PROPAGATION OF SHORT RADIO WAVES

PROPAGATION OF SHORT RADIO WAVES

..

MASSACHUSETTS INSTITUTE OF TECHA’OLOGY

RADIATION LA130R.4TORY SERIES

Board of JZditors LOUIS X. RIDENOUR, ~&for-in-CJzief GEORGE B. COLLIXS, Dtpufu Ildifor-in-C’&ef

BRITTOX CHANCE, S.,l. GOUDSMrT, IL, (;. HERFI, HUBERT JI. JAMES, J[LZAX K. ~NIFP, ,J.4MEsL. LAWSON, LEONB.LINFORD, CAROL G. JIOXTCOMEEY, C. NE\vTo~, ALBERT lf. STONE, 1,01.-Is .1. TURNER, C,EORGE I;. \’.\LLEY, JR., HERBERT H. \YRE.iTON

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27. ~O\{PCTt XG 31Ec1[.4X[SUS AND I.l NKAGES—&obOda 28. Ix Dxx—Hrnne!/

PROPAGATION OF SHORT RADIO WAVES

Edited by DONTALD E. KERR

ASSISTANT PROFESSOR, DEPARTMENT OF PHYSICS JOHNS HOPKINS UNIVERSITY

I

OFFICE OF SCIENTIFIC RESEARCH AND DEVELOPMENT NATIONAL DEFENSE RESEARCH COMMITTEE

FIRST EDITION

iVEW YORK . TORO.VTO . LONDON

McGRAW-HILL BOOK CO MPAA’Y, I.NTC.

1951

PROPAGATION OF SHORT RADIO WAVES

COPYRWJHT, 1951i BY THE MCGRAW-HILL BOOK COMPANY, INC.

PRINTED IN THE UNITED STATES OF AMEH.l~A

All rights reserved. This book, or parts thereoj, may not bereproduced in ally form tci!hot(l permi$sioh I1j

thepublishers.

SCIEI$NX LIBRARY

m

PROPAGATION OF SHORT RA J910 WAVES EDITORIAL STAFF

DONALD E. KERR LEONB. LINFORD

S. A. GOUDSMIT ALBERTM. STONE

CONTRIBUTING AUTHORS

ARTHUR E. BENT RICHARD A. CRAIG WILLIAM T. FISHBAIX

JOHN E. FREEHAFER WENDELL H. FURRY

HERBERT GOLDSTEIN

ISADORE KATZ

DONALD E. KERR

R. B. MONTGOMERY EDWARD M. PURCELL

PEARL J. RUBENSTEIN

A. J. F. SIEGERT J. H. VAN VLECK

Foreword

T

The Radiation laboratory of MIT, which operated under the super- vision of the National Defense Research Committee, undertook the great task of preparing these volumes. The work described herein, however, is the collective result of work done at man,y laboratories, Army, Navy, university, and industrial, both in this country and in England, Canada, and other Dominions.

The Radiation Laboratory, once its proposals were approved and finances provided by the Office of Scientific Research and Development, chose Louis N. Ridenour as Editor-in-Chief to lead and direct the entire project. An editorial staff was then selected of those best qualified for this type of task. Finally the authors for the various volumes or chap- ters or sections were chosen from among those experts who were inti- mately familiar with the various fields and who were able and willing to write the summaries of them. This entire staff agreed to remain at work at MIT for six months or more after the work of the Radiation Laboratory w-as complete. These volumes stand as a monument to this group.

These volumes serve as a memorial to the unnamed hundreds and thousands of scientists, engineers, and others who actually carried on the research, development, and engineering work the results of which are herein described. There were so many involved in this work and they worked so closely together, even though often in widely separated labora- tories, that it is impossible to name or even to know those who contributed to a particular idea or development. Only certain ones who wrote re- ports or articles have even been mentioned. But to all those who con- tributed in any way to this great cooperative development enterprise, both in this country and in England, these volumes are dedicated.

L. A. DUBRIDGE

Prejace

M~st of the volumes of the Radiation Laboratory Series are devoted to specific radar subjects such as components, systems and their applica- tions, or measurement techniques. This volume, however, treats the phenomena associated with the propagation of short radio waves between terminal points, whether they betheradar antenna serving a dual purpose or the antennaa of a communications system. The intention is to present a summary of the state of knowledge in the microwave-propagation field at the close of the war. There has been no attempt to produce either a handbook or textbook, but only an interim report on a rapidly changing subject. An attempt has been made to survey all relevant information that was available, from whatever source, and to summarize as much of it as was feasible.

The preparation of the book was undertaken primarily by the Pro- pagation Group (Group 42), and all of its thirty-odd members contributed either directly or indirectly to the material given here. In addition, sub- stantial contributions have been made by authors who were not members of this group but who worked closely with the group during the war. The division of authorship was to a certain extent arbitrary. The principal criterion was, of course, familiarity with the subject matter, and where possible the people who had made original contributions were favored.

There were limiting factors, however, such as the degree of availability of possible authors and the fact that it was impractical to have a large number of writers. Unfortunately, it is impossible to give adequate recognition to all those who have contributed directly or indirectly or even to represent the correct proportion of the contributions of those whose names appear here.

A vast amount of material was available for consideration—much more than could have been presented in one volume, Consequently, some topics have been omitted completely, as, for example, diffraction by trees, hills, and obstacles other than the earth or objects used as radar targets. In this case, as in some others, no significant original work on the subject was done at the Radiation Laboratory, and reviewing work done entirely by others did not appear desirable. Other subjects that have been omitted are the numerous attempts at application of radio- meteorology to forecasting of radio and radar propagation performance and the climatological studies needed to make such knowledge useful on

ix

x PREFACE

a world-wide scale. In this case, authors were not available to undertake the work. In choosing the meteorological material that was to be pre- sented, it was decided that in the limited time available it was feasible to present only the material considered to have the soundest fundamental background and to eliminate material that involved an appreciable amount of speculation or that would require reworking or further research to put it into the desired form, In general, throughout the book when similar decisions were necessary, they were nearly always made in favor of an exposition of selected material rather than a sketchy, uncritical report of a large amount. We are aware that despite our attempts to include data from many sources our own work tends to predominate; knowing it most thoroughly, we have treated it in greatest detail.

Much of the wartime work was necessarily done in haste without adequate preliminary planning, care in execution, or sufficient analysis of results. If we appear to be overly critical or pedantic here, the reader is asked to understand that this arises, at least in part, from the reaction of the authors to the nature of much of the source material from which the following chapters are formed. We have not hesitated to point out the need for criticaf examination of the data reviewed here, for such an exami- nation must certainly be one of the first steps in further research in the field. We have also made numerous suggestions for future investigations.

The methods employed in recent propagation research are, we believe, rather important, and we have described them in some detail when it appeared that the description would aid others in future plans. Apparatus details involving radio-frequent y techniques are omitted, as most of them are covered in other volumes of this series, but methods of planning experi- ments and of analyzing results are emphasized. The meteorological in- strumentation and new measurement techniques are also emphasized, as they are of utmost importance in investigations of the effects of atmospheric refraction on microwave transmission.

Nomenclature and symbols were matters about which positive decisions were necessary if the book was to be readable, The present choice is the result of considerable deliberation and compromise among several well- established but highly conflicting systems, It embodies as much as possible of the best or of the most firmly established features of each system.

A serious attempt has been made to avoid undue overlapping use of sym- bols but at the same time to adhere to uniform usage throughout the book;

some inconsistencies appear inevitable, however.

We have attempted to acknowledge the sources of all our information, even though, unfortunately, these sources are frequently in the form of reports that possibly will never be gencrall y available. Some of the re- ports cited here are beginning to appear in the literature as this material goes to press, however, and the appropriatec footnot c references have been inserted wherever possible. When t.hc source of cxpcrimcntal material is not specifically stated, it may be assumed to be the Rndi:li ion T.:~borutor:/,

PREFACE xi

but because of the high mobility of ideas, it is not always possible to be certain of their origin. Except for the measurements on oxygen and water- vapor absorption, ship and aircraft cross secti ens, and a few miscellaneous items, almost all of the Radiation Laboratory material is the work of the Propagation Group or of its close associates.

The information summarized here represents a large investment of effort by many persons and agencies, and it is impossible to acknowledge fully our indebtedness to all of them. Our principal indebtedness is to the remainder of the Propagation Group, whose work contributed so much to this volume. Second, we must acknowledge particular indebtedness to the several authors who at considerable inconvenience to themselves con- tributed their services long after the termination of the activities of the Radiation Laboratory Office of Publications.

We should like to acknowledge specifically the very great assistance rendered by the several branches of the armed services, who contributed generously in both man power and in equipment such as boats, aircraft, housing facilities, and the many other items necessary to carry on field operations on a large scale. We should like to thank the members of the U.S. Weather Bureau and its several branch offices, whose personnel not only contributed information but in some cases participated in our research program. We are also greatly indebted to Dr. Charles Brooks of the Blue Hill Observatory of Harvard University for his meteorological advice.

Most of the aircraft soundings in Chapter 3 were obtained by Robert H.

Burgoyne and Earl G. Boardman, who contributed his aircraft and his services as skillful pilot. This work deserves special mention because of its hazardous and highly exacting nature.

In an attempt to ensure accuracy in reporting the work of other groups, we have submitted portions of the manuscript for review to several indi- viduals and organizations. Particular thanks are due to the following people: Sir Edward Appleton, Dr. R. L. Smith-Rose, and the other mem- bers of the Tropospheric Wave Propagation Committee in England;

Dr. John B. Smyth of the U.S. Navy Electronics Laboratory; A. B. Craw- ford of the Bell Telephone Laboratories; Professor Paul A. Anderson of Washington State College; Dr. H. H. Beverage of RCA Laboratories;

K. A. Norton and Dr. T. J. Carroll of the Central Radio Propagation Laboratory, Bureau of Standards; and Professor C. R. Burrows of Cornell University. The corrections and suggestions offered by these men have been of great value in integrating the descriptions of the work with which they are most familiar. Thanks are also due Norma W. Donelan for her aid in final preparation of the manuscript.

CAMBWDGE, MASS.

.lV@, 1947

DONALD E. KERR

Contenk

FOREWORD . . . . . . . . . . . . .

PREFACE . . . . . . . . . . . . . . . . . CHAP. 1. ELEMENTS OF THE PROBLEM

(JOHN E. FEEEHAFER AND DONALD E. KERR) EVOLUTION OF THE PRESENT PROBLEME+.

1.1. The Ionosphere andthe Transmission of Long Waves 1,2. Optical Properties of Short Waves

TROPOSPHERIC REFRACTION

13. The Effects of Variable Gradlentsof Refractive Index 1.4. The Meteorological Elements andthe Modified Index 1,5. The Modified Index and Field-strength Distribution ATMOSPHERIC SCATrERINQ AND AmENUATION . . .

.,. . . . vii . . . . . . ix

. . . 1

. . . . 1 . . . . . . 1 . . . . . . . 3

,.. . . 9 . . . . 9 ,.. . . . 12 . . . . 15

. . . 22

1.6.

1.7.

18.

CHAP. 2.

Radar Echoes from Precipitation . . . . 22

Scattering and Absorption by Particles . 23

Absorption by G~es . . . . . . . . 25

THEORY OF PROPAGATION IN A HORIZONTALLY STRATI- FIED ATMOSPHERE . . . . . . . . . . . . . . . . ...27

(JOHNE.FREEHAFER, WILLIAM T. FISHBACK, WENDELL H. FURRY, AND DONALD E. KERR)

FUNDA~ENTAL CONCEPTS . . . . . . . . . . . . . . . ...27

2.1. Transmission in Free Space . . . . 27

22. The Transmission Medium and the Pattern-propagation Factor . . . 34 GEOMETRICAL OPTICS . . . . . . . . . . . . . . . . . . . . . . . ..41

2.3. Ray-tracing Formulas . . . . . . . . . . . . . . . . . . . . . .41 24. The Modified Index . . . . . . . . . . . . . . . . . . . .5o 25. Limitationa of Ray Methods.. . . . . . . . . . . . . . ...53 PEYSICAL O~CS . . . . . . . . . . . . . . . . . . . . ..58

2.6. The Field from a Dipole in a Stratified Atmosphere near the Earth . . 58 27. The Fundamental Theorem. . . . . . . . . . . . . . . . . ..65 28. Phsse-integral Methods . . . . ., . . . . . . . . . . . . ..7o TEE LWEMMODIFIEO-INDEXPEOFILE . . . . . . . . . . . . . . . . . 87

29. The Properties of Solutions of @y/df’ + w = O . . . . . . . S7 210. The Field Integral . . . . . . . . . . . . . . . . . . . . . . .95 211. The Int-erference Region.. . . . . . . . . . . ..98 212. The DMractionRegion. . . . . . . . . . . . . ..109 Mmmom FOR CALCULATING Fmrm STRENQTH wrmi STANDARD REFRACTION 112

213. The Interference Region... . . . . . . . . . . . . . . . . . .113 214. The DbhctionRegion.. . . . . . . . . . . . . . . . . .. ,122

. .. .

xiv COYTE.VTS

215. The Intermediate Region 125

216. Contours of Constant Field Strength 130

Tm BILINEAR lIODIFIED-I~DEX PROFILE 140

2,17, Definition of the Problem and Preliminary Formulation 140 2,18. }Iethods for (’calculating Characteristic Y:ducs 146 2.19. Behavior of Characteristic Values and Charwtcristic Functions for the

First Mode 161

2.20. The Problem of Calculating Field Strength for the Bilinear Profile 168

NONLIIWME klODIFIED-INDEX PROFILES 174

221. The Linear+ xponent ial and Power-1aw Profiles 174

CHAP. 3. NIETEOROLOGY OF THE REFRACTION PRO131.E;X1 181

(RICHARD A, CRAIG, ISADORE I<ATZ, R. B. J1ONTGOMEEY, AND PEARL J. RULrENSTICINi

HUMIDITY AND REFRKTIW INDEX 1s1

3.1. T’apor Pressure and Saturated Vapor 182

32, \f’ater-vapor Concentration 18-I

3,3. Saturation Temperatures on Isobaric Cooling 186

3.4. Refractive Imiex of Air at Radio Frequencies 1~{)

VERTICALLY IICIMOG~~~OUS AR AND .4n1.iBAT1c CH.~FiLiES Ig;j 3.5. .kdiabctic Temperature Lapse Rate and l’otcntial Tcmper:lttlr(, 1{)~

36. Humidity Lapse in Homogemmus .\ir 1!)(;

3-7. Gradimt of Refractwc J[od!ilus in Homogc,ntwus Air, I’ot{,ntial \l,vd(il[,s 198

38. Characteristic Curves and J[ixiug 200

REFRM~YT.iTIO~ AND D~W~TFTtO~ OF SoU?MINGs 202

39. Approximate Formula for Refractive Modulus 203

3.10. Representation of Soundings 206

EDDY DIFFUSION 208

311. Eddy Viscosity and Eddy Diffusiwty 208

3.12. Layer of Frictional Infiuenrc in Neutral Eq(lilibrium 213 3.13. I.ogarithmic Distributions in the Turhnlcnt 130uMI:wJ Lay~,r 215 VERTICAL DISTRIBUTIONS m XWTR.AL AND UNST.i RLE EQUILIBRIUM 2[9

314. Heating from Below 220

3.15. Application of Logarithmic Distrlhution ~~3

3.16. Rate of ilfodification of T_inst:lhlc .Mr Wfi

VERTICAL DISTRIBUTIONS IN STABLE EQC-ILIDRIVM 2X

3.17. Cooling from Below y~~

3.18. Shear in Stable E,luilibrium 2:14

3.19. Initially Hon~ogencous \Varm .\il over Cold \\:itt,r 237

3.20. Complex Over-water LImhficati{]lls 2,io

3.21. h’octurnal Cooling Wd Diurnal (’w-1~~ 23:3

OTHER ATMOSE’HERICPROCW+ES AIiD TH~IR F;FFWT ox 11-IWOFJM.S ’260

3.22. Subsidence and Subsidence Invcrsmns 2tio

323. Fronts and Frontal Inversion 2{i3

324. Sea-breeze Circulations 26+

CONTENTS xv

3.25. Horizontal Gradients .267

3.26, Local Variations with Time 268

INSTRUMENTS TO MEASURE TEMPERATURE AND HUMIDITY IN THE LOWER

ATMOSPHERE . . . . . . . . . . ...272

327.Psychrograph . . . . . . . ..272

328. WiredSonde. . . . . . . . . ,283

329. Aircraft Psychrometer 287 3,30. Resistance Thermometer and Humidlometer . 289 3.31, Thermocouples . . . . . . . ..290

3.32. General Problems Associated with Low-level Soundings 291 MET~OKOLOGICALCONSTANTS. . . . . . . ,292

3.33. Useful Meteorological Constants . . . . . 292

CHAP. 4. EXPERIMENTAL STUDIES OF REFRACTION 294 (PEARL J, RWBENSTEIN, Domrm E. KERR, AND WILLIAM T. FMHBACK) ONE-WAY TRANSMISSION OVER WATER . 294 Transmission overMassachusetts Bay 4.1. Radio Me~urements Program . 296 4.2. Meteorological Measurements and Analysis. 297 4.3. General Characteristics of Transmission ... 301

44. Comparison wrth Theory 307 4.5. Tranermssion under Complex Conditions 315 4.6. Some Statistical Results . 319 Transmission Experiments in the British Isle8 4.7. The Irish Sea Experiment .322 4.8. South Wales to Mt. Snowdon .328 Transmission akmgthe California Coast 49. San Diego to San Pe&o. .328 Transmission over an Inland Lake 4.10. Flathead Lake . . . . . . . ..335

ONE-WAY TRANSMISSION OVER LAND 336 4.11. Early Experiments 336 4.12. Summary of General Characteristics 340 4.13. Addhional Observations. .343 4.14. Discussion . . . . . . . . . ...350

RADAR TRANSMMSION . . . . . . . . . . . . . ...353

4.15. New England Coast.... . . . . . . . . ..354

4.16. California Coast . . . . . . . . . ..361

4.17. Weleh Coast . . . . . . . . . . . 363

4.18. The English Charnel Region. ! 367 4.19. Other Regions . . . . . . . . .369

SPACE VARIATIONS IN FIELD STBENQTH 373 4.20. Shallow Surface M-inversions 374 421. Deep Surface M-invemione . .378 4%2.ElevatedM -aversions.. . . ...382

ANGLE MEASUREMENTS ON SHORT OPTICAL PATJiS 385

4%1.Measurement.ao fAngleo fArrival . .386

4%LTheoreticalDis cession.. . . . . . . . . . . . . . . . . . ..39 I

xvi CONTEA’TS

CHAP. 5, REFLECTIONS FROhl THE EARTI1’S SURFA(’I? 396

(DONALD E. KEW, WILLIAM T. FISH BACK, AND IIMI{IIEKTGOLI..TLINI

THEORY OF SPECULAR REFLECTION 39fj

5.1. Fresncl’s Equations fora Smooth I’lanc S(]rfacc 3!16 5.2. Geometrical Interpretation of the Divc>rgcnm Factor 404

53. Effcctsof Reflections on Field Strength ‘m

5.4. Surfacc Roughness 411

REFLWTION CO~FFICI~NT OF THE OCEAN 418

5.5. Measurements of Short-time Variations 4]9

5.6. Interference hfeasurcmcnts over I.ong Itangcs 421

57. Intcrfercmce Me&surements at Short lt:ingcLs 4’27

5.8. Interpretation of Measurcmnts 4~\)

REFLECTION COEFFICIENTOF LAND 430

59. Measurements over Long Ranges .430

510. Me=urements at Short Ranges 4:33

5,11. hleasuremcnts of Time l’ariations 434

5.12. Interprctat ion of Measurements 435

EIIRORS IN RADAR HEIGIIT LIEASUREMENTS 436

513. Qualitative Discussion 437

514. Illustrative Examples 441

CHAP. 6. RADAR TARGETS AhTD Et ‘II( )1;S 44,5

(DONALD E. KERR AND lIRKtiLRT Gormsw,w

THE RADAR CROSS S~CTION OF ISOLATMI T~ IR,CTS 445

6.1. Scattering from a Sphere 445

6,2. Vector Form of Huygcns’ Prmclph, 4.5-I

6.3. Scattering from Planes and Curved Surfaces 456

COMPLEX TARGETS . . . . . . . ...469

6.4, Radar Cross Section of Aircraft 470

6,5. Radar Cross Section of Ships 472

SEA ECHO . . . . . . . ..481

66. N’ature of the Problem .4s1

67. Nature of the Sea Surface 486

6.8. Validity of the Fundamental Assumptions 490

6,9. Frequency Dependence of Sea Echo 494

6.10. Measurements of the Properties of Sea-echo Cross Scrtlon 499

6.11. The Fluctuation of Sea Echo 514

612. Theories of Sea Echo 518

THE ORIGINS OF ECHO FLUCTUATIONS 527

6.13. The Limitations of System Stability 527

6.14. Atmospheric Variations 531

6.15. Fluctuations in the Space Interference Pattern 535

6.16. Isolated Moving Targets 539

6.17. Interference Phenomena m Complex Targets 547

THE FLUCTUATIONS OF CLUmEE ECHOES 550

6.18. The Nature of Clutter Echoes 55o

6.19. The Theory of Clutter Fluctuations 553

CONTENTS xvii 620. Experimental Techniques inthe Study of Clutter Fluctuations 562 6.21. Experimental Resulte . . . . . . . . . . . . . . . . . . . ...571

CEAP. 7. METEOROLOGICAL ECHOES . . . 588

(HERBEET GOLDSTEIN, DONALD E. KErLR,AND ARTHUR E. BENT)

ORIQLNOF THE ECHO . . . . . . . . . . . . . 588

7.1. The Echo from IncoherentscatterersDistribu@d in Volume 589 72. Evidence of Direct Correlation between Meteorological Echoes and

Precipitation . . . ...591 7.3. The Approximate Magnitude of Rain Echoes on the Drop Theory. 596 74. Possible Alternative Theories to Scattering by Drops 598

7.5. Modifications of the Drop Theory 604

THE INTENSPPY OF METEOROLOGICALECHOES 607

7.6. The Radar CrossSection of Single Drops 608

77. Drop-size Distribution .615

7.8. Echoes from Solid Precipitation 618

GENERAL PROPERTIES OFPRECIPITATIONECHOE~ 621

7.9. Identifying Characteristics 621

7.10. Confusion and Maskkg of Other Echoee 625

PRECIPITATIONECEO PROPERTIESAND METEOROLO~lCALSTFWTCTURE 626

7.11. Cksshication ofEcho Types 626

712. Thunderstorms . . . ..627

713. Other Forms of Localized Precipitation 632

7.14. Widespread Precipitation 633

7.15. Cyclonic Storms of Tropical Or@n 636

CRAP. 8. ATMOSPHERIC ATTENUATION 641

(J. H. VAN VLECK,E. M.PURCELL, AND HERBERT GOLDSTEIN)

8.1. Properties of the Complex Dielectric Constant 641

THEORY OFARSORFIYONBY UNCONDENSEDGASES 646

tY2. Oxygen . . . ...648

8.3. Uncondensed Water Vapor. .656

MEASUREMENT OF ATMOSPHERIC ABSORPTION . 664

84. Direct Measurement of Absorption by Oxygen 665

85. Measurements of Water-vapor Absorption 666

ATTENUATIONBY CONDENSEDWATER. . . . . . . . . . . . . . . ...671 8.6. Phenomenology of At@nuation by Precipitation 671

87. Calculation of Attenuation by water Drops 674

88. Calculation of Attenuation by precipitation in Solid Form 685

89. Mee=ementa of At@nuation by mln 688

APPENDIX . . . 693 (DONALDE.KERR, A.J. F. SIEGERT,ANOHERBERTGOLDSTEIN) Application of the Lorentz Reciprocity Theorem to Scattering 693 Coherent and Incoherent Scattering from Assemblies of Smtterers 699

NAME INDEX . . . 707

SUBJECT INDEX . . . 713

CHAPTER I

ELEMENTS OF THE PROBLEM BY JOHN E. FREEHAFERAND DONALD E.

This first chapter is intended to serve two purposes:

KERR

(1) It introduces anumberof definitions and concepts that will be useful to orient the reader before reembarks upon the detail of the following chapters. (2) It pre- sents a thumbnail sketch of the type of information that is to follow, but free from the specialized terminology and methods that are frequently necessary in later chapters. We attempt only to indicate a few of the high lights rather than to give a complete outline, as the latter course would lead to undue repetition.

EVOLUTION OF THE PRESENT PROBLEMS

It is interesting to observe that the workers in the present microwave field have returned to the wavelength region in which major features of Maxwell’s electromagnetic theory received confirmation through the r~

searches of Hertz. The details of the trend from short to long waves and back again are of no interest here, but some features of the trend are suffi- ciently relevant to be considered briefly. In addition, the complete identity of many of the problems considered here with well-known prob- lems in optics is emphasized, as experience has shown that familiarity with concepts and methods of optics is of great utility in the microwave field.

1.1. The Ionosphere and the Transmission of Long Waves. ~Interest in the propagation of electromagnetic waves over the surface of the earth first became active when Marconi demonstrated in December 1901 that signals could be transmitted across the Atlantic Ocean. During the next 18 years the most eminent mathematicians and physicists of the time con- tributed to a lively discussion of the quantitative aspects of the four mechanisms that had been proposed to account for the reception of signals beyond the horizon. These were diffraction, reflection from an elevated layer of ionized gases, atmospheric refraction, and transmission by means of surface waves that follow the interface between media of different dielectric properties.

Formulation of the diffraction theory by considering the field due to an oscillating dipole near an imperfectly conducting sphere readily yields an infinite series that unfortunately converges at a hopelessly slow rate

I By John E. Frwhafer.

2 ELEMENTS OF THE PROBLEM [SEC.11 when the radius of the sphere measured in wavelengths is large. The attempts of various investigators to approximate this series led to widely divergent results, none of which agreed with the empirically determined Austin-Cohen formula,’ which seemed adequate to summarize the experi- mental data available at the time.

Meanwhile other possibilities were being examined. The presence of ionized gases at great heights in the atmosphere was inferred from obser- vations of the spectrum of light from the night sky. These showed the continuous presence of lines associated with the aurora. Furthermore the existence of a conducting layer aloft had been postulated to explain the observed variations of terrestrial magnetism. In 1902 both Kennelly and Heaviside’ published articles suggesting that long-range radio transmission might be accounted for by reflecting from an elevated conducting surface.

Ten years later 12ccles3contributed an investigation of the mechanism of ionization by solar radiation and presented the fundamental theory of ionic refraction.

The effect of the decrease of refractive index of the air because of the decrease in its density with height was also examined by several inves- tigator.’ On the assumption that the temperature is constant, it was shown that the radius of curvature of the rays would always be many times the radius of the earth. It was therefore concluded that bending by the atmosphere, although acting in the right direction, is unimportant.

Finally the idea that radio waves could be guided along the surface of the earth much as electromagnetic energy can be conducted along wires was examined theoretically and experimentally. It was observed that high antennas were not necessary to receive signals and that ranges were usually longer over water than over land. Zenneck5 was able to explain some of the observations by applying to the radio problem the results of his demonstration that Maxwell’s equations admit a solution that repre- sents a surface wave guided along the interface between two different

1Accordingto the Austin-Cohenformula

whereI is the currentinducedin s receivingantennaof heightZ*per unit currentin the transmittingantennaof heightz,, The rangeand wav(,lcngthare denotedbyrand A.

The parametera = 1.5 X 10-3 (kilometers)-‘f for transmissionover sea water. For the data from which this formulawas deducedsee L. W. Austin, Vat.BUT.Stmruiards Bull. 7, 315 (1911).

2.4. E. Kennelly,Elec.Worki,39, 473 (1902);O. Heaviside,EncyclopediaBritannica, 10thcd., Vol. 33.

? W. H, Eccles,Proc, ROV.Sot. A,8’7,79 (1912); E[ectriczan,79, 1015 (1912).

t J. A. Fleming,Principles of Electric Wave Telegraphy andTelephony,1919,p. 660.

5J. Zenneck, Ann. Physik, 23, 846 (1907); Lehrbuch der drahtlosen Telegraphic, 2’” Aufl., Stuttgart,1913.

