《RADIO WAVES IN THE IONOSPHERE》

CHAPTER 1．INTRODUCTION1

1．1The composition of the ionosphere1

1．2 Plane waves and spherical waves．The curvature of the earth1

1．3 Effect of collisions and of the earth＇s magnetic field2

1．4 Relation to other kinds of wave-propagation2

1．5 The variation of electron density with height．The Chapman layer3

1．6 Approximations to the electron density profile5

1．7 The variation of collision-frequency with height6

1．8 The structure of the ionosphere7

1．9 Horizontal variations and irregularities10

CHAPTER 2．THE BASIC EQUATIONS11

2．1Units11

2．2 Harmonic waves and complex quantities12

2．3 Definitions of electric intensity E and magnetic intensity H13

2．4 The current density J and electric polarisation P14

2．5 The electric displacement D and magnetic induction B15

2．6 Maxwell＇s equations16

2．7 Cartesian coordinate system16

2．8 Progressive plane waves17

2．9 Plane waves in free space19

2．10 The notation ？and H20

2．11 The energy stored in a radio wave in the ionosphere20

2．12 The flow of energy．Poynting＇s theorem21

2．13 The Poynting vector22

CHAPTER 3．THE CONSTITUTIVE RELATIONS24

3．1Introduction24

3．2 Free,undamped electrons24

3．3 Electron collisions．Damping of the motion25

3．4 Effect of the earth＇s magnetic field on motion of electrons26

3．5 The effect of the magnetic field of the wave on the motion of electrons28

3．6 The susceptibility matrix29

3．7 The Lorentz polarisation term30

3．8 The effect of small irregularities in the ionosphere31

3．9 The effect of heavy ions31

3．10 The energy stored in a radio wave in the ionosphere(continued)33

3．11 The principal axes35

CHAPTER 4．PROPAGATION IN A HOMOGENEOUS ISOTROPIC MEDIUM38

4．1Definition of the refractive index38

4．2 The Maxwell equation derived from Faraday＇s law38

4．3 Isotropic medium without damping39

4．4 Isotropic medium with collision damping40

4．5 The physical interpretation of a complex refractive index41

4．6 Evanescent waves41

4．7 Inhomogeneous plane waves42

4．8 Energy flow in inhomogeneous plane waves44

4．9 The case when X=1,n=045

CHAPTER 5．PROPAGATION IN A HOMOGENEOUS ANISOTROPIC MEDIUM．MAGNETOIONIC THEORY47

5．1Introduction47

5．2 The wave-polarisation47

5．3 The polarisation equation48

5．4 Properties of the polarisation equation49

5．5 Alternative measure of the polarisation．Axis ratio and tiltangle51

5．6 The Appleton-Hartree formula for the refractive index52

5．7 The longitudinal component of the electric field53

5．8 The flow of energy for a wave in amagnetoionic medium54

5．9 The effect of heavy ions on polarisation and refractive index56

Examples57

CHAPTER 6．PROPERTIES OF THE APPLETON-HARTREE FORMULA59

6．1General properties．Zeros and infinity of the refractive index59

6．2 Collisions neglected60

6．3 Frequency above the gyro-frequency60

6．4 Longitudinal propagation when Y＜160

6．5 Transverse propagation when Y＜161

6．6 Intermediate inclination of the field when Y＜162

6．7 Frequency below the gyro-frequency64

6．8 Longitudinal propagation when Y＞164

6．9 Transverse propagation when Y＞165

6．10 Intermediate inclination of the field when Y＞165

6．11 Effect of collisions included66

6．12 The critical collision-frequency67

6．13 Longitudinal propagation when collisions are included69

6．14 Transverse propagation when collisions are included70

6．15 Intermediate inclination of the field72

6．16 The＇quasi-longitudinal＇approximation76

6．17 The＇quasi-transverse＇approximation77

6．18 The effect of heavy ions78

CHAPTER 7．DEFINITION OF THE REFLECTION AND TRANSMISSION COEFFICIENTS85

7．1Introduction85

7．2 The reference-level for reflection coefficients85

7．3 The reference-level for transmission coefficients87

7．4 The four reflection coefficients and the four transmisson coefficients88

7．5 The sign convention88

7．6 The reflection coefficient matrix90

