The propagation of radio waves : the theory of radio waves of low power in the ionosphere and magnetosphere

This book is concerned with the ionosphere and the magnetosphere, and the theory of their effect on radio waves. It includes accounts of some mathematical topics now widely used in this study, particularly W. K. B. approximations, Airy integral functions and integration by steepest descents. The subject is divided into ray theory and full wave theory. Ray theory is useful for high frequencies when the ionosphere is treated as a horizonally stratified medium. The discussion of the magnetosphere, whose structure is more complicated, includes an account of whistlers and ion cyclotron whistlers. The book has been planned both for final year undergraduates and as a reference book for research. It is suitable as a course book on radio propagation for students of physics or electrical engineering or mathematics. Some of the topics are presented from an elementary viewpoint so as to help undergraduates new to the subject. The later parts are more advanced. Because the subject is so large and has seen many important recent advances, some topics have had to be treated briefly, but there is a full bibliography with about 600 references. Frontmatter......Page 1 Contents......Page 5 Preface......Page 13 1.1 The earth's atmosphere......Page 17 1.3 Waves in ion plasmas......Page 19 1.4 Relation to other kinds of wave propagation......Page 21 1.5 Height dependence of electron concentration: the Chapman layer......Page 23 1.6 Collision frequencies......Page 26 1.7 Observations of the ionosphere......Page 28 1.8 The structure of the ionosphere......Page 30 1.9 The magnetosphere......Page 33 1.10 Disturbances of the ionosphere and magnetosphere......Page 35 Problems 1......Page 36 2.1 Units and symbols......Page 38 2.2 Definitions of electric intensity e and magnetic intensity h......Page 39 2.3 The current density j and the electric polarisation p......Page 40 2.5 Harmonic waves and complex vectors......Page 41 2.6 Maxwell's equations......Page 43 2.7 Cartesian coordinate system......Page 44 2.8 Progressive plane waves......Page 45 2.10 The notation [SCRIPT CAPITAL H] and H......Page 47 2.11 The power input to the plasma from a radio wave......Page 48 2.12 The flow of energy. The Poynting vector......Page 49 2.13 Complex refractive index......Page 50 2.15 Inhomogeneous plane waves......Page 51 Problems 2......Page 53 3.2 Free, undamped electrons......Page 54 3.3 The Lorentz polarisation term......Page 56 3.4 Electron collisions. Damping of the motion......Page 58 3.5 The Debye length......Page 60 3.6 Effect of the earth's magnetic field on the motion of electrons......Page 61 3.7 Effect of the magnetic field of the wave on the motion of electrons......Page 62 3.8 Electric neutrality of the plasma. Plasma oscillations......Page 64 3.9 The susceptibility matrix......Page 65 3.10 Complex principal axes......Page 66 3.11 Properties of principal axis elements of the permittivity. Effect of ions......Page 69 3.12 Collisions. The Sen--Wyller formulae......Page 73 3.13 Electron--electron collisions. Electron--ion collisions......Page 77 Problems 3......Page 78 4.1 Plane wave and homogeneous plasma......Page 82 4.2 Isotropic plasma......Page 83 4.3 Anisotropic plasma. The wave polarisation......Page 84 4.4 Properties of the polarisation equation......Page 87 4.5 Alternative measure of the polarisation. Axis ratio and tilt angle......Page 89 4.6 Refractive index 1. The dispersion relation......Page 90 4.7 Longitudinal component of electric polarisation and electric field......Page 91 4.8 The flow of energy for a progressive wave in a magnetoplasma......Page 92 4.9 Refractive index 2. Alternative derivations and formulae......Page 93 4.10 Zeros and infinity of refractive index. Equal refractive indices......Page 95 4.11 Dependence of refractive index on electron concentration 1. Y 1......Page 100 4.14 The transition collision frequency......Page 102 4.15 The terms `ordinary' and `extraordinary'......Page 104 4.16 Dependence of refractive index on electron concentration 3. Collisions allowed for......Page 105 4.17 Approximations for refractive indices and wave polarisations......Page 110 Problems 4......Page 115 5.1 Introduction......Page 119 5.2 Refractive index surfaces......Page 120 5.3 The ray. Ray surfaces......Page 124 5.4 Properties of ray surfaces......Page 127 5.5 Crystal optics......Page 129 5.6 Classification of refractive index and ray surfaces. C.M.A. type diagrams......Page 132 5.7 Dependence of refractive index on frequency......Page 140 5.8 Group velocity......Page 144 5.9 Properties of the group velocity......Page 147 5.10 Effect of electron collisions on the group refractive index......Page 153 Problems 5......Page 155 6.1 Introduction......Page 157 6.2 The variable q......Page 158 6.3 The Booker quartic. Derivation......Page 160 6.4 Some properties of the Booker quartic......Page 162 6.5 Some special cases of the Booker quartic......Page 167 6.6 The discriminant of the Booker quartic......Page 168 6.7 The Booker quartic for east--west and west--east propagation......Page 169 6.8 The Booker quartic for north--south and south--north propagation......Page 171 6.9 Effect of electron collisions on solutions of the Booker quartic......Page 176 6.10 The electromagnetic fields......Page 178 Problems 6......Page 179 7.1 Introduction......Page 181 7.2 The differential equations for an isotropic ionosphere......Page 182 7.3 The phase memory concept......Page 183 7.4 Loss-free medium. Constancy of energy flow......Page 184 7.5 W.K.B. solutions......Page 185 7.6 The W.K.B. method......Page 186 7.7 Discrete strata......Page 188 7.8 Coupling between upgoing and downgoing waves......Page 190 7.9 Liouville method and Schwarzian derivative......Page 191 7.10 Conditions for the validity of the W.K.B. solutions......Page 193 7.11 Properties of the W.K.B. solutions......Page 194 7.12 W.K.B. solutions for oblique incidence and vertical polarisation......Page 196 7.13 Differential equations for anisotropic ionosphere......Page 197 7.14 Matrix theory......Page 199 7.15 W.K.B. solutions for anisotropic ionosphere......Page 203 7.16 The matrices S and S[MINUS SIGN]1......Page 206 7.17 W.K.B. solutions for vertical incidence......Page 208 7.18 Ray theory and `full wave' theory......Page 209 7.19 The reflection coefficient......Page 210 Problems 7......Page 211 8.2 Linear height distribution of electron concentration and isolated zero of q......Page 213 8.3 The differential equation for horizontal polarisation and oblique incidence......Page 215 8.4 The Stokes differential equation......Page 216 8.5 Qualitative discussion of the solutions of the Stokes equation......Page 217 8.6 Solutions of the Stokes equation expressed as contour integrals......Page 218 8.7 Solutions of the Stokes equation expressed as Bessel functions......Page 220 8.9 Zeros and turning points of Ai([GREEK SMALL LETTER ZETA]) and Bi([GREEK SMALL LETTER ZETA])......Page 221 8.11 Asymptotic expansions......Page 222 8.13 Stokes lines and anti-Stokes lines......Page 225 8.14 The Stokes diagram......Page 227 8.16 Furry's derivation of the Stokes multipliers for the Stokes equation......Page 228 8.17 The range of validity of asymptotic approximations......Page 229 8.18 The choice of a fundamental system of solutions of the Stokes equation......Page 230 8.19 Connection formulae, or circuit relations......Page 231 8.20 Stratified ionosphere. Uniform approximation......Page 232 8.21 The phase integral method for reflection......Page 234 8.22 The intensity of light near a caustic......Page 239 Problems 8......Page 243 9.2 Some properties of complex variables and complex functions......Page 245 9.3 Saddle points......Page 247 9.4 Error integrals and Fresnel integrals......Page 249 9.5 Contour maps......Page 253 9.6 Integration by the method of steepest descents......Page 254 9.7 Application to solutions of the Stokes equation......Page 257 9.8 The method of stationary phase......Page 263 9.9 Higher order approximation in steepest descents......Page 265 9.10 Double steepest descents......Page 267 Problems 9......Page 268 10.1 Introduction......Page 270 10.2 The ray path......Page 271 10.3 Wave packets......Page 273 10.4 Equations of the ray path......Page 275 10.6 The reflection of a wave packet......Page 277 10.7 An example of a ray path at oblique incidence......Page 279 10.8 Poeverlein's construction......Page 280 10.9 Propagation in magnetic meridian plane. The `Spitze'......Page 282 10.10 Ray paths for the extraordinary ray when Y 1......Page 286 10.12 Lateral deviation at vertical incidence......Page 289 10.13 Lateral deviation for propagation from (magnetic) east to west or west to east......Page 291 10.14 Lateral deviation in the general case......Page 292 10.15 Calculation of attenuation, using