Paraboloidal Wave Functions 1231961257

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P A R A B O L O I D A L

WAVE

F U N C T I O N S

by

Kathleen Mary Urwin, M.Sc.

University of Surrey.

November, 1968

ABSTRACT

The Whittaker-Hill equation arises when Helmholtz’s equation v 2^ +

=0

is separated in general paraboloidal coordinates.

Paraboloidal wave functions are (certain) solutions of the Whittaker-Hill equation, with period 7T or 2 7f . Chapter I is introductory:

the general paraboloidal coordinate

system ani the separation of Helmholtz’s equation are discussed. 2 Chapters II and III deal with the case k 0:

the theory for k

2

0.

1.

Introduction.

82

2.

Formal infinite series solutions.

83

3.

Determination of the even solution of period 7T .

,90

4.

Tne other.three solutions.

112

Special cases.

115

• 5* 6.

Continuity of y

as a function of u) or

.

118

7.

Orthogonality properties.

124

8.

Simple relations.

128

9.

Degenerate cases.

134

10. Numerical results.

Chapter V.

139

2

Perturbation solutions for k

> 0.

1.

Tne perturbation method.

147

2.

The series solutions for 0 .

150

3.

Tne characteristic numbers.

159

4.

Convergence. ’

160

5.

Degenerate cases.

169

6.

Tne form of the trigonometric series

7.

An alternative derivation of X .

182

8.

Perturbation solutions of real fractional order.

188

References.

^ for small m s

. 173

191

NOTE

Equations are numbered in each section:

if a number only is

quoted the reference is to an equation in the current section.

If

the reference is to an equation in a different section of the current chapter, the section number precedes the equation number; if the reference is to an equation in another chapter, both chapter and section number are quoted. Numbers in square brackets refer to the list of references at the end of the thesis.

CH AP TE R

I.

Tne coordinate system and separation of Helmholtz's equation. The general paraboloidal coordinate system has two families of elliptic paraboloids and one family of hyperbolic paraboloids as its coordinate surfaces.

The system has been fully discussed by

Arscott [2,3]; we give here those details which are important when discussing the solutions of Helmholtz’s equation by separation of variables in this system, and its application to boundary value problems. 1.

The coordinate surfaces. General paraboloidal coordinates ( sinhY,

(2)

z

= 2c sinhd sinyS coshY,

(3)

where d ,p , V

are all real and

O £ JL
03 z > 0;

< fa A

IfO< fb

5

< °

(10)

y < 0 $ z > 0.

/>

c < -~7T 3 then the four surfaces /3 - ' t .

5

- i (7f - fo 0) together make up the complete hyperbolic

paraboloid given by equation (9). are degenerate:

The surfaces 1p>c = 0,’t k-r, 7T

= 0 is that part of the half-plane z = 05 y > 0

lying outside the parabola

whilst

= vr is the part of the

half-plane z •= 05 y < 0 lying outside the same parabola. A

Thus'

= 05 /3C = 77■ taken together make up an infinite plate with a

"parabolic" hole.

In a similar way the surfaces

= -fc

together give the whole region in the plane y = 0 outside the parabola

Apart from the regions inside the parabolas

and

in the

planes z = 0, y = 0 respectively, the correspondence between (

3y3 3 Y ) an^ (x,y,z) is one-one for ^ $ /3 , Y

the ranges given by (4).

restricted to

The region inside P^ is covered twice by

the surface J. = 0, the paraboloidal coordinates (0, ± y l , Y ) giving the same point.

Similarly, the region inside P^ is covered

twice by Y = 0. V/e note the following important lines: