reflection relation with respect to the point

(r,O,n/2).

(iii) ~(r,e,~) = ~(r,2n-e,n-~), reflection relation with respect to the point (r,n,n/2).

It is evident that if F(x,y,z) is a single-valued function in the whole]R3 space, then

f(r,e,~) := F(x(r,e,~) ,y(r,e,~),z(r,e,~»

obeys the following relations:

(i) f(r,e,~) (ii) f(r,e,~) (iii) f{r,e,~) f{r,e,~+2n), f(r,-e,n-~) , f(r,2n-e,n-~).

Now let f (r,e ,<p) be continuously differentiable in r ~ 0, ..00 < 6,<p < 00.

Then, after some analysis, it turns out that the relations (i), (ii) and (iii) are sufficient conditions to guarantee that f corresponds to a conti-nuously differentiable function F(x,y,z) in the whole JR3 space.

We observe that the function f(r,6,<p) is doubly periodic with respect to

e

and <p, that is, periodic in both 6 and <p with period 21T. Further, the points (r,O,1T/2) and (r,1T,1T/2) are centres of symmetry of f(r,6,<p) in the extended domain _00 < 6,<p < 00.

THEOREM 1.1. Let f(r,6,<p) be a continuously differentiable function in the domain r ~ 0, _00 < 6,<p < 00. Then f is a single-valued continuously

diffe-rentiable function of the point (x,y,z) if and only if f satisfies the following conditions: (i) (ii) (iii) f(O,6,<p) f(r,6,<p) f(r,6,<p)

is independent of 6 and <po

f(r,6,<p+21T), periodicity condition.

= f(r,-6,1T-<p) } reflection conditions.

= f (r ,21T-6 ,1T-<p)

o

1.5. Re6eJLwc.e6

[1] Blaschke, W. and K. Leichtweiss, Elementare Differentialgeometrie. Berlin etc., Springer-Verlag, 1973.

[2] Buck, R.C., Advanced calculus. New York, McGraw-Hill, 1956.

[3] Hildebrand, F.B., Advanced calculus for applications. Englewood Cliffs, Prentice-Hall, 1962.

[4] Kong, A. C., The propagation and radiation properties of waveguides and horns of elliptical cross-section (doctoral dissertation). Guildford, University of Surrey, 1971.

[5] Kraus, Lo, Diffraction by a plane angular sector (doctoral disserta-tion). New York, New York University, 1955.

[6] Kraus, L. and L.M. Levine, Diffraction by an elliptic cone. New York, New York Univ. lnst. Math. Sci., 1960. EM Res. Report, EM-156. [7] Morse, P.M. and H. Feshbach, Methods of theoretical physics. New York,

[8] Page, C.H., Physical mathematics. Princeton, D. van Nostrand Comp., 1955.

CHAPTER 2

THE SCALAR HELMHOLTZ EQUATION IN THE CONICAL COORVINATE SYSTEM

2.0. I~duction

There are a number of problems in physics and engineering defined in a co-nical domain and formulated in terms of potentials satisfying the scalar Helmholtz equation. For a simple mathematical description of these problems i t is recommendable that the boundary of the domain is a coordinate surface, in order that separation of variables may be successful.

From now on we take the origin of coordinates at the apex of the cone. It should be understood that our cone is actually a half-cone in the sense of mathematics. Thus the cone C is defined by a set of straight half-lines from the origin through the points of a simple closed piecewise-smooth

*

curve on the unit sphere. It is natural to define the interior G of the cone corresponding to the interior of the curve on the unit sphere. We

de-.

fine G as part of the cone C between two concentric spheres with radii rO and r

1 (0 < rO < r1) centred about the origin. The domain G is part of G* between and on the two concentric spheres, and

G

:= G U

G.

In this chapter we shall investigate the scalar Helmholtz equation (2.1)

with boundary conditions, either (2.2) or (2.3) u = 0, ~ € G (Dirichlet condition) , au an

=

0, ~ € G (Neumann condition) •

Here n is the outward normal, k* is the wave number defined by k* which W/27T is the frequency and c is the phase velocity.

2.1. SepaJr.a,t[on 06

the

vaJLi.able r

w/c in

Let r,e,~ be general conical coordinates in the sense of section 1.1. Be-cause the boundary conditions (2.2) and (2.3) are independent of r we shall first separate the r dependence. For that purpose we suppos'e

The Helmholtz equation (2.1) is then transformed into

1 *2

v(e'~}~rR(r} +:2 R(r}~tV(e,~) + k R(r)v(e,~) 0 ,

r where as in chapter 1. It follows that and (2.4) (~ R(r) + k*2 R(r» 2 r r R(r) ~tv(e,~) v(e,~)

=

v(v+1) ).l

*

*

in which v (v+1) =).l is the separation constant.

For the r dependence we now obtain the following equation: d 2 dR *2 2

dr'(r dr) + (k r - v(V+ 1) ) R

=

0 •

We observe that this is the differential equation of the "spherical" Bessel functions with the linearly independent solutions

and

h

V

·(2} (k*r)

~ ~

H(2) (k* ) 2k*r v+~ r •

Here, H

(1~

and H

(~~

are the Hankel functions of the first and second kinds;

(1) v+.(2) V-r-,

h

v and hv are called the "spherical" Hankel functions [1;437J.