the Booker quartic......Page 293 10.16 Phase path. Group or equivalent path......Page 294 10.17 Ray pencils......Page 295 10.18 Caustics......Page 297 10.19 The field where the rays are horizontal......Page 301 10.20 The field near a caustic surface......Page 302 10.21 Cusps. Catastrophes......Page 303 10.22 The skip distance......Page 304 10.23 Edge focusing......Page 305 Problems 10......Page 308 11.2 The reference level for reflection coefficients......Page 311 11.3 The reference level for transmission coefficients......Page 313 11.4 The four reflection coefficients and the four transmission coefficients......Page 314 11.5 Reflection and transmission coefficient matrices......Page 315 11.6 Alternative forms of the reflection coefficient matrix......Page 316 11.7 Wave impedance and admittance......Page 317 11.8 Reflection at a sharp boundary 1. Isotropic plasma......Page 320 11.9 Properties of the Fresnel formulae......Page 322 11.10 Reflection at a sharp boundary 2. Anisotropic plasma......Page 323 11.11 Normal incidence. Anisotropic plasma with free space below it......Page 324 11.12 Normal incidence. Two anisotropic plasmas......Page 325 11.13 Probing the ionosphere by the method of partial reflection......Page 327 11.14 Spherical waves. Choice of reference level......Page 329 11.15 Goos--Hönchen shifts for radio waves......Page 331 11.16 The shape of a pulse of radio waves......Page 336 Problems 11......Page 341 12.1 Introduction......Page 344 12.2 Vertically incident pulses......Page 345 12.3 Effect of collisions on phase height h(f) and equivalent height h[PRIME](f)......Page 346 12.4 Equivalent height for a parabolic height distribution of electron concentration......Page 348 12.5 Effect of a `ledge' in the electron height distribution......Page 352 12.6 The calculation of electron concentration N(z), from h[PRIME](f)......Page 353 12.7 Ray paths at oblique incidence......Page 358 12.8 Equivalent path P[PRIME] at oblique incidence......Page 361 12.9 Maximum usable frequency, MUF......Page 364 12.10 The forecasting of MUF......Page 365 12.11 Martyn's theorem for attenuation of radio waves......Page 368 Problems 12......Page 369 13.1 Introduction......Page 372 13.2 Reflection levels and penetration frequencies......Page 373 13.3 The calculation of equivalent height, h[PRIME](f)......Page 375 13.4 Ionograms......Page 378 13.5 Topside sounding......Page 381 13.6 The calculation of electron concentration N(z) from h[PRIME](f)......Page 384 13.7 Faraday rotation......Page 388 13.8 Whistlers......Page 392 13.9 Ion cyclotron whistlers......Page 396 13.10 Absorption, non-deviative and deviative......Page 407 13.11 Wave interaction 1. General description......Page 409 13.12 Wave interaction 2. Outline of theory......Page 411 13.13 Wave interaction 3. Kinetic theory......Page 413 Problems 13......Page 414 14.1 Introduction......Page 416 14.2 The eikonal function......Page 418 14.3 The canonical equations for a ray path......Page 419 14.4 Properties of the canonical equations......Page 421 14.5 The Haselgrove form of the equations......Page 423 14.6 Fermat's principle......Page 425 14.7 Equivalent path and absorption......Page 428 14.8 Signal intensity in ray pencils......Page 430 14.9 Complex rays. A simple example......Page 433 14.10 Real pseudo rays......Page 438 14.11 Complex rays in stratified isotropic media......Page 440 14.12 Complex rays in anisotropic absorbing media......Page 441 14.13 Reciprocity and nonreciprocity with rays 1. The aerial systems......Page 444 14.14 Reciprocity and nonreciprocity with rays 2. The electric and magnetic fields......Page 447 14.15 Reciprocity and nonreciprocity with rays 3. Applications......Page 449 Problems 14......Page 452 15.1 Introduction......Page 454 15.2 Linear electron height distribution......Page 455 15.3 Reflection at a discontinuity of gradient......Page 457 15.4 Piecewise linear models......Page 459 15.5 Vertical polarisation at oblique incidence 1. Introductory theory......Page 462 15.6 Vertical polarisation 2. Fields near zero of refractive index......Page 464 15.7 Vertical polarisation 3. Reflection coefficient......Page 466 15.8 Exponential electron height distribution......Page 469 15.9 Parabolic electron height distribution 1. Phase integrals......Page 472 15.10 Parabolic electron height distribution 2. Full wave solutions......Page 476 15.11 Parabolic electron height distribution 3. Equivalent height of reflection......Page 480 15.12 The differential equations of theoretical physics......Page 482 15.13 The hypergeometric equation and its circuit relations......Page 483 15.14 Epstein distributions......Page 486 15.15 Reflection and transmission coefficients for Epstein layers......Page 488 15.16 Ionosphere with gradual boundary......Page 489 15.17 The `sech2' distribution......Page 491 15.18 Other electron height distributions......Page 492 15.19 Collisions. Booker's theorem......Page 493 Problems 15......Page 495 16.1 Introduction......Page 496 16.2 First order coupled equations......Page 498 16.3 Coupled equations near a coupling point......Page 501 16.4 Application to vertical incidence......Page 505 16.5 Coupling and reflection points in the ionosphere......Page 508 16.6 Critical coupling......Page 511 16.7 Phase integral method for coupling......Page 515 16.8 The Z-trace......Page 518 16.9 Additional memory......Page 521 16.10 Second order coupled equations......Page 523 16.11 Försterling's coupled equations for vertical incidence......Page 525 16.12 Properties of the coupling parameter [GREEK SMALL LETTER PSI]......Page 526 16.13 The method of `variation of parameters'......Page 530 16.14 The coupling echo......Page 533 Problems 16......Page 534 17.1 Introduction......Page 536 17.2 Further matrix theory......Page 537 17.3 Coalescence of the first kind, C1......Page 539 17.4 Coalescence of the second kind, C2......Page 541 17.5 Ion cyclotron whistlers......Page 546 17.6 Radio windows 1. Coalescence......Page 548 17.7 Radio windows 2. Formulae for the transparency......Page 550 17.8 Radio windows 3. Complex rays......Page 556 17.9 Radio windows 4. The second window......Page 558 17.10 Limiting polarisation 1. Statement of the problem......Page 559 17.11 Limiting polarisation 2. Theory......Page 562 18.1 Introduction......Page 566 18.2 Integration methods......Page 568 18.3 Alternative methods 1. Discrete strata......Page 569 18.4 Alternative methods 2. Vacuum modes......Page 572 18.5 Alternative methods 3. The matrizant......Page 574 18.6 Starting solutions at a great height......Page 576 18.7 Finding the reflection coefficient......Page 578 18.8 Allowance for the earth's curvature......Page 579 18.9 Admittance matrix as dependent variable......Page 582 18.10 Other forms, and extensions of the differential equations......Page 585 18.11 Numerical swamping......Page 590 18.12 Reciprocity......Page 592 18.13 Resonance......Page 595 Problems 18......Page 597 19.1 Introduction......Page 599 19.2 Vertical incidence and vertical magnetic field......Page 600 19.3 Oblique incidence and vertical magnetic field......Page 603 19.4 Resonance and barriers......Page 607 19.5 Isolated resonance......Page 609 19.6 Resonance tunnelling......Page 612 19.7 Inversion of ionospheric reflection measurements......Page 618 19.8 Full wave solutions at higher frequencies......Page 622 Answers to problems......Page 625 Bibliography......Page 628 Index of definitions of the more important symbols......Page 659 Subject and name index......Page 668 This book is largely concerned with the ionosphere and the magnetosphere, and gives the theory of their effect on radio waves. It includes accounts of some mathematical topics that are now widely used in this study, particularly W.K.B. approximations, Airy integral functions and integration by steepest descents. The subject is in two main parts, usually called ray theory and full wave theory. The first is useful for high frequencies and includes ray tracing and caustic surfaces. The second is used for lower frequencies when the approximations of ray theory fail. Much of the book is concerned with the ionosphere treated as a horizontally stratified medium, but there is some discussion of the magnetosphere whose structure is more complicated. This includes an account of whistlers and ion cyclotron whistlers.The book has been planned both for final year undergraduates and as a reference book for research. It is suitable as a course book on radio propagation for students of physics or electrical engineering or mathematics. Some of the topics are presented from an elementary viewpoint so as to help undergraduates new to the subject. The later parts are more advanced. The subject is very large and has seen many important recent advances. Some topics have had to be treated only briefly, but there is a full bibliography with about 600 references.