7．7 Alternative forms of the reflection coefficients90

7．8 Spherical waves91

Examples94

CHAPTER 8．REFLECTION AT A SHARP BOUNDARY96

8．1Introduction96

8．2 The boundary conditions96

8．3 Snell＇s law97

8．4 Derivation of the Fresnel formulae for isotropic media98

8．5 General properties of the Fresnel formulae100

8．6 The Fresnel formulae when the electric vector is in the plane of incidence101

8．7 The Fresnel formulae when the electric vector is horizontal105

8．8 Reflection when X=1,Z=0,n=0106

8．9 Normal incidence108

8．10 Homogeneous ionosphere with parallel boundsries108

8．11 Normal incidenceon a parallel-sided slab112

8．12 Reflection at normal incidence when the earth＇s magnetic field is allowed for114

8．13 Earth＇s magnetic field horizontal．Normal incidence115

8．14 Earth＇s magnetic field vertical．Normal incidence116

8．15 Reflection when the earth＇s magnetic field is included．Approximate formulae for oblique incidence116

8．16 The validity of the approximations119

8．17 Reflection at oblique incidence．The Booker quartic120

8．18 Some properties of the Booker quartic123

8．19 Reflection at oblique incidence for north-south or south-north propagation124

8．20 Reflection at oblique incidence in the general case126

Examples127

CHAPTER 9．SLOWLY VARYING MEDIUM．THE W．K．B．SOLUTIONS128

9．1Introduction128

9．2 The differential equations128

9．3 The phase memory concept130

9．4 Loss-free medium．Constancy of energy-flow131

9．5 Derivation of the W．K．B．solution131

9．6 Condition for the validity of the W．K．B．solutions133

9．7 Properties of the W．K．B．solutions134

9．8 The reflection coefficient136

9．9 Coupling between upgoing and downgoing waves137

9．10 Extension to oblique incidence138

9．11 The differential equations for oblique incidence140

9．12 The W．K．B．solutions for horizontal polarisation at oblique incidence140

9．13 The W．K．B．solutions at oblique incidence when the electric vector is parallel to the plane of incidence142

9．14 The effect of including the earth＇s magnetic field143

9．15 Ray theory and＇full wave＇theory144

CHAPTER 10．RAY THEORY FOR VERTICAL INCIDENCE WHEN THE EARTH＇S MAGNETIC FIELD IS NEGLECTED146

10．1The use of pulses146

10．2 The group velocity147

10．3 The equivalent height of reflection h′(f)149

10．4 The＇true height＇and the＇phase height＇150

10．5 The equivalent height of reflection for a linear profile of electron density150

10．6 The equivalent height of reflection for an exponential variation of electron density151

10．7 Equivalent height for a parabolic profile of electron density152

10．8 Equivalent height for the＇sech2＇profile of electron density156

10．9 Two separate parabolic layers157

10．10 The effect of a＇ledge＇in the electron density profile158

10．11 The calculation of electron density N(z),from h′(f)data160

10．12 Solution when N(z)is monotonic161

10．13 Partial solution when N(z)is not monotonic163

10．14 The shape of apulse of radiowaves166

10．15 The effect of electron collisions on group refractive index170

10．16 The effect of collisions on equivalent height h′(f)and phase height h(f)171

10．17 Relation between equivalent height,phase height and absorption172

Examples174

CHAPTER 11．RAY THEORY FOR OBLIQUE INCIDENCE WHEN THE EARTH＇S MAGNETIC FIELD IS NEGLECTED175

11．1Introduction．The ray path175

11．2 Wave-packets177

11．3 The equation for the ray when the earth＇s magnetic field is neglected178

11．4 The ray path for a linear gradient of electron density179

11．5 The ray path for exponential variation of electron density180

11．6 The ray path for a parabolic profile of electron density182

11．7 The skip distance183

11．8 The equivalent path P′ at oblique incidence185

11．9 Breit and Tuve＇s theorem．Martyn＇s theorem for equivalent path186

11．10 The equivalent path at oblique incidence for a linear gradient of electron density188

11．11 The equivalent path at oblique incidence for a parabolic profile of electron density188

11．12 The dependence of signal on frequency near the maximum usable frequency190

11．13 The prediction of maximum usable frequencies．Appleton and Beynon＇s method191

11．14 The curvature of the earth192

11．15 The prediction of maximum usable frequencies．Newbern Smith＇s method194

11．16 The absorption of radio waves．Martyn＇s theorem for absorption195

11．17 The effect of electron collisions on equivalent path197

Examples197

CHAPTER 12．RAY THEORY FOR VERTICAL INCIDENCE WHEN THE EARTH＇S MAGNETIC FIELD IS INCLUDED199

12．1Introduction199

12．2 Magnetoionic＇splitting＇199

12．3 The group refractive index—collisions neglected200

12．4 The effect of collisions on the group refractive index204

12．5 The equivalent height of reflection h′(f)—collisions neglected205

12．6 The h′(f)curves when collisions are neglected206

12．7 The penetration-frequencies for the ordinary and extraordinary waves208

12．8 The equivalent height for a parabolic layer209

12．9 Two separate parabolic layers210

12．10 The effect of a＇ledge＇in the electron density profile212

12．11 The effect of collisions on equivalent height h′(f)212

12．12 The polarisation of waves in a wave-packet214

12．13 The calculation of electron density N(z)from h′(f)215

12．14 Example of the use of the method218

12．15 Other versions of the foregoing method221

12．16 Failure of the method when N(z)is not monotonic222

12．17 Theuse of h′x(f)for the extraordinary ray223

Example224

CHAPTER 13．RAY THEORY FOR OBLIQUE INCIDENCE WHEN THE EARTH＇S MAGNETIC FIELD IS INCLUDED225

13．1Introduction225

13．2 The variable q225

13．3 Derivain of the Booker quartic226

13．4 The transition to a continuous medium228

13．5 The path of a wave-packet229

13．6 The reversibility of the path230

13．7 The reflection of a wave-packet230

13．8 A simple example of ray paths at oblique incidence231

13．9 Further properties of the Booker quartic233

13．10 The Booker quartic for east-west and west-east propagation236

13．11 The Booker quartic for north-south and south-north propagation238

13．12 The Booker quartic in the general case when collisions are neglected244

13．13 Lateral deviation at vertical incidence246

13．14 Lateral deviation for propagation from(magnetic)east to west or west to east248

13．15 Lateral deviation in the general case250

13．16 Calculation of attenuation,using the Booker quartic250

13．17 The＇refractive index＇surface in a homogeneous medium252

13．18 The direction of the ray253

13．19 The ray velocity and the ray surface255

13．20 Whistlers256

13．21 Determination of ray direction by Poeverlein＇s construction258

13．22 Propagation in the magnetic meridian．The＇Spitze＇260

13．23 The refractive index surfaces for the extraordinary ray when Y＜1262

13．24 The refractive index surfaces for the extraordinary ray when Y＞1266

13．25 The second refractive index surface for the ordinary ray when Y＞1268

Examples270

CHAPTER 14．THE GENERAL PROBLEM OF RAY TRACING271

14．1Introduction271

14．2 Equations of the refractive index surface and the ray surface272

14．3 The Eikonal function274

14．4 The canonical equations for a ray,and the generalisation of Snell＇s law276

14．5 Other relations between the equations for the ray surface and the refractive index surface278

14．6 Fermat＇s principle279

14．7 Equivalent path and absorption279

14．8 The problem of finding the maximum usable frequency281

Examples282

CHAPTER 15．THE AIRY INTEGRAL FUNCTION,AND THE STOKES PHENOMENON283

15．1Introduction283

15．2 Linear gradient of electron density associated with an isolated zero of q283