2.2. The ~veJt6e dependenc.e

From (2.4) we obtain for the e,~ dependence

with boundary condition, either

v

=

0, (e,~) €

n

(Dirichlet condition) , or

Clv

e

~

}

Here

n := {(S,Ip)

I

(S,Ip) is on a simple closed piecewise-smooth curve on the unit sphere}

,

n := {(S,Ip)

I

(S,Ip) is the interior of

n

on the unit sphere} and

n

:=

nun

Consequently, we have to investigate the eigenvalues and the eigenfunctions of the Dirichlet and Neumann problems for the Beltrami operator 6

t in a do~ main

n

on the unit sphere. It is easy to see that the Beltrami operator with either the Dirichlet or the Neumann boundary condition is a Hermitian opera-tor with respect to the inner product

(2.5) (u,v) =

If

U(S,Ip)v(S,Ip)h;(S,Ip)h;(S,Ip)dSdlp

n

*

*

Moreover, if h

S and h, are both positive and bounded in

n,

6t is uniformly elliptic here. From the spectral theory of elliptic Hermitian operators [4] it is not difficult to see that the following theorem holds.

THEOREM 2.1. Consider the two eigenvalue problems

with boundary condition, either

v

=

0, (S,Ip) E

n

(Dirichlet condition)

,

or

av

0, (S, Ip)

n

(Neumann condition)

an

=

E

.

* h*

-If h

S and Ip are continuous and nonzero in

n,

either eigenvalue problem admits a denumerable set of eigenvalues having the following properties. The eigenvalues ~* are real and form an infinite sequence (with ~ as the

n

*

*

only accumulation point) such that 0 < ~1 $ ~2 $ ••• in the Dirichlet

*

*

case, and 0 $ ~O $ ~1 $ ••• in the Neumann case.

The corresponding eigenfunctions vk(S,Ip) can be chosen such that they form a complete set of orthogonal eigenfunctions with respect to the inner

To indicate a proof of this theorem we shall first transform the Beltrami operator with the aid of stereografic projection and conformal mapping into the two-dimensional Laplace operator multiplied by a function positive on the unit disk and show that this operator has a compact inverse. The theo-rem then follows from the well-known theory of compact Hermitian operators. We do this in the following steps.

(i) The unit sphere is given by the equation

2 2 2 x +y +z 1

and we identify the north pole with the point (0,0,1). We choose the parameter representation of the unit sphere so that the north pole lies outside

n.

We then consider the stereographic projection [2;20J from the north

*

pole on the complex z plane that coincides with the x, y plane in the Cartesian coordinate system. This transformation is given by

*

z x

*

+ iy

*

~_{l-z ' ~E}

n

* * _e

where x and y are functions of and cp. * ax ax 1 x az

w=

- - - - +

_ae

,

ae 1 - z ₍₁ 2 - z)

*

~=~_1_+ ae ae 1 - z y az 2

ae '

(1 - z) ax* ax 1 x az ~=~~+ 2~' (1 - z)

*

~=~_1_+ y az acp acp 1 - z (1 _ z)2 ~ Let **

I

az*1 and hcp := ~

then, after some calculation,

and

~

(ii)

The operator 8

t transforms into the operator

*

.*

defined in the domain n bounded by the simple closed curve n

_.

as stereographic projection of nand n, respectively.

This operator can also be written as

*

in which R is the distance from the north pole to the point z .

*

According to the Riemann mapping theorem [2;172] we can map n with the aid of a conformal mapping

*

, =

g(z )

=

~ + in on the unit disk

B := {(~,n)

rt

is mapped on

B

:= {(~,n)

I

~2 + n2 = 1} •

We have now shown the equivalence of the eigenvalue problems

with boundary condition, either

v(e,ep)

=

0, (S,ep) €

n

or

;n v(S,ep) = 0, (S,ep) €

n

and the eigenvalue problems [2;175]

-2

*

f (~,n)8,v + II v

=

0, (~,n) € B ,

with boundary condition, either

v(~,n) = O,(~,n) €

B

(Dirichlet condition) ,

or

a

-2

Here f (~,n) is defined as

*

-*

Z E n

We observe that the inner product (2.5) becomes

(u,v) ..

J

f

f2

(~,n)u(~,n)v(~,n)d~dn

• B B

(iii) Invoking theorems (2.28) and (2.35) of the appendix the proof of our theorem is complete.

2.3. Se.paJtation

06

the. vaJUa.bteA 6 and cp

In the previous section we considered the spectral properties of the trans-verse Laplace operator on a domain

n

of the unit sphere with either Dirich-let or Neumann boundary conditions. The choice of the coordinates 6 and cp was relatively unimportant there, so long as the operator ~t remains uni-formly elliptic in

n.

Now we shall be more specific. If

n

corresponds to a cone with elliptic cross-section, we want to choose 6 and cp such that the boundary

n

becomes a curve 6

=

6

0 constant, so that we are able to sepa~ rate the coordinates 6 and cpo Hence we choose for 6 and cp those of the sphero-conal coordinates introduced in section 1.3, and consider the domain

n

corresponding to the parameter values 0 < 6 < 6

0, -~ < cp < ~. Since we want to consider only functions that are regular in fl, we now have to ad-join the regularity conditions of section 1.4.

We now reconsider the eigenvalue problem (v,~*) (which we will call the Bel-trami problem) :

(2.6)

0, 0<6<60,-~<cp<~,v1-0

(2.7) v(6,21T) v(6,0), _~(6,21T)ov ov (6 0)_{ocp ,}

(2.8) V(O,1T-cp) v(O,cp) , ae-(O,1T-cp)ov - ai(O,cp)ov either

(2.9)

or (2.10)

o

(Dirichlet condition)

o

(Neumann condition)

The conditions at ~

=

O,2n may be replaced by v(e,2n+~}

=

v(e,~} if we ex-tend v to a 2n-periodic function and v(e,n-~} = v(-e,~} if we extend v be-low the line e

= o.

It should be stressed that the existence of eigenvalues and eigenfunctions for Q has already been shown1these functions when considered as functions of e and ~ of the sphero-conal system certainly satisfy the regularity con-ditions. We shall now, by separation of variables, construct a set of so-called separable solutions of the above boundary-value problem and show that from these solutions in this way all eigenfunctions of the transverse Laplace operator in Q (with either Dirichlet or Neumann conditions) can be obtained by finite linear combinations.

The selection of the sphero-conal coordinate system and the boundary and regula-rity conditions leads to separating the eigenfunctions v(e,~} as

v(e,~}

=

tl(e}~(~} •

Then the equation (2.6) separates into the following two Lame equations:

(2.11)

and (2.12)

~_{1-k cos (e}de( 1-k cos (e) de) + (ll}2 2 d ~ 2 2 dtl

*

_k2 . 2_{s~n (e) -),}tl}

*

_{= 0}

o

where ),* is the separation constant.

For simplicity we put), ),* + k,2₁₁* are transformed into

Then equations (2.11) and (2.12)

(2.13) and (2.14) /1

_k,2sin2(~}~(A_k,2sin2(~} d~)

+ d~ d~ * 2 2 * + (ll (1 - k' sin (~)) - (ll - ),)}~ = 0

o

respectively.

DEFINITION 2.2. A function ~(~) is called

periodic

if ~(~) ~(2TI+~).

A function ~(~) is called

even symmetric

if ~(TI-~) = ~(~). A function ~(~)

is called

odd symmetric

if ~(TI-~) = -~(~) •

0

DEFINITION 2.3. A function v(e,~) is called

separable

if v(e,~) = 8(e)~(~}.

A function v(e,~} is called

strongly separable

if v(e,~} = 8(eH(~} with

~(~) symmetric (that is, even or odd symmetric) • 0

LEMMA 2.4. If ~n eigenfunction v(e,~) of the Beltrami problem is separable

then v(e,~} is either strongly separable or the sum of two independent

strongly separable eigenfunctions.

PROOF. If v(e,~} is an eigenfunction of the Beltrami problem then, since the coefficients of the Beltrami operator are even functions of ~ that have period TI, v(e,TI-~) is also an eigenfunction. It follows that

w₁(e,~) :=!:l(v(e,~) + v(e,TI-~»

and

are also solutions of the Beltrami problem. These functions are independent, unless one of them is identically zero. we observe that w

1(e,~) =w1(e,TI-qJ)

and w2(e,~}

=

-W2(e,TI-~}:

If v(e,~} is separable, i.e., v(e,~} = 8(e)~(~} then

w

1(e,~) = !:l8(e}(~(~) + 4>{T[-~» and

and these functions are, obviously, strongly separable.

o

We now have to find appropriate auxiliary conditions for 8 and ~.

From the periodicity condition (2.7) it follows that v(e,~)

=

v(e,~+2TI),

hence the solutions of the Lame equation (2.14) must satisfy the periodici-ty condition ~(~)

=

~(~+21r) or, equivalently, ~(O)

=

~(2TI) and ~I(O) =~,(2TI). The right-hand boundary condition belonging to equation (2.13) follows di-rectly from (2.9) and (2.10):

or

o

(Neumann problem) •

In order to find a boundary condition for the 6 equation at the left-hand end point 6 0, we observe that from the regularity conditions (2.8) it follows for a separable eigenfunction v(6,~) = 8(6)~(~) that

and

Hence for strongly separable eigenfunctions we have

(2.15)

(2.16)

d:~O)

=

0 if

~(~)

is even sYmmetric, 8(0)

=

0 if ~(~) is odd symmetric •

Conversely, we can find eigenvalues and strongly separable eigenfunctions of the two-dimensional Beltrami problem by looking for non-trivial solutions

*

of the ~ and e equations (2.13) and (2.14) with the same values of ~ and

h and satisfying the conditions

(i)

(H)

(Hi)

~(~) is periodic and sYmmetric

{

~:(O)

= 0 if

~

is even sYmmetric, 8(0)

=

0 if ~ is odd sYmmetric •

{

s(e

_o)

=

0 in the Dirichlet case , d8(6 0 )

--d6 0 in the Neumann case •

The above considerations may be summarized in the following theorem:

THEOREM 2.5. v(6,~) = 8(e)~(~) is a strongly separable eigenfunction

cor-*

responding to the eigenvalue ~ of the Beltrami problem if and only if there exists a h such that

(i) ~(~) satisfies (2.14) and is periodic and sYmmetric, (ii),8(6) satisfies (2.13) and the boundary conditions

d8(0) = 0 if ~ is even sYmmetric , d6

a(o) = 0 if ~ is odd symmetric, a(B

O) = 0 in the Dirichlet case ,

da

de(e_O) = 0 in the Neumann case.

o

From lemma 2.4 it is obvious that the strongly separable eigenfunctions span the space of all separable eigenfunctions; we will show presently that they even span the space of all eigenfunctions.

