zenneck 2 sommerfield

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621.396.11:538.566 Paper No. 873RADIO SECTION

A RELATION BETWEEN THE SOMMERFELD THEORY OF RADIO PROPAGATIONOVER A FLAT EARTH AND THE THEORY OF DIFFRACTION AT A STRAIGHTEDGEBy Prof. H. G. BOOKER, M.A., Ph.D., Associate Member, and P. C. CLEMMOW, M.A.

{The paper was first received 12//i April, and in revised form 4th July, 1949.)Sommerfeld,1 and, since then, other ways of deriving th e theoryhave been given,2.3> 4> s each bringing out some special featureof the problem. It is the object of this paper to present yetanother point of view, which promises to be of practical impor-tance for dealing with situations where the path of propagat ionlies partly over land and partly over sea.When first approaching the problem of radiation from a pointsource T (Fig. 1) in th e presence of a flat, imperfectly reflecting

SUMMARYA new way of visualizing the Sommerfeld theory of propagationover a flat, imperfectly reflecting earth is presente d. The Sommerfeldtheory arises because the ray theory of propagation from a source inthe presence of a flat, imperfectly reflecting earth is only an approxi-mation. The ray theory involves the assumption that the Fresnelreflection coefficient of the earth does not vary rapidly with angle ofincidence, and this assumption is not satisfied for glancing incidenceof vertically polarized waves on the earth's surface at broadcastingwavelengths. The main object of the new presentation is to facilitatethe solution of problems involving propagation near the surface ofthe earth partly over land and partly over sea, but these applicationsare not included in the paper.

It is convenient to think of a two-dimensional problem in whichthe transmitter is a line source parallel to the earth's surface, havinga vertical polar diagram of circular shape. Such a source may beFourier analysed into plane waves whose directions are distributedin a vertical plane of propagation perpendicular to the line source;the amplitudes of all the plane w aves are the same and they are in thesame phase at the source. When these waves are reflected from theearth they produce an angular spectrum of reflected waves, the ampli-tudes and phases of which are determined by the Fresnel reflectioncoefficient. This angu lar spectrum could be thought of as arising, inthe absence of the earth, from an aperture distribution on the verticalplane through the primary line source. The aperture distribution thatproduces the angular spectrum of reflected waves in this way is theexact image of the primary source in the imperfectly reflecting earth,an d is given by the Fourier transform of the Fresnel reflection coeffi-cient. For a perfectly conducting earth this aperture distributionreduces to a line source identical with the primary source and locatedat the optical image line. The correction required to this when theearth is not perfectly conducting is mainly the following. An aperturedistribution extending indefinitely downwards from the image linemust be introduced, and this consists essentially of the aperturedistribution produced by diffraction of the Zenneck wave under ascreen extending from the image line upwards. The field producedby the primary source in the presence of the imperfectly reflectingearth is thus thefield hat would be produced with an almost perfectlyconducting earth, together with the field arising from diffraction ofthe Zenneck wave under the image line.If diffraction of the Zenneck wave under the image line is calculatedby the edge-wave approximation we merely arrive at the ray theoryof reflection from the earth of radiation from the primary source: theedge wave from the image line, together with the wave from an imagein an almost perfectly conducting earth, makes up the wave from theFresnel image for the imperfectly reflecting earth. But, at broad-casting wavelengths, points close to the earth are often too close tothe shadow edge, formed by diffraction of the Zenneck wave underthe image line, for application of the edge-wave approximation. Wethen have to apply the full theory of edge-diffraction based on theCornu spiral, and this gives the Sommerfeld theory.

(1) INTRODUCTIONThe theory of radiation from a radio transmitter over a flat,imperfectly reflecting earth was worked ou t originally by

Written contributions cm papers published without being read at meetings areinvited for consideration with a view to publication.Prof. Booker is at Cornell University, New York, and Mr. Clemmow is at ImperialCollege, London University.