15．3 The differential equation for horizontal polarisation and oblique incidence285

15．4 The Stokes differential equation286

15．5 Qualitative discussion of the solutions of the Stokes equation287

15．6 Solutions of the Stokes equation expressed as contour integrals288

15．7 Solutions of the Stokes equation expressed as Bessel functions291

15．8 Tables of the Airy integral functions291

15．9 The W．K．B．solutions of the Stokes equation292

15．10 The Stokes phenomenon of the＇discontinuity of the constants＇292

15．11 Stokes lines and anti-Stokes lines293

15．12 The Stokes diagram294

15．13 Definition of the Stokes constant295

15．14 Furry＇s derivation of the Stokes constants for the Stokes equation296

15．15 Asymptotic approximations obtained from the contour integrals297

15．16 Summary of some important properties of complex variables297

15．17 Integration by the method of steepest descents300

15．18 Application of the method of steepest descents to solutions of the Stokes equation302

15．19 Integration by the method of stationary phase307

15．20 Method of stationary phase applied to the Airy integral function309

15．21 Asymptotic expansions310

15．22 The range of validity of asymptotic approximations310

15．23 The choice of a fundamental system of solutions of the Stokes equation312

15．24 Connection formulae,or circuit relations313

15．25 The intensity of light neara caustic313

CHAPTER 16．LINEAR GRADIENT OF ELECTRON DENSITY319

16．1Introduction319

16．2 Purely linear profile．Electron collisions neglected319

16．3 Application to a slowly varying profile322

16．4 The effect of electron collisions．The height z as a complex variable326

16．5 Constant collision-frequency．Purely linear profile of electron density327

16．6 The slowly varying profile when collisions are included．Derivation of the phase integral formula329

16．7 Discussion of the phase integral formula331

16．8 Effect of curvature of the electron density profile333

16．9 Reflection at a discontinuity of gradient334

16．10 Linear gradient between two homogeneous regions336

16．11 Symmetrical ionosphere with double linear profile340

16．12 The differential equation for oblique incidence applicable when the electric vector is parallel to the plane of incidence343

16．13 The behaviour of the fields near a zero of the refractive index for＇vertical＇polarisation at oblique incidence346

16．14 The generation of harmonics in the ionosphere347

16．15 The phase integral formula for＇vertical＇polarisation at oblique incidence348

16．16 Asymptotic approximations for the solutions of the differential equation for＇vertical＇polarisation349