SEC.12] OPTICAL PROPERTIES OF SHORT WAVES 3 media. Further work by Sommerfeldl seemed to show that a surface wave of the type discussed by Zenneck would actually be excited by an oscillating dipole at the boundary between a dielectric and imperfectly conducting plane. Sommerfeld’s results were later questioned by several investigatore,z and there is still disagreement concerning the proper nomen- clature for and interpretation of the surface wave.

The initial exploratory period was brought to a close with the appear- ance of two important papers by Watson in 1919. In the first of these~

he exposed the errore that were responsible for the confusion in the diffrac- tion theory and gave the correct solution. His results showed that dif- fraction failed completely to account for the long ranges observed. In the second paper4 he derived the Austin-Cohen formula on the assumption that transmission takes place in the space between two concentric conducting spheres. This was a clear-cut triumph for the reflecting layer hypothesis and directed future effort toward the fruitful field of ionospheric research.

During the next 14 years’ most of the known phenomena concerned with the transmission of radiation at frequencies up to 30 Mc were observed and satisfactorily explained in terms of measured properties of the iono- sphere. Except under conditions of unusually dense ionization, ionic refraction cannot account for the reflection of energy at wavelength much below 9 m. Thus when new developments in experimental technique extended the usable spectrum from 10 down to 1 m and below, interest in the effects of diffraction and atmospheric refraction was revived.

1s2. Optical Properties of Short WavesY—The wavelength range of interest in this book is from about 3 m down to about 1 mm, but no attempt will be made to fix definite limits. The material presented here applies particularly to the region in which the wavelength is small enough that

“surface waves” and reflections from the ionosphere are absent but where at the same time it is long enough that large numbers of atomic or mole- cular resonances in the gaseous components of the atmosphere do not occur in a small wavelength interval. The limitations imposed in this way are flexible and depend, for instance, upon whether the transmission path under consideration is long and near the earth’s surface or is short and high in the atmosphere. Much of the material presented here is valid far outside this loosely defined wavelength range, but outside this range it must be applied with a full knowledge of other effects that may mask those under present discussion.

L.A, Sommerfeld,Jahrb. drahtlosen Telegraphic, 4, 157 (1911); Ann. Phyw’k, .81, 1135(1926).

zSee,for instance,F. Naether,E. N. T., 10, 160 (1933).

3G. N. Watson,Proc. Roy. Sot., 95, 83 (1919).

4Ibid., p. 546.

5For a reviewof the activityinthis field see H. R. Mimno,Rev. Mod. Phys., 9, 1 (1937).

sBy Donald E. Kerr.

4 ELEMENTS OF THE PROBLEM [SEC. 1.2 Within this wavelength regionl the relation of the wavelength to the size of material objects at the transmission terminals and along or near the transmission path is of primary importance to the properties of the trans- mission. In the microwave region most of these objects are comparable in size to the wavelength, and many aie much larger than the wavelength.

As a consequence it is desirable to discard some of the concepts employed in radio engineering at longer wavelengths and to substitute some of the language and techniques of optics, which have been devised to handle pre- cisely this situation in a clifferent part of the electromagnetic spectrum.

It is convenient to recall that for purposes of classification, optics is divided into the two broad fields of geometrical optics and physical optics.

Geometrical optics is, in general, the simpler of the two, as it predicts propagation of waves along rays according to simple geometrical laws without regard to wavelength or phases. Physical optics, on the other hand, results from a solution of the wave equation and thus automatically introduces wavelength, phases, and penetration of waves into shadow regions in which geometrical optics gives no information. As Baker and Copson point out, “There is a general theorem due to Kirchhoff which states that geometrical optics is a limiting form of physical optics. More precisely, the difTuse boundary of the shadow in diffraction phenomena becomes the sharp shadow of geometrical optics as the wavelength of the light tends to zero.’” We shall find that both physical and geometrical optics are necessary in the study of microwave propagation and that important cases arise in connection with certain types of refraction in which the transition from one to the other may become poorly defined.

In general, geometrical optics is used wherever possible because of its com- parative simplicity; unfortunately, however, it may easily yield highly erroneous or meaningless results where it is needed most, and in this same region physical optics is likely to be so difficult as to be useless. In such cases one resorts to numerous ad hoc artifices and special devices and often bold interpolation in an attempt to fill the gaps left by straightforward theory.

There are many classical experiments of optics that have useful counter- parts in the microwave field. One of the best known of these experiments is that of Lloyd’s mirror.3 It is ordinarily performed with a smooth glass plate or a mirror illuminated by a point or line source of light. A screen or photographic plate is arranged to indicate the intensity of the light in the region illuminated both by the source and by reflection at nearly graz- I The rangeof wavelengthsbelow about 1 m will also be referredto frequentlyas the microwaveregion.

zThe Mathematical Theory of Huygens’ Printi”pZe,Oxford,New York, 1939,pp.79.f.;

alsoJ. A. Stratton,Electromagnetic Theory,McGraw-Hill,New York, 1941,p. 343. For a very completediscussionseeM. Born,Optik,Springer,Berlin,1933,Chaps.2, 3, and4.

3Jenkins and White, FundamentalsofPhr@cal Optics, hIcGraw-Hill,New York, 1937,pp. 66 and 407.

SEC. 1.2] OPTICAL PROPERTIES OF SHORT WAVES 5 ing angles from the mirror. A sketch of the experiment is shown in Fig. 1.1. In the shaded region the light intensity results from a super- position of the light from the source and that reflected from the mirror.

The light reflected from the mirror appears to come from an image of the source indicated by the broken lines,

phase with the source by the amount of the phase shift upon reflection.

For grazing incidence this phase shift is just T, as shown by the fact that the intensity is always a mini- mum (zero for most practical pur- poses) at the surface of the mirror.

It will be shown later that if absorp- tion losses in the mirror are neglected the intensity at a height h on the screen-above the surface is given by

Z = 410 sinz ah,

and the image appears to be out of Free-space

intensity ‘1

..;-,,. -~_---

Reflectlng Intensity surface

FIG. I. I. —Interference of light wavea M illustrated by the Lloyd’s mirror experiment.

The vertical scale on the right has been expanded for the sake of clarity,

where 10 is the intensity from the source in the absence of the mirror (denoted hereafter as the free-space intensity) and a is a constant. This variation is indicated by the curve at the right of the figure.

If the surface of the earth is sufficiently smooth, it produces essentially Interference

region

~

Fm. 1.2,—Crosssection of the earth of radius a. A source of radiation is at P,.

and energyreachesl’, along the two in- dicatedraypaths when P, is in the inter.

ferenceregion. WhenP, is inthe diffrac- tionregion,energyreachesit by diffraction aroundthe bulgeof the earth.

the effect shown above upon micro- wave radiation at distances for which the earth’s curvature is unimportant.

If the surface is so rough that the reflection is diffuse rather than spec- ular, however, the interference pattern disappears and the intensity is essen- tially 10, with additional minor irregu- lar variations from point to point in space (for discussion of the details see Chap. 5).

As the earth is spherical rather than plane, the Lloyd’s mirror phe- nomenon is modified considerably at large distances. Figure 1”2 shows a cr~ss section of the earth, above which a bource of radiation at PI sends energy along the direct and reflected-ray paths to the point P2. The horizon for the point P, is at T, and thetangent Tay is the ray through P, and T, extended indefinitely.1 The region above bT and above the tan- 1EarlierterminologydesignatedP,T asthe Lineof sig)d. This is not an appropriate term when both terminalaare elevated, When line oj sightis used in this volume, it appliesto the direct-raypathP,PZ.

6 ELEMENTS OF THE PROBLEM [SEC.12 gent ray beyond T is called theinterference region, as the intensity can be described in terms of waves following the two paths and adding vectorially to produce a resultant intensity having periodic fluctuations roughly similar to those above the Lloyd’s mirror.1 The region beyond T and below the tangent ray is called thediffraction region, as energy penetrates this region by ditlraction, which is the process of principal interest in physical optics. The ease with which energy penetrates into this region depends upon numerous factors to be discussed in Chap. 2, but it is of interest here to observe that as the wavelength decreases, the “shadow”

cmt along the tangent ray by the bulge of the earth becomes more sharply defined. This is another way of saying that the rate of attenuation of intensity with distance or height increases with decreasing wavelength.

As most transmission paths require propagation along a line very close to the tangent ray, where the diffraction shadow begins to be very pro- nounced, the exact position of the transmission terminals with respect to the tangent ray is important and is usually stated in describing a path.

The unobstructed path P,P2 in Fig. 12 is called an “optical” path; whereas if Pz is on the tangent ray to the right of T, the common term is a “grazing”

path. If P2 is in the diffraction region, one refers to an “extra-optical”

path. An additional correction is almost always made in which an eflectioe value of earth’s radius of $a is used in calculating ray trajectories to allow for refraction effects to be described later. In this case a grazing path may also be referred to as “radio-optical.” Both terms will be employed in this book, with an effective earth radius of $ the true value always being implied.

The effects of the index of refraction of the atmosphere n will receive considerable attention in this book. It will be found that not the absolute value of n (which is roughly 1.0003) but rather its vertical gradient is very important in determining the intensity in the vicinity of the tangent ray and in t,he diffraction region. In the first steps of analysis of the effects of refractive index gradients it is convenient to resort to geometrical optics, invoking Fermat’s principle and Snell’s law of refraction derived from it, in order to trace the rays describing the wave paths in the atmosphere.

In the interest of precise statement of the problem, wedefine the rays as the normals to the surjaces of constant phase oj the wavefronts.z When the transmission medium is homogeneous, the rays also give the direction of propagation of the energy of the waves. In the vicinity of sharp corners or in regions in which ray patterns exhibit certain peculiarities as a result of refraction, the energy no longer follows the rays and geometrical optics fails to give meaningful results. Physical optics is then required to con- tinue the analysis. The reader should bear in mind that although I The true situationis considerablymore complicatedthan suggestedabove. The detailswill be discussedfully in Chaps. 2 and 5.

2% Sec. 2.3 for derivationof the equationfor the rays.

SEC.1.2] OPTICAL PROPERTIES OF SHORT WAVES 7 raytracing procedures are very useful in the region in which they are valid, they are easily misused and numerical results based on them should be used with caution. This point will recur frequently in later sectiom.1

In order to include the specific problems of radar in a study of the propagation process it is necessary to investigate the scattering of micro- wave radiation by objects ranging from raindrops to battleships. More specifically, the process of interest is the diffraction of plane waves by these objects. Here again the procedures of physical optics supplY the results in the cases in which an exact solution is possible at all, but most practical cases are so complicated that analytical methods usually consist of a series of desperate artifices that lean heavily on geometrical optics where pos- sible. We shall consider here only two simple cases to illustrate the fundamental principles, leaving the details to Chap. 6.’

The simplest radar target (and the only one for which the scattering has been calculated with complete rigor) is a sphere. The radius of the sphere is denoted by a, and the wavelength of the incident radiation by A.

The ratio a/h and the dielectric constant and conductivity of the sphere are sufficient to define the scattering problem. (The effect of the latter two quantities will not be considered here.) We begin with a very small sphere, that is, one for which a/A <<1. The wave incident upon the sphere excites currents in it which in turn act as the source of a new wave- the scattered wave that we are seeking. These currents radiate a wave that is identical with the field from a classical Hertzian dipole with a suitably chosen electric dipole moment; thus the scattered wave may be computed in a simple manner from well-known formulas. The ratio of the apparent scattering cross sectionz of the sphere to its geometrical cross section (as noted by an observer at the radar measuring the back-scattered wave) is proportional to (a/A) 4. This fourth-power dependence is an example of the well-known Rayleigh scattering law, used by Lord Rayleigh to explain the blue color of the sky and in fact applying to any scattering object with dimensions sufficiently small in terms of wavelength.