Fig. 1.Direct and reflected rays.earth, we are inclined to think that the field strength at a point Pmay be calculated by combining the results for a direct ray TPand for a ray TRP reflected from the earth, allowing for theFresnel reflection coefficient of the earth at the appropriate angleof incidence. This is largely true but is misleading in theimportant practical case where T and P are close to the surfaceof the earth and the waves are vertically polarized. If T and Pare actually on the surface of the earth, the correspondingFresnel reflection coefficient is 1, which means that the directand reflected waves neutralize each other and there is no field.This result is true only in the sense that, at a sufficiently largerange r, there is no field to the order of I jr. But a contributiondoes exist of the order of 1/r2 which is of practical importance.This contribution arises because the Fresnel reflection coefficientreferred to above applies only to incident plane waves, and inusing it for an incident spherical wave, radiated from T, anapproximation is involved. Thus a correction has to be appliedto the ray theory, and for communication between points closeto the earth's surface the correction is frequently of paramountimportance.

The difficulty just described is aggravated in practice bythe fact that the (complex) permittivity of the earth is usuallylarge compared with that of the atmosphere. As a result theFresnel reflection coefficient for vertically polarized waves ispractically + 1 for all directions of incidence on the earth,except nearly glancing directions. As glancing incidence isapproached the reflection coefficient swings rapidly from + 1to 1. Now the approximation upon which the ray theory isbased involves the assumption that the reflection coefficientvaries sufficiently slowly with angle of incidence in the neigh-bourhood of the particular angle of incidence concerned. Thisassumption, therefore, breaks down for nearly glancing incidenceupon the earth of vertically polarized waves, and this is justwhat is commonly involved in communication between twopoints close to the earth. As a result, the variation of fieldstrength with range for vertically polarized waves close to the

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BOOKER AND CLEMMOW: SOMMERFELD THEORY AND STRAIGHT-EDGE DIFFRACTION THEORY 19earth's surface often shows no sign of obeying the ray theoryfor some considerable distance from the transmitter, and evenat greater ranges does so only in the sense that it vanishes tothe order of 1//-, leaving correction terms of the order of \\Apredominant.The fact that propagation near the surface of an imperfectlyreflecting earth involves important corrections to the ray theoryimplies tha t the prob lem is one of diffraction. As a diffractionproblem, however, it is of an unusual type. It is therefore alittle surprising to find that it is possible to reduce the problemto the well-known one of diffraction "at a straight edge. Thatthis is possible and, indeed, convenient is the main point of thepaper.For the most part we shall discuss a two-dimensional insteadof a three-dimensional problem, replacing a point source by aline source parallel to the earth's surface. This simplifies thetreatment without restricting practical application, because theratio of field strength in the presence of the earth to that in theabsence of the earth is the same for both p roblems, except withinthe first wavelength from the source. We shall therefore dealwith the two-dimensional problem, express field strength as aratio to what would be obtained in the absence of the earth, andthen use the result in three-dimensional applications.If T in Fig. 1 now represents a line source parallel to theearth's surface, T' is the image line. Then T' in fact forms thediffracting edge involved in the theory. Imagine that the earthis removed and that a plane vertical screen is introduced extend-ing from T' upwards. Then the theory of radiation from T inthe presence of the earth is intimately bo und u p with diffractionunder the edge T' of a certain plane wave.The plane wave to be diffracted under T' is in fact the Zenneckwave. It will be remembered that, prior to Sommerfeid'soriginal paper, Zenneck6 had published a theory of propagationover the ear th's surface. This theory dealt essentially with avertically polarized plane wave incident upon the earth at theBrewster angle. Fo r this angle there is no reflected wave, butonly a wave transmitted into the earth. Fo r the large (complex)dielectric constant usually possessed by the earth, the waveincident in the atmosphere at the Brewster angle is travellingalmost horizontally, while the wave refracted into the earth istravelling almos t vertically downward s. The Zenneck theorythus presents a wave travelling over the surface of the earth,with energy being abstracted sideways from it by the earth, inthe same way as series resistance abstracts energy from a wavetravelling along a transmission line. We shall refer to the waveabove the earth in the Zenneck theory as the Zenneck wave,which is thus simply a wave incident upon the earth at theBrewster angle. It was no do ubt though t at the time of Zenneck'spaper that the Zenneck wave represented, to a useful extent,propagation over the earth's surface at some distance from atransmitter. That this is hardly true was one of the points ofSomm erfeid's 1909 paper. The Zenneck wave represents radia-tion from a transmitter so different from those used in practicethat simple use of the wave is seriously misleading.

It is a feature of the presentation of the Sommerfeld theorygiven in this paper that it focuses attention on the extent towhich the Zenneck wave is actually involved in propagation overa flat earth. In calculating the field to the right of TT', inFig. 2, we remove the earth and insert the vertical screen withits lower edge at T' as previously described, and then allow theZenneck wave, with appropriate amplitude, to be incident onthe screen from the left. By diffraction unde r T' the Zenneckwave makes an important contribution to the required fieldstrength, the remaining contribution being essentially that fora perfectly conducting earth . Fo r an imperfectly reflecting earththe Zenneck wave thus plays an impo rtant role. But the wave

Screen--Primary source

Earth's surfaceT AImage-line

Fig. 2.Diffraction of Zenneck wave under image line.is never "s een " direct, and it is only by diffraction under theimage line that it makes its contribution.The Zenneck wave is diffracted at a simple straight edge inFig. 2, because a two-dimensional problem is being considered.To describe in the same way the three-dimensional p roblem, thescreen in Fig. 3(a) (now viewed normally) would be replaced bythat in Fig. 3(6). The Zenneck wave on the far side of the

Fig . 3.Diffracting screens for Zenneck wave.(a) For two-dimensional problem.(6) For three-dimensional problem.

screen would be seen by diffraction throu gh the slot in Fig . 3(A),extending from the image point downw ards.If it is true tha t the field strength d ue to a line source parallelto an imperfectly reflecting earth may be calculated by addingto the field strength for an almost perfectly conducting earththa t due to the Zenneck wave diffracted under the image line,as in Fig. 2, it may be wondered how, in certain circumstances,this is equivalent to the ray theory depicted in Fig. 1. Tounderstand this point, it should be noted that the region in whichwe have to calculate the field strength in Fig. 2 is to the rightof TT' and above the earth. This is wholly within the shadowof the Zenneck wave cast by the screen extending upwardsfrom T' . Well within the shadow cast by a screen, the diffractedwave may be represented as a wave radiating from the diffractingedgethe so-called edge wave. When the edge wave from T ',due to diffraction of the Zenneck wave, is added to the wavefrom T' produced by the image of T in an almost perfectlyconducting earth, the combined wave from T' is simply theordinary Fresnel image of T in the imperfectly reflecting earth.Thus the ray theory of Fig. 1 is reproduced when diffractionof the Zenneck wave under T' in Fig. 2 is represented by theedge-wave approximation. However, when the Zenneck waveis travelling almost horizontally, as is usually the case, the edgeof the shadow (shadow edge) cast by the screen in Fig. 2 is notfar below the position of the earth 's surface. Consequently,when evaluating the field near the earth's surface, we are oftentoo close to the shadow edge to use the edge-wave approxima-tion, and diffraction of the Zenneck wave under the image linemust then be evaluated by means of the Fresnel integral (Cornu



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20 BOOKER AND CLEMMOW: A RELATION BETWEEN THE SOMMERFELD THEORY OF RADIO PROPAGATIONspiral), and this gives precisely the usual Sommerfeld formulafor propagation over a flat earth.The approach to the Sommerfeld theory outlined above ismade merely by piecing together some quite well-known resultsin Fourier analysis. Use of Fourier analysis in the problemwas, of course, emphasized originally by Weyl.2 But in recenttimes facility with Fourier analysis has become more widespread,particularly in connection with directional aerials, for which thepolar diagram and the aperture distribution are Fourier trans-forms. As a result, the derivation of the Sommerfeld theorygiven in this paper is now much more obvious than it wouldhave been ten years ago.In Section 2 the features of Fourier analysis required for thediscussion are briefly stated . Fro m these results the exact imageof a source, in an imperfectly reflecting earth, is derived inSection 3, and is related to the ray theory of reflection from theearth in Section 4. In Section 5 the special case when thetransmitter and receiver are on the earth's surface is studied,and the usual Sommerfeld formula is derived. The advantagesof regarding the Sommerfeld extension of the theory for aperfectly conducting earth as arising from diffraction of theZenneck wave under the image line are illustrated in Section 8,by the difficult problem of propagation along a path partly overland and partly over sea.(2) SOME RESULTS CONCERNING APERTURE DISTRIBU-TION AND ANGULAR SPECTRUM