LEMMA 2.6. If a(e)~(~) is a strongly separable eigenfunction with

eigen-*

value ~ then the separation constant A satisfies

o

< A < ~

*

= v (v + 1) •

PROOF. If a

(F

0) satisfies equation (2.13), then

a

(B)

0

(B) de +

11 -

k2cos 2e

By integrating by parts and using the boundary conditions (2.15) and (2.16) respectively, as well as (a x

~:)B=BO

= 0, it follows that A<

~*.

If ~ satisfies the equation (2.14), then

211

J

d

It

? 2 d~ - 2

*

- ( 1 - k"'sI.n (ql)-)~(ql) dql -IeI ~ d~ dql

o

dql + + A

o .

By integrating by parts and using the periodicity conditions, it follows that A > O.

*

We now investigate the spectrum of the cp problem for a given 1.1 '"v (v+1) •

Let ~(cp;A) be a solution of equation (2.14) that satisfies the periodicity

conditions (2.17) and ~(O;A) ~(211;A) (2.18) d~(O'A) '" d~(211'A) dCP' dCP'

Then (since the coefficients of the differential equation have period 11)

~(CP+1I;A) is also a solution of equation (2.14) that satisfies the

periodi-city conditions (2.17) and (2.18). It follows that

and

also satisfy (2.14), (2.17) and (2.18); at least one of them is non-trivial. We observe that ~1(cp;A) '" ~1(cp+1I;A) and hence ~1(cp;A) is a solution with period 11. Also, ~2(CP;A) is a solution with period 211, for which ~2(CP;A) ",. '" -~2(cp+1I;A). From this we may conclude that (2.14) with the periodicity conditions (2.17) and (2.18) is equivalent to (2.14) on the interval (0,11) with the boundary conditions

or

~(O;A) '" ~(1I;A) and d~(O.A) '" d~(1I.A) dCP' dcp'

d~ d~

and dcp (O;A) '" - dcp (1I;A) •

Hence we have to investigate the following Sturm-Liouville eigenvalue pro-blems.

PROBLEM A.

with the periodicity conditions

~

(0) '"

~

('IT),

~(O)

'"

d~

(11) dcp dcp

PROBLEM B.

with the periodicity condition

o

-_{4>(0) = -4>}- _{(IT), - ( 0 )}d4> dlp d4> - -(IT) dlp

With the aid of lemma 2.1 and theorem 3.1 from [3;214J we can formulate the following theorem:

THEOREM 2.7. For any

-

lJ

*

>0 the eigenvalues ~., i <: 0, of problem A and the

~

eigenvalues Ai' i <: 1, of problem B, form infinite sequences (with co as the

sole accumulation point) such that

For A

=

A_Othere exists a unique eigenfunction without any zero in [O,lTJ. For A

=

A

2i+1 and A

=

A2i+2, i <: 0 there exist eigenfunctions 4>2i+l(lp) and 4>2i+2 (lp) respectively with precisely 2i + 2 zeros in [O,lT). For A

=

A

_2i+₁ and

X

=

A

_{2i+2 there exist eigenfunctions} ~2i+l(lp) and ~2i+2(lp) respectively with precisely 2i+1 zeros in [O,lT). The eigenfunctions 4>i (lp), i <: 0 and

~i(lp), i <: 1 together can be chosen such that they form a complete set of

orthonormal eigenfunctions with the inner product

(u,v)

IT

I

o

o

LEMMA 2.8. If 4>(lp) is an eigenfunction corresponding to the eigenvalue A of

the lp problem then 4>(lp) satisfies either 4>(lT+lp)

=

4>(lp) or 4>(lT+lp)

=

-4>(lp) •

PROOF. The functions Wi (lp,A) and W

2(lp,A) are independent, unless one of them is trivial. Consequently, if the eigenvalue Ai is simple then one of these functions must be trivial. If, however, Ai has multiplicity 2 then from theorem 2.7 it follows that the corresponding eigenfunctions both be-long either to problem A or to problem B; in the first case W

2(lp;A) is zero,

LEMMA 2.9. Each eigenspace of the ~ problem has an orthonormal basis consis-ting of symmetric eigenfunctions.

PROOF. If ~(~) is an eigenfunction of the ~ problem with eigenvalue A then, since the coefficients of the differential equation are even and have period

TI, ~(TI-~) is also an eigenfunction of the ~ problem.

It follows that

and

are also solutions of the ~ problem. It follows from substitution of

~ + (TI-~) at appropriate places in the integrand that (X

l,X2)

=

O. Hence Xl and X

2 are orthogonal and independent unless one of them is trivial. We observe that Xl(~) Xl(TI-~) and X2(~) = -X2(TI-~).

From theorem 2.7 we know that an eigenvalue A has at most multiplicity 2.

I f A is simple then one of the functions Xl and X₂ must be trivial. Hence

~ (~) is symmetric.

A

Let now A have multiplicity 2 and let ~ (~)

,

~(~) be an orthonormal basis for the corresponding eigenspaces. I f for one of the functions ~ and ~ both Xl and X2 are non-trivial then because (X l ,X2) 0 we can choose these functions Xl and X

2 as an orthogonal basis for the eigenspace corresponding to A. In the other case the functions ~ and ~ are both symmetric, and since they are orthogonal they can be chosen as a basis.

The results arrived at above may be summarized in the following theorem:

o

*

THEOREM 2.10. For any 1J >0 the eigenvalues of the ~ problem form an in-finite sequence AO,A

l, ••• (with as the only accumulation point) such that

The eigenfunctions ~i' i ~ 0, can be chosen such that

(i) they satisfy either ~i(TI+~)

=

~i(~) or ~i(TI+~)

=

-~i(~)'

(ii) they are symmetric,

(iii) if two eigenvalues (Ai A

~i+l(cp) , or

Moreover ~i is even symmetric and ~i+l is odd symmetric, or vice versa, (iv) they form a complete set of orthonormal eigenfunctions with respect to

the inner product

(u,v)

211

I

o

o

REMARK. In comparison with theorem 2.7 the eigenvalues are numbered so that the following statement about zeros of the eigenfunctions holds: ~O

has no zero in [0,211). ~4i-l and ~4i have precisely 4i zeros in [0,211), i 1,2, ••• ~4i+l and ~4i+2 have precisely 4i+2 zeros in [0,211), i 0,1,2, .••

THEOREM 2.11. Independent strongly separable eigenfunctions of the Beltrami

problem are orthogonal with respect to the inner product (2.5);

(u,v)

*

PROOF. Let ~ be an eigenvalue of the Beltrami problem to which one or

*

more strongly separable eigenfunctions belong. If the eigenvalue ~ is simple then the corresponding strongly separable eigenfunction is

orthogo-*

nal to all other eigenfunctions, a consequence of theorem 2.1. Let ~ now be multiple and let u

=

8i~i and v

=

8j~j be independent strongly separable eigenfunctions then 211

_{~i (CP)~.(CP)k·2c6s2(cp)}

60 8 i(6)8. (6) (u,v)

J

] dCP

J

] d6 + /1 - k,2 sin2cp /1 - k2cos 2 (6) 0 0 211 ~i(cp)~j(cp) 6 0 8. (6)8. (6) +

J

dcp

J

J. J k2sin2(6)d6 •

Ii -

k· 2sin2 (cp) /1 - k 2cos 2 (6) 0 0

From theorem 2.7 it follows that two independent eigenfunctions ~i and ~j

of the ~ problem satisfy 211

I

o

o .

o .

If Ai ~ A

j then it follows from (2.13) and the boundary conditions that 6

0

I

o

If A.

=

A then according to theorem 2.10

~ j

odd symmetric, or vice versa. Consequently

~i is even symmetric and ~j is

211

I

o

o •

Hence also in this case ei~i and ej~j are orthogonal.

0

*

THEOREM 2.12. The eigenspace corresponding to an eigenvalue ~ of the Bel-trami problem can be spanned by a finite number of mutually orthogonal strongly separable eigenfunctions •

. PROOF. Let v (6 ,~) be an eigenfunction corresponding to~

*.

Since the functions

~n(~) constitute a complete orthonormal set (see theorem 2.10), we can

ex-pand v(6,~) in a Fourier series

co

L

n=O

e*

(6)~ (~) n n with (2.19)

e*

(6) := n 211

I

o

From the equation 21T

I

o

o

it follows after some calculations that e*(a), n

=

0,1, ••• is a solution of

n

the equation

*

I 2 2 d I 2 2 den * 2 2 *

{l-k cos (a)da({l-k cos (a)da ) + (Jl (l-k cos (a» -An)e_n

o.

From the boundary conditions (2.9) or (2.10) and the definition of e* it_n

*

follows that en satisfies the right-hand boundary condition

en(a

_o)

=

0 (Dirichlet condition) or

de

dan(a_O)

=

0 (Neumann condition)

*

The appropriate boundary condition for en at a

v(a,~) satisfies the regularity conditions

and

o

can be found as follows:

We know, however, that ~n(~) is symmetric. Combining these facts it follows from (2.19) that e* satisfies the left-hand boundary condition

n

en(o)

=

0 if ~n is odd symmetric or

de

dan(O)

=

0 if ~n is even symmetric.

Comparison with theorem 2.5 shows that each eigenfunction v(a,~) of the Beltrami problem is a (possibly infinite) linear combination of strongly separable eigenfunctions e*~ (~). We shall show in two independent ways

n n

has finite multipli-the expression 8*~ (~) is a non-trivial

n n

problem corresponding to the eigenvalue

*

states that each eigenvalue ~

For each n, for which 8*

F

0,

n

eigenfunction of the Beltrami

*

~. Theorem 2.1

(i)

city, however.

(ii) From lemma 2.6 i t follows that if 8*

F

°

then the corresponding A

sa-n n

tisfies

°

< A < ~* Since ~ is the only accumulation point of the

n

sequence A

O,A1, ••• only a finite number of 8~ are not identically zero.

*

Let now ~ have multiplicity M with independent eigenfunctions

Let ~.e., j = 1, ••• ,N be strongly separable Beltrami solutions, occurring

J J

with nonzero coefficient in at least one of the Fourier expansions of v₁, ••• ,v

M• Then these functions, which by theorem 2.11 are mutually

ortho-*

gonal, span an N-dimensional eigenspace corresponding to ~ which contains the space spanned by v

1, ••• ,vM• Hence N= M. D

COROLLARY 2.13. The strongly separable eigenfunctions of the Beltrami pro-blem span the same space as the collection of all eigenfunctions of the

Beltrami problem.

D

Consequently, when in the future we consider eigenfunctions of the Beltrami problem we shall restrict ourselves, without loss of generality, to the strongly separable eigenfunctions of the Beltrami problem.

2.4.

Appe.ndix.

2.4.0. Inttoducto~y ~~kQ

Let throughout this section

B := {x _{I Ixl} < 1} ,

.

_{x

I

Ixl 1} B :=

.

B

=

B U B , f(x) E CO(B) A f (x) F

° .

DEFINITION 2.14. We note the set of all square Lebesgue integrable complex valued functions on B as the space L

2(B), which will be considered as a Hilbert space with the inner product defined by

(u,v) :=

f

f2(x)u(x)v(x)dx

B

and the norm by

lIuIl2 := (u,u)

B

o

DEFINITION 2.15. L;(B) := {u(x)

I

u(x) E L

2(B) A (u,l)

oJ.

o

o

LEMMA 2.16. With the inner product and the norm such as defined in

defini-*

tion 2.14, L

2(B) is a separable complex Hilbert space [5;27J.

Now we consider the two-dimensional eigenvalue problems

-2 0 - 2

f tm+Au

=

0, x E B, u E C (B), u E C (B), u ~ 0 wi th the boundary condition, ei ther

.

u

=

0, X E B (Dirichlet condition)

or

au •

an

=

0, x E B (Neumann condition) •

I:i is the two-dimensional Laplace operator normal.

1 - 2

LEMMA 2.17. Let u E C (B), u E C (B) then

• Here n is the outward

B

ft.u

dx

_J

au ds • an

B

.

o -

.

1 1 - 2

LEMMA 2.18. Let u € C (B), u € C (B) and v € C (B), V € C (B) then

J

(ut.v + (grad(u),grad(v»)dx=

.I

u :: ds

B

This is Green's first identity.

B

o

B

J

(ut.v - vt.u)dx

f

(u dn - v dn)dsdV dU

.

B

This is Green's second identity.

0

DEFINITION 2.20. Let S(x;y) be a fundamental solution of the Laplace equa-tion with unit source at y, then

-t.XS(x;y) = o(x - y) •

o

LEMMA 2.21. Let u € C1(B), u € C2 (B), Y € Band S(x;y) be a fundamental so~

lution of the Laplace equation with unit source at y, then

u(y) = -

f

S(x;y)t.u(x)dx +

B

This is Green's third identity.

f

(S(xoy)dU(X) - u(x)dS(x;y»ds

• ' d n dn x

B

IJ

LEMMA 2.22. Let u € C1(B), u € C2(B) , UFO be a solution of Dirichlet's

eigenvalue problem

wi th the boundary cond1 tion

u = 0, X €

Ii ,

then

I

(grad(u),grad(u»dx B

1 - 2

LEMMA 2.23. Let u € C (B), u € C (B), UFO be a solution of Neumann's

eigen-value problem

f -21'.u + AU = 0

wi th the boundary condition

au

•

an = 0, x € B ,

then

I

(grad(u) ,grad(u»dx

B

Hence, if u constant, then A 0, else A> O.

o

The last two lemmas are a consequence of Green's first identity.

2.4.1. V-i.JU.chf.e:t'~

ugenvalu.e.

pMb£.em

DEFINITION 2.24. Green's function G(x;y), x €

E,

y € B is defined as

fol-lows:

(1) G(x,y) is a fundamental solution of the Laplace equation with unit source at y.

.

(2) G(x;y)

=

0, x € B.

LEMMA 2.25. For the domain B, ~reen's function is given by

~(x;y)

= -

~1T

log<lx - yl) +

;1T

10g(lyl·lx - y*l)

o

where x €

E,

y € Band y* _:=

j;j2

1 _y.

o

We now consider Dirichlet's eigenvalue problem

-2 1 2

f l'.u + AU = 0, U € C (a), u € C (B)

with the boundary condition u

=

0, X €

B.

With the aid of Green's third

identity and the property of symmetry of G(x;y) we obtain

u(x) = A

f

G(x;y)f2(y)u(y)dy, x € B • B

This representation also applies if u E CO(B), u E C2(B) [4;225]. Let

A=~,

then

B

J

G(x;y)f2(y)u(y)dy ~u(x), X E B • Let now u E L

2(B), then, with the aid of Weyl's lemma [4;225-226,199], the following equivalence theorem holds.

THEOREM 2.26. Dirichlet's eigenvalue problem

with the boundary condition u = 0, XE

a,

is equivalent to the eigenvalue problem

B

J

G(Xiy)f2(y)u(y)dY ~U(X), X E B, ~ =

r'

1 u E L₂(B), UFO.

D

For the sake of convenience we shall write this eigenvalue problem in the . operator notation

in which the integral operator T is defined by

(Tu) (x) :=

f

G(Xiy)f 2 (y)u(y)dy • B

LEMMA 2.27. The integral operator T is a linear, Hermitian, compact opera-tori He~tianwith relation to the inner product from definition 2.14.

PROOF. Linearity is trivial. Because G(XiY)

=

G(YiX),

(Tu,v)

=

(u,Tv)

=

f f

B B

2 2

f (x)f (y)G(xiy)u(y)v(x)dxdy

Moreover f(X)G(Xiy)f(y) is square integrable over B x B, which means

J

f

B B

2 2 2

f (x)G (xiy)f (y)dxdy < 00 •

With the aid of the spectral theorem of compact Hermitian operators [5i202J, and from lemma 2.27 and theorem 2.26 we obtain

THEOREM 2.28. Dirichlet's eigenvalue problem

-2 0 - 2

f t.u+AU = 0, X€ B, u € C (B), u € C (B), uf.O

with the boundary conditions u

=

0, X€ B has denumerably many positive

eigenvalues A., j = 1,2, ••• with corresponding orthonormal eigenfunctions

o -

J 2

u

j € C (B), uj € C (B).

0

2.4.2. Neumann'.6 e-igwvai.ue pltObl.em

To solve Neumann's eigenvalue problem we shall introduce two Neumann

func-*

'tions, namely N(XiY) and N (XiY).

DEFINITION 2.29. Neumann's function N(XiY), x € B, Y€ B is defined as

fol-lows:

(1) N(XiY) is, as a function of x, a fundamental solution of the Laplace equation with the unit source at y.

(2) _an-(XiY)aN

_{= -}

1_2~ , X € B•

x

(3)

.J

N(Xiy)ds_x

B

o •

o

LEMMA 2.30. Neumann's function N(XiY) for the domain B is given by

N(XiY) -

~~

log(\x - yl) -

~~

10g(lyl·lx - y*l) where x € B, Y € Band _Y

*

_:=~Y1

lyl

o

DEFINITION 2.31. Neumann's function N*(XiY), x €

a,

Y€ B is defined as follows:

*

(1) N (XiY) is, as a function of x, solution of

aN*

.

(2) _{ari'""'(x, Y)} = 0, X € B

.J

* ₀ (3) N (Xly)ds x

a

B

Since the Neumann function as defined above is perhaps not conventional, the the following explanation may be given.

With the aid of lemma 2.17 we obtain

J

*

aN ari'""'(Xly)ds_x

.

B

J

!J.N*(Xly)dx B

J

f2(x) dx - 1

11111

2

B

a

*

aN •

and therefore it is possible to postulate a;-(x1Y)

=

0, x € B. We observe

*

that N (XIY) is determined uniquely but for a solution of a Neumann problem.

**

For convenience we suppose that N (XIY) satisfies the first two conditions of definition 2.31. Then

*

N (XIY) N**(XIY) + g(xIY)

where g(xIY) is a solution of Neumann's problem !J.g(xIY)

=

0, x € B

with the boundary condition

*(XIY)

=

0, x € 13

.

For fixed Y, this problem has as solution g(x/y)

=

c(y). This constant is

*

defined by the third condition of definition 2.31. Hence N (x;y) is unique-ly determined.

*

LEMMA 2.32. Neumann's function N (XIY) for the domain B is given by

*

N (x/y) N(x/y) -

o

Now we consider Neumann's eigenvalue problem

-2 1 - 2

f !J.u + AU

=

0, U € C (B), u € C (B), u 1

a

with the boundary condition

au

=

a

•

an ' x € B •

u(xl

From lemma 2.17

J

*

2

J

i.!.'LL

A N (xiylf (y)u(y)dy + 2 u(y)dy.

B B 1I11r

A

f

f2(ylu(yldy

o.

B

We shall now exclude A

=

0 with the corresponding eigenfunction u - 1. Hence

B

and it follows that

f

*

2

u(x)

=

A N (xiy)f (y)u(y)dy .

B

THEOREM 2.33. The "restricted" Neumann eigenvalue problem

-2 1 - 2

f 6u + AU

=

0 , X E B, u E C (B), u E C (Bl, u

t

constant

with the boundary condition

~~

=

0, X E

B

is equivalent to the eigenvalue

problem

J

N (xiy)f (ylu(yldy

*

2

=

P

1 u, X E B B \.I . 1

*

=

r'

u E L2 (Bl, UFO.

For simplicity we write this eigenvalue problem in operator notation

in which the integral operator S is defined by (Sul (x) _:=

J

B

*

2

N (xiylf (y)u(yldy •

LEMMA 2.34. The integral operator S is a linear,Hermitia~compact operator.

Using the spectral theorem of compact Hermitian operators we infer:

THEOREM 2.35. The "restricted" Neumann eigenvalue problem

-2 1 - 2

f Au + AU

=

0, X E B, u E C (B), u E C (B), u t constant

au •

with the boundary condition an

=

0, x E B, has denumerably many positive eigenvalues Aj ' j ;= 1,2,... wi th corresponding orthonormal eigenfunctions

1 - ~ C2 (B) •

u

j E C (B), uj ~

2.5. ReOvr.el1c.u

o

[lJ ---, Handbook of mathematical functions1 with formulas, graphs

and mathematical tablesl ed. by M. Abramowitz and LA. Stegun. Washington, National Bureau of Standards, 1964. (Applied Mathema-tical Series, nr. 55.)

[2J Ahlfors, L.V., Complex analysisl an introduction to the theory of ana.-lytic functions of one complex variable. New York, McGraw-Hill,

1953.

[3J Coddington, E.A. and N. Levinson, Theory of ordinary differential equations. New York, McGraw-Hill, 1955.

[4J Hellwig, G., Partial differential equationSl an introduction. New York, Blaisdell Publ. Comp., 1964.

[5J Helmberg, G., Introduction to spectral theory in Hilbert space. Amster-dam, North-Bolland Publ. Comp., 1969.

[6J Ince, E.L., Ordinary differential equations. New York, Dover Publ. Comp., 1956.

CHAPTER 3

LAME

EQUATIONS

3.0.

I~oductlon

This chapter is most important to our whole work. In fact, we here show that the solutions of the ~ Lame equation are related to those of the

e

Lame equation.

Results about solutions of the ~ Lame equation are known since a long time (Ince, [4], [5]). For the

e

Lame equation very little has been published, however.

Sometimes Levine is quoted as to have obtained various results on the solu-tions of the

e

Lame equation, but his report [8] although announced (in [7]) has not appeared.

Kong [6] in his doctoral thesis says that for lack of a better method of computing the

e

Lame solution he used a numerical approach, viZ., a 4th-or-der Runge~Kuttamethod.

In 1945, Erdelyi investigated the ~ Lame solutions by representing them by a series, infinite in general, of associated Legendre functions of the first kind [3].

In 1965, Sleeman expressed the Lame solutions associated with the corres-ponding Lame polynomials by means of series of associated Legendre functions of the second kind [11].

These results for the ~ case have led us to the idea of representing the solutions of the

e

Lame equation in terms of series of Legendre functions. The result obtained in this way, viZ., the connection between the coeffi-cients of the solution of the ~ equation and those of the solution of the corresponding

e

equation seems to be new (but similar to existing results for the periodic and non-periodic solutions of the Mathieu equation [9]).

3.1. Soi.uti.OtL6 06

the

q> Lame

equa:tion

In the previous chapter we were led to Lame's equation

(3.1)

o

o

< k' < 1, 0 ~ q> < 2n ,

with periodicity condition ~(q»

=

~(cp+2n) for the q> solutions.

Here, \I is a fixed parameter and A is the eigenvalue. Let (A,~(q») be a

solution of this eigenvalue problem. In this section we shall show that to each such eigenfunction ~(cp) there corresponds uniquely an eigenvector u of a certain infinite tridiaqonal matrix, corresponding to the same eigenvalue

A.

In the previous chapter we proved that we can restrict ourselves, without loss of generality, to 2n-periodic eigenfunctions ~(cp) satisfying

(i) either ~(n+cp) (ii) either ~(n-cp) ~(cp) or ~(Hcp) ~(q» or ~(n-cp) -~(cp) -~(cp) •

Starting from these properties we can divide the eigenfunctions into four classes and we may expand these eigenfunctions into trigonometric Fourier series, namely I:

'"

L~ A(2n) 2r cos(2rcp), n

=

0,1,2, ••• r=O {~(q»

=

~(Hcp) ~(n-q»} • II: L (2n+1)_C\l (m)_.,. :=

'"

~L A(2n+1l 2r+ 1 cos ( (2r + 1)q», n

=

0, 1 ,2 , ••• r=O -~(n+cp) -~(n-cp)} • III: L_S\l(2n) (m)_.,. := ~L B(2n). 2r sLn(2rcp)~ n

=

1,2,3, ••• r=1 {~(cp) = ~(Hq» -~(n-cp)} • IV: L~~n+1)(cp) :=

'"

~L B(2n+1). 2r+1 sLn«2r+_1)q», n = 0,1,2, •.• r=O { <l> (cp) -<l>(n+cp) <l>(n-cp)} •

We remark that the upper indices of the four classes of periodic Lame solu-tions are related to the ordinal numbers of the corresponding eigenvalues. Substitution of these formal series into the differential equation (3.1) shows that the coefficients must satisfy the following recurrence relations in which A := A -

~v(v

+1}k,2 (for the sake of convenience we have omitted the upper index)

I:

o ,

k,2 2 2 - -(v-2r+2}(v+2r-l}A + [(2r) (1-~k' ) - A]A -4 2r-2 2r k,2 - -4-(V - 2r - 1} (v+2r + 2}A 2r+2 0, r 2,3, • .. •

II: [(l-~k'2) - v (v +41)k ,2 - A]A₁ - -4-(V -2} (v+3}Ak,,2 3 = 0 , k,2 2 - T(v-2r+1} (v+2r}A 2r_1 + [(2r+1)2(1-~k') - A]A2r+1 -k,2 - T(V - 2r - 2} (v+2r + 3}A 2r+3 0, r 1,2, • .• . III: [4(1-~k'2) - A]B₂ - -4-(V-3) (v+4}Bk ,2 ₄

o ,

2,3, • •• . 0, r k,2 2 - -(v-2r+2}(v+2r-l)B + [(2r) (1_~k,2) - A]B -4 2r-2 2r k,2 - -4-(v - 2r - 1) (v + 2r + 2) B 2r+2 IV: [(1-~k'2) + V(V+1}-4-- A]Bk,2 k,2 1 - T(V-2}(v+3)B3 0 , k,2 2 2 - -4-(v-2r+1}(v+2r}B 2r_1 + [(2r+1) (l-~k' ) - A]B2r+1 -k,2 - T(V - 2r - 2} (v+2r + 3}B 2r+3 0, r 1,2,3, ••