16．17 Application of the phase integral formula350

CHAPTER 17．VARIOUS ELECTRON DENSITY PROFILES WHEN THE EARTH＇S MAGNETIC FIELD IS NEGLECTED353

17．1Introduction353

17．2 Exponential profile．Constant collision-frequency354

17．3 The phase integral formula applied to the exponential layer357

17．4 The parabolic layer358

17．5 Partial penetration and reflection363

17．6 The equivalent height of reflection for a parabolic layer365

17．7 Electron density with square law increase366

17．8 The sinusoidal layer368

17．9 Circuit relations．Introduction to Epstein＇s theory369

17．10 The hypergeometric differential equation370

17．11 The circuit relations for the hypergeometric function372

17．12 Application to the wave-equation375

17．13 The reflection and transmission coefficients of an Epstein layer377

17．14 Epstein profiles378

17．15 Ionosphere with gradual boundary380

17．16 The＇sech2＇profile380

17．17 Fixed electron density and varying collision-frequency383

CHAPTER 18．ANISOTROPIC MEDIA．COUPLED WAVE-EQUATIONS AND W．K．B．SOLUTIONS385

18．1Introduction385

18．2 The differential equations385

18．3 The four characteristic waves387

18．4 Matrix form of the equations389

18．5 The differential equations for vertical incidence391

18．6 The W．K．B．solution for vertical incidence on a loss-free medium392

18．7 Introduction to W．K．B．solutions in the general case394

18．8 Introduction to coupled wave-equations394

18．9 F?rsterling＇s coupled equations for vertical incidence396

18．10 Coupled equations in the general case,in matrix form398

18．11 Expressions for the elements of S,S-1,and -S-1S′399

18．12 The W．K．B．solutions in the general case401

18．13 The first-order coupled equations for vertical incidence402

18．14 The W．K．B．solutions for vertical incidence405

18．15 The first-order equations in other special cases406

18．16 Second-order coupled equations408

18．17 Condition for the validity of the W．K．B．solutions410

Example411

CHAPTER 19．APPLICATIONS OF COUPLED WAVE-EQUATIONS412

19．1Introduction412

19．2 Properties of the coupling parameter ψ412

19．3 Behaviour of the coefficients near a coupling point417

19．4 Properties of the coupled differential equations near a reflection point and near a coupling point418

19．5 The use of successive approximations421

19．6 The phase integral formula for coupling423

19．7 The Z-trace424

19．8 The method of＇variation of parameters＇426

19．9 The＇coupling echo＇428

19．10 The transition through critical coupling429

19．11 Introduction to limiting polarisation432

19．12 The free space below the ionosphere433

19．13 The differential equation for the study of limiting polarisation434

Examples436

CHAPTER 20．THE PHASE INTEGRAL METHOD437

20．1Introduction437

20．2 The Riemann surface for the refractive index438

20．3 The linear electron density profile440

20．4 The parabolic electron density profile443

20．5 A further example of the method446

20．6 Coupling branch points and their Stokes lines and anti-Stokes lines450

20．7 The phase integral method for coupling452

20．8 Further discussion of the transition through critical coupling(continued from§19．10)455

Example457

CHAPTER 21．FULL WAVE SOLUTIONS WHEN THE EARTH＇S MAGNETIC FIELD IS INCLUDED458

21．1Introduction458

21．2 The differential equations458

21．3 Vertical incidence and vertical magnetic field459

21．4 Exponential profile of electron density．Constant collision-frequency460

21．5 Exponential profile(continued)．Incident wave linearly polarised462

21．6 Other electron density profiles for vertical field and vertical incidence464

21．7 Vertical magnetic field and oblique incidence．Introduction to Heading and Whipple＇s method464

21．8 Regions O,Ⅰ and Ⅰ(a)465

21．9 Reflection and transmission coefficients of region Ⅰ467

21．10 Regions Ⅱ and Ⅱ(a)468

21．11 The reflection coefficients of region Ⅱ469

21．12 The combined effect of regions Ⅰ and Ⅱ471

21．13 The effect of an infinity in the refractive index472

21．14 Isolated infinity of refractive index474

21．15 Refractive index having infinity and zero476

21．16 The apparent loss of energy near an infinity of refractive index479

Example481

CHApTER 22．NUMERICAL METHODS FOR FINDING REFLECTION COEFFICIENTS482

21．1Introduction482

22．2 Methods of integrating differential equations483

22．3 The size of the step484

22．4 The choice of dependent variable484

22．5 The three parts of the calculation of reflection coefficients486

22．6 The starting solutions at a great height487

22．7 Calculation of the components of the reflection coefficient489

22．8 The wave-admittance in an isotropic ionosphere491

22．9 The wave-admittance matrix A for an anisotropic ionosphere493

22．10 The starting value of A495

22．11 Relation between the admittance matrix A and the reflection coefficient matrix R496

22．12 The differential equation for A498

22．13 Symmetry properties of the differential equations499

22．14 Equivalent height of reflection499

Example501

CHAPTER 23．RECIPROCITY502

23．1Introduction502

23．2 Aerials502

23．3 Goubau＇s reciprocity theorem505

23．4 One magnetoionic component506

23．5 Reciprocity with full wave solutions508

Appendix．The Stokes constant for the differential equation(16．98)for＇vertical＇polarisation510

Bibliography512

Index of definitions of the more important symbols525

Subject and name index530