Suppose that, in a system of Cartesian co-ordinates, the regionx > 0 is filled with a homogeneous medium h aving prop agationconstant k and characteristic admittance T\? For a plane wavein the medium, k is the increase of phase lag per unit distancein the direction of propagation, and r) is the ratio of magneticto electric intensity. Suppose that , in the plane x = 0 anelectromagnetic field is maintained having compon ents along theplane which vary w ith time, an d w ith po sition in the plane, in aprescribed manner. The field maintained in the plane x = 0is propagated into the region x > 0, and we may think of theplane x = 0 as the aperture plane of an aerial system radiatinginto the region x > 0.

The following simplifications will be adopted:(i) The electromagnetic field will be supposed to vary har-monically in time with a prescribed frequency corresponding toa wavelength A, and the c6rresponding complex oscillation-function will be suppressed.(ii) The field will be supposed two-dimensional and will betaken as independent of the z-co-ordinate.(iii) The magnetic field will be taken parallel to the z-axis andthe electric field parallel to the xy-p\ane.As explained in another paper,8 the field at any point (x, y)in front of the aperture plane x = 0 may be expressed as anangular spectrum of plane waves:{x, y) - (I/A) f>O S)(- S, C, 0) exp [ - jk(Cx + Sy)]dS/CJ . . . . (1)

P ( S ) = \ S X O , y ) w p < j k S y ) d y . . . ( 4 )

H(x, y) - , 0, 1) exp [ - jk(Cx + Sy)]dS/C. . . . (2)where k(C, S, 0) is the propagation vector of an individual waveof the spectrum.The aperture distribution is

#,(0 , v) = (I/A) fp(S) exp ( - jkSy)dS . . (3)and in terms of this the angular spectrum is given by the Fouriertransform:

An example of an aperture distribution in which we shall beinterested is an infinite straight slot cut in an infinite, perfectlyconducting, metal sheet, with an alternating voltage of ampli-tude V applied across the slot. If the sheet is in the planex = 0 and the slot along the z-axis, wo have

y (5 )where S(y) is the unit impulse-function. The correspondingangular spectrum is (6)At a distance r from the slot, appreciably greater than A, thewave radiated by the slot is

Z(x,y) = V[~ (ylr), (x/r), 0] exp [-j(kr - 7r/4)]IV(rX) . (7)\ . . (8)

Another example of an ap erture distribution in which we shallbe interested arises in connection with diffraction of a plane waveby a semi-infinite, "blac k" screen with a straight edge. Supposethat the screen occupies the portion of the plane x = 0 for whichy > 0, and that the incident wave is given by& = A(S0, C o , 0) exp [ - jk(Cox - Soy)] . . (9)H = >qA(Q, 0, X) e x p [ - jk(Cox - Soy)] . . (10)

so that the aperture distribution in the plane x = 0 may betaken as #,.(0, y) = AC Q ex p


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OVER A FLAT EARTH AND THE THEORY OF DIFFRACTION AT A STRAIGHT EDGE 21(3) THE EXACT IMAGE OF A SOURCE IN AN IMPERFECTLYREFLECTING PLANE EARTHSuppose that the atmosphere and the earth may each beregarded as homogeneous media separated by a plane interfacey 0, as indicated in Fig. 4. Let k, 77 be the propagation

Fig. 4.Reflection of a vertically polarized plane wave by a plane earth-constant and characteristic admittance of the atmosphere andk', t] ' those of the earth. A vertically-polarized plane waveincident upon the earth at an angle of elevation 6 (S sin d,C cos 6) is

= (S, C, 0) exp [-jk(Cx - Sy)] . . . (17)H =TJ(O, 0,1) exp [-jk(Cx-Sy)] . . . (18)

This gives rise to a reflected wave which may be writtenW = P(S) ( - S, C, 0) exