6663-1-T Copy THE UNIVERSITY OF MICHIGAN COLLEGE OF ENGINEERING DEPARTMENT OF ELECTRICAL INGINRING Radiotion Lobfeoroy UNIFIED APPROACH TO EXCITATION PROBLEMS IN COMPRESSIBLE PLASMA Y-K Wu July 1965 Prepared for Contract No. AF 30(602)-.3381 Project 5579 Task 557902 Contract Monitor, L. Strauss EMASA 6663-1 -T = RL-2144 Contract With: Rome Air Development Center Griffiss Air Force Base, New York 13442 Administered through: OFFICE OF RESEARCH ADMINISTRATION. ANN ARBOR

A CKNOWLEDGMENT It is a pleasure to acknowledge the assistance of Professor C-M Chu, who brought this problem to my attention and provided guidance and support for the investigation. Thanks are also due to Mr. J. J. LaRue and Dr. D. B. vanHulsteyn for reading and correcting the manuscript, to many other colleagues of the Radiation Laboratory for stimulating discussions, and to the computation group of the Radiation Laboratory for assistance in numerical calculations.

ABSTRACT This investigation is concerned primarily with a general, unified approach to the solution of excitation problems in a compressible plasma which may be anisotropic and inhomogeneous, and which may include different types of sources, e. g. electric current sources, magnetic current sources, fluid flux sources and mechanical body sources. A macroscopic, hydrodynamic approach is chosen and is based on the linearized Euler equations of motion and the Maxwell equations. The Maxwell-Euler equations are reformulated through linear operator and generalized transform techniques into an equivalent matrix integral equation. When the medium is homogeneous, this integral equation has an ideal kernel and the explicit solution can be easily obtained. A thorough study is given for the excitation of disturbances due to different types of sources in a homogeneous electron plasma immersed in a constant magnetic field. Collisional damping effects are neglected and an adiabatic condition is assumed in the present study. As a preliminary requirement, the dispersion relation in the form of a cubic equation for the propagation constant square is analyzed as exactly as possible. Some illuminating graphs showing the propagation constants as functions of the normalized plasma frequency are employed for the above analysis and they are explained in conjunction with the so called Clemmow-Mullaly-Allis diagram. In due process, a proper terminology is introduced for the three types of waves involved in an electron plasma. The radiation field is then solved for both two- and three-dimensional excitation problems. Exact solutions are obtained for two-dimensional problems, and asymptotic solutions are obtained for three-dimensional problems by direct utilization of the dispersion curves. Some dispersion curves and the radiation field from a point current source oriented in the direction of a constant magnetic field are presented in graphical form, which are obtained numerically by a computer.

A proper ionospheric model is used for this calculation, which indicates comparatively strong excitation of modified plasma waves. Also, equivalence relations between different types of sources are obtained, which can be employed to express the fields excited by one type of source in terms of the fields excited by another type of source. An illustration is given for the application of the operator transform formulation employed in this report to a three fluid plasma problem, and its application to the excitation problems in an inhomogeneous tnedium is also discussed.

PREFACE In this work, a general, unified investigation of the excitation problems, considering different types of sources, in a compressible plasma with an externally impressed constant magnetic field is presented. A macroscopic, hydrodynamic approach is chosen, and the linearized Maxwell-Euler's equations are reformulated through linear operator and generalized transform techniques into an equivalent matric integral equation. The candidate is indebted to his committee chairman, Professor C. M. Chu, who brought this problem to his attention and provided guidance and support for this investigation, and to other members of his committee for helpful suggestions and correcting the manuscript. The author is thankful to Mr. J. J. LaRue and Dr. D. B. vanHulsteyn for reading and correcting the manuscript, and to many other colleagues of the Radiation Laboratory for stimulating discussions. Also, the author would like to express his appreciation to Dr. A. A. Nagy of Space Physics Research Laboratory, The University of Michigan, for his assistance in collecting latest data on ionospheric properties. For computer calculations the author is thankful to the computation group of the Radiation Laboratory. This investigation was supported by the Rome Air Development Center under Air Force Contract No. AF30(602)-3381. ii

TABLE OF CONTENTS PREFACE LIST OF ILLUSTRATIONS LIST OF APPENDICES CHAPTER I INTRODUCTION CHAPTER II GENERAL FORMULATION 2.1 Basic Equations 2.2 Operator Transform Method 2. 3 One-Fluid Plasma A Operator Form B Generalized Telegraphist's Equations C Fredholm Integral Equation D Formal Solution in a Homogeneous Plasma CHAPTER III WAVE PROPAGATION IN ONE-FLUID PLASMA 3.1 Introduction 3. 2 Dispersion Relation 3. 3 Basic Types of Waves A Waves in Absence of Magnetic Field B Cold Plasma C Warm Plasma 3. 4 Characteristics of Waves A Electromagnetic Waves in a Cold Plasma Page ii vi xi 1 4 4 7 11 12 16 19 21 25 25 26 30 30 31 34 41 42 iii

CHAPTER B C D IV 4.1 4.2 A B C 1 2 3 4 D 4. 3 A B V 5.1 5.2 A Plasma Waves 2 2 2 Propagation Constants vs. = w / o p Effect of Direction WAVE EXCITATION IN ONE-FLUID PLASMA Introduction Two Dimensional Problems Field Solutions in Transform Space for All Types of Sources Physical Interpretations Comparison of the Excitation Effects of Different Types of Sources Electron Fluid Flux Source Mechanical Body Source Line Magnetic Current Source Transverse Electric Current Source Equivalence Relations Three Dimensional Problems Basic Derivation and Analysis Actual Calculation NUMERICAL RESULTS IN IONOSPHERIC PLASMA Ionospheric Model Radiation Fields At 400 Kilometers 42 46 69 75 75 76 76 81 83 83 86 87 89 91 95 95 99 CHAPTER 106 106 107 161 iv

B At 250 Kilometers 162 C At 100 Kilometers 163 D At 70 Kilometers 163 NOTATIONS 206 BIBLIOGRAPHY 210 v

LIST OF ILLUSTRATIONS Figure 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13. 14. 15. 16. 17. 2 2 2 (Propagation Constant) vs. w2 /W P 2 2 2 (Propagation Constant) vs. 2 /w P Phase Velocity and Group Velocity 2 2 2 (Propagation Constant) vs. 2 /W2 P 2 2 2 (Propagation Constant) vs. Wt /2 P 2 2 2 (Propagation Constant) vs. 2 /2 P (Propagation Constant) vs. 2 /w P (Propagation Constant) vs. 2 /2 P 2 2 2 (Propagation Constant) vs. w /Wi P 2 2 (Propagation Constant) vs. w / P 2 2 2 (Propagation Constant) vs. w /w P 2 22 (Propagation Constant) vs. t / t0 P P Q < 1, Q2> 1, 2 vs. o 0 2= 0 0 > i, Q < 1, lQ> 1, < 1, Q< 1, Qu> 1, Q = 1, Q = 1, Q = 1, 0<0 < r/2, U -o 0< <1r/2, U -- 0 = o 0 = 0 0 = 7r/2 0 < e < 7/2 0 < 0 < X/2 e = 0 0 = 7/2 0 < 0 < /2 Page 43 44 46 48 49 50 51 52 53 54 55 56 57 73 108 Dispersion Curves for Three Waves Dispersion Curve 400KM, = 3 x 105 El, El vs. p 400KM, t=3x 10, Point Current Source (O = 0 ) Dispersion Curve 400KM, w=3 x 106 109 110 vi

Figure Pag( 18. ET vs. f 400 KM, x=3 x 10, Point Current Source (0 = 0~) 111 19. E~ vs. ~ 400 KM, c=3x 106, Point Current Source (0 = 0~) 112 20. Dispersion Curves for Modified Ordinary and Extraordinary Waves 400 KM, =3 x 107 113 7 21. Dispersion Curve for Modified Plasma Wave 400KM, uw=3x10 114 22. E'l, E' vs. p for Modified Ordinary Wave 400 KM, w=3x10 0 M Point Current Source (0 =0 ) 115 23. E, El vs. 0 for Modified Extraordinary Wave 400 KM, 7 M = 3 x 10, Point Current Source (0 = 0 ) 116 24. ElE, El vs. p for Modified Plasma Wave 400 KM, = 3 x 10 Point Current Source ( =0 ) 117 25. Dispersion Curves for Modified Ordinary and Extraordinary Waves 400KM, w=3x 108 118 26. Dispersion Curve for Modified Plasma Wave 400 KM, w=3x10 119 27. El, El vs. f for Modified Ordinary and Extraordinary Waves 8 M 400 KM, w=3 x 10, Point Current Source (0 = 0 ) 120 28. Elg, E' vs p for Modified Plasma Wave 400 KM, 8 M =3 x 10, Point Current Source ( = 0~) 121 29. Dispersion Curve 250 KM, w= 3 x 10 122 30. EV vs. f 250 KM, Lc=3xl0 Point Current Source (0 = 0~) 123 vii

Figure Page 31. E vs. 250 KM, =3x 105 M Point Current Source (0 = 0 ) 124 32. Dispersion Curve 250 KM, = 3 x 10 125 33. Ei vs. 250 KM, j=3x 10, Point Current Source ( = 0) 126 6 34. ET vs. 250 KM, =3x10, M Point Current Source (0 = 0) 127 35. Dispersion Curves for Modified Ordinary and Extraordinary Waves 250 KM, =3 x 10 128 36. Dispersion Curve for Modified Plasma Wave 250 KM, u =3 x 08 129 37. E', E1 vs. 0 for Modified Ordinary Wave 250 KM, M =3 x 108, Point Current Source (0 = 0~) 130 38. El, E' vs. 0 for Modified Extraordinary Wave 250 KM, 8 M o= 3 x 108, Point Current Source (0 = 0~) 131 39. EP, E' vs. P for Modified Plasma Wave 250 KM, 8 M L=3x 108, Point Current Source (0 = 0~) 132 40. Dispersion Curve 100KM, w=3x105 133 41. E0 vs. 100 KM, w=3 x 10 Point Current Source (0 = 0~) 134 42. E vs. 0 100KM, o=3x05, M Point Current Source ( = 0) 135 viii

Figure Page 43. Dispersion Curve 100 KM, = 3 x 10 136 44. El vs. 100 KM, Lj=3x106 Point Current Source (p = 0~) 137 45. E1 vs. v 100 KM, 3 x 106, M Point Current Source (0 = 0~) 138 46. Dispersion Curves for Modified Ordinary and Extraordinary Waves 100 KM, w=3 x 10 139 7 47. Dispersion Curve for Modified Plasma Wave 100KM, w=3x10 140 48. El, El vs. f for Modified Ordinary Wave 100 KM, o=3 x 10, Point Current Source (f = 0 ) 141 49. E, E vs. f for Modified Extraordinary Wave 10 KM, w= 3 x 10, Point Current Source (f = 0 ) 142 50. E E, vs. f for Modified Plasma Wave 100 KM, 7 M w= 3 x 107, Point Current Source (f = 0 ) 143 51. Dispersion Curves for Modified Ordinary, Extraordinary and Plasma Waves 100 KM, w=3 x 10 144 52. ElP, El vs. f for Modified Ordinary and Extraordinary Waves 100 KM, w=3 x 10, Point Current Source (f = 0~) 145 53. E, El vs. f for Modified Plasma Wave 100 KM, 8 M w= 3 x 108, Point Current Source (f = 0 ) 146 54. Dispersion Curve 70 KM, w=3 x 105 147 55. ElE, El vs. f for Modified Extraordinary Wave 70 KM, = 3 x 105, Point Current Source (f = 0~) 148 ix

Figure Page 56. Et vs. A for Modified Ordinary Wave 70 KM, -= 3 x 10, Point Current Source (0 = 0~) 149 57. E' vs. p for Modified Ordinary Wave 70 KM, M w = 3 x 10, Point Current Source (, = 0 ) 150 58. Dispersion Curve 70 KM, w= 3 x 106 151 59. El, El vs. 0 for Modified Ordinary Wave 70 KM, 6 M = 3 x 10, Point Current Source (f = 0~) 152 60. El, E1 vs. 0 for Modified Extraordinary Wave 70 KM, 6 M = 3 x 10, Point Current Source (0 = 0~) 153 61. El, Et vs. for Modified Plasma Wave 70 KM, 6 M w = 3 x 10, Point Current Source (f = 0 ) 154 62. Dispersion Curve 70 KM, w=3 x 10 155 63. El, Et vs. f for Modified Ordinary and Extraordinary Waves 7 M 70 KM, w=3 x 10, Point Current Source (f = 0~) 156 64. EL', Et vs. 0 for Modified Plasma Wave 70 KM, 8 M = 3 x 10, Point Current Source (0 = 0~) 157 65. Dispersion Curve 70 KM, = 3 x 108 158 66. El, E' vs. 0 for Modified Ordinary and Extraordinary Waves 70 KM, w=3 x 10, Point Current Source (0 = 0~) 159 67. Elk, Et vs. 0 for Modified Plasma Wave 70 KM, 8 M w = 3 x 10, Point Current Source (0 = 0~) 160 x

LIST OF APPENDICES Appendix Page A GENERAL FORMULATION FOR THREE-FLUID PLASMA 165 B EVALUATION OF INVERSE TRANSFORMATION FOR TWO-DIMENSIONAL PROBLEMS 182 C THREE ROOTS FOR THE CUBIC EQUATION 186 D SOME ANALYSES FOR THE HYPERBOLAS 190 E DERIVATION OF X 194 F PROBLEMS OF INHOMOGENEOUS PLASMA 196 xi

CHAPTER I INTRODUCTION This investigation is concerned primarily with the excitation of wavelike disturbances in a plasma, and is based on the linearized, coupled Euler's equations of motion and the Maxwell equations. The propagation of plane waves in a plasma has been studied extensively (1) (2) (3) (4), by many investigators (e.g. Spitzer, Ratcliffe, Oster, Ginzburg Budden, Pai, Tanenbaum and Mintzer, and Denisse and Delcroix (). In these studies the dispersion relation for the various waves was of primary concern and the excitation of these waves was not considered. Recently, excitation problems in a plasma have attracted the attention of many investigators. Ginzburg and Kolomenskii considered the special case of the radiation of a point charge moving in a transparent anisotropic medium. (11) Bunkin studied the radiation field of a given distribution of external currents in an infinite homogeneous anisotropic medium and seems to be the first investi(12) (13) gator to give a general solution to the excitation problem. Kogelnik, Arbel(1) (14) (15) (16) (17) (18) Kuehl, Mittra, Mittra and Deschamps, Clemmow, Wu, Motz (19) (20) (21) and Kogelnik, Arbel and Felsen, and Chow, analyzed similar radiation problems in an infinite homogeneous anisotropic medium. In all these works a single fluid plasma was considered and was assumed to be incompressible and thus could be characterized by a tensor dielectric constant. With the assumption of incompressibility, the longitudinal plasma wave (22) does not appear. Whale discussed the importance of the radiation of energy as an electron plasma wave, and has shown that the calculated power radiated by this 1

2 type of wave yielded results in good agreement with rocket observations. Hessel (23) and Shmoys ) have considered the excitation by a point current source in a compressible plasma in the absence of a static magnetic field, and found that most of (24) the power goes into the plasma wave. Seshadri( treated the radiation from a line magnetic current source in a compressible plasma. However, a general, unified investigation of the excitation problem, considering different kinds of sources (e.g. electric current source, magnetic current source, fluid flux source, and mechanical body source) in a compressible plasma with a constant magnetic field is not available. The objective of the present investigation is to find this general solution by using a unified and.systematic formulation. In principle, the linearized Euler equations of motion and the Maxwell equations, including sources, may be considered as a linear operator relating the field quantities to the sources. Due to the large number of variables involved, the solution of the excitation problem, i. e., finding the inverse of the operator, in a medium which may be snisotropic and inhomogeneous, is very involved. In general, analytical solutions for such high-order systems can be obtained only in special cases. In this work, the formal operator method is used as a systematic approach to the excitation problem. Operator methods are a well-known and potent tool in quantum mechanics. The introduction of the operator method into electromagnetic (25X26) (27) fields has been explored by Bresler and Marcuvitz, Moses, and others. (28) Recently, Diament) has introduced the formalism of an operator method combined with a generalized transform method in obtaining the formal solutions of Maxwell's equations for general linearized media. Because of the compact notation,

3 systematic approach and convenience for numerical analysis his formal operator transform techniques are extended in the present Work to the system of linearized equations describing the excitation of disturbances in a plasma. In Chapter II, the general operator transform formalism for the linearized equations of plasma disturbances is developed and applied to a homogeneous, compressible electron fluid plasma immersed in a uniform magnetic field. In Chapter mI, the propagation of the three types of waves involved in an electron fluid plasma is studied carefully by analyzing the dispersion relation in the form of a cubic equation in propagation constant square obtained from Chapter II. Also, a consistent and general terminology for these three types of waves is developed, since there is no standard terminology available. In Chapter IV, the wave characteristics obtained from Chapter III are utilized to solve the excitation problems in a compressible electron fluid plasma with a constant magnetic field. Equivalence relations between different types of sources are derived which can be applied to both two- and three-dimensional problems. In Chapter V, a proper ionospheric model is used to calculate the dispersion curves, and then the radiation fields in the forms of asymptotic solutions. The numerical results are presented in graphical form. The application of our formalism to a three-fluid plasma problem is illustrated in Appendix A. Its application to the excitation problems in an inhomogeneous medium is discussed in Appendix F. As far as perturbation problems and general numerical solutions of the problems are concerned this method seems promising, but it does not look too promising to obtain exact solutions by using this method.

CHAPTER I GENERAL FORMULATION 2.1 Basic Equations In this section the basic equations governing weak disturbances produced by various kinds of sources in a neutral plasma composed of electrons, ions and neutral particles will be presented. The parameters and assumptions applicable to the undisturbed plasma are as follows: (a) The number densities of electrons, ions and neutral particles are denoted by N, N. and N, respectively. Assuming the ions are e i singly charged and a neutral plasma, the electron and ion number densities are equal and will be denoted by N i, e. N EN.iN. o, e 1 o Negative ions are not considered in this investigation. (b) The electron mass and the average mass for the ions and neutral particles are, respectively, m, m. and m n e 1 n (c) The effective collision frequencies for momentum transfer between different types of particles is denoted by v ab where the subscripts a and b refer to the types of particles. It is to be noted that these collision frequencies for momentum transfer satisfy the relations: N m v xN Vmb a a ab b ba (d) The acoustic velocities for ion, electron and neutral particle gas under adiabatic conditions are U., U and U, respectively. 1 e n (e) The plasma is constantly under the action of a d-c magnetic field B (f) The plasma as a whole is stationary. (g) Each gas obeys the ideal gas law. 4

5 In addition to the preceeding assumptions, it will be assumed that the sources of the disturbances are weak and thus the second order terms, such as the products of the perturbation terms, and the thermal and viscous effect can be neglected. In addition, it is assumed that the properties of the disturbed medium are nearly the same as the properties of the ambient medium. In this case, a set of linearized equations is usually considered to be adequate to relate the disturbances to their respective sources. Considering just one Fourier component of the disturbances in the form of e, and employing the rationalized mks system of units, this set of equations is the following linearized inhomogeneous Maxwell and Euler equations o(3) (7) (29) (30), (31), (32) (Oster, Tanenbaum and Mintzer, Watanabe, Cohen ), and Pai(33)) - (a) The Maxwell equations: VxE -ipu h =-K (2.1) Vxh+i wE E-eN (V.-V ),J (2.2) o o 1 e (b) The momentum transport equation and the mass transport equation for the electron gas: -2 - - ] -ioN m V +m U Vn + eN E+V xB o e e e e e o e o +N m v.(V -V.)+N m v (V -V )MF (2.3) o e el e I o e en e n e N V- V +V *VN -icon aQ (2.4) o e e o e e

6 (c) The momentum transport equation and the mass transport equation for the ion gas: 2 r- - - -kiN m.V +m.U.Vn.-eN +V.xB J o 1 i1 1 1 1 0 1 0 +N m.v. (V.-V )+N m.v. (V.-V )F. (2. 5) o 1 le 1 e o 1 in 1 n 1 N V. V.+V.' VN -iUn.-Q. (2.6) 0 1 1 0 1 1 (d) The momentum transport equation and the mass transport equation for the neutral particle gas: -ijN m V +m U Vn +N m v (V -Ve) In n n n n n n ne n +N m v.(V -V.)=F (2. 7) 1n n n 1 n i n N V. V +V * VN - in xQ (2.8) 1 n n 1 n n The following notation has been used in the above equations: h: varying component of the magnetic field e: dielectric constant of free space E: varying component of the electric field (constant component is not considered in this investigation) V.: fluid velocity of the electron, ion, or neutral particle gas e, i,n J: electric current source P~: permeability of free space K: magnetic current source n: varying component of the number density of the electron, e, 1, n ion or neutral particle gas e: absolute value of the charge of an electron

7 F.: mechanical body source for the electron, ion or neutral e, i, n particle gas Q. fluid flux source for the electron, ion or neutral particle e, 1, n gas The set of Eqs. (2. 1) through (2. 8) represents a system of partial differential equations relating 18 scalar functions. In the following section we shall present a formal operator transform method, which is convenient to solve this set of equations with source terms present. 2.2 Operator Transform Method A formal solution to the set of Eqs. (2. 1) through (2. 8) can be obtained by an operator transform method. This method is an extension of that used by (28) Diament() for the formal solution of Maxwell's equations in general linear media. The procedure for obtaining the formal solution is as follows: FIRST: For the purpose of exhibiting a general solution to a system of basic equations, it is convenient to reformulate them in the following single operator equation '(V(r)^ (r) (2.9) where O(r) is a field vector composed of the field variables such as the electric field E, the velocity field V, etc., 0 (r) is the source vector containing various excitation sources such as the electric current source J, the mechanical source F, etc., and %f is the system matrix differential operator relating the field to the sources. lVcontains all the properties of the medium and is a function of

8 the space coordinate r. in general, without loss of generality, Use' systam of basic equations can be rearruaged so that some of ti identity jaatrices. SEOOND: Here, we Introe the gemeralised rmeru teei~use which aoat to choosing same convenient basis of represeatation fw tile seition and transforming the eperstU dierential equation in real spaa -to an ewator integral equation in trsform space. Th generic suarm ation symbol Dgh as used in Quantum Me bmni..o~, will be usd, whiab eqrW tex-t t Oxpression fellowing this symbol be Integrated or summed ever ft We e 1e of the repeated variable. Fo mally, for azy quantity a0r). we nay nihaos the followng tbransform pr. Transftrm A(s) A / Es, r)Oas) 1mverse aitr) rav, t..)A(s) (2I.1 with the property that o(r. s)Om, P) (r,p) f and ~d(a,.r)(r, a) aIN. a )L The Ldemfactoirls. s) cmprisesi a Dirac delta ti or a wo"M ke Iita a unit dyadic. as requiredI. variables may be cnsidered S (a1, a The rage of thereal- so ad trasfo-rm space Variablms is nw to + w. in this case a Fouirer trsr is appropriate ur d(ss r) and O. a) are

9 I -ir s ds, r) e (2f)3 (2.12) ir* s c(r, s) e If the nature of the problem requires the cylindrical coordinate system, we can apply a Fourier-Bessel transform given by d(s, r) a e-i( + z)J (qp) (2.13) c(r, 1 i(no + Bz) cdr,a) " — e i J (qp) (2)2 The ranges of p and q are 0 to oD, with weight functions p and q, respectively; the range of c is 0 to 2r, and that of n is all integers; the ranges of z and f are -oa to oo. The real space is expressed by the cylindrical coordinate system (p, p, z), and the transform space is expressed by (q, n, 8). Now, we proceed to the transformation of the operator Eq. (2.9). Let i(s) and (s) be the transforms of the vectors <(r) and f (r), respectively, i. e., (s) $ d(s. r)(r) P(r) * o(r, s) (2.14) f() s) d(s, r)O(r) (2.15) 0 (r) $c(r, s)(s) Also, we take the transformation law for the matrix operatorlVas V' (, s) p d(u, r)ic(r, s) (2.16) Premultiplying both sides of Eq. (2.9) by d(u, r), and then substituting the expansion for O(r) as given by the transform pair in Eq. (2.14) and summing or integrating over the complete r-space the operator Eq. (2. 9) in the real space

10 b-eo —m the operator integral equation in the transform space 0W(u, )i(s) 'f (u) (2.17) This equation has the charaer of a generalized integral equation of the first kind, with f<) as forcing funetien, Js) as the unknown function, andu/tu, a) as the kernel. )V(n, a) is a function of two composite variables of the tranform space and retains all the pertinent information about the system. THIRD: Because of the earlier rearrangement and dagonalization, the dyadic kernel lI(u, a) can be properly partitioned so that the order of the matrices to be manipulated may be reduced, by introducing coupled integral equations of the second kind which in turn may be recombined into one integral equation of the second kind. For example, we can have for Maxwell's equations 1i( ). (2.18) L-Y(ui, s)I(U, s)j Then, the partitioning of j(s) and(u) Into two vectors V(s) Wdul i.,s), (') L j (2.19) L LJ(uU produces the following ooupled integral equations V(u)'-W(u)+l Z(u, ) I(s) (2.20) I(u)^J(u)+ Y(u, s)V(s) (2.21) Let V(s) and Ki) correspond, respectively, to the transform of the electric field and the transformof the magnetic field, then these Eqs. (2.20) and (2.21) have the generalized forms of the telegraphist's equations of Schelkunof if s is taken to Indicate different modes in the waveguide.

11 The elimination of either the field vector V or I in the Eq.. (2.20) and (2.21) gives the general form of the Fredholm integral equation of the second kind, e.g., V(u)(u)+t K(u, s)V(s) (2.22) where F(u)sW(u)$ Z(u, )J(s) (2.23) K(u, s)O Z(u, v)Y(v, a) are both known tmction. For homogeneous media the kernel has the ideal form K(u, s)- N(s)I ( u, ) (2.24) aad the integral Eq. (2. 22) an be explicitly solved as Vt) [-N(sM)] F(). (2.25) Another dgenerate e exists when the dyadic kernel can be expressed in ftored form as K(u, us)A(u)B(s) (2.26) ad again we have explicit solutions. For inhomogeneous media, in general, an explicit formal solution to the Integral equation can be obtained recursively by (28) the application of the general theory studied by Diament(28). If approximate solutions are sufficient, a general kernel can be approximated by a degenerate kernel. 2.3 ne-FLid Plasma The operatorftransform method presented in the previous section will be applied to othe-fluld plasma problems. Its application to three-fluid plasma problems is very compliated,, and is illustrated in Appendix A. By one-fluid

12 plasma we will consider only the electron gas, and the motions of ions and neutral particles will be neglected. Our intention is to study the applicability of our method to simpler problems thus paving the way to more difficult threefluid problems. Besides, there are many practical situations in which we can neglect the effect of heavy particles, e. g., the ratio propagation in the most part of the ionosphere except, maybe, D-region. Although electron fluid plasma problems have been studied by many investigators (Ginzburg, Budden, Bunkin( 1) Arbel (13)), there are still many important problems to be solved. One of such problems is the excitation problem in the compressible electron fluid plasma immersed in a constant magnetic field. This problem will be given a full treatment in the subsequent chapters. FA] Operator Form The basic equations for the electron fluid plasma can be obtained from Eqs. (2.1), (2. 2), (2. 3) and (2. 4) as VxE-ipu Ah -K (2.27) Vxh+ie (lE+eN V-J (2.28) o o -iWN mV+mU Vn+eN E+VxB =F (2.29) N V. V+V. VN -iLn Q (2.30) o o where all subscripts e have been dropped, and also we have neglected the collisional dissipation effects. In order to be able to obtain a proper matrix form of the Eqs. (2.27) through (2. 30), we will express E and V in terms of h, n, J and F by employing Eqs. (2. 28) and (2. 29). Firstly, E is eliminated between these two equations

13 to get 2 2 2 -e -U e 1 ( V+w Vxb= e Vxh - - Vn - J+ F (2. 31) io c i E m N iw m mN where b is the unit vector in the direction of the externally applied constant magnetic field, and also use is made of the conventional electron cyclotron frequency, 2 2 wo eB / m, and the electron plasma frequency, w 2e N / e m. Secondly, Eq. c o p 0 0 (2. 31) is explicitly solved for V by taking scalar product and vector product of A - Eq. (2. 31) with b, and then E can be solved from Eq. (2.28). Their results are A h + A12n + E S (2. 32) A21h +A22n + V 2 (2.33) 2 2 2 2 o /ie ei (2 -2 ) A b bx(Vx )+ V- Vx 1 - bb. VxI - 11 2 22 2 2 2 2 2 e w (W - ) -_W W W W J o p c p 2 2 (2. 34) eU to /iE (to -to ) 2 (- ) -to (E LO iVxI O A A + (2 36) 21 11 + o ieN O - O E W A A 0 (2.37) 22 12 ieN 0 2 2 2 t/ie i, - (W -k )- iww - '' - 0 p C - bxF+ -- F+ bb F 1 m c 2 2 2 2 2 2 (W -to) to (to - ) P c p 2 2 2 (W -tW ) ~ J +to ii bxJ+ J- bbJ -i (2.38) 2 2 2 b 'i Po (o-to ) p

14 S i- 0 S2 S +i J! o (2.39) 2 e ieN 0 0 In the expressions given by Eqs. (2. 34) through (2. 37) we have employed some dyadic operations with their associated matrices as follows: 1 0 0; A A A^ A A, -0xx+yy+ zz!> 0 1 0| (2.40).0 0 1 T (^ ^A ^ a ^ A\, A A Vx1 Vx +y +Z ) x(xx + yy+ zz) axx + y + z / a a az ay (2.41) -a a 0 d Ly dx V. l ( xx +y + zz) > (2.42) bb( b +yb + zb) (xb + b + b ) x y z x y z b b b b > x x y (2.43) b z bx (b + b + zb )x(xx +yy+ zz) i 0 -b b x y z z y - ) i b 0 -b (2.44), z x!-b b o L y x

15 Eqs. (2 r where ( Now the original Eqs. (2.27) and (2. 30), together with the rearranged '. 32) and (2. 33), can be put into the following desirable matrix form: 1 0 i- 0 h -iK/u iN 0 1 0 w(V.)+- VN 1 ijn iQ/N A1 A12 1 0 E S 21 22 L | 2,V 1)' is the transpose of the matrix given by Eq. (2. 42). This matrix equation can, then, be put into an operator equation / - (r) = 0 (r) (2.45) (2.46) where h / h -iK/0UoiU n iQ/u (r)= - I, 0 (r)= - I E S (2.47) V S2 _ Thus, the basic Maxwell-Euler's Eqs. (2. 27) through (2. 30) have been reformulated into a single abstract relation between the sources and the resultant fields. /(r) is a ten-vector containing the field quantities, P (r) is a tenvector representing the source quantities, andf/is the system matrix differential operator relating the fields to the sources. Two identity submatrices of u/, as can be seen from Eq. (2. 45), are highly significant in deriving an integral equation of the second kind in an inhomogeneous medium.

16 B Generalized Telegraphist's Equations Generalized Fourier transform as given by Eqs. (2.10) and (2.11) will be used to transform the operator Eq. (2.46). Then, Eqs. (2.14) and (2.15) give the transform pairs for the field vector and the source vector given in Eq. (2.47), and Eq. (2.16) gives the transform for the matrix differential operator, l<. The resultant integral equation of the first kind as given by Eq. (2.17) is $ J(u, s)(s) (u) (2. 48) This equation may be put into the generalized forms of the telegraphist's equations by partitioning the transform of the field vector, (s), the transform of the source vector, j (s), and the transform of the matrix differential operator,lf (u, s), as follows: h It(s) n V (s) e -(s)= d(s, r e (2.49) E Vt(s) V I(s) e -iKt/0/u Jt(s) iQ/U W (s) (s) = d(s,r) (2.50) S1 Wt(s) 2 J (s) 2 e where It(s), V (s), Vt(s) and I (s) are, respectively, the transform of the magt e t e netic field, h, the transform of the density variation, n, the transform of the electric field, E, and the transform of the fluid velocity, V, and also J (s), W (s), t e W (s) and J (s) correspond, respectively, to the transform of the magnetic current t e

17 source, K, the transform of the fluid flux source, Q, the transform of the threevector source function, S1, and the transform of the three-vector source function, S2. Taking advantage of the orthonormality property of the transformation kernels as given by Eq. (2.11), the ten-dyadic kernel,2L(u, s), can be partitioned as (u,s) 0 -Yt(u,s) 0 0 l(u, s) 0 -Z (u, s) A/ (u, s)= e -Zt(u, s) -Tte(, s) (u, s) 0 -T et(u,s) -Y (u,s) l(u, s) et e (2.51) where l(u, s) is a Dirac or Kronecker delta function which is the same as the scalar form of the idemfactor 1(u, s). The three-dyadic immittance functions Yt(u, s) and Zt(u, s); the three-dyadic transfer function T et(u, s); the three-rowvector impedance function Z (u, s); the three-column-vector admittance function Y (u, s); the three-column-vector transfer function Tte(u, s) are defined as: e te Vx I -Yt(u, s) $ $ d(u, r)i i- c(r, s) (2. 52) -Zt(u, s)$ ^ d(u, r) A1 c(r, s) (2. 53) -T t(u, s) $ d(u, r) A21 c(r, s) (2. 54) -Z (u, s) d(u, r) - (V- 1)' + VN - c(r, s) (2. 55) e W W o -Y (u, s) X ( d(u, r) A22 c(r, s) (2. 56) e d(u, r) A2 c(r, s). (2. 57) -T (u, s) $ d(u, r) A c(r, s). (2.57) te 12 In Eqs. (2. 52), (2. 53) and (2. 54) both transformation kernels, d(u, r), and inverse

18 transformation kernels, (r, s), are three-d4iagonal-dyadics; In Eq. (2.55) d(Eu, r) can be taken as a scalar and c(r, a) taken as a three-diagonal-dyadic; in Eqs. (2. 56) and (2. 57), three-diagonal-dyadics can be used for d(u, r) and scalars can be used for c(r, s). Substitution of the partitioned matrices as given by Eqs. (2. 49), (2. 50) and (2. 51) into the transform integral Eq. (2. 48) decomposes this equation into the following set which, due to their form, will be called the generalized telegraphist's equations. It(u) Jt(u) + ^ Yt(u, ) Vt(s) (2.58) Vt(u) - W(U) + ^ Zt(u, ) It(s) + T (u, s) Ve(s) (2.59) I (u) - J (u) + $ Te (u, ) It(e) + Ye(U. ) V () (2.60) (u)W(u)+$T(u,)I(s) Y )V ) (2.60) e e et t e e V (U) NW (u) + $ Z (u.-s) I (s) (2.61) e e e e Equations (2.58) to (2.61) contain in a compact form, in the transform space, the laws governing the excitation and propagation of "fields" in the linearized medium. The set of immittnees and transfer functions, which can be evaluated from Eqs. (2.52) through (2.57), contain all the intrinsic properties of the medium, while four components of the source vector, W (u), Jt(u), J (u), W (u) represent all the sources. For Maxwell's equations only, if c and p in Eqs. (2. 58) and (2. 59) are replaced by appropriate tensor permittivity and permeability, the pertinent equations are It(u) Jt(u) + 0 Yt(u, s) Vt(s) (2.62) Vt(u) - Wt(u) + $ Zt(u. s) It() (2.63) ^ t(2.63)

19 (28) This set of equations is the original form given by Diament. They may be (35) compared with the telegraphist's equations of Schelkunoff) or the network (36) equations of Marcuvitz Zt may be interpreted as an impedance function while Yt may be interpreted as an admittance function. Hessel, Marcuvitz and (37) Shmoys have explored some aspects of the application of transmission line equations to a problem involving a compressible plasma and air and the associated boundary between the plasma and air. However, they did not consider the effect of a constant magnetic field. The results of their investigation yield some versions of Eqs. (2. 58) to (2. 61). Equations (2. 58) and (2. 59) give the transmission line system for the transverse electromagnetic wave, and Eqs. (2. 60) and (2. 61) give the transmission line system for the electron acoustic type of wave. LC] Fredholm Integral Equation The general Fredholm integral equation of the first kind, Eq. (2.48), which is equivalent to the original Maxwell-Euler's equations, will now be reformulated into a general Fredholm integral equation of the second kind which is more amenable to analysis. At the same time we have reduced the order of the matrices to be manipulated from 10 x 10 to 4 x 4. This step can be easily performed for Maxwell's equations in the form of Eqs. (2. 62) and (2. 63), but for our MaxwellEuler's equations we can not directly reduce the generalized telegraphist's Equations (2. 58) through (2. 61) into the integral equation of the second kind. Thus, in order to effect this reduction (s), f (s) and Zf(u, s) will be partitioned in the following way.

20 ~K()i f(s)9 (s)=, P(S) (2. 64) (s).2(si where It(S) Vt1 Jt(S) W (s) 1(S) (S)= I 1( 2(s)= L (s)= LV SJ Lw(s (S (s) e e e e (2.65) and, (u, s) -W2 (U s) lxf (u,s) (2.66) - 1(u, s) 1(u, s) 21 where KYt(u s) 0 1 Zt(u, s) Tte(u s)' 12 21 L o Z (u, s)j T (u, s) Y (u, s)1 e et e (2.67) The introduction of these partitioned matrices into the integral Eq. (2. 48) gives the following coupled integral equations ( (U) +(u)+$ i2(u, s)2(s) (2.68) 2(u)f2(u)+ 2 1(us)1(s) (2.69) and the substitution of Eq. (2. 69) into Eq. (2. 68) gives rise to the desired integral equation of the second kind;l(u)=F(u)+$ K(u, s)j(s) (2.70) where the compound source is F(u)~f (u)+$ 12(u, s)2(s) (2. 71) 1 12 - S

21 and the four-dyadic kernel is K(u, s)=$ t12(u, v) -L2(v, s) (2. 72) [D] Formal Solution in A Homogeneous Plasma The integral Eq. (2. 70) can be easily solved for a homogeneous plasma because the kernel has the ideal form K(u, s)=N(s) 1 (u, s). Choosing a Fourier transform and thus using the transformation kernels as given by Eq. (2.12), we can obtain from the defining Eqs. (2. 52) through (2. 57) S -Yt(u, s)= - 1 (u, s) (2. 73) o N -Z (u, s)= -- s' 1 (u, s) (2. 74) e to W /iE 2 2) o c b+i -zt(us 2 2 2 22 L 2 W( -0 ) -o t o P c 2 -i 2 2 bb l(u, s)+ 1 (u, s) (2. 75) (o -w ) o ewU /iE (to-to) ( -o ) - o P c 2 ] 2 2 bb' s (u,s) (2.76) 2 26 _W -O )( p E W o2 7 -Tet (U, s) (2.77) o 0 Eo -Y (u,s)- T (u,s) (2.78) e ieN te 0

22 where 0 b= b z y iJ S= S 2J -bb z 0 b b x j F b 11 b = by~ IbJ L1 b Ib b b -X y zJ S 1 ~2 S3 1-:0 -S3s I S~x s30 jS )i $ Y(u,v)T (v, s) t te I $ Z(u, V)Ye(v, sJ The kernel of the integral Eq. (2. 70) is K (u, s)=$lt(' (u, v) V (v, s 12 2 1 $ Y (u,v)Z z(v, S) $ Ze(u, v)T et (V, S) (2. 79) (2. 80) with $Y t(u, v)Z z(v,, S) p C F - -w Ito~cb + -C p 2 -tww 22bb~l ~1s1(u, S) (to-to) __j p $Y t(uv)T te (v, s) (2. 81) iec2U 2 (u2W2)2- 2w2 p 2 tw 7 s itoc b- 2 bb'I sl1(u, s) L-. (to -toI p (2. 82)

23 Z (u, v)T (v,s) e etw 2 s ic b bbtl S 1 (u,s) (2.83) 2 22 22 2 2 (to -t) ) -t (o -L ) - p c p Z (u, V)Y (v, s) e ' e 2 22 2 2 =s' i b+ - P —I c bb' sl(us) 2 22 22 c t 2_ 2 (- ) - ) - (- -J ) P c p (2. 84) where cx 1/ is the velocity of light in free space. Thus, the kernel has the ideal form Z$ A2(u, v) tf1(v s) = N(s) 1 (u, s) (2. 85) where N(s) is a 4x4 matrix. Substitution of this ideal form of the kernel given by Eq. (2. 85) into the integral equation will produce the solution of the integral equation directly as -N(s)1 F(s) (2.86) iV (S) 1 -In real space the magnetic field and the density fluctuation field are given by F-h(r - - J which is usually evaluated by the method of residues at the zeros of the determinant det. t-N(s) 0) (2.88)

24 Equation (2. 88) is the conventional dispersion relation when s is interpreted as the propagation constant. Thus, the importance of the dispersion relation in finding the excited fields is obvious. 2(s), which is a six-column-vector composed of the transform of the electric field, V (s), and the transform of the velocity field, I (s), can now be obtained from Eq. (2. 69). be obtained from Eq. (2. 69).

CHAPTER III WAVE PROPAGATION IN ONE-FLUID PLASMA 3. 1 Introduction The close relationship existing between the dispersion relation, which describes the propagation characteristics of waves, and the excited fields is apparent from the fact that various poles of the inverse transformation integrals give the dispersion relations for the different types of waves. Lighthill(3) and Felsen ), all stressed the importance of the direct application of wave surfaces obtained from the dispersion relation in finding radiation fields. Thus, in this chapter we will analyze the dispersion relation and discuss the propagation characteristics of those waves existing in an electron fluid plasma, which is the preliminary requirement for solving the excitation problems. Collisional dissipation effects are neglected in order to show the salient features. This should be practically permissible for high frequencies and in higher ionospheric regions. To facilitate the analysis it is convenient to give a proper terminology to the waves whose propagation constant squares are given by the roots of the dispersion relation. For an electron plasma a standard terminology has not yet been established for the three waves corresponding to the three roots. Judging waves by their frequency characteristics, Denisse and Delcroix(8) have used the terms, ordinary waves, extraordinary waves and electron waves. Allis, Buchsbaum and Bers(4) have adopted the optical criteria of judging waves by their local characteristics, ice., by the shapes of the phase velocity 25

26 surface and the polarization, thus they have used the terms, ordinary waves, extraordinary waves, right-handed circularly polarized waves, left-handed circularly polarized waves, and plasma waves. The terminology used in this work will be developed in a manner similar to that of Denisse and Delcroix(8) However, since the motion of the ions has been neglected, a more exact analysis of the roots is possible and the result can be related directly to the work of Allis, Buchsbaum and Bers, and Stix. Thus, the names, "modified ordinary wave", "modified extraordinary wave" and "modified plasma wave" have been associated with each branch of the root of the dispersion relation for the intermediate inclination of the constant magnetic field to the propagation direction. The point of view adopted here is that the ordinary and extraordinary electromagnetic waves in magnetoionic theory (Ratcliffe (2)) and the plasma wave are coupled together and modified by each other due to the constant magnetic 'field. 3.2 Dispersion Relation Without loss of generality, the coordinate axes can be chosen such that the externally applied constant magnetic field is in the ^ direction and given by B yB. o o (3.1) Thus b -b = 0, b = 1. x z y Applying Eq. (3.1) to Eq. (2. 81) through (2. 84) the following matrix for the kernel function N(s) is obtained:

N(s)= 1 N11 iN2 N 31 N i141 2 7 N N N 12 13 14;1 N N N 22 23 24,1 N N N 32 33 34 1 X 2 2 2 2 2 (to -to ) - w t p c (3. 2) N42 N43 N44 i with 2 r N r~c w2W 2)2 _2 2 2 p c + (w2-W c W2 3 2 2 2 p c to -to p (3. 3) (3. 4) 2 2 itwcw 222 1p i 2 2L 22 -.2 2 — 2 -N 2 P- 2 p c N13 x L o s2 1s3 2 2 to -to p (3. 5) 2 2 N z iec Utos 14 c 2 2 2 N21 xc toi 32 2 to -to p 2 2 2. 3 -i (to -to -tc)~ p c (3. 6) (3. 7) (3. 8) 2 2 2 22 2 N 22xc (s 1+ s ) (to-t -to ) 22 1 3 p N 2 I itoto2 23 S2 toU 1 3 N x -ec2Uto(s2+s 2) 24 ci1 3 2 22 (to -W o p cj (3. 9) (3. 10) (3. 11) N ~(ito w2 2 N31 toCI Li -s 2 2 2 2 2 (to -t ) -to to +Ss p c 1 3 2 2 - to -to p

28 2 2 iw 2 2 2 N -c s < s -sw( -to ) 32 2 t 1 3 p c 2 2 22 2 2 (to -t) - 2 2 2 2 N ^c s + s (o - -o ) 33 1 2 2 2 p C [ t -to p (3.12) 22 2 c N E-iec U os 5s +iS 34 iec U ~cS2 Is1 2 2 + is3 34 c i W -p p 2 41 e -I 3 2 2+ t w -t p 2 isl) (3.13) (3.14) (3.15) (3.16) (3.17) N (s +s 43 e 2 1 2 i3 2 W. -W J cp 2 2 2 44 2 2 2 p 2 2 2 2 where s 2 +s2+s 3 1 2 53. (3.18) The required dispersion relation is obtained from the determinant of the four by four matrix [ -N(s)j as given by Eq. (2.88). After some manipulation the dispersion relation can be expressed as

4 2 2 2 2 2 w (v-*> ) - 2 +U 8 is 2et. Ll-N~sIJ (2 2[2 2)2 2 2]E2 1 +3 x p p c + 2 2 22 2 2 2 4 22 2 2 222+2 ~(.1 f2 4 22 +U (s +s g)( — ) -c a - c (. +2s +s ) 1...(. 2+2 2 2. 2 12 p 2. ID1. 2 3 L 2 281 82 83 4929 2 2 2 c CC 2 2 2 p p c CS2 2 /42 4 2 222 (3.19) 2 C2 2 2c2 2 2 2 - 8 9 2 4\. p p c

30 To facilitate the analysis, the following notation will be used: s+ s s s sinO, s s cos 0 ~s1 63 x c p-, -=/, -=-, -=W c o U e t t o where 0 is the angle between the direction of the magnetic field and the propagation direction. Using this notation, the general three-dimensional dispersion relation as given by Eq. (3. 19) becomes s (Q cos 0 -1)+s L(1-Wo )( +230 )2 ( +22 cos 0 - W cos 0) 22 22 2 2 2 2 2 20 2 2 2 2 22 - +s o (1-W) (2P +fl )+Q (2( +fl cos W-13 o cos &-(3 to o o e o e o e o e o 24( 2) 22 2 0. (3.20) e o o o 3.3 Basic Types of Waves Simple systems are considered first, thus introducing plasma waves, ordinary waves and extraordinary waves. Next, Cardan's solution for a cubic equation will be used to obtain an exact solution to the dispersion relation, Eq. (3.20), and these three roots are identified as modified plasma waves, modified ordinary waves and modified extraordinary waves. [AI Waves in Absence of Magnetic Field Since the externally applied constant magnetic field is the main reason for complicating the nature of the waves, the case without magnetic field is considered first. The dispersion relation, Eq. (3. 20) reduces to

31 6- 4(l -S2 2+- 2+ 2) -22 2 21- 2)3 24 2 0 oe 0 0 e 0 e0o This relation can be factored into f (1- 2 _e (l- 2 ] 0 (3.22) This first factor yields a* 3 7 -, (3.23) which is the propagation constant for the electromagnetic wave modified by the effect of space charge. The second factor yields s = 2el-, (3.24) e 0o which is the propagation constant for the plane plasma wave. Some authors would prefer using the terminologies, "electro-acoustic wave", "electron acoustic wave" or "electron wave" (Denisse and Delcroix ) instead of "plasma wave", but we use the term "plasma wave" which has more historical importance. In the event that the ion motions can not be neglected, we can still use the modified terms, "electron plasma wave" and "ion plasma wave". [B] Cold Plasma The complicated effect of the static magnetic field is considered for the simple system, where I-. (3.25) e Physically this situation corresponds to the case that the electron gas temperature is so low as to be negligible, and the plasma wave does not propagate. In general, the major effect in this system is to split the electromagnetic wave into two

32 (2) components which are well known in the field of Magnetoionic Theory (Ratcliffe Budden (5). 2 The dispersion relation, Eq. (3.20), reduces to a quadratic equation in s s 1- )- Q(1-W cos 0) o o 4 2 22 212 + s22 \-2 (1- 2)2+Q2(2- 2COS2 -O2) + 2 4(1- ) l- )- 0. (3.26) o o o This equation shows that s has one root equal to zero at W2 1, 2j = 1+ (3.27) o o 0 and also one root goes to infinity at 2 -1f2 Oj = 2 2 (3.28) 1-f2 cos 0 The two roots which characterize the two components of the electromagnetic wave are 2 2 2 LO s S m f 1- 2 2 thn 4 12 (3.29) 00 1 sin0 + [sino 0 2 2 1/2 + + - cos 0 2(1 -u ) 4(1- ) 0 0 2 s2 =2 1- 2 (3. 30) 0 22 F 4 1/2 2sin0 L sino 2 2 1 L 1- 2 - 2 + cos 0 2(1-u ) 4(1-u ) 0 0 These are the Appleton-Hartree formula for the collisionless case. If the propagation direction is perpendicular to the direction of the constant magnetic field, 0, then the formula (3. 29) becomes 2' s = -, (3.31) o l- -

33 and the formula (3. 30) becomes 2 2 2 2 1 -s2= 1- - o j (3.32) s1,2 L l - 0 The wave characterized by Eq. (3. 31) yields the same propagation constant given by Eq. (3.23) which is unaffected by the magnetic field, and is for this reason, usually called the ordinary wave. The other wave chara cterized by Eq. (3. 32) is usually called the extraordinary wave. Equations (3. 31) and (3. 32) go to zero and infinity at those points given by Eqs. (3. 27) and (3. 28). (2) f Conventionally (Ratcliffe ), for e the wave characterized by Eq. (3. 29) Conventionally (Ratcliffe(2), for0 2' is also called the ordinary wave, and the wave characterized by Eq. (3. 30) is called the extraordinary wave, but this definition breaks down at 0 a 0, since for the propagation parallel to the magnetic field, i. e. 6 - 0, the formula (3. 29) yields 2 S 2 - 1- (3. 33) and the formula (3. 30) yields 2 s =2 L- 1 J (3. 34) which do not display the zero at Lo -1, nor the infinity which is given by (3.28) 0 2 also at w x 1. This is because the process of obtaining Eqs. (3. 33) and (3. 34) 0 2 from Eqs. (3.29) and (3. 30) by setting 0 E0 is not valid at w l 1, since the dis0 persion relation (3.26) has a common factor (1- w) for 0 x0. Still, it is true that 0 Eq. (3. 33) corresponds to the ordinary wave and Eq. (3. 34) corresponds to the 2 extraordinary wave for Lt <1. What happens near 0 "0 is that the ordinary wave 0

34 2 is taken over by the extraordinary wave completely at Li 1, and only the 0 2 extraordinary wave is present for Li > 1. The true state of affairs is illustrated 0 more clearly by Fig. 1 and Fig. 2. [ Warm Plasma The three roots of the dispersion relation, Eq. (3.20), will be expressed by Cardan's formula. These three roots characterize three types of waves. Or, more specifically, these three roots are the squares of the propagation constants for three types of waves. One of these roots will reduce to the propagation constant square of the ordinary wave at Q J, thus, we will call the wave characterized by this special 2 root the "modified ordinary wave". Another root will reduce to the propagation constant square of the plasma wave at 0 x0, and we will call the wave characterized by this root the "modified plasma wave". Then, the wave characterized by the remaining third root should be called the "modified extraordinary wave". Since the electron plasma is assumed to be dissipationless, the three roots of Eq. (3. 20), that is the squares of the propagation constants, are either positive or negative real numbers, corresponding to propagating or evanescent waves respectively. This fact can also be proved from the original ten-by-ten system matrix obtained (8) from Eqs. (2. 27), (2. 28), (2. 29) and (2. 30) (Denisse and Delcroix ). The mathematical details follow next. The dispersion relation (3. 20) can be rewritten in the following standard form 6 4 2 r=, (3.35) 5 +ps +qs +r=0, (3.35)

35 where 2 2 2 2 e oo 2 2 2 e (1-a )-2o P P= -2Po +fe oO + e2 2 (3. 36) SQ cos 0-1 2 2 2 2 2 2 2 2 2 2 2.2 e 0e P o e o o o eo 0 0 e o 2 2 - 2 cos 0-1 (3.37) 2 4( 2). 2)2 -2 P p (1-to ) (1-w ) -n e o o o r" 2 * (3.38) 2 cos 0-1 The three wave propagation constants are given by the three roots of Eq. (3. 35) as 2 P kl A+B- (3.39) 2 A+B A-B P k - 2 -3 (3.40) 2 A+B A-B r- P k " a- 3 — (3.41) 3 2 2 3 (3.41) where K'b b2 a3-1/3 A:-'2- + 27 (3.42) 2 4 27 b [ b2 a3 1/3 B \|4 - -(3.43) 1 2 1 3 ax (3q-p2), bE-(2p3-9pq+27r). (3.44) Also, there is the relation 3ABx-a, which determines the choice of one of the three roots for A and B to be used in order to make the function "a" real. The expressions for the three roots as given by Eqs. (3. 39), (3. 40) and (3. 41) in terms of the original coefficients of Eq. (3.20) are given in Appendix C.

36 For the case of transverse propagation, 0 = -, and so we have 2 p - 3(1-Q -co )+2(3 (1-w ) e o o 0 22 2 224 22 q ^ 3 (- 2)(1-c 2 2)+(1- w2)2 l+ 3 (1-22 o e _ o o o o o 4 2 2 2 2 r -( 3 2(1-c 2) d(1-2)-c 2(2-; 2) o e 0 o o and a 2 2 (l-2)2-Q2 22224 2 2)2- 4 2 2 23 (1-co ) 4~] -(3c 9 -(3 4 (1- ) -( (1 -co ) 2 3 o eL o oe o e 0 0 0 21 4 2 2 2 4 9 292 2 2 b - -3 (3 (1-co )(1 6-c)-( ( (1-co ) (1 -co ) o o e o o e o o 2222 2 22 b -Q -( ) [- 2 + ) (1]-c o27 27 o e 2 2 o e L ~o o e Thus we have

3 b b2 a3 A W. —+ -.+ 2 4 27 32 222 2924 3 2 2222 22 22 27 e 0 0 2oe 1L WoJ0t oL +3 234w2& 2 [(29 2) + (2 02]22 9 [ )4 ] I 2 2 0 0 2 o0 00 0 V4"oe Lo 2 2p2~ LoJ' 0000

38 Similarly we can obtain (+i I,1 2 2 1o2 2:I 2 1 1 -o,.)~ 4 o e o /2 2 0 e 2 2] 0 13 4 4 i" 2 1 +7\J/3/3jl-LO )( Q2 - 2 1 22 -0 e 0 2 2 2I (1-w ) I~ 0I I Finally three propagation constants will be given by sxA + B -3 12 3 0 e + 1 /32 e 2- ai 2 /32/32L(10 2)22 /3 0 0 (3.45) 2 A+ B A -B. P + -3 ---S11 2 2 3 1 02 2 2 1 2L 0 /2 1 /3 2 e 02 2 0 194 4 F1 2X 1 +1 -~4 oe L 0/32 /32 0 e 2 A+B A -B P SIII 2 - 2 3 x 2 (_ 2) 0 0 22 22 22 2"2 /32 I o e i (3.46) (3.47)

39 Equation (3.47) gives the dispersion relation for the ordinary wave, and (A+B A-B P since Eq. (3,47) is obtained by applying the same formula 'T- - |-3 -- 2 2 3 2 which is used to define k in Eq. (3.41), the wave with propagation constant kg will be called the "modified ordinary wave". Also, the Eqs. (3.45), (3.46) and (3.47) display all the zeros of the dispersion relation, (3.20), at w2 31 and 0 W = 1 + Q. Thus, this process of defining the modified ordinary wave is similar to that of defining ordinary wave in magnetoionic theory, and there is no contradiction of this process in obtaining Eq. (3.47) from Eq. (3.41). In the case of propagation along the magnetic field, 0 =0, and we have 2 22 2 2 2 2P to PX - (1-w - )-2, + e 0 o 2 0 0 2 2 (1-2 22 2 4 2 (-1- -t -_ _ _ 3 142 -( (1 4 2)

40 2 6 2 3+ 6 o0 o -=_ ~-/ (1- w ) +32 3 27 e 0 0 (14~) o e o e (1 -cw2) (I ~- w 22 324 o L 0 ) -3.~ 2 2 2 2 (1 -w ) (1.4~ - w ) Q2 (1I~) 2 3 b a 3 - - 4 2 7 x 27 0 ow 21~2~ (14~)L ) 42 2(10-W~i 2M-wo,) -2// 2220 0 e (14~Q)2 o7w 2(1-w22 + 2 04 0 0 o e (IS2) A 12 (,1 2 p2 A 3 e ow o 2 14~ IJ 2O 2 0 0 -3 1 _S2) o 0 + - -'(1 2 2 2 1 2 2 2 1-0 -W0 -.D -.0 (I-W ) -0 3 e 0 0 2 i-Q Thus, the three propagation constants for the three waves are given by 2 pA+ -P 2 (-L2) SI =AB~~3 e1Wo 2 A+B AB 2 Li sII 2 +2 ~ - 3=0 ' (3.48) (3.49) (3. 50) 2 s III Al-B A -B 2-~P 24' 22 3 0k +Q-~

41 The wave characterized by Eq. (3.48) is called the plasma wave as shown before, and since Eq. (3. 48) is obtained by applying the same formula, (A+B —), which has also been used to define k in Eq. (3. 39), the wave with propagation constant equal to k1 will be called the "modified plasma wave'.'. The remaining propagation constant k3, then, should be identified as the propagation constant for the "modified extraordinary wave". The close relation between the wave characterized by Eq. (3. 39) and the plasma wave can also be shown by making 0 — > 0 in Eq. (3.45). This equation then reduces to the dispersion relation of the plasma wave as given by Eq. (3.48). 3.4 Characteristics of Waves In Part A some characteristics of two types of waves involved in magnetoionic theory will be reviewed briefly which will be helpful in discussing the general case later. Magnetoionic theory has been investigated quite thoroughly by various authors (Ratcliffe(, Budden (5)), and liberal use will be made of their results. In Part B the characteristics of the plasma wave have also been analyzed carefully, which should be helpful in understanding the modified plasma wave which is closely related to the plasma wave. In Part C a detailed analysis of the propagation constants as a function of the normalized plasma frequency will be given and the results presented in graphical form. The purpose of this analysis is to determine the conditions for which the various waves propagate and also to assist in clarifying the terminology developed in Section 3. 3. In Part D the surfaces showing the variations of propagation constants with respect to propagation direction will be sought. These surfaces will be used to obtain the asymptotic solution for the radia - tion fields in the next chapter.

42 [A~ Electromagnetic Waves in a Cold Plasma The ordinary wave as characterized by Eq. (3. 31) does not depend on the static magnetic field because it is linearly polarized with its electric field parallel to the static magnetic field. Hence, the electrons are forced to move only parallel to the static magnetic field, and the wave behaves as if the field were absent. The extraordinary wave as given by Eq. (3. 32) is affected by the static magnetic field. For intermediate inclinations of the static magnetic field with respect to the propagation direction, 0 is different from 0 or 2, and the 2 variations of the propagation constant as given by Eqs. (3. 29) and (3. 30) with 2 (5) t are shown by Fig. 1 and Fig. 2 (Budden ). The dotted lines show the limiting 0 positions for 0=0 and 0= ~, and the thick lines are typical curves which always lie in the shaded regions bounded by the dotted lines. The dotted lines are the curves 2 for Eqs. (3. 31), (3. 32), (3. 33) and (3. 34), and the line w = 1. The thick curve 0 marked 0 would deform continuously into the straight line for ordinary wave at = -, and the thick curve marked x would deform into extradordinary wave at 2j 7r 2 0. One value of s is infinite when 2 1-2 2 - 12 2 (3.51) o 2 2 o 1- cos 0 Physically, both to and s must be real, thus graphically, the region of interest to wave propagation is confined to the first quadrant. [B] Plasma Wave The restoring force of the plasma wave is electrostatic, and the limiting (42) case of very low electron temperature was studied by Tonks and Langmuir (42) They have derived the plasma frequency, ao, with which the electrons oscillate

43 s 21 I Il o F IG I. P O A A INC NS A T s.t2 I I I 2 0o S2< 1, O<O'(<ir/2. U-*~O

44 Ig I ~ 2 FIG. 2: (PROPAGATION CONSTANT) VS. 2 2 w V p >1, 0 < e<7r/2, U —O0

45 (43) regardless of wavelength. Bohm and Gross have given the microscopic analysis of plasma oscillation and obtained the dispersion relation for the plane plasma wave as given by Eq. (3.24). Or, more exactly, the sound velocity, U, in Eq. (3.24) is given by the expression L[^kT/m] /, where v is the ratio of the specific heat at constant pressure to that at constant volume (V' 3 for electron gas), and kB is Boltzmann constant. The plasma wave is a longitudinal wave, in which E and V are parallel to the direction of propagation. It resembles the sound waves that propagate in a neutral gas, but there exists a fundamental difference (8) between the two waves (Denisse and Delcroix ). The former is supported by short range incoherent collisions, while the latter is supported by the coupling between the charged particles provided by the electrostatic field. The range of the forces due to the electrostatic field is limited only by the Debye length. Since the phase velocity v of the plasma wave is given by p v - U/(l- ) 2, (3.52) ps 0 and the group velocity v is given by g dw 2 1 v = U(- 2) /2, (3.53) g ds o 2 their variations as a function of w are given by Fig. 3. Both velocities will 0 approach the sound velocity in the electron gas, U, when the frequency w approaches infinity. Also, the two velocities have the order of magnitude of the sound wave in the electron gas, and are related by v v U2 (3. 54) P g

46 FIG. 3: PHASE VELOCITY AND GROUP VELOCITY VS. w2 0 [C] Propagation Constants vs. 2 2 2 In this part the dispersion relation will be analyzed as a function of the normalized plasma frequency squared for various values of the normalized gyrofrequency. 0f0 is considered first which corresponds to the situation where there is no magnetic field.l 0< i and P > 1 are considered next which corresponds respectively to the case when the gyro-frequency is less than and larger than the

47 angular frequency of the waves. Finally, Q= 1 is considered which corresponds to the case when the angular frequency is equal to the gyro-frequency. The results of this analysis are presented in graphical form in Fig. 4 through Fig. 13. In these graphs the square of the propagation constants are 2 displayed as functions of u. As explained with respect to Fig. 1 and Fig. 2, 0 we are interested in only the first quadrant in these graphs since only those waves, whose propagation constants are in this region can propagate, and physically 2 2 2 2 there is no negative w. We will plot and analyze k1, k2 and k3 in a manner similar to that used to obtain Fig. 1 and Fig. 2. Thus, the two limiting cases, o 0 and 0 = - corresponding respectively to longitudinal and transverse propagation, will be analyzed first. The curves for these two limiting cases provide the boundary lines for the shaded areas where k, k and k always lie. a) Q x0: Physically, this is the limiting case with no static magnetic field which was discussed in Sec. 3. 3, and the dispersion relation is given by Eq. (3. 22) as 2 2 2 2 2 21 2 2( 2) [s2 - (1- ) = 0. 0o e o 2 2 2 p Variations of s with respect to w = can be seen from Fig. 4. o 2 tO Practically, the acoustic velocity is, of course, much less than the velocity of light, thus 2 2 >32 >> 2. (3.55) e o

48 s21 ' - \3 oo i\ -- FIG. 4: (PROPAGATION CONSTANT) VS. u2/, =0 p.. I'

49 2 1 FIG. 5: (PROPAGATION CONSTANT)2 vS. w 2/2 2>1, 0=0

50 sI 0 FIG. 6: (PROPAGATION CONSTANT) VS. w /2 nQ<1, 0=0

51 I I I I 24) 0'J 1 I-S2 1+ 'R I I I I I I s2 1 n I I I I I I I I I 1 20 -11 2) 1 e I I FIG. 7.4, (PROPAGATION CONSTANT) 2 VS. w 2/W2 Q> 1 a = X/2 p J

52,2( -,2) te~t FIG. 8: (PROPAGATION CONSTANT) VS. 2/w2 <1, 0 =r/2

53,2?2 c - 3iA e sza.,: b r7 dd7 U t eezj 2 k Modified Plasma Wave k2 Modified Extraordinary Wave 3 FIG. 9: (PROPAGATION CONSTANT) VS. w2 /2 <1, 0< 0<7Tr/2

54 / \ I I I I I I I 1-'-,_ Q2. k Modified Plasma Wave - 2 ii:;*,.;:. 'k Modified Extraordinary Wave 2 - n __ ____ k Modified Ordinary Wave j3 FIG. 10: (PROPAGATION CONSTANT)2 VS. W2p/2 Q >1, 0 < < r/2 r / ); L

55 S 2 2 13 e s2 222 FIG. 1 1: (PROPAGATION CONSTANT)2VS. w22 p ~~=1,9 0=0 rr.,- I , / 0 I i (."

56 I FIG. 12: (PROPAGATION CONSTANT) VS. u2/& 2 1 = 1, = 7r/2

57 -[ r/ I Zl/./ fl:-.~ ~-H-l 7- 7,- 'r rang, Modified Plasma Wave Modified Extraordinary Wave Modified Ordinary Wave 2 0 FIG. 13: (PROPAGATION CONSTANT )2 VS. 2 /2, 0<0<Pr/2 2 = 1, O<9<r/2 bl*. Id Ra f I Y., M c9

58 However, in order to illustrate the important features of the analysis of the waves propagating in an electron gas on the graphs the value of B will be e chosen such that 2 2 e o b) ft> 1 andft<l: The two most general situations are for 2> 1 and n< 1. The three propagation constants for the modified plasma wave, modified ordinary wave and modified extraordinary wave for these two situation are shown in Fig. 9 and Fig. 10 respectively. Two special cases will be discussed first. The third case is the general case. Case I: Propagation Along the Magnetic Field (0 = 0) Three propagation constants are given by Eqs. (3.48), (3. 49) and (3. 50). Their graphs are plotted in Fig. 5 and Fig. 6. sI is the propagation constant for the plasma wave and s and sII are the propagation constants for the two electromagnetic waves. As mentioned in Sec. 3. 3, the distinction between the ordinary and extraordinary wave is not clear in this case, but in keeping with the definition (8) used in this work (Denisse and Delcroix () si is the propagation constant for the modified extraordinary wave and sII the propagation constant for the modified ordinary wave. Case II: Propagation Across the Magnetic Field (0 = -) The three propagation constants in this case are given by Eqs. (3. 45), 2 2 (3.46) and (3.47). The two roots sI and sn given by Eqs. (3. 45) and (3. 46) can

59 2 2 be combined to yield an equation of second degree in s and 02 which represents 0 a conical section. The equation is 22 2 2 2 2 22 2 2 2 2 2 2 (s )+( + )s w +f 6 ( ) +s I (_-1) -2 o e 0 o e 0o e oi -2w 22 2 + 3 (1- 2 ) 0O. (3.56) o o e o e Equation (3. 56) can be analyzed by considering it as the general equation of second degree Ax 2+Bxy+Cy +Dx+Ey+F= 0, (3.57) 2 2 where x u and y= s. Since the following inequality 0 B -4AC ( -2 )2> 0 o e is satisfied, this conical section is a hyperbola. 2 When 0, 2 2 2 o when 2 <1- 2- 1 2 o 2 s 2 e e p 2 2(1 -Q2) when Q2 1 --- 0 1 e 2 2 2 2 0 i (1-n2) whenn2 <1 — e 2 2 2 e 2 2 0 Two points of intersection with the Li - axis are obtained at t 1 + Q. If 0 0 - 2 1-t 2

60 44 2 2 2 o e Q ( Q + o e c 1 ' 4 2 2 2 ( 2 1 2 + o e o o e o 2 2 and so s 0 at this point. At another point 1-w = 1 2 - 2 2 / 1 1 o0 i2 2 e 3,2 2- ( 2 o e o o e and we can see that 2 s 5 0 when < 1 2 2 o s = 0 when > 1 + - 1 I 2 e The detailed analysis of the slope of the hyperbola, and the coordinate transformation applied to Eq. (3. 56) in order to reduce it to the standard hyperbolic equation, is given in Appendix D. The standard hyperbolic equation expressed in the new coordinate system (i -s1 )is O

2 2 4 4+.4 4 4i4 4 4 oe e 0 0 o e 0e 4 fe4 4 4 o e 0 e 404 A4 04 2 2 44 4 4 00e 0 e 00 o 0 0 2 4 4 4 21+0Oo0e +O0+Oe f 2 [/e(n 21)/3j01 -0/3/32 1+0 $ +0 +13 -2/3 /3 -+13 + 1+134 /74 + (a oe oe -oe o e e 0 C.,, I(512 ) - /3 -1-~ 1+ /3+ +0/3 0 0 +/3131-/31 + 1+ 43 +4 +4i 0 o e +00 Pe+0 oe o e o e o e 0 e [1+0 40+ ~+A4_ (1+p 2o2] 2 +04 + + 4 00e 0 e ej o eo0e A0n 2 2 (02-0 2)2 0 0 r~0J (3. 58)

62 2 The transverse axis of the hyperbola is parallel to s' - axis when 132 _>i - 1 12 and the transverse axis is parallel to a' - axis when 32 Q <i -1 2H 1 are itemized as follows is parallel to 0 92 (i) when n < 1 + ~21, the center of the hyperbola translates 22 e toare itemized as follows: 2 -axis. 2 o (ii) When Q > 1 + -4 1, the center of the hyperbola translates 2 e to positive s' - axis. 2 2 (iii) The slopes of the asymptotes in hy' -s' plane have the absolute value to positiv e s 1+ axis. + (1+3 21 2) + V 1+13 +o +13 2 _ o e o e o e 2 2 I and the propagation constant for the ordinary wave sI are plotted in Fig. 7 and Fig. 8.

63 Case III: Intermediate Inclination (0 < 0 < - ' The numerical values of three propagation constants given by Eqs. (3. 39), (3. 40) and (3. 41) could, of course, be obtained directly from these equations. However, such a procedure is tedious and the important features for this case can be obtained by an approximate analysis using the well known results for the two types of waves in cold plasma as shown in Fig. 1 and Fig. 2. The main difficulty in this analysis is to determine the points of discontinuity for three propagation constants, where the transition of modified plasma wave to modified extraordinary wave takes place in Fig. 9 and the transition of modified plasma wave to modified ordinary wave takes place in Fig. 10. These points can be found approximately from the fact that the appearance of modified plasma wave is associated with the disappearance of the discontinuities in the graphs of Fig. 1 and Fig. 2 (Ginzburg (4). Thus, these transition take place at 2 1-f2 o 2 1-2 cos 0 This result can also be found approximately from the original Eq. (3.20). Since it 2 is true that in most of the range of cO 2 2 k1 >> k2 2 2 1 >> k3 2 and so k1 can be obtained approximately by first two terms of Eq. (3.20) as ( 2 2 2 2 2 2 l- -w +n W cos 0) k 2 2.e (3.59) (1-Q cos 0)

64 4 2 2 (the coefficient of s is simplified by using the relation 2 >> ). The exe o pression in Eq. (3. 59) assumes the minimum number in the neighborhood of 2 2 2 2 2 1-1t - - +Q to cos O= 0 0 0 or 2 1-n2 = 2 2 1- cos e The bounding curves for the shaded areas in Fig. 9 and Fig. 10 are obtained from Figs. 5, 6, 7 and 8. The curves for the three propagation constants, 2 2 2 k, k2 and k always lie in these shaded areas. c) Q = 1: This condition is satisfied when the angular frequency of the wave is equal to the electron gyrofrequency, and the Eq. (3. 20) reduces to s sin + (2 + sin -23 2sin 26)+s 2 Ll-O ) (2 +2 ) o o o e o o o e o 2 2 2 2 2 2 2 2 2 4 2 2 -(2 +3 cos 0e-3 cos e0-3 w ) + /13 (t -1)(Wc -2) = 0 e o e o e oJ e o o o (3.60) 2 One propagation constant becomes zero at to = 0, 1 and 2, and also one propagation constant becomes infinite at 0 = 0. Case I: Propagation Along the Magnetic Field (6 = 0) In this case Eq. (3. 60) can be simplified to 2s +s CL (2/3 + )23 + 3 )23 -()(-2 0222 (3.61) e o e o o o and the two roots of Eq. (3. 61) are given by s = (l- ), (3.62) I e o

65 and 2 2 o 2 III (2 - w (3.63) Equation (3. 62) gives the propagation constant for the plasma wave, and Eq. (3. 63) characterizes one of the electromagnetic waves which can be reduced from 2 2 2 Eq. (3.50). sI and s versus w are plotted in Fig. 11. Case II: Propagation Across the Magnetic Field ( = 90 ) Equation (3. 60) will give 6 4 22 22 22 22 2 2 22 s +s 23 2(w21)+w 1 +s (1- ) (23 +3 )-(2-w 2)2 2 o o o o e o o e o o e +2 4 2(c 2- 1) ( 2-2) = 0 (3.64) e o o o o and 1 2 2 22 2 222,. 1L3o! wo)+1 \W o13eJ8 +3Wo3e1o 1 F2 2 22 3 2 2 42 424 3 o o e o e o Thus we have -+ < for all w and so we have three real and unequal 4 27 -o 2 roots for all to except at two negative points; - - = - ) p +4u 2 4 e 4-o oo e e (132-132)2 o e 2 b a 2 and also at w = 0, where we have - + - = 0 which means at least two real roots are equal.

66 Actually, we can separate the ordinary electromagnetic wave from Eq. (3. 64) and obtain 72 2 2] 4 2 2 22] 2 2 2 Is -3 (1-w ) s +ss ( -1)+w c. + 13 w (o -2) =O o o L 0 0o e o e o o (3.65) which gives the propagation constants of the three waves as s2 1 2 2_ 1 I 2+ - - 2 2 2 2 22 sI = w 2-1 )+ - 1) + / (+ -1) -t /3 +/ 13 w I 2Lo o ei 4 o 0 o e o e o (3.66) 1 2 2 2] - I 2 2 22n2 22 s I2 L - 2 o ( - )+W J - 1 /3 (wo -1)-wte ie +/3 3 0o (3.67) s I 3 (1-w ) (3.68) II o 0 Eq. (3. 66) and Eq. (3. 67) together give 4 22 24 2 2_2 22 2 2 2 s +s (/ + )+w / / -s13 -2/ / 0 (3.69) o o e o o e o which represents a hyperbola. Equation (3. 69) can also be deduced from Eq. (3. 56), and an analysis similar to that applied to Eq. (3. 56) can be used for the analysis of Eq. (3. 69). Some of this analysis is given in Appendix D. The standard form of the hyperbolic equation obtained by application of coordinate transformation to Eq. (3. 69) is

+ 3 ) 4 4 4 1/2 (1+ e)(1 / +43 ) + (1+ /3 + + 4) x o e e e o e o e o e 4 2 2 4 4 - 4 2 21/2 2 4r, 4 4. 1 224( 4 + 4+ 4 1 2(1+3 2 - - + l+^ 4 o/e 2(1+3 o e o e oe o e (2 d 3 2 1 +1 3 413 +3 ) 21 4(10 41 3 70) 4) o e o e e o e 2 2 24_ 4 44 2/] 2 f- 4 4 -(1+1 3 )+1 3 +1 ) 2(1+3 1 3+ +3) o e o e o e o e e n bo\ o e ole ov e o opes fo1 1e +s 1o3 is +3 ea/t j 4 4. 213.2....... / 2 42 4 12 22 131 1- (1344 -13) )(3.70) o e e 0 e 2 The transverse axis of the hyperbola is always parallel to the s' -axis. 2 2 than one. This hyperbola intersects the o -axis at t = 0 and 2, and the s -axis at s =2 and 0. Figure 12 shows the graphs for Eqs. (3. 66), (3. 67) and (3. 68). 0 Case III: Intermediate Inclination (0 < 0 < ) This case is similar to the case given in Fig. 10. The three propagation constants obtained from Eq. (3. 60) are very tedious to plot directly, so some

68 singular points are discussed first and the boundary lines for their variations are introduced from Fig. 11 and Fig. 12. At u =0, i s = 0 2 2 2 o(3.71) s (s -2) 2 = 2 At = 1, 0 s sin e +s (2 2cos20 +B 2sin 2) - 2( 2 cos2 +0 2sin20) 0 o e o o e s2 = (3. 72) which gives s = 23 /2 and s = o at 0 = 0, and 2 e - e e o k s = at.2 2 2 At =J 2, t4 2 2 2 2 2 2 2 2 ) / s sin +2s +) cos 0 + ( sin ) + 2( 2sin2 + 2 2cos2e+ 2 2) 0 (3. 73) o o e e 2 ' S:0. Rearrangement of (3. 73) gives 2 (s2+{ 2) tan 0 = (3. 74) s +2(B~ +2)s + (OB +20 ) o e

69 Thus at 2 2 i S =-d 0 0: 2 S = 00 and at 2 2 -p2- 2 s -23 -/ e o 2 2 2 S = -/3 \ 0 Three propagation constants are plotted in Fig. 13. This is a limiting case and the identifications as modified plasma waves, modified extraordinary waves and modified ordinary waves can not be seen clearly from the graph. As it stands, we do not know how the thick dotted curve for the modified extraordinary wave changes to the straight line for the plasma wave. More elaborate analysis is necessary in this case. [D] Effect of Direction In order to discuss the dependence of the propagation constant on direction, 2 tan 0 is derived from Eq. (3. 20) as 2 \62 47 2 2 222 2 2 227 tan 0 (s (() )(L3e +20 )-2 (Oe +203 e 0o ) 22 -(n - 2,2(22 2+ 2,+ s +2 2 2 22) 2s 2 22 2 2 2 2 2 ) +s/ (_(- )(23 +/32 )-27 (2( 2 -/ c 0 0 e 0 e e o e +2 4 2 2 2 2w 2) e o o L o I l- e o e o0 -+ d s (-(1)- ( 3 2)-+2(2 +2-f 2 2) 2f (1-w ) (1-w2)2 -n2] s -1 (3.75)

70 Equation (3. 75) can be factored as 2 2 2 2 '2 2 2o 2 2 o 2 2 tan 0 = (2 -1) s - (l-w ) s - (1 — ) s - (1 — x e 0o o 1- o 12 2 2 r2 2 2 oe 2 1 s -3 (1-w ) - (1 ( + ) I 0 0 2 0 2 2 2 o e o 4 4 131 27/2 -1 o-e 2 1 1 )( 2 2 2 2 2) 4 o 2) 2 12 o e o o e o 2 2 2 2 2 1 1 - /3 x s 1- - 1 2 0 2 23.. 2 o e o 4 4 -2 22 s - 4 2 0 e o and 4.cutoff respectively by A is 2 (3.77) o ) 0 2 Now we want to investigate which values of the parameters (o, ) and give propagation, and which values give attenuation. The boundaries of these regions are the lines along which s = oo and s = 0, which are called "resonance" and "cutoff' respectively by Allis The principal cutoffs are given by (1-u2) =0 (3. 77) 0 (1) = 0 (3.78) (1 — =

71 2 (1- +)= (3. 79) The principal resonances are given by Q = + 1 (3.80) Resonance occurs also at the angle 0 which satisfies the condition 2 2 tan = (2 -1) (3.81) This can happen only for Q>a 1 and is obtained from the original dispersion relation 2 2 (3.20) by setting the first coefficient equal to zero, i.e., Q cos 0 -1 = 0. There is one more boundary line existing at 2 2 i +Q = 1. (3.82) o In the case of a cold electron plasma this condition gives resonance for the extraordinary wave as can be seen from the following expression: (1-2) 2 2 22(1 ) _2 s2=2 2 (3.83) 1-2 - 0 This is also quite apparent from Fig. 9, where the transition between the modified plasma wave and the modified extraordinary wave takes place at the angle satisfying W 2 (3. 84) 1- cos e Equation (3. 82) is the lower boundary of (3. 84) corresponding to 0 = 2. Equation (3. 84) can be rewritten as 22 1- (2 + 2) 2 o cos 0 22 (3.85) 0 2 2 and so this transition angle exists only for Q +w > 1. 0

72 In Fig. 14 sample plots of (s-0) curves are given for eight regions in the 2 2 (c -2 ) plane. These curves will be called dispersion curves. The surface of 0 revolution obtained by the rotation of the dispersion curve around the s2 axis is known as the Fresnel phase surface. The boundaries of the eight regions in Fig. 14 are given by Eqs. (3. 77), (3. 78), (3. 79), (3. 80) and (3. 82). The direction of mag2 netic field is assumed to be in the n - axis direction. The propagation constant of light in free space, 3 = -, is given by the dotted circle in the figure as a reference. o c These dispersion curves are deduced from some wave normal surfaces calculated by Allis, Buchsbaum and Bers, and also confirmed by our numerical results in Chapter V. The wave normal surface shows the variation of the phase velocity v with respect to the direction in space, thus it has the inverse relation with the P Fresnel phase surface, which can be seen from the definition of the propagation constant, i.e., s --. v p 2 In region 1, corresponding to 0 < w < 1- Q of Fig. 9, three distinct dis0 2 2 persion curves exist. In region 2, corresponding to 1- t < 0 < 1- Q of Fig. 9, only a modified ordinary wave and a modified plasma wave exist. In region 3, 2 2 corresponding to 1-0 <u < 1 of Fig. 9, the transition takes place between the o modified plasma wave and the modified extraordinary wave at the angle 0t satist 2 fying Eq. (3. 85). In region 4, corresponding to 1 <o < 1 + Q of Fig. 9, only a 0 2 modified extraordinary wave exists. In region 5, corresponding to Lo > 1 + of Fig. 9, none of these waves exist.

\C - >/ \%- -O p: Modified Plasma Wave x: Modified Extraordinary Wave o: Modified Ordinary Wave c: Light ol FIG. 14: DISPERSION CURVES FOR THREE WAVES

74 The dispersion curves in regions 6, 7 and 8 can be explained in con2 junction with Fig. 10. In region 6, corresponding to 0 < co < 1 of Fig. 10, 9 0 there are three distinct dispersion curves. The resonance angle 0 for the modir fied plasma wave is given by Eq. (3. 81). In region 7, corresponding to the range 2 1 < w < 1 + Q in Fig 10, the transition between the modified ordinary wave and the modified plasma wave takes place at the angle 0 satisfying Eq. (3. 85), and the t resonance angle 0 for the modified plasma wave is given by Eq. (3. 81). Between 0 and 0 there is the relation t r - -12 _2___ 0 =tan K2 -1 > 0 =tan1 (2-1) 1-S 2 ) (3.86) r t 2 z o 2 Region 8, corresponding to w > 1+Q in Fig. 10, has the same features as region 7 0 except that the modified extraordinary wave does not exist. In the limiting case of Q= 1, it can be seen from Fig. 13 that there are two 2 waves which exist in the range between 0 < w < 1 with closed dispersion curves, 0 2 while there is only one wave in the range 1 < w < 2 which also has a closed dis2 persion curve. No wave is possible for w > 2. 0 When Q = 0, only the plasma wave and the ordinary wave exist. Their dispersion curves are circles. In general, these three wave constants do not become equal except at the transition angle 0 in regions 3, 7 and 8.

CHAPTER IV WAVE EXCITATION IN ONE-FLUID PLASMA 4.1 troduction Ih this chapter the general excitation problem in a homogeneous electron fluid plasma, which is compressible and uniformly impressed by a constant magnetic field, will be treated by applying the formal solution obtained in Chapter II. The dispersion relations analyzed in Chapter III will be utilized directly in calculating the excited fields. This type of problem, as far as is known, has never been treated in the literature. The excitation problems discussed in the literature so far maybe divided into three categories: (1) Cold plasma problems with a uniformly impressed constant magnetic field. In this type of work the longitudinal plasma wave does not come into picture. Typical examples of this type of problem are the works of Arbel(, and Arbel and Felsen (20) They start their formulation with "ordinary" and "extraordinary" modes. (2) Compressible plasma without an externally applied constant mag(30) netic field. With these assumptions Cohen has shown that the field can be separated into two types of modes; one mode is transverse in nature and has all the fluctuating magnetic field, and another mode is longitudinal in nature and has all the fluctuating density field. The radiation of this acoustic-type of wave has been investigated by Hessel and Shmoys, Whale, Chen(45) and Wait(46) (3) Two-dimensional problems in a compressible plasma with externally. (24) impressed constant magnetic field. Seshadri has investigated the radiation 75

76 characteristics of a line magnetic current source in a homogeneous compressible plasma of infinite extent with an externally impressed uniform magnetic field. The problems studied in this chapter include and extend the problems of the third category. First, the unified and systematic formulation developed in Chapter II will be applied to two-dimensional problems, and Seshadri's solution will emerge as a special case. Next, general three-dimensional problems will be (38) treated where Lighthill's method will be used to obtain the asymptotic solutions for the excited fields. The equivalence relations obtained between the various types of sources, by means of which the fields excited by one type of source can be expressed in terms of the fields excited by another type of source, are some of the highlights of the unified operator transform method. 4.2 Two-Dimensional Problems Here we consider those excitation problems where the fields are not varying in the direction of the constant magnetic field, which was assumed to be in the y-direction. A] Field Solution i Transform Spafor All Types of Sources The transforms of the magnetic field and the density fluctuation field are given by Eq. (2. 86), namely L Ve(s) LI - N(s)] - F(s) To find the explicit expressions for It(s) and Ve(s) we have to find the inverse of the matrix [l-N(s], and the Fourier transform of the general source function JV())+ Js Wt(s) F(s8) (s)+ U2 (s, V (v) N. (4.1) 12 2 N_ E-SOIJc, w () 0 1 i(S

77 The inverse matrix for the two-dimensional problem is given by 0 0 -1 1-N(s) 1 det.!1 -N(s)] 0 M22 0 M42 42i 0 (4.2) 0 M33 0 24 0 M 44 where det. I-N(s) can be easily obtained from Eq. (3.19) by setting s2= 0 and is det. L1-N(s) 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 = ( -t — s c ) ( - -s c )(W -o -s U )-t (- -s c) I p L p p C 2 2-1 x (2 -t ) P r 2 2)2 ( - )p p 2 21 -1 C and 2 2 2 22 2 2 2 2 2 2 2 2 2 s s c (t - - )+s U ( -to -s c ) M1 ~1- 1- P c p M12 2( 2 2 2 to -t, (u, -t ) -t to L p c 2 22 2 2 2 2 2 2 2 22 s s c s c (t -t -t )+s U (t -w -s c ) M - 3 c1- c 13 2 2 2 22 22 2-t I - ( - ) - P K p c (4.3) (4.4) (4.5) 2 2 2 2 s U to - ) P: 22 1-2 22 2 2! ( -to ) -t to 1 '' P c J 2 2 t s/e i c p M - 1 -24 2 22 22 (t -o ) -U P c 22 2 2 s c } 2 2 I t -to P J (4.6) M31 =M13 (4. 7)

78 2 2 M ~1 -3 C 33 2 2 to -W p I 2 2 ec U to s m42= 2 2 2 (to -to ) - tj p 2 22 2 2 2 22 22 2 s c (to -o -to ) + s U (to -to -S c ) I-p C p 1- 2 - i22~ 2 2 I6 ~p 1- c 2 2 1-2 2 ) t o t-to) (4. 8) (4. 9) (4.10) C p 2 22 2 27 -s c (to -to -to ) M 1-p C 44 2 22_ 2 2 (uo -to ) -t to c - p CL 2 2 2 2 to-to I 2 2 2 Here s s 1 +5 The four components of the four-vector general source function are found from Eq. (4. 1) to be 2 s ec F1( = tx tom + o2 -t2f ) wM~w _W) p2 2 is c 2 + 2 2 2 2 2 (t -to ) -to to p c w2 _2 (t P-o 1 tot tow f (5) + i z +is 2 2 isys to -to p i 2(2 W2- 2) 2 p c j (s) 2_ 2 2 2 2 Iz (to -t ) -to to p c (4. 11) F(s) = J (S 2 ic e f (S) + s f(s)]+ - 2 — f S S 2 ty 2w 22_w2o2)m L c3 z Ilx 3 1 p 2 sji(s) +sj(s)!-(to -to -to ) is j (s)- sji(s; to L3Z lx — i p c ~3 x 1 z - (4.12)

79 2 slec (som(W -W) )Y 2 2 2 is2c 2 2 e ( -e! i 2 +' e 2 22 2 2 2m ~cfz(s)+ f (s) j (s) P c. 2 + ( 2 - 2 -o )ix(S) _ -- -- I () p c x 2 2 y() (4.13) w - t W _ P s- r.r- w2 /e es2 - a -q F4(s) W (s) f (s) i 4Te f2 es2 m y2 2e y P c +s w 2 2 2 (W 2 c " 1 es-W C ZU + "p L 2 2 +s 2 L x((s)+ * (4.1 4) The transform of the electric current source, j(s) d(s, r)J, and the transform of the mechanical body source, f(s) $ d(s, r)F, have been used in Eqs. (4.11) through (4.14). In deriving Eqs. (4.11) through (4.14), without loss of generality, the constant magnetic field is assumed to be directed along the positive y-axis, thus these equations may be used in three-dimensional problems. After setting s2= 0 in Eqs. (4.11), (4.12), (4.13) and (4.14) and using the result in Eq. (2. 86), the three components of the transform of the magnetic field

80 and the transform of the density fluctuation field may be expressed as follows: 11 tx 2 2 2 2 ( -i -S c p k (2 2) [ )+ 2 s ec 2 2 wm(w2 -W ) p is c2 f (a) + 3 j (a) p - as c Jt(s)- ss3c2 tz(s) 1 tx 1 3 tz } (4.15) I () - I( W 2)(w2- 2-2U 2) 2 2 F (s) ty l P p e c 2 2 2 2 ec U s 2 c A F es | e() 2 22 2 2 2 m (< -p ) -W ~ p C 2 2 (' (s) + -— f(=)) \ z Uw x 2 + s3 3 p 3 p '( (c (0 2 2 s )+es ( +2 X m (0-to \O h ~.^yy.} (-< f (a)+ C x 2 2 =_-.- f_ f( 2 2 (-i J(s)+ -— 2 — jz( (4.16) 1 ')' 2 2 2 22 w- - s8 C p 2 slec0 2 -S- fy(s) - lsc () tom y I 2 1 I3 tx() 2 2 Ve(a) = ~ — e - e euu [2 2X2i 2-22c 2> 2( 2 C F<...)... p..... c (a 9 9 (4.17) _u l/e... N j (s)+ go 2s2X2 wZ e I (~fs(8) + =-._=.R_ f (s 8)+ (2W 22W2w L c t(sz)'i s x Ip tp zli) p C 2 2 2 -t 2 (0 2 2 es3 / ( 0 -m- f(s)+ -- - - fz(8 CX io z/ 3p / ( (-1C i - - j> 2 2 (s)+ 2..i ( w- - (4.18) 2 22 2 2 22 2 2 222 where Aa ( -p -s c ) p( -W -s U (C -sc) P p c O

81 B l Physical Interpretation The types of waves excited by the different types of sources can be determined from the original Maxwell-Euler's Eqs. (2. 27) through (2. 30), or from the explicit solutions in transform space, Eqs. (4.15) through (4.18). Although it can be shown directly from Eq. (2.27) through (2. 30) that for the magnetic field in the y-direction and no field variation in the y-direction the fields can be separated into two independent sets of components, such a separation can also be seen clearly by examination of Eqs. (4.15) through (4.18). In particular, since h and n are the inverse transforms of It(s) and V (s) respects e tively, their characteristics depend upon the two types of poles of I (s) and V (s). The poles of I ty(s) and V (s) are determined by ty e 2 22 2 2 22 2 2 222 A=(2-LJ -s c )(L2 -W -s U )-Wo (W -s c ) = 0 (4.19) p p c which is the dispersion relation for the coupled extraordinary wave and plasma wave, or the modified extraordinary wave and the modified plasma wave. The poles of Itx (s) and It (s) are given by 2 2 2 2 2 -ac -c s = 0 (4.20) which is the dispersion relation for the ordinary wave. Thus the propagation constants determined by (4.19) are given by Eq. (3.45) and Eq. (3.46) and the corresponding field components are E, E, h, V, V x z y x z and n. Similarly, the propagation constant determined by (4.20) is given by Eq. (3.47) and the corresponding field components are E, h, h and V. y x z y

82 Explicitly, the results can be summarized as follows: (a) A line magnetic current source K= AK 6(x)6(z) will excite the modi0 fied extraordinary wave and the modified plasma wave, but not an ordinary wave. This can be seen from the fact that the y-component of the transform of the magnetic current source, Jty(s), exists only in F (s), which will produce It (s) and V (s) as given by Eqs. (4.16) and (4.18). e (b) A line electric current source J= ^J 6(x)6(z) will excite only ordinary 0 waves, since only It (s) and Itz(s) are different from zero as can be seen from Eqs. (4.15) and (4.17). (c) The transverse components of the electric current source will excite the modified extraordinary wave and the modified plasma wave, but not the ordinary wave, since (s)and j(s) exist only in the Eqs. (4.16) and (4.18). (d) The transverse components of the magnetic current source K will excite only the ordinary wave, since Jt (s) and Jt (s) exist only in the Eqs. (4.15) and (4.17). (e) The transverse components of the mechanical body source F will excite the modified extraordinary wave and the modified plasma wave, and the longitudinal component of F will excite only the ordinary wave. This can be seen from the existence of f (s) and f (s) in the Eqs. (4. 16) and (4. 18), and the existx z ence of f (s) in the Eqs. (4.15) and (4.17). y (f) The electron fluid flux source Q will excite only the modified extraordinary wave and the modified plasma wave. Since W (s) exists only in the Eqs. (4.6) and (4.8). Eqs. (4.16) and (4.18).

83 LC] Comparison of the Excitation Effects of Different Types of Sources Applying the Eq. (2. 69), and the transform of the magnetic field given by Eqs. (4. 15) through (4. 17), the transform of the density fluctuation field given by Eq. (4.18), and also the expressions given by Eqs. (2. 75) through (2. 78), the solutions to four typical, simple excitation problems will be presented in this section. The source terms are chosen such that only the modified extraordinary wave and the modified plasma wave are excited. If necessary, we can always superpose the ordinary wave which is decoupled from the other two waves. 1. Electron Fluid Flux Source 2 Let Q = Q 5(x)6(z) (2), then we can obtain its transform W (s) d(s, r) -o (4.21) e w a From Eqs. (4.15), (4.16), (4.17) and (4.18) we have It(s) = 0 (4.22) -ec U W ()(sa2 +s32) Ity(s) Xo4 (4.23) Itz () = O (4.24) as c (w -w )-c _ +(2 W )2W (s) Ves) - P c c v e (4.25) e a and then from Eq. (2. 69) we can obtain

84 [ 2 2 2 2 e 2 E1 ( - ~2 "C s )+ C 3 (a) o vt. () ~ a y o (2- 2-c2$2) - i * ' w (a) V (a)... tu2 - -— c J eW tz e A 0 (4.26) E 2 2 222 1 2 ae B3il (W -w a-c 8 )-iww3 * Iexi() = N W (8) - - 0 ( I (a) C O I (s)a ez ] (4.27) c 2 2 2 22) 1 -- Omm-o1Sll s -s3( J - -C 8 )+ ia 21 c 3 p c Ia) 2 WU w (8) N e 0 A In order to obtain the expressions for the fields in real space, the inverse of two different functions must be known. The details of the evaluation of the inversion integrals are presented in Appendix B and the results given in Eqs. (4.28) and (4.29), respectively Ef i(s fd dss 1 1 iJ 2 22 2 [2 ( - H(5(r)J15) c U (. ) 1 1 I L16)H'~~(.8 O - co (2+ s)2 ~i(slx+sbz) A,. dslds3 rs 2 (1) 2 (1)2 U2 2 2 2 2U o (l0I- () C U (s -5) II-I~ (4.29)

85 Application of Eqs. (4.28) and (4.29) to Eqs. (4.22) through (4.27) yield the following field solutions: 2.h e (- — ) I-I o- )(8Ir r) (4.30) 2 2 2 2 2 22 2 2 2 2 2 _ _ ( _ _a c ( a - - ELxc Q2 2_ 2 22 2 (W -W W \, ) _E 2 1 a.. (^ (4 31) 2 +iC (43) x N p ax 1 caz 1 ax 2 -eU Qo that h a 2V a a. x z y y - (4.33) ie cvet P aioI c ax I af kind of o rz x N p ax I ct 1 ' z 2 — 2 - 2 C) a 2 2 a -+i N p az I ax 1 w f+ 2 a_ where is and sH are the propagation constants for the modified plasma wave and the modified extraordiry wave as given by Eqs. (3. 45) amd (3.46), and H (1) is the conventional Hankel function of first kind of order zero. Also it is clear th =h =E -V.0. x z y y

86 2. Mechanical Body Source )2 Let F = x F 6 (x) 6 (z)( 2r), then its transform is 0 f(s) d(s, r) F xF (4.36) Applying Eqs. (4.15) through (4.18), the following expressions are obtained: 2 2 2 2 2 2 2 /ec F \ -Ws (LW -_ )-ic s +cj (s +s ) m2 ) 3 p c 1 3 1 3 (4.37) It(s) It(s) = 0 2 2 2 2 2 2 / iF -s X -W )+WC s s +iW s3(o -c s ) V(S)= -1 p c3 (4. 38) e m(/ A The transforms of the electric field and the fluid velocity field can then be obtained from Eq. (2. 69) as 2 2 2 2 2 2 2 2 /eF w( -c s1 )+ i c ss -o +U s o.) 1 c 1 3 p 3 Vty (s)= 0 (4.39) 2 2 2 2 /eF c S S3+iu (u -c 3 )-WU s s ) _ 3 c 3 13 Vtz(S). E /~ myo aM 0 3 2 3 2 2 2 22 2U 2 2 2 2 iF w sls-iw s +iU ss3(s ) P 3 ) (s I (s) c LiF J+LU+ c ex 2 \mN/ w o I (s)=O 2 (4.40) WW2 F ss -iw w s +J i +uU ss(s ez 2 (;P N A to o

87 By using the relations given by Eqs. (4. 28) and (4.29), all the inverse transforms for Eqs. (4. 37), (4. 38), (4. 39) and (4. 40) can be obtained as follows: eNoF F2 2 2 -7 o o 2 2 a 2 2 2 2, + E --- ( -w ) c -iw c +cc - +WU 2 -- 1 x \ /mN/ p 1 c axaz 1 2 1 2 l1: o o ax 3z E = 0 (4.41) y eN F 2 2 = 0( o 0 i w 2 2 2 a d. t 2 a E L- i -+-; w-icc-U) \ / \mN / i c 1 axaz 1 c 2 o o z 2 2 2 c eF r-(W - ) a2 K yh = I --— ( ) _ (4.42) 2 a ax 1 i xz 21 y-U2 2 I -~ o.. W c Fc a a iF 2F 4 2 2 2 2 -iccoc " - 2 -2 U a2- a i 2/ M A N L c 2 1c 2 2 - axaz I ixa z W 0 c c 2 2 n Li W 2L _^+! - W c -^ (4.44) W m c 3z ax 1 3ax 2 c z 2z 3. A Line Magnetic Current Source A line magnetic current source can be expressed as K K 6 (x)6(z)(2ir)2 (4.45) 0

88 then its transform is - iK 6 (x)6 (z)(2j)r -iK J (s) = $d(s, r) O (4.46) ty co t 0 0 As before, the application of Eqs. (4. 15) through (4. 18) gives 2 2 2 2 22 22 i\ (-o )(w-w -s U )-W c I (S)= pC ty 4 / 0 (4. 47) I t(s); = I(s) = 0 tx Itz S iKw 2 V ) )c. (4. 48) e \Se eqt and then, the application of Eq. (2. 69) gives 2 22 22 2 22 2 /ic K -S W (c -w -c )+iuO w 2S +W U s s 3 c P 1p 3 tX 2 Vy (s) EO (4. 49) 2 2 2 2 2 2 2 2 2 ic K s W (w -w -W )+iww wc s -w U s s K z V(S) 2~ o(o( p WE co I (S)~ -1 I(s ---2 (S) ex eN53ty eN 0 tX o 0 I (S) =0 (4. 50) ey 1 WE I (s)=-s I (s)-iV (S) ez eN 1 ty eN 0 tz o 0

89 The field solutions in the real space are obtained from Eqs. (4. 47) through (4. 50) by applying the inverse transformation and using Eqs. (4.28) and (4.29) as ' K\ h2 22 2 2 2 2 2 h = —! )(w -o U -(w ) -Li w \ y c / p 2 p c 1 (4.51) h =h =0 x z 2 iK w 'c J e =iu 2 (4.52) 2 (E =ic K c p a +i(w2 a iU 2 a 2 2 - 2 x L +x I p c az 1 az 2 E = 0 (4.53) 2 Y.2K LiL a 2 2 2 a 2 E = ic K - (Li -L ) - +iU 2 z to z 1 p c 3x 1 ax 2 2 iV = 0 (4.54), y a J 2 a iU2 a K. -a ) -— 2 a - -- 2 z e7 N w L c az 1 p ax 1 ax 2 ' 0 0 4. Transverse Electric Current Source Assume that the electric current source is given by 2 J = xJ 6 (x) 6 (z) (2 Y), (4. 55) then, its transform is given by j (s)= d(s,r)J 6(x) 6 (z)(2)2 =J. (4.56)

90 The solution for the electric current source can be obtained by the same methods used in the previous examples. The results in the transform space are 2 - iJ c C 0 I (s) ty 2 WLA * I (S) = I (S) =0 tx tz 2 2 2 2 2 22 2 -s1 iLww t +s3to (w -W -to )-s 3s U to cp c p 3 (4. 57) J I0 ~2 22 2 2 22 -v (S) - (to -)+1 L W w to Lj + s s c to e ewA - lp p c p 3 1 p (4. 58) I 412 2 2 2 2 -2 2 2 2 2122 2 i 0 2 2 2 2 c Us 5 i 1 I ity(S (4. 59) iJ 0 0 J (I (s);:= 0 ex eN A ~I (S) =0 Iey 0 2o 2- c2(t2 2) 2u 2 -j c p 13 L c p -c U 7-2 22 2~ 2 2 2 2' 2 2 Lto (o -to )+ic i 4. 25 s +c to s2+U to j - pto p 13 p1I p (4. 60) 7. 2 2 2 2 -i2 (s) 2 2 - Llctp +to (cU 3 to ph1 - and in the real space -iJ c to2 h = h = 0 to w o w-to-to 2) a4 www2 1+ c p ) c p ax 1 a U2 2 * i dz (4.61)

91 2 2 2 a 2 2 2 J ) 3 c (4. 62) ew i ax c p dz i dx 2 iJ; 02~ 0 2 2 2 2 2 2 2 2 2 1 2 2 dx J E = (4. 63) x y 2 2 /E itcxp cUpi W Uxdz z WE ep 1Ix.z axa 2 2 2 2 (c) 2 2w2 1 2 2 1 iV to ( - ic -- -c -U - 1 c pz y0 (4. 64) -V \eN i LcUp -w2 (c -U2) a + ic2 ( ) a2 z eNp p 3xaz p 2 D2 Equivalence Relations Equivalence relations between different types of sources in real space are obtained from the equivalence relations in transform space, which are derived from Eqs. (4.11), (4.12), (4.13) and (4.14). These relations can be used to obtain the excited fields due to one type of source from the solutions obtained for another type of source. In deriving Eqs. (4.11) through (4.14) for the four components of the source function F(s), the assumption of = 0, was not used ay

92 so the equivalence relations obtained from these equations can be applied to the three-dimensional problems in an electron fluid plasma as well as to the twodimensional problem. First Relations: In the transform space s, the magnetic current source Jt(s) and the mechanical body source f(s) are related by the following equivalence relations: 2 2 e 2 2 sec is c -- L O -C (s)8)+ 2 wf (s)+ i f (s) (4.65) tx 2 2 f (y 2 2 2 2 2 w z (cx z -m (c -w ) (jW - ) -W - P p c 2 e 2 2 -secty s 2 Ix j i3 3x 2 2 ty (W -) ) -W W - - om( ) -o ) ( w) - - ) - P p c Second Relations: The following equivalence relations exist between the magnetic current source J t(s) and the electric current source j(s) in the transform space: 2 2 is2c r w 2 2 ( 2 2 2 t(S) = 2 22 j (s)-(W -W -w )j ( t o 2 2) 2 2 W xp c z (W - ) - P c 2 is3C + i-2 jy(s) (4.68) 2 2 2 ( -w p

93 2 2 2 (S)-ic 2 2 2 J (s) = 2 2i c 2 j(s)+ ( j - -x )-( -s) (t -_ ) -. x P c (4.69) 2 2 is2.. 2 2.2 I Jt(S) = i tz 2 22 2'2 w (s)+( - )J (S) p c is1 c - 2 2 (S) (4.70) 9 -O p Third Relation: In the transform space, the following equivalence relation exists between the electron fluid flux source W (s) and the mechanical body e source f(s): s 2 2 W (S) f (S) s)+ P- f S) e 2 = y 2 2 2 2 2 1 c z ico x m(u -u ) (J -o ) -uo o p p c 2 2 ' -t -- - P + s -u f (s)+ 2 f (s) (4. 71) 3 cx icA zJ Fourth Relation: In the transform space, the following equivalence relation exists between the electron fluid flux source W (s) and the electric current e source j(s): 2 2 ( 2 2 s W W -W W (s):S - 2 e ( Viu )- L Jx(s e(u -w ) (2 -u ) -u 2 2 L p p c ( 2 2 2 r u -u - + s3 l -iW j (s)+) (4.72) 3 c x t z L)~ -

94 Equivalence relations in the real space can now be obtained easily from Eqs. (4. 65) through (4. 72) by applying the inverse transformation. First Relation: 2 2 ec 2 2 2 i c — o m ( - i )Vx F + yxVF 2)2- p2(w-w ) p c -. p -c yV. F+ww L (4. 73) Second Relation: 9 2 2 i- - 2 2 2 w2 2 2)VJ w ) y - iK = r~5 ---(-T-2 -W - )VxJ- c P yxVJ - 2 2 P c 2 2 yx V -iw a -2yV +i. 2 dJ (4.74) c p c p y Third Relation: aF - ( 2F 2x - aF iQ 1 m p 2 2 ay 22aY 22 2 2 2 cax w ax m(W -j ) (a) -a) ) -( cW P p c c aF W2- 2 aO -io -x+- ---- z (4.75) c az W az Fourth Relation: 2.2 2 2 i e aQ - ai i -e ax w 2 2 22 y 2 2 22 c ax w ax e( -w ) (w -w ) - w P p c a 2 2 F c -w aF - iZ 'z - (4.76) c z w az

95 4. 3 Three-Dimensional Problems As can be seen from the two-dimensional problems treated in the last section, any type of source excitation problem can be solved with equal ease by using the formal solution derived in Chapter II. In order to show the salient features of this technique when applied to a three-dimensional problem, the radiation fields due to a point current source will be obtained. LA] Basic Derivation and Analysis The point electric current source can be expressed as J=y(2r)3J 6(x)6(y)6(z) (4.77) and its transform given by j (s) = d(s,r)(2 r) J 6(x)6(y)6(z) = J (4.78) y o From Eqs. (4.11), (4.12), (4.13) and (4.14), the source function can be expressed as 2 is3c F1 (s) = 2 J F2 (s)= ) (4. 79) is c F (s) = - J X 3 2 2 O! p / 2 s2W p F,(s) - 2 J F4(s) = 2 2 o e(w -uw ) Next, the inverse matrix [-N(s)J is obtained by applying Eqs. (3.2) through (3.19). The transforms of the magnetic field and the density fluctuation field can now be obtained from Eq. (2. 86), and the transforms of the electric field and the

96 fluid velocity field obtained from Eq. (2. 69). In order to obtain solution in real space from the resulting expression it is necessary to evaluate the following integral,oo 0o a coo i(slx+s2Y+S3Z) i = L(s)e dseds ds ds 1 ds2 ds3 (4.80) G 1 G(s) 1 2 3 The asymptotic solution to the integral (4.80) can be obtained by applying the principle of stationary phase. For the present problem it is convenient to use the theorem developed by Lighthill. The solution of (4. 80) satisfying the radiation condition is asymptotically given by 2 i(sx+S 2y+S 3z) 4ir CLe- + 0( 1) (4. 81) r L IVGI IK! r as r -> o. The summation is over all points (s1, s2, s3) of the surface G(s) = det. L1-N(s)j = 0 where the normal to the surface is parallel to the direction of observation and (F- VG)/(3G/ a)< 0. At each of these summation points the Gaussian curvature K can not be zero. C is + i where K < 0 and VG is in the direction of +_r, and +1 where K > 0 and the surface is convex to the direction of + VG. Usually, threefold Fourier integrals with axial symmetry are treated by conversion into Hankel transforms, but such a conversion complicates the asymptotic evaluation and also loses sight of the close relation existing between the radiation fields and the dispersion relation. Thus, we will use the asymptotic solution given by Eq. (4. 81), which is directly expressed in terms of the characteristics of the phase surfaces.

97 In the present problem the axis of symmetry is in the y- direction and 2 2 2 so the dispersion relation is a function of s2 and (s +s ) i.e. 1 3' - 2 2 2 det. l-N(s) = f(s2, s + S). 2 2 2 Using the simple notations b-s2 and c-s1 +s3 the Gaussian curvature reduces to f ( f 2bc(f f -2 2f ff+f f )+ f (bf +cf) c L bb c bc b c cc b b c (4. 82) (bfb + cf ) and VG | = 2 bf f (4. 83) If the surface of revolution can be expressed by ' 2 2 s2 = '(s1 +S3 ) (4.84) the Gaussian curvature has the simple form U tt K = (4.85) /s + l,2)2 2+s (1+ cq) Because of the axial symmetry, it is only necessary to find the field variation in a plane containing the y- axis. Stationary points will be given by the equation l' = - tan 4(4.86) or equivalently by the equations af af f -I tan f 0, f=0 (4.87) P or s f- tan p l s2b= 0 f = (4.88) p c 2 b where 2 s = s + s =- C. p - 1 3 -

98 At an inflexion point of the plane curve given by (4. 84) the Gaussian curvature is zero and Eq. (4. 81) can not be used. Instead of (4. 81), the following expression for J must be used: 3 (2 ) i( 1 )L is r+ gnK proportional to K is the principal curvature for the parallel section 5 1 e r I (4.89) where c': + 1 depending on whether VG is in the direction of +r, and A is aK and K is the principal curvature for the meridian section. m On the surface of revolution the principal directions are on meridians and parallels, and the principal curvatures are given by 2c _ _ K = = (4. 90) P 2 2 2 2 2 1 4 +2 s (( +S ) c cc 1 3" K =4. 91) m 3 (4.91) 2 2 2 -2 -!1+4(s +s ) q 2 (1+ (')2 where C( is given by (4. 84). Equations (4. 90) and (4. 91) are obtained as inverse of the two roots of Eq. (4. 92) solved for the radius of curvature, R; 4 +8 4 scc ( +s2) R - 4 q+8 3(s +s3 )+4 ( s +s) x + s)(C Is 1 3C3 cc3 (492) - 2 2 22 2S(4.92) +S ) T R+s +4 (s + s3 )0= 1 3 cs

99 The product of (4. 90) and (4. 91) gives the Gaussian curvature (4. 85). As for X, it is shown in Appendix E that OK - = 6 as cos (tan ' |) (4.93) P BI Actual Calculation For the actual calculation of the stationary points it is very complicated to solve either Eq. (4. 87) or (4. 88) directly. Instead, we will find the radiation direction p corresponding to each point on the dispersion curves in the following way. First, the form given by Eq. (3.20) is used for the dispersion relation, det. 1l-N(s) = 0. Then, for each dispersion surface f(s, 0)=s-F(0)= 0, (4.94) there are relations of the form f sin cos ds --- = sin0 - as s dO P (4.95) f f sine ds - cos 0 + - Os2 s dO Substitution of (4. 95) into (4. 87) gives tan (O )= d. (4. 96) From Eq. (3.20) the right hand side of the above equation can be expressed as s sin2 -2 2 22 2_ 2 2 (sin3 )+e o3 (o o e2)] I ds, 0 eo 00 e 0(4.97) s de 4 2 6s A'+4s B'+2C' where Eq. (3.20) is written as

100 A's +Bs + Cs +D' = 0 (4.98) with 2 2 A' - 2cos 0-1 2 2 2 2 2 2 2 22 2 B' (1-c)(/3 +21 )-2 (1 + 23 cos 0-B o cos 0) o e o e o e o 2 2 2 2 2) 2 2 2 2 22 2 2 0 2) (Ii ) (2P +~ (223 +3 Cos 0-3 t0 COs 0-3 ~o ) C' -eo o e o e o e o 2 4 2 2 2 2 -D'I F o4 (l-to ) [(l- ) e o o L Then for each angle 0, corresponding to each point on the dispersion curves, we can calculate p from Eqs. (4. 96) and (4. 97). Next, the Gaussian curvature must be evaluated at each stationary phase point from Eq. (4. 85). The following two expressions will be used for this purpose: af cos ds ds Os sinO + --- o s dO -= 2s-de (4.99) ds 3f sin ds ' ( P cos + -- ds2 s dO and 2, ds / 2 sin 2 (4. 100) cos 0 + - d) sos d)d Equation (4. 99) is obtained from (4. 95). Substitution of (4.99) and (4.100) into (4.85) gives the following expression for the Gaussian curvature: / cot0 ds 1+2 2 (ds 2 1 d si s dO/ L s \d0 / d K -+ 2 (4.101) L 1 - (ds\2 2

101 where (6s4 A'+ 4s2 B+ 2C')2Q sin20 5- 3s2(2 2 -22 ) 2 de L 0 e o 0 0 e o dO + 2 K4_2 2 2 2 2+ 2 2 2 2) 2 +2sn2cos20 s -s (2 - )/ - ) -2s sin2O x L o e o o o e o 2 2 2 22 2_ 23 ds 2 Ls2 - (20s -2 - )+3 ( - 2o {(12s3A'+4sB') - sin2 L o e o o o e o 2 dO 4 2 2 2 4A +4s2B+2C1 -2 -2s (23 - )+3 ( - (6sA'+4s B+2C'). (4.102) o e o o o e At the limiting angle 0 = 0, the Gaussian curvature is given by K 1 d2s )2 (4.103) At an inflexion point where q"' = 0, K as given by (4. 90) and X as given by (4. 93) must be calculated. For this purpose it is necessary to know m (t"' (1 + (4 ) - 3'( ") (4. 104) '(1 2(4.104) as 5 P (1+ ( '2)2 q " is obtained from (4. 100) as

102 dO da d 6 1d33 2 43d0 3 dp.adO3 Isd do2 a3 +[+241 2 + +25 (a os - 1.sin60) + It I ddo2(2 F12~ 2.dOs +ssn eW 22 OS+2 ow-a dd x AIiNs bi jrn wm0h uL VQ, Ui is edNW.e.....o $T* (T (4qr.dIe~sIn tdo of( q (4.31.O am w be.0~~d1 tiL&d -iaAd& M som olctclo bold wre sonto~ be 02 (110/010(10 lW aU 2 2 1. 29 10S) x 5 2 2 1 p p p 2 2 2 22 2 2 2' 2 U 202 e 2 l~2(lj j1.2+ 3 2 2 op 3 ~ ) w (w -w) ~(~~~~31 eiSl+3+ d1 c2 d3(4( D7P)

103 z '400 ~.00 00_O -00 J e 0.00 (E 0 det. 1-IN(s2) -00 2 s U 2 2 2 22 22 2 (5 L~ S+10 -10 ) (c s + 0 ) 5 2 2 S 3 4XC S 10 (w -10 ) c p 1 3 2 pj U2 2 c 2 2 2 2 10 10 2 c~w 2s c 2 102 30(30210 (is3+ i0LiS e i(six+ S2 Y+53 Z) ds1 ds2 ds3 J I (4.108) 0 OD]D00 det11E j E = dtB- N(s)' -0 -0 0 2 2 2 w2( 2 _w2 _22s 401 2_10) -- p 2 22 2 27 1 22 22 2 22 2 2 22 227 -c s (c s _0)I - 10 (210 -10 -o s -c 5 ) - U s (c s + w -W0)1 p j w2(2 W)-p p 2 p p I C 2 + I 10 2 2 4 c2 2 c 22i w w 5I 4 (4.109) The sum of x~E and ~E will be expressed in terms of two vectors in the direcx z A A tion of p and by using the fact that A A XsI1 +Zs3 '= j s I P and A A\ A -XS3 +Zs, = 5 xy = IsP Ip These two vector components are (4.110) is- r 42 2 2 - — c j O ) rZI VGJ VI KI U2c4 \i0 22 3 2 2 10 (10 1c ) 3 22 10(10-10)) 2 Ce is 10r10 w - 4K - - oc p s s (c2- ~2 r7 G\FiKI 2 4 2 p) VG l IU c E (4. 111)

104 and is r 4 2 2 2 2 2 2 -Ce - ) J c s2 s2U 424 2 p 242 2 P / 2 4 V i p 2 2 2 4 2 2s 2 2 2 L p P p p 2 2 2 2 2 2 co W CO C p and 0 are unit vectors in the cylindrical coordinate system with the y-direction taken as the direction of the axis. The asymptotic solution for the component of electric field in the y-direction is obtained from (4.109) as is ~r 4 2 2 - 2 42 v2Ce -W _ J s2 U 2 2 ( 22 2 2 2(2 2 1 2 2 2 22 2 2 \i 2 2 -W 2) P P 2 2 22 2 2 2 2 2 -s s c o i s pon (S +s ) l -aso (4.113) p Jo:2c 2 2 2 2 2 2 2 2 2 2 -u s (c s2p -2] +(1- s2) L- <s2+s2)+" s^2i (4.113) Finally, it will be shown that the radiation condition expressed by the requirement (r- VG)/(aG/ aw) < 0 is not essential to the calculation of amplitude variations. First of all, it is easy to see from Eq.. (4. 88) that if the point (s, s ) satisfies this equation, its symmetrical point (-s, -s ) will also satisfy this equation. r VG of these two stationary points have different signs, and the radiation requirement selects the one which has a sign opposite to that of aG/ aw. But, the amplitude of each stationary point contribution in Eqs. (4. 111), (4. 112) and (4.113) is the function of s2 and s which appear only in the form of their product p

105 or squares, thus it is not necessary to select one of two stationary points, by calculating VG and dG/ dc, in order to plot the amplitude variations.

CHAPTER V NUMERICAL RESULTS IN THE IONOSPHERIC PLASMA 5.1 Ionospheric Model Because of the great variations in ionospheric properties depending upon time and geographic location, attempt is made to use those data corresponding to day-time, mid-latitude and late 1962 ionosphere. For electron density pro(47) files use is made of those ionograms utilized by Stone, Bird and Balser, and (48) some topside sounding ionograms analyzed by Bauer and Blumle. From these profiles it seems convenient to divide the ionosphere into four regions: (i) Above F-peak region; (ii) Around F-peak region; (iii) E region; and (iv) D region. As for the electron temperature use is made of the United States Standard Atmosphere, 1962, and it is assumed that thermal equilibrium condition prevails throughout the ionosphere. Thermal non-equilibrium seems to be ascertained in certain altitude regions by Spencer, Brace and Carignan(, Nagy, Brace, Carignan and Kanal, and others, but for our linearized treatment the assumption of thermal equilibrium should be able to represent general characteristics. The variation of the magnitude of the earth's magnetic field is rather small and thus, it will be assumed constant with a value of 0. 5 Gauss, which will give 06 2 13 the electron gyrofrequency as w = 8. 79 x 10 and L = 7. 73 x 10 c c The following altitudes are chosen for the four regions: (i) 400 Km: T = 1,487~K, U2 = 2.25 x 1010 5 2 14 N = 1.7x10 electrons/cc, o 5.4 x 10 o p (ii) 250 Km: T = 1,357K, U2 = 2.05 x 1010 5 2 15 N =6x 10 electrons/cc, =1. 9 x 10 o P 106

107 (iii) 100Km: T 210~K, U =3.11 N - 3 x 10 electrons/cc, o x 109 = 9.54x 10 p (iv) 70 Km: o 2 T 2 220 K, U = N = 10 electrons/cc, 0 3.34x 109 2= 3.18x 101 P 5.2 Radiation Fields The magnitudes of the electric field components excited by a point current source oriented in the direction of the earth's magnetic field are calculated from Eqs. (4.111), (4. 112) and (4.113) for each stationary point contribution. | EM is the magnitude of the projection of the electric field on the meridian plane, which is obtained as 1 IEM (E l2+ 1Ep )2 (5.1) 5 6 Calculations are made for four frequencies, which are ow.3 x 10, w= 3 x 10 7 8 ow 3x10 and o 3x108. The dispersion curves and Ed', EM' versus 0 are given in Fig. 15 through Fig. 67, where and 36w' r' IE, 0 JO 0 36v c1r IK EM'.... ~E '0 1FcrP (5.2) (5.3) Due to the symmetrical nature of the physical system and the source involved, E p', EM' vs p are plotted only for the range p 0 to P=-90. The angle ps and O = 00 corresponds to the direction of the earth's magnetic field. According to our numerical results, the dependence of the radiation patterns of the excited fields on the altitude of the ionosphere is not too conspicuous.

FIG. 15: DISPERSION CURVE 400KM, a 3 x105

109 0 'I -8 -0 I 2 3X10-7: =90~ FIG. 16: E, E' VS. s 400 KM, w = 3 x 105 POINT CURRENT SOURCE (o = 00)

0 0 -- a4 r tr 01TT

111 0 0 I! '-e 0 0.5 I 1.5 X 105: =900 FIG. 18: EE VS. 0 400 KM, = 3 x 106 POINT CURRENT SOURCE (0 = 00 )

112 0 0 I 2 3XI0-5 <=90~ FIG. 19: E' VS. p M 400 KM, w = 3 x 106 POINT CURRENT SOURCE (0 = 0~)

113 FIG. 20: DISPERSION CURVES FOR MODIFIED ORDINARY AND EXTRAORDINARY WAVES 400KM, = 3x107

FIG. 21: DISPERSION CURVE FOR MODIFIED PLASMA WAVE 400KM, = 3xl10

3 Xl 0 0 9-G 0 = 90~ FIG. 22: Eb, E VS. FOR MODIFIED ORDINARY WAVE 400KM, = 3x107 POINT CURRENT SOURCE (P = 0~)

6X10T' 5 0=90~ FIG. 23: Ej6, Em. VS- pb FOR MODIFIED EXTRAORDINARY WAVE 400 KM, uS 3x10 POINT CURRENT SOURCE ( II 3 FIG. 23: El Ej VS...; FOR MODIFIED EXTRAORDINARY WAVE 400KM, cLi 3x10 POINT CURRENT SOURCE (p = 0~)

117 l0 9 8 6 0 o. -=90~ FIG. 24: E, E' VS. p FOR MODIFIED PLASMA WAVE 7 400KM, = 3 x10 POINT CURRENT SOURCE (p = 0~)

FIG. 25: DISPERSION CURVES FOR MODIFIED ORDINARY AND EXTRAORDINARY WAVES 400 KM, w = 3 x 10

FIG. 26: DISPERSION CURVE FOR MODIFIED PLASMA WAVE 400 KM, w = 3 x 108

400 KM, U = 3 x 10 POINT CURRENT SOURCE POINT CURRENT SOURCE (o = 0~)

I O 8 'I 6 E, X I0'? 2 0 = 900 FIG. 28: El, E' VS o FOR MODIFIED PLASMA WAVE P M 400KM, y 3x108 POINT CURRENT SOURCE (0 = 0~)

['3 FIG. 29: DISPERSION CURVE 250 KM, w = 3 x 105

123 4=90~ FIG. 30: E' VS. o 0 250KM, w=3x105 POINT CURRENT SOURCE (o = 0 )

124 0 SI -0. 0 1 2 3 4 5 6X10-6 < =90~ FIG. 31: E4 VS. p 250KM, = 3 x 105 POINT CURRENT SOURCE (p = 0~)

125 C0 - _q 1 It 0L0 C\I

126 0 0 0 I 2 3X: =90~ FIG. 33: EI VS. 0 250 KM, = 3 x 106 POINT CURRENT SOURCE (0 = 0~)

127 o / 0 - -.= 4437 / 0 I 2 3 4 5 6X1: =90~ FIG. 34: E' VS. p M 250KM, = 3x 106 POINT CURRENT SOURCE (: = 0 )

FIG. 35: DISPERSION CURVES FOR MODIFIED ORDINARY AND EXTRAORDINARY WAVES 250KM, c =3x108

FIG. 36: DISPERSION CURVE FOR MODIFIED PLASMA WAVE 250KM, x 3 x 108

0 0 '3 c 0 0 IX10-2 2X10-2 = 90~ FIG. 37: El, E VS. P FOR MODIFIED ORDINARY WAVE 250KM, = 3x10 POINT CURRENT SOURCE (0 = 0~)

131 0 (' II 0 -~ 00 O o x 0~- CY) 0 > C N#. 0 ISA 0 0 oo=#0

0 I! 0 =90~ FIG. 39: E', E' VS. 0 FOR MODIFIED PLASMA WAVE 250 KM, w = 3 x 10 POINT CURRENT SOURCE (0 = 0 )

FIG. 40: DISPERSION CURVE 100KM, co 3x105

134 0 He0 0.5 1.0 1.5 X IC -=90~ FIG. 410 El VS. p 100 KM, w = 3 x 10 POINT CURRENT SOURCE ( = 0 ~) )-8

135 O 0 1.0 2.0 3.0X10'8 = 900 FIG. 42: EM VS. 0 M 100 KM, o = 3 x 105 POINT CURRENT SOURCE (P = 0~)

136 '-4 co C) 9~ 3

137 O 'I 0 0.5 I 1.5X10' =.90~ FIG. 44: E VS. 0 100 KM, U = 3 x 106 POINT CURRENT SOURCE (0 = 0) -6

138 0 0 I 2 3X106 = 90~ FIG. 45: E' VS. p 6 100KM, = 3 x 10 POINT CURRENT SOURCE (- = 00)

CICD FIG. 46: DISPERSION CURVES FOR MODIFIED ORDINARY AND EXTRAORDINARY WAVES 100KM, 100KM, w=3x1O

0 FIG. 47: DISPERSION CURVE FOR MODIFIED PLASMA WAVE 100 KM, = 3x 107

-eIt 3 4 5 5 6 7X 10-6 = 90~ FIG. 48. E, IG. 48:, E' VS. FOR MODIFIED ORDINARY WAVE 100KM, C= 3x107 POINT CURRENT SOURCE (P - 0~)

0 II 0 1 2 3 4 5 6X10-6 -=90~ FIG. 49: E', E' VS. p FOR MODIFIED EXTRAORDINARY WAVE 100KM, w=3x10 POINT CURRENT SOURCE (o = 0~)

12 10=0 FIG. 50: El', EM VS. B FOR MODIFIED PLASMA WAVE 100KM, =3x10 POINT CURRENT SOURCE (P = 0~)

FIG. 51: DISPERSION CURVES FOR MODIFIED ORDINARY, EXTRAORDINARY AND PLASMA WAVES 100KM, w=3x 108

0 0 -../ 0 1 2 3 4X10-2 4 =90~ FIG. 52: E., EM VS. FOR MODIFIED ORDINARY AND EXTRAORDINARY WAVES 100 KM, = 3 x 108 POINT CURRENT SOURCE (p = 0~)

146 '.5 m 0 II 1.0 0.5 0 = 900 FIG. 53: E, ElM VS. p FOR MODIFIED PLASMA WAVE 100KM, = 3 x 108 POINT CURRENT SOURCE (0 = 00)./ -, I{

FIG. 54: DISPERSION CURVE 70 KM, = 3 x 105

=90 0 II 0 I X I0-14 2 X I0-14 FIG. 55: E., EM VS. 0 FOR MODIFIED EXTRAORDINARY WAVE 70 KM, = 3 x 105 POINT CURRENT SOURCE (p = 0 )

149 0 so 0 0.5X I0-13 IX 10-13:= 90~ FIG. 56: E VS. 0 FOR MODIFIED ORDINARY WAVE 70 KM, U = 3 x 105 POINT CURRENT SOURCE (p = 00)

150 0 0 -9 - 2 =90~ FIG. 57: EM VS. 0 FOR MODIFIED ORDINARY WAVE 70KM, w = 3 x 105 POINT CURRENT SOURCE (0 = 00)

FIG. 58: DISPERSION CURVE 70KM, w=3x106

0 O1 0 0.2 0.4 0.6 0.8 1.0 1.2X10-'0 < =900 FIG. 59: E, E' VS. P FOR MODIFIED ORDINARY WAVE 70KM, w = 3 x 10 POINT CURRENT SOURCE (p = 0~)

O 0O 0 FIG. 60: EI, EM y M IX I0-" 2X10 -=90~ VS. 0 FOR MODIFIED EXTRAORDINARY WAVE 70 KM, = 3 x 106 POINT CURRENT SOURCE (0 = 00)

154 O 0/ = 200 If — G. 0 1 2 3 < =90~ FIG. 61: El, EM VS. 0 FOR MODIFIED PLASMA WAVE 70 KM, = 3 x 106 POINT CURRENT SOURCE (p = 0~)

-An C~y] FIG. 62: DISPERSION CURVE 70KM, w=3x107

<9:901 0 10 20 30 40X 10-9 FIG. 63: EVLJ E' VS. A FOR MODIFIED ORDINARY AND EXTRAORDINARY WAVES 70 KM, ( = 3 x 10 POINT CURRENT SOURCE (o = 0~)

157 4 E I/ lo-II 3 0 2 o-e t =900 FIG. 64: El, E' VS. 0 FOR MODIFIED PLASMA WAVE 70KM, = 3 x10 POINT CURRENT SOURCE (y = 0~)

I I._1. 5.1189 X 103 8- 90~ - I ' 5.19 X 10: FIG. 65: DISPERSION CURVE 70KM, co 3x108

0 oko 70KM, w= 3x108 POINT CURRENT SOURCE (0 = 0~)

6 o 4 0 M '3 -6.-t 0 0 = 90~ FIG. 67: El, El VS. p FOR MODIFIED PLASMA WAVE 70 KM, w = 3 x10 POINT CURRENT SOURCE (0 = 0~)

161 However, there is quite a variation in the magnitude of the excited fields. Modified plasma waves are essentially composed by EM components, as can be expected from their longitudinal nature. The relatively large magnitude of the modified plasma wave compared with the modified electromagnetic waves should be comparable with the result of (23) Hessel and Shmoys. They consider the excitation by a point current source without static magnetic field. Wait(46) has studied the radiation from a slottedsphere antenna immersed in a compressible plasma, without static magnetic field, and concludes that the relative power in the acoustic type of wave is increased as the dimension of the antenna is reduced. The assumption of an infinitesimal point source might have given an unrealistically large contribution of modified plasma wave, but comparatively strong excitation of this type of wave seems to be possible, because these fields are basically decided by the dispersion relation. Other interpretations of the figures will be given in the following. LAJ At 400 Kilometers When wc= 3 x 10 the dispersion curve belongs to the region 8 of Fig. 14. The transition angle 0 for modified ordinary wave to modified plasma wave is -1 equal to tan (29.18). The resonance angle 0 for the modified plasma wave is r -1 tan (29. 3). Due to a turning point in Fig. 15, there are two rays existing inside the cone f < 17. 75 as shown in Fig. 16. This terminology "ray" has been used by Arbel(13) for each stationary phase point contribution. Large contribution due to modified plasma wave is restricted to a very narrow region near the axis = 00.

162 When Lo= 3 x 10 the dispersion curve as given by Fig. 17 still belongs to the region 8 of Fig. 14, but now 0 =68. 7 and 0 = 70. Due to a turning point, t r there are three rays existing inside the cone ( < 4 as indicated in Fig. 18. At 0 20 corresponding to 0 - 700 the stationary point goes to infinity and the asymptotic solution can not be applied. There are two rays existing in the regions 4~< < 20. A large modified plasma wave contribution exists near and inside the boundary cone = 20. 7 When xc= 3 x 10 the dispersion curves given in Fig. 20 and Fig. 21 belong A to the region 1 of Fig. 14. The patterns of E'- are similar to that of F.-0 of (13) Arbel for modified ordinary and extraordinary waves, but E,- p patterns (13) A are somewhat different from G.-0 of Arbel. Actually, G. is not the true mag1 1 nitude of the projection on the meridian plane. The radiation field of the modified plasma wave as given in Fig. 24 is much greater than those of other two waves. The modified plasma wave is elliptically polarized with a small Ed component, except in the direction 0= -0 where the wave is linearly polarized in the direction of earth's magnetic field. 8 When w= 3 x 10 the dispersion curves are given by Fig. 25 and Fig. 26 and E', EM vs 0 given by Fig. 27 and Fig. 28. Everything is the same as in 7 the case w x 3 x 10 except for the similarity of the propagation constants of modified ordinary and extraordinary waves, which are nearly equal to the propagation constant of light in free space. F At 250 Kilometers Most of the features are the same as in the case of[A~ at 400 Km. Dispersion curves and El, E'Mvs 0 patterns are given by Fig. 29 through Fig. 39. P M

163 7 When o- 3 x 10, the criterion corresponds to region 5 of Fig. 14 and 8 there is no radiation field existing. At w= 3 x 10, the modified ordinary and extraordinary waves have almost equal magnitude of EM and E, everywhere M except in the direction of earth's magnetic field where there are no waves of these types existing, and also in the direction p = 90 where these waves are linearly polarized in y-direction. These features can be seen from Fig. 37 and Fig. 38. [C XAt 100 Kilometers All features are the same as in the case of altitude 400 Km, except for 0 6 the appearance of second turning point near 0 = 60 at w= 3 x 10. The dispersion curves and E, E' vs p patterns are given by Fig. 40 through Fig. 53. 6 At w= 3 x 10 the second turning point on the dispersion curve causes one ray to appear outside the cone = 20. Actually, there is another ray ex0 isting outside the cone = 20 with a large intensity which can not be seen in Fig. 44. D] At 70 Kilometers 5 When La 3 x 10 the dispersion curves given by Fig. 54 belong to region 7 of Fig. 14. The patterns nearp = 0 in Fig. 55 are not indicated because of the turning point very close to 0 = 0. As for the modified ordinary wave, there are two turning points, one near = 0 and another one near p = 29. 5 as shown in Fig. 56 and Fig. 57. 0 and 0 are still the same numbers given for the rs t 5 case of 400 Km and L = 3 x 10

164 When w= 3 x 10 the dispersion curves given by Fig. 58 belong to region 6 of Fig. 14. There are no turning points for any dispersion curves. E, EM vs 0 patterns are given by Figs. 59, 60 and 61. The main features at ej 3 x 10 and s = 3 x 10 as given by Fig. 62 through Fig. 67 are the same as in the case at 400 Km.

APPENDIX A GENERAL FORMULATION FOR THREE-FLUID PLASMA A. 1 Operator Form In order to be able to obtain a proper operator form of the basic Eqs. (2.1) through (2.8), we will express E, V, V and V in terms of h, n, n., n, J, e 1 n e i1 n F, F. and F by employing Eqs. (2.2), (2.3), (2.5) and (2. 7). The procedure e 1 n is as follows. First, E is eliminated between Eqs. (2.2) and (2. 3) to obtain an equation involving V, Vi and V. Next, E is eliminated between Eqs. (2.2) and (2. 5) to obtain another equation involving V, V. and V. Then, V will be eliminated e 1 n n from these two new equations and (2. 7) to get the following pair of equations: - A - AV +BV.+w V xb = S (A.1) e 1 ce e e -- A - CV +DV. -w.V.xb S. (A.2) e 1 ci 1 1 where 2 A- -iw+ip +v.+v en ne (A.3) to ei en -il+v.+V ni ne 2 v env ni B -i - - (A. 4) W ei -iW +v.+v ni ne 2 t. V. v pi - in ne (A. 5 C- -i pl - - + (A.5) t ie -ito+v.+v ni ne 2 t v. V. in m D- -io+i -i +v +v - m (A.6) e ie in -iO+ v + ni ne 165

166 2 2 - U Vn U v Vn - eVxh e e n en n e iwE m N N (v.+v - i) o e o 1 nl ne - F v F eJ e + en n (A. 7) iw m m N m Nm.++v - iw) o e eo 1 n ni ne 2 2 - U Vn. U v Vn - eVxh i 1 n in n i ie m. N N (v.+v -io) o 1 o 1 ni ne - F. v F eJ F1 v in F J + J + i n n (A. 8) itomNm m Nm(v.+v -ij) 01 0 o i 1 n ni ne uo and w in Eqs. (A. 1) and (A. 2) are the electron and ion cyclotron frequencies ce ci ' ' eB eB O O given by t = and w =-, respectively, and to and w. in Eqs. (A.23) ce m ci m. pe pi 2 e i eN 2 0 through (A. 6) are the electron and ion plasma frequencies given by w =2 pe m c 2 eN eo and t. =-, respectively. Pl m.e 1 0 The pair of Eqs. (A. 1) and (A. 2) can be easily solved for V and V. in terms e 1 - -AA - AA - A - Aof S, Si, bb - S bb - S, bxS and bxS. Then E can be found from Eq. (2.2), and e e i e i V can be found from Eq. (2 7). Utilization of the expression of S and S. as n e 1 given by Eqs. (A. 7) and (A. 8) will, then, give the solutions for E, V, V. and e 1 V in terms of h, n, n., n, J, F, F. and F. These four equations, each of n e 1 n e 1 n them involving E, V, V. and V separately, together with the original Eqs. (2.1), (2 4), (2. 6) and (2. 8) can now be put into the following desirable matrix form. (2. 4), (2. 6) and (2. 8) can now be put into the following desirable matrix form.

0 0 0 VX1 0 0 0 IN 0 (V La o 1 0 0 o 0 1 0 o 0 0 1 A11 A12 A13 A14 A21 A22 A23 A24 A31 A32 A33 A34 A A A A 41 42 43 44 =? j w a= 1)+-~4 I La) 0 0 =1 I I) +-iVN. 1 (0 0 0 0 0 -iN0 Li) 0 11 e n. 1 n 11 E iN1.= I 0 -~(V* 1)+-VN 41 La La 1.a I 0 0 I~ (A0 iQ1 iQ U I S 2 3 4 = 0 0 0 I 0 0 0 V e 0 0 V. I -j 0 0 0 V~ n (A. 9)

168 where the matrix components A.. and the source vectors 8, 8, 8 are given by and 84 4 eN A 11 ie w o (A31 -A21 e 0 eN A12 = ie (A -A22) 12 jE w 32 22 o eN A13 — '(A -A ) 13 iw (A33 23 o eN A14 iw (A34 A24) o A 1 = 1 / D 21 A1 oe (A. 1 0) (A. 11) (A. 12) (A. 13),a2+A)+ /ABC -- L0 2BCD -ce ~ce 'ci D3 1 ce (D2 ci BC ce ci + 1) a A -eVxl B bx We -a - i m AI O e l le'% wce,~ c BC+AD cewcl A 2 tAA bb. c1 ce c A \w ce D - c ) A bx eVx ioi m o i (A. 14) A 1 r 1 22 a1 24 -1 I-Q&e D i - + 1) i /ABC ' I 2BCD D3 ce ci ci U2v. e N o +D) bb u2.1 'D2 ce ( BC ce ci (A. 15)

169 Br I1 1. I +)+- 1 19BC+AD D012 2.A1 A 24 4d 0 A 2 -% ^ -M No I 9 (A. 16 24 W w7 a I D BC 1BEsNAD D A2-(~ - *o1 1 D I ABC NM 40 ( 42+A+ 'I 25mD3 -a- + 3,u ven. (03Lawl 1+41"m' 1 A2+.^ALI A D A I + aw o(m VAM viny. (A. 17) A 31 zA1 4* cei e ()ci +2 Kwoew ci w2 ci A 2 2 0O A4A.-1)bbo + Wce-c (A~ D o e 1 (CD 2ABC A 3 ~+allmm Wod. qm2~ 2ci w 1T +h L Wi QA 2 ce 4 2+D) A tb* 1 (01 ci Ki2BC+1 ce ci / ix] 3x oiex (A. 18) A32 A F-I — ce ci eci ce ci 1 (C+ ADD 2A2 A2 w~eci 2 2 ' ci ce + 1 (w l U2 N 0 (A. 19)

170 3 1 BCD 2ABC A A33 A 2 2 A2 + D)+ - -+A bb o1 (. o 2 B. ce ci o L ci e ci ce ce 2 U n(A2 B c 2l A - un2 C- -1 ( +1 34 N.(v.+ -iU).. 2o i ni ne 1 ce ci ce ci A^ \) t. W 2 2 w/ i. \w. 2 ce ci o ce ci ce ci ci ce 1 1 A 1 BCD 2ABC A V I - +D- +- +A )bien A 2 2 22. 2 1 L(. 2 2 (. ce ci o ci ce ci ce 1 ( A BC +1 bx. V. T (A.21). 2. in ci L ce ci 2 ce (n A +v niA31 4 1 ne321/\ i 3 A = (A. 22) 41 (v +v.-iw) ne ni 42A (ne 22+ ni 32 (A 23) A^(v 2 +v A i) ne ni ne 23 n(i 33 (A. 2 A3 (v +v.-i ) i4 ne ni 2ABC+ v U - 12 'neA24 niA34 N1 i A (A.25) 44 (v +v.-i ) ne ni

171 eN - Si to w 3 2/ Eto 0 0 (A. 26) i F ~v S+v S + jne 2 ni 3 N1mJn (vne +vni -iW 4 (A. 27) S 2 1 liD Ao 2\, 2ow. ce to. cecil Ci B -1 A A iLtoi w %eLtoci 3 -T1(IABC 2BCD DF 2 wt ce cito. ce ci D) ~;f; -j F e ve Fn ___+ e + e WEo e No me 1 n ni +vne -U 1 BC+AD 2 2 A 2 ce ci o. to ci ce eJ+F in Fn LE0 N01 1 n Vni +Vne.w wcewci wce ci A bxI (A. 28) e ci ce ci 2 ce ci to. to ci ce __ A'F v F A A -eJ e + en n 1t) ce ci ce Ci l 0E me 0 me 1 n(Vni + ne IW 1 1A 1 BCD 2ABC A A 2 \2 2/A 2 to L t. 2 1....to. to 2 to. ce cl to el ce el ce.A. bY 1 tci ( - BC'\1 to ce ci ce A- F. v. F bx eJ + 1+ mn -w KitocJEM. N m N M(v.+v -t) __ i 1 i 1 n ni ne ' (A. 29)

-i In the above expressions 2 2 2 ( 2 ) + A.2 + 1 (A.30) 1 \ + - / 2 2 eoe leci w e i oe e i ci and A2' AD-BC. (A.31) The matrix Eq. (A. 9) can be put into an operator form as Ir P(r) 2 J(r) (A. 32) where h -iK n o e n iQ. ~(r) - V (r) | iQn (A.33) e u \ 2 S3 54 Equation (A. 32) can be considered as an abstract relation between the sources and the resultant fields. P (r) is the eighteen-vector representing the field quantities, p (r) is an eighteen-vector representing the source quantities, and If is the system matrix differential operator relating the field to the sources.

1 T,3 A. 2 Generalized Telegraphst's Equations Generalized Fourier transform as given by Eqs. (2.10) and (2.11) will be used to transform the operator Eq. (A. 32). Then, Eqs. (2.14) and (2.15) give the transform pairs for the field vector and the source vector as given by Eq. (A. 33), and the Eq. (2. 16) gives the transform for the matrix differential operator, t/, which can be obtained from Eq. (A. 9). The resultant integral equation of the first kind as given by Eq. (2.17) is repeated here as follows: (Zf(us) ip() - (u) (A. 34) Equation (A. 34) may be put into the generalized forms of the telegraphist's equation by partitioning the transform of the field vector, (s), the transform of the source vector, f(s), and the transform of the matrix differential operator, t' (u, s), as follows: h It(s) ne ve(8) n V (s) e e ni' IVi(s) n V (s) J(s) d(s,r) - n (A. 35) E Vt(s) e e V. I.(s) 1 1 I (8) n _n

174 Q e 2 We(S) -iKQ J(s) iQe w W W(s) three by one column matrices, and V (s), Vi(s), V (e), W (s), Wi(B) and W (e) e S Ji (8) J4 J(s) where It(s), Vt(s), I (s), I(8), I (s), J (s), W (s), J (s), Ji(s) and J (s) are e n t t e 1 n three by one column matrices, and V (a), V.(s), V n(), We (a), W (s) and W (s) are scalars. In view of the orthonomality of the transformation kernels, c(r, s) and d(s, r), the eighteen-dyadic kernel l/(u, s) can be partitioned as

1 75 r --- a I o 0 0 0 N 0 0 0) 0 I II -. "-4 1%w 0 0 0 0 0 s f"'-4 0 0 0 Go 0 0 0 %o '-4 ft c -% I I I co U, "-V0 0 0: V-4 0 Hz no "-: I I.ON 4 0 0 H6. '-4 I4 0 0 0 N. cc cc.$cc m f "'-% f-p cc: cc f"v.O( I I cc o cc >, >, >, 1 " V 0c AR* H EH- HI I I I ft OOP U,

176 In Eq. (A. 37), 1 (u, s) is a Dirac or Kronecker delta function, which is the same as the scalar form of the idemfactor 1 (u, s). The three-dyadic immittance functions Yt(u, s) and Zt(u, s); the three-dyadic transfer functions T t(u, s), Tit(u, s) and T nt(u, s); the three-row-vector impedance functions Z (u, s), Z.(u, s) and Z (u, s); the three-column-vector admittance functions Y (u, s), Y.(u, s), Y (u, s), n e el en Y. (u, s), Y.(u, s), Y. (u, s), Y (u, s), Y.(u, s) and Y (u, s); the three-columnle 1 in ne ni n vector transfer functions Tt (u, s), Tti(u, s) and Tt (u, s) are defined as follows: Vx 1 -Yt(u, s) -d (u, r)i c (r, s) (A. 38) t =t0 iN, - -Z (u, s) d(u,r) (V. i +-VN ~* c(r,s) (A.39) e LW w oj - Z.(u, s) -Z (u, S) (A. 40) 1 e -Z (u,s) - d(u, r) (V + - VN1- 1 c(r,s) (A.41) n L j 1 -Zt(us u,s) d(u,r) A c(r, s) (A. 42) - Tte(u, s) - d(u, r)A12 c(r, s) (A. 43) -Tti(u, s) $ d(u,r)A13c(r, s) (A. 44) -Ttn (u, s) $ d(u, r)A14 c(r, s) (A. 45) -T t(u, s) - d(u,r)A21 c(r, s) et 21 (A. 46)

177 -Y (u, s) $ d(u, r) A22 c (r, s) (A. 47) -Y.(u, s) $ d(u, r)A23 c(r, s) (A. 48) -Y (u, s) $ d(u,r)A 24c(r, s) (A.49) -Tit(u, s) - d(u, r)A31 c (r, s) (A. 50) -Y. (u, s) - d(u, r)A c(r, s) (A. 51) -Y.(u, s) - d(u, r)A33 c(r, s) (A. 52) 1 33 -Y. (u, s) - d(u, r)A34 c (r, s) (A. 53) in 34 - Tnt(u, s) $ d(u, r)A41 c (r, s) (A. 54) -Y (u,s) $ d(u,r)A42 c(r,s) (A. 55) -Y.(u, s) - ~ d(u,r)A 43c(r, s) (A.56) -Y (u, s) - d(u, r)A 44c(r, s) (A. 57) Although the same notation d(u, r) and c(r, s) have been used for the transformation kernels and the inverse transformation kernels in Eqs. (A. 38) through (A. 57), it should be clear that they are different; e.g., in Eqs. (A. 39) and (A. 41) the transformation kernels are scalars and the inverse transformation kernels are

178 three-diagonal-dyadics, and in EqB. (A. 38)., (A.- 42),. (A. 46),, (A. 50) and (A. 54) both the transformation kernels and inverse transiormatims kernels are thr~ee diagonal -dyadlcs. By substitutin Eqis. (A. 35), (A. 36) and (A. 37) into the integral Eqs. (A. 34) the following generalized telegraphist'is equations can be obtained: I (U) J(U) +$ Y(u, s) V(s) t t t' t V (U) zW (u) + Z (u. s)1I(s) + $Tt, a~) V e(s) t t ti tu s te~)+ e u ) a Ie(U) J e(u) +$Y (u,sa)V (s) + $T (u S) I (s) e e e e et t +$Y ei (ujis) Vi(s)+$ Yen (U~s ) V (s1) V e(U) sw (u)+o ze (u,8) Ie(s I i I~u 8)i(8+ it u )It S +$Y. (u, s) V(s >$i(~)nS Vl(u) = W1( )+$ u1(u)s)1n(s) I1(U) J n(u)0Yn u )v (s)+$ T t(u~,s8) I(s) +$n Y(u, s) ve( )+$ Y ni(u, S) V 1(s) (A. 58) (A. 59) (A? O60) (A. 61) (A. 62) (A. 63) (A. 64) vn(U n(U) +$ Zn(u,0) In(S) (A. 65)

179 A. 3 Fredholm ntegral Equation By properly partitioning I(s), f (s) and 1W(u, s) as given by Eqs. (A. 35), (A. 36) and (A. 37), the basic Eqs. (2.1) through (2. 8) can be reformulated into the general form of the Fredholm Integral Equation of the second kind. First of all, the transform of the field vector is partitioned into three column vectors each with six components as follows: 1(5) 3wher where -1 It(8) Vi(s) v (8) V (s) n the transform V (s) (s) ns) Similarly, vectors where J1(8) 12(s) / 3(s) Jt(s),~r II % 7 of the source vector is partitioned as three six-column(A. 67) k w s8) e Wl(s) w (s) n L. w t(S) f2(B)) J (S) J(s),.3(s) Jn (S) Ln

180 Next, the transform of the matrix differential operator as given by Eq. (A. 37) is partitioned in the following form r(u, s) -2(u,) - 13(u, s) Wo (u, s) - (us) (u,) s) 0 (A. 68) -3(UI. s) o i(u,,) Substitution of Eqs. (A. 66), (A. 67) and (A. 68) into the integral Eq. (A. 34) gives three coupled integral equations which are Il(U) li(u)+ tJ12(u, s)2(s)+$ a3(u,)~3(8) (A.69) P2(u) ~ f2(u)+ 2 t (u, s) ) (a) (A. 70) 3(u) = ~3(u)+j /31(u, s) () (A. 71) where Y (u, s) 0 I 112(u, ) 0 e(u,s) (A. 72) 0 0 0 0 0 0 0 0 '1 3(u|' (s) (A. 73) Zi(u, s) 0 0 Z (u, s) n

181 Zt(u,. te(u, s) Tti(u,) T ) Tn(u 8s) 'U (u ) T (A. 74) Tet(u, s) Y (u, s) Y i(u, s) Y (u. s) et e ei en T (u,s ) Y. (u, ) Y (u, 8) Y in(Us) it ie i in ie,)(u, ()) (A. 75) 31 LTnt(U a) Y (u,) Yn(u) Y u,) _ nt ne ni n Finally, the substitution of Eqs. (A. 70) and (A. 71) into Eq. (A. 69) gives the desired Fredholm integral equation of the second kind for the field variable l(as) as al(u) a F(u)+ K(us) <(s) (A. 76) where we have defined F(u) # l(U)+ 12(u, )2(s)+^ 13(u, )() (A. 77) and K^ 2(u,) (uv) lUI(v, s)+$ 23(u, v) J(v, s) (A.78) which are both known functions. Thus, we have reduced the order of the matrices to be manipulated from 18 x 18 to 6 x 6, and also, we have all the advantages of solving the integral equation of the second kind.

APPENDIX B EVALUATION OF INVERSE TRANSFORMATION FOR TWO-DIMENSIONAL PROBLEMS Basically, the following integral must be evaluated 2 2 i(S x+s z) 0 00 (s+s )e 3 Ads1 ds3 -00 -00 where 2 2 22 2 2 22 22 2 2 2 Ai (UW - -c s )( -W -U s )-o (j -c s). P p C From A=0, we can obtain (B.1) U 2 2 2 2 2 (U -O )(c +U )-c to PD 2 2 2c U J-2 22, ) 22 - 2 2 2 2 2 2 -( - )4cc+ )-c - -4c 2 U ) -) _ L p P 4c4U4 (B.2) and and 2 2 2 2 2 2 W -w )(c+U )-c I a 2c2U2 L I - III~- I f 2 2 2 2 2- 2 2 2 w -W )(c +U )-c -4cU 4 4cU I -- - - - 1 2 22 2 2 IL(t - - ) 2 P c - (B.3) where sII and sI are the propagation constants for the coupled waves. It is easy to see that in the limit as C -- 0 I S -U 5 p-J-. II C which is the propagation constant of the ordinary electromagnetic wave, and 2 2. P 2 U which is the propagation constant of the plasma wave. 182

183 Equation (B. 1) will be integrated with respect to the s -plane first. There 3 are simple poles at + 2 2 s3- II and s3- V si-s1 If these poles are on the real axis, the integration path must be indented in such a way so as to produce outgoing waves, i. e., the contour should go above the negative pole and below the positive pole. The contour should be closed in the upper-half plane for z > 0 and in the lower-half plane for z < 0. The result is -i.oo - 2 -2 22- 2 2expi (slX+si2-lsl2 z z) c u(s21 -s2) Js W I I 2 sI2 - s expi (X+ s -s |z|) ds1 (B.4) S -s The integral in (B. 4) has branch points at s + sI and s1 - +sI. The choice of branch cuts and contours are shown in the following sketch, which ensures outward traveling phase fronts but not necessarily outward energy propagation.

184 Iml s Im Il +sHu -S +sI 1 Branch Cuts The remaining part of the integration is best performed by the following substitution: s = s cos(9+p) (B.5) for the first part, and s = sI cos (O+) (B.6) for the second part, where = tan-1 z[ (B.7) x Only the first part of the integration indicated in (B.4) will be carried out in detail, since the second part is of the same form. From (B. 5) -s sin(O + ) -dT ^n^^

185 The integration limits for the new variable 9 can be obtained by expanding cos(9 +0) as coB (8 +0)+iOj -cos(e6+O)cosh6.-isin(O +O)sinhO. Ir i 1 r O. + wD. The path of integration in the 0 - plane is given in the following sketch. er Integration Path This yields the original definition of the Hankel 'functions given by Sommerfeld ~10 i -n 2 c2u2(2 2 IIII Isfcos' e dO e sros dej I-0 _______ 2 (1) s l - 2H(1) s ) c 2 s - 2) L-sIIo Isr-1 )(o ) BI I (B. 8)

APPENDIX C THREE ROOTS FOR THE CUBIC EQUATION 2 2 2 To express the three roots k, k2 k3 as given by Eqs. (3. 39), (3. 40) and (3. 41) in terms of the original coefficients in the dispersion relation, it is,2 3 a only necessary to find the expressions for a, b and + as follows. 4 27 In terms of original quantities a (3q-p ) 21 2 2 2 72 1 (1- )-2 2: o o 3 2 1)2 3 C (O cos 0-1) 3 l-Q)(3w -2)+2w (3-)3 W (2+3w )-23 S (1-Q ) 3 13 (14)(3w -_2)+2 c (3 L'23 2 4 2 2 e 0 o o ooo '. oe o 2 2 (2cos e-1) ( 32 W22 2)+ 2 (C.2 o e o e o o- 1) 186

b= 1(2p 3_9pq +2 7r) 2 7 2(23 2_ 2 22 22 2f 2 4F 2 2 22 2 42 62 2 2 24 2 2 42t 224 6 + - ~2 2 )22 2 0 314w2+32 21 2132 B~2)(.32 )2 2_21 2 04 2L12(2 2 2 132 + (e 20 1) (C. 2)

422 b a 4 22 22 4 2 2 4 22 b-[ 2)2+ a o e.1+W - X 2 — 2 |- + (+t)o 1 —fI )l 4 27 27e 4 o 2o2 o 4 o o o (l Cos 00-) +6 6_ 2 7224 -2o4) 42_ (2+717 2-22 24+7 4 + 12(2+4 2_ 12) o o o e o o o o o o o Cos 1)Co2t'd B ) p2ge r 12) 2 24 2546 3B4 l-I(I_{2)2 +(4-15w 2+4( 4X14T)_1+23W 2 —w +20 -2w ) o e o o o 4 o o 0 0 10 2 (5~ 2 2 2 4 61 12 4 2 O L l 2+17w_ -2+-+wj + (2w -4w 10

1 4138( 1 2 2 2 2 2 2 2 2 2... 3 eJI41)z-l+w )(I -Q?) _(1-w (1-a )-2w (1-w ) (~?OS2 _1)1 66V 122312 2 6 4 2 22 2 4 6 -(2w-_1)(1-_?)3+-I(w 2+2)2(l _E )2+(9w 6-22w 4+15w -2)(1-{~ )+4w 2(2 -5w +4wJ -w) '2 02 0 0 0 0 o 0 0 o' 8 4 12 4 22 2 31 4 6 2 2( 6 Lj4 2 +/ 1 21 -(-w +6, )(14~0) +(4 -2 2w + -wu -4w )(I- 4~) -2w( +6w -17w +10) 0 e L 2o 0 2 o 0 0 0 0 0 j 10 2 V24 6 2 2 2 54 6~ 12 2 2 4 6 +11 -29+4w-3w )(1 -Q )2 (8-6w - +2w )-( w (4+2 w -4wL +w) 0 e 0 0 0 20 0~ 0 0 0 0 0) - - 24f1 22:222 L e 0 4wo1 o2i V3 +I6[32L 0(1I{42)+(2 _wj2)(1Iw2)7 I 1(2-w)22138j 2~ 209o_1)4J 0 e 4 0 0 e 20 0 4 0 (C. 3)

APPENDIX D SOME ANALYSES FOR THE HYPERBOLAS Some supplementary analyses are given here for the hyperbolas shown in Fig. 7, Fig. 8 and Fig. 12. First, Eqs. (3.56) and (3. 69) are transformed to the standard hyperbolic equation by the following coordinate transformation: =\ cosO -s' sinO \ ~ (D. 1) ( s U= sine+s' cosO where 2+ 2 231 2 0 e o e tan 2 = e cos 20 = 2/ 2_1 30 e-1 /t+ 4 4+ 4 o e 1+1313+13+1 o e o e 1. — r -^\ ^ —~i ---72" 2 2+ 4 4 T 2 sin -1-cos20 o o e o e o e smiO = -- 2 + 4 4 o e o e 1 2 2 4 4-i+p 4 4 ji^ \^\^ n 2' cos l+cos2 e 1+ e 13 e 2 + o e o e J The results of this transformation are Eq. (3. 58) and Eq. (3. 70). Next, the slopes of the hyperbolas in Fig. 7 and Fig. 8 are analyzed as follows. From Eq. (3.45) and Eq. (3.46) we can obtain 190

191 2 2 2 ds /3/31 2 __ 2 (x~ 2 -v 2 / dw I3 /3 0 0 e 4 4 /3/3,i 0 e 2 1 1 2 2 2 (1 -c ) + -+2/3/3(1 -W) 0 e 0 + I") 7 0 2 27r 2 2 2 -' -/ / (1 -c ) 47~ o e L 0 - (D. 2) This slope will become infinite when 2 2 7-2,11\2 I-= 4 2 2 2 0eL0\/232//2 Lo0 - 0.e 0 0 0 11 2 *".. (1-co ) = 0 e ~o e 1 1)2 (~ /2)2. e V U 2 ds2 /3 Thus, there are two points where 2!1-~ co if Q7> 1-2 '- 1. Since dw /3 0 e 1 1 -~ these two points can be given approximately as /3 / 0 e 2 2 co~ 1-~7 0 (D. 3) This slope will become zero at + L2 2)2 2j 2 2 - /3 0 1 ___________ 2 1 e 0 \22/ 0 e

192 that is 222 22 2(2 2 2 22 2 2 2) (1-c2)2(/2 - _ ) ( - 2 1 +3 )(1-w )+2 )= 0 O O e e o e o o e, I..- - - 22 n2f3P2(B2+2)+_ 2(P2+2) JQ2 - 2 e o e e o e \ e n2 2 (22_/32)2 2.. (1- )= 0 2 2 0 ds o So, again there are two points where 2 = 0 if Q> 1- 2 1. If the condition d2 2 o e 2 2 /3 >> 3 is applied, these two points are approximately given br e o 2 n2,2+ (D.4) o and also if >> 1 2 1 _-22 (D. 5) The slope of the hyperbola in Fig. 12 is analyzed as follows. From Eqs. (3. 66) and (3. 67) ds 2 2 2 o e 2 - -(0 + )+ d2e O 2 K ( 2 2 2 2 2 2 L2( -1) e2] (/3o2-3e2)+o2/3 o eJ o e e 22 2-2-2 2 2 O o/ e o e o (D. 6) thus 2 ds 2 dw 0 at -2 (332 2)+ 22/ 2382 (2 (-32+2 2 /2)32 2 e o oe e o e e o. W =, —,- < O., (D. 7) o (32 _2)2 (/32/2)2 o e o e

193 and 2 ds2 0 3 2(232f22 2 2 2 ~ 2 2 03 -0 +0 (O e/s He0 oe0 oe 20 0 2 -22 2 2-2 /2 0 e 0 e (D. 8)

APPENDIX E DERIVATION OF X The expression for X given by Eq. (4. 93), which is repeated here for convenience, will be proved in the following: aK X am cos (tan' '| ). (4.93) P First, by choosing tangent plane as the coordinate plane, the phase surface at the point (s 0 S 0 s3 ) can be expressed as 1 2 30+ 2 2 1 2 I S2 2 a2s21, ~S - (SI +2 20 2 1 10 as,'as, 1 10 3 30 s- as, 1 3 1 3 2 2 - a2 m ast 3 194

195 3 and the coefficient of the term with (s 3- 0) is 1 w2 1 8m 6 3aa' (E. 3) 6 3 3 6 a To transform back to the original coordinate system, use is made of the relation (see sketch below) a I 0.a 8 +8 nine +. * 5- OOSe +5. cosne +m and2 2 m *nnd aK 8s3 aK aK m 3 m a m as 57 ' coss 8 - (E.4) 0 = tan | S3 Traslation of Coordinate in s2-s3 Plane Substitution of (E. 4) into (E. 3) oompletes the proof.

APPENDIX F PROBLEMS OF INHOMOGENEOUS PLASMA [1 i Introduction The unified operator approach for excitation problems in an ionized medium formulated in this thesis can be, in principle, used for a general linear medium either homogeneous or inhomogeneous. However, due to the complexity of the resulting equations, direct application of the formulation to an inhomogeneous medium has not been tested. In this appendix, some general thoughts regarding the solution of excitation problems in an inhomogeneous medium are discussed. This phase of work should be a subject of further research.!- -i K2 2 Perturbation Solution The integral formulation of the problem is naturally useful in finding approximate solutions for a medium with weak inhomogeneities so that the classical Born approximation may be applied. For example, if a medium is homogeneous except for a small region where the density is somewhat different from the homogeneous region, the kernel of the integral equation to be solved, i. e. j(u) = F(u) + $ K(u, s) (s), (F. 1) can be represented by K(u,s) =Kh(u, s)+K (F.2) where Kh (u, s) is an ideal kernel corresponding to a homogeneous medium, 196

197 and KA is a small operator caused by weak inhomogeneity. If o (u) is the solution for the homogeneous medium, we may set (u) = 0O() + DAn (F. 3) and the substitution of Eqs. (F. 2) and (F. 3) into Eq. (F. 1) gives () + (u)+ F(u)+ Kh(u,s)+K [ (s)+^ (F.4) By neglecting higher order small terms, and using the relation i (u) = F(u)+$ Kh(u,s)I(s), (F.5) the perturbed field can be approximately given by l $ K (s), (F. 6) which can be thought of as classical approximation of scattered field applied to both electromagnetic and plasma waves. For arbitrary inhomogeneity, if the kernel can be approximated by a degenerate form such as K(u,s) =A(u)B(s)+K (u,s), (F. 7) where the small perturbation is expressed by K (u, s) whose resolvent kernel H (u, s) can be given by a rapidly convergent Neumann series, the resolvent kernel of this integral equation can be expressed exactly as follows. The integral equation is given by Eq. (2. 70) (l(u) = F(u)+$ K(u,s)1i(s). (2.70) Its solution is given in terms of a resolvent kernel H (u, s) as il(u) = F(u)+$ H(u,s) F(s). (F.8)

198 Upon substituting Eq. (F. 8) into Eq. (2. 70), the following resolvent equation is obtained H(u, s) = K(u, s) + $ K(u, v) H(v, s). (F. 9) Similarly, the following equation can be obtained H (u, s) =K (u, s) + $ K (u, v) H (v, s). (F. 10) Now, let A (u) =A(u) + H (u,s) A (s) (F. 11) B (s) = B (s) + $ B (u) H (u, s) (F. 12) R = $ B(s)A (s) = $ B (s) A (s) (F. 13) then, from Eqs. (F. 10) and (F. 11) it can be seen that A (u) = A (u) + $ K (u, v) A (v) (F. 14) Also, by assuming H (u, s) = H (u, s) + H (u, s), (F. 15) 0 X then, substituting Eqs. (F. 7) and (F. 15) into Eq. (F. 9), and employing the relation given by Eqs. (F. 10) and (F. 11), we can obtain the following integral equation H (u,s) = A(u)B (s)+$K (u, v) H (v, s). (F. 16) X 0 X Thus, the solution of Eq. (F. 16) will completely define the desired resolvent H (u, s). By subtracting and adding $ A(u) B(s)A (s) in the right-hand side of Eq. (F. 14) we have A (u) = A(u)fi-R 1+$ rA(u)B(v)+K (u, v) A (v). (F.17) 0- 0 0

199 - -1 Post multiplication of -1-R B (s) to Eq. (F. 17) gives o o A (u) l-R B (s) =A(u)B (s) + K(u,v)A (v) 1 -R 1 B (s). (F.18) O O~ O O O O o 0 Upon comparing Eq. (F. 18) with Eq. (F. 16), it is seen that H (u, s) =A (u)!l-R - B (s) (F.19) X 0 O0 0 and H(u, s) = H (u, s) +A (u) [i-Ro B (s). (F.20) 0 0 0-~ 0 3| General Numerical Scheme In principle, the integral equation formulated in this work, i. e. i1 (u) = F(u) + X K(u, s) j(s), (F.21) can be formally solved in terms of the resolvent kernel, where we may reformulate the kernel by making X as a frequency dependent parameter or use the original kernel with X = 1. The general mathematical techniques of obtaining the resolvent kernel in the one-dimensional case have been discussed by Smithies(52) and the formal extension of this method to the case of a dyadic kernel has been discussed by Diament(). In principle, if we write the resolvent kernel as H (u, s; X) c(u s; X) (F. 22) the necessary computation for the resolvent can be expressed in the form of a power series in A co P) W P n, (F. 23) n= n=0

200 00 n=0 and the coefficients, P and c (u, s), can be obtained from a set of recursion n n equations. Although this formal solution is feasible in principle, it has not been tested numerically as to the complexity of numerical operations necessary. It is felt, however, that for some problems where the c 's are decreasing rapidly as n increases, the evaluation of the first few terms of the series will be sufficient to obtain an approximate solution of the problem. If the kernel is reformulated in terms of a parameter X which is directly proportional to frequency or wavelength an approximate solution, appropriate to low frequency or high frequency, respectively, may be obtained by evaluating the first few terms of the series. 4 One-Dimensional Problem For simple configuration of the inhomogeneity, such as one-dimensional excitation problem, the general integral equation can be reduced further. As an illustration, consider a plasma medium having a density variation with respect to the z-direction, and neglecting the static magnetic field. If the Fourier transforms of the following functions are given as 1 — ir- s $ - 3 e N (z) = N(s) 6(s1)6(s2), (F.25) (22 - $ (- ir. s -N )(z) =g(s3)6(s1)6(s2), (F.26) (2r)3 e2 3 1

201 $13 e (2ir) -ir. s N (z) 0 - 2 - -1!ci oz m 0 i - N (ZY i 2 0 e — i 1-7 00 = 6(s 1)6 (s 2) 2 O N (-I )g(s 3+i) di =p(s 3)6 (sI )6(s2) (F. 27) $13 e (2wY) $13 e (2-x) -ir. s -1 ~ —1 N z) =N 0 0 (s 3 ) 6(s ) 6(s2 ) (F. 28) -ir- s NZ)-1 0 r-cm 1-1 L2 0 s1 )6(s2 ) 00D - OD N 0 (-I? )g(s3 +IldI = q(s3) 6 (s1) 6 (s2) (F. 29) Equations (2. 52) through (2. 57) reduce to -Y (u, s) = s- (u, S) t 0o 0 0 1= -T (u, S) - - s6 (u s )6 (u 3U et e 1- 1 2 52)g 3s ) (F. 30) 6 (u I-s1)6u 2- s2)u 3- s ) (F. 31) (F. 32) -Z (u, s) = - N (u -s )6u -s )6 (u -s e to0 o33 1 1 2 2 (u3 -s -L 0 1! 6(u -s )6(u -S ) N (u -s ) - 1 1 2 2 to 0 3 3 (F. 33)

202 2 -~ mWU s -Y (u, s) - 6(u -s )6(u -s )q(u - ) (F. 34) e 2 11 22 3 33 e -imU 2s -T (u, s) = 6(u -s)6(u -s2)g(u3-s3) (F.35) te e 11 22 33 The matrix components of the kernel are obtained from Eqs. (2. 81) through (2. 84) and Eqs. (F. 30) through (F. 35) as 2== - u Y t(u v)Zt(v 2 6(u -s )6(u-s2 ) p(u3 -s3)+ 6(u3-s3) (F. 36) 2 Yt(u, v)Tt(v, ) l us6(u1-s )6(u2-s )g(u3-s) (F.37) $Z (u, v)T (v, s) e et f700 e s3-u3)s2 (u3-s3) s 1 0 N (u3-v3)g(v3-s3)6(ul-S )6(u2 -s2)dv3 (F. 38) Z (u, v)Y (v, s) e e 2 oo e Thus the kernel K(u, s) has the partially ideal form as given by the following equation, Eq. (F. 40), and the multiple integrations in Eq. (2. 70) can be reduced to a single integration with respect to s3.

1'2 Aii3s3+u191 ) ll2a3 123 A4u2s2+ul1 ) 8U3+u253.a13'132 L US1 Kiurd 2 x a [(3-s3) p (a3-s3)j JXi (u38o3) e0w (s s2 2+s3u3) 6(u1- 1 )64(u2 — 2 0] (F. 40).&N(I4 -v )g(v3sdv me U2 C 0 e - 00%.w N u -V Mv -is )dv i 3 3 3 3 3iI

204 Upon substituting this kernel, given by Eq. (F. 40), into the integral Eq. (2. 70), four coupled integral equations are obtained which are given by Eqs. (F. 41) through (F. 44). It is to be noted that in these equations the integrals correspond to the contribution due to inhomogeneity of the medium, and the last terms of the integrands in Eqs. (F. 41) and (F. 42) and the first term of the integrand in Eq. (F. 44) are the coupling terms between plasma wave and electromagnetic wave due to inhomogeneity. The source functions F (u), F2(u), F3(u) and F4(u) are given by Eqs. (4. 11) through (4.14) by setting co =0. c2 2 2 tx 1 2 3 2 tx 1 2 ty 1 3 tz to - - _2 2 iU m +p U (su )g(u 3-S) V(s3) ds (F. 41) Iu(+ u u2uy(U+ u u3 iz(u7 It(U) F 1(U) - - B I)tx (U)-(U 1u -- 00 + u(u(3-s3)g(u3-s )v (s) ds3 (F. 42)

205 2 2 2 I (u) F u-~~uuI()uI()(u+ )L (U) tz 3() -2 u1 3 tx( U2 3Ityu)( 1 + 2 t z w 2 2 2. - u I (s )-u s I (s )+u +u )It (s)p(u -s d (F. 43) Ve(u) =F4(u)+ ew~u 2 (s 3-u 3)I tx(s 3 )+u1I (u 3-s 3 )I ty(s 3 / *o0 J-oD m0(u 3 -v3)gV3-s3)dv3 MEU2 OD 0 2 2 + 2 (u +u +s ii ) V (s ) N (u -v )qdv-s dvA ds 2 1 2 33 e 3 0 3 e -00 (F. 44)

NOTATIONS N =N =N.: o e I m and m: e U and U: e B w o h: E - V and V: e n and n e E o o J K e - F and F e Qe and Q (r) (r) $ " I P Undisturbed electron and ion number densities. The electron mass. The acoustic velocity for electron gas. Externally applied constant magnetic field. Radian frequency. Varying component of the magnetic field. Varying component of the electric field. Fluid velocity of the electron gas. Varying component of the electron number density. Dielectric constant of free space. Permeability of free space. Electric current source Magnetic current source Absolute value of the charge of an electron. Mechanical body source for the electron gas. Fluid flux source for the electron gas. Field vector. Source vector. Matrix differential operator, Generic summation symbol. 206

207 d(s, r) c(r, s) ff (u, s) i(s) i(s) K(u, s) A b w and w ce c 2 2 w and w pe p i(u,s) It(s) I (s) e J(s) w (s) e Wt(s) J (s) e Yt(u, s) Transformation kernel. Inverse transformation kernel. Transform of 7/. Transform of /(r). Transform of p (r). Kernel of the Fredholm integral equation of the second kind. Unit vector in the direction of the externally applied constant magnetic field. Electron cyclotron frequency. Electron plasma frequency. A Dirac delta function or a Kronecker delta and a unit dyadic. Transform of h. Transform of n. Transform of E. Transform of V. Corresponding to transform of K. Corresponding to transform of Q. Transform of the vector source function S1. Transform of the vector source function S2. Three-dyadic admittance function.

208 Zt(u, s) Tet(u, s) T (u,s) et Z (u, s) Y (u,s) T (u,s) te S1' 2' S3 c 0 C d = w /u u = u/U o p j(s) f(s) H (1)(sr) 0 1 2 K 0 Three-dyadic impedance function. Three-dyadic transfer function. Three-row-vector impedance function. Three-column-vector admittance function. Three-column-vector transfer function. Rectangular coordinates of the propagation constant s. Velocity of light in free space. The angle between the direction of a propagation constant and the direction of the static magnetic field. Transform of the electric current source J. Transform of the mechanical body source F. Hankel function of the first kind and order zero. Inversion integral given by Eq. (4.28). Inversion integral given by Eq. (4.29). Gaussian curvature. Angle between the direction of a ray and the direction of the static magnetic field.

209 2 2 2 S =S + s p 1 3 E, E E: p 0' y Three components of the electric field in a cylindrical coordinate system.

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