As a/A approaches unity, the situation becomes far more complex.

The scattered wave no longer behaves like the radiation from a simple dipole, but rather from a group of electric and magnetic dipoles, quad- ruples, and more complicated charge and current distributions within the sphere. The ratio of apparent cross section to geometric cross section increases ‘at a rate less than (a/A) 4, finally reaches a maximum, and oscil- lates with a substantially constant period and with slowly decreasing amplitude of oscillation about the value unity, which it approaches in the limit as a/A approaches infinity (see Fig. 6”1). This limiting case, for which the back-scattering cross section is precisely equal to raz, is the

LIn particular,see Sees.1’5, 2.5, and 4.24.

2A precisedefinitionof scatteringcrosszcction is deferredto Sec. 2.2.

8 ELEMENTS OF THE PROBLEM [SEC.1.2 value predicted by geometrical optics, which can give only results inde- pendentof wavelength. Because of itssimplicity andits resemblance to sections of practical radar targets the sphere has received a great deal of attention.

As an example offering a sharp contrast to the sphere, we consider a flat plate, for convenience assumed to be made of metal and having a fairly simple shape. Whereax the scattering properties of the sphere are independent of orientation of the sphere because of its perfect symmetry, the cross section of the plate depends upon its orientation. If the plate is very small in terms of wavelength, the cross section again follows the (a/~)4 Rayleigh law, where u is now some suitably determined average dimension of the plate. As the plate becomes large, the scattering cross section continues to be a function of wavelength (instead of becoming independent of wavelength as does the sphere), and it also becomes criti- cally dependent upon orientation. For a wave normally incident upon

Reflected wave~

\ F“

pattern o

Normal tosurface e

a

\

ln;:~ent Plane

reflector

Fm. 1.3,—Scattering from a flat metal plate for which aIk >>1 (not to scale).

the plate the ratio of back-scattering cross section to geometrical area A is just 47rA/X2. This expression will be recognized as the gain of an antenna of area A having a uniform field strength over its mouth; thus a large plate viewed at normal inci- dence behaves like an antenna excited by the currents across its mouth that are actually excited on the plate by the incident wave.’ As the orientation is changed, the cross section fluctuates rapidly as a result of the multiple lobes of the diffrac- tion pattern of the plate, as shown in Fig. 1“3. The width of the main lobe is roughly A/a radians. Most of the energy of the incident wave is reflected specularly in the main lobe in a direction such that the angles of incidence and reflection are equal, and the remaining small amount , of energy is distributed in the side lobes throughout the remaining solid angle. Similar diffraction phenomena are inherent in every optical instrument and must be considered in detail in design of the instrument.

The preceding discussion sketches qualitatively some of the broad fea- tures of the behavior of microwaves in relation to their environment, with particular emphasis on the usefulness of concepts and techniques of optics.

From here on we consider details of individual sections of the propagation field, borrowing these optical methods freely when it appears advantageous to doSO.

1This statementia in~”ndedto convey a qualitative idea, not to state an exact equivalence(aceSec. 6,2).

SEC.1.3] THE EFFECTS OF VARIABLE GR.4DIENTS TROPOSPHERIC REFRACTION

BY JOHN E. FREEHAFER

1“3. The Effects of Variable Gradients of Refractive Index.-Although waves shorter than about 6 m are seldom and those in the centimeter range never reflected bytheionospherc jthere were many observations prior to the opening of hostilities in 1939 to show that the horizon does not always limit the range of ultrahigh-frequency radio facilities. For instance, in 1932 Trcvor and Carter reported reception of 69-cm signals over a path from New York to Rocky Point, Long Island, the path being 1.2 times radio-optical. In 1934 Hershberger reported transmission of 75-cm waves over a range of 87 miles, \vhich was five times the sum of the horizon dis- tances from the antennas. Durin~ the war, many striking observations of “anomalous” ranges were made. A radar beacon on a frequency of approximate y 200 M c/see at Bathurst, Gambia, West Africa, was seen on several occasions by aircraft flying below 6000 ft at ranges exceeding 500 miles. Echoes from N-ova Scotia were seen at Provincetown, Mass., by a 10-cm radar at a height of 150ft above sea level. The range in this case was limited to 280 miles by the length of the sweep of the indicator, Per- haps the longest terrestrial ranges observed were obtained on a 200-Mc~sec radar at Bombay, India, which received echoes from points in Arabia 1700 miles away. On the other hand propagation conditions can give rise to strictly limited ranges. For instance, there were occasions when for periods of several hours, centimeter radars in good operating condition on Fisher’s Island, New York, were unable to see Block Island 22 miles away although it was optically visible.

As the frequencies involved rule out the ionosphere, attention is directed to refraction by the atmosphere, and in fact, because of the limited vertical extent of the antenna patterns involved in many of the observations, to the lower portion of the atmosphere, the troposphere. It is readily shown, a-s we shall see in Sec. 1“5, that when meteorological conditions are such that a layer of atmosphere exists in which the index of refraction decreases rapidly with height, radiation of sufficiently short wavelength may be trapped in the layer and guided around the curved surface of the earth by an action analogous to that of a waveguide. Conditions are especially favorable to the formation of such trapping layers or ducts in the first few hundred feet of air above the surface of the sea when the temperature of the air is greater than that of the water or aloft at the boundary between air masses of contrasting temperatures.

Experimental evidence of the effect of a surface duct on radio trans- mission is shown in Fig. 1.4. (The experiment is described in detail in Sec. 4.20.) A 3-watt 10-cm c-w transmitter was located at a height of 25 ft above the surface of the sea. The receiving antenna, a paraboloid with a beamwidth at 10 cm of about 25°, was carried in the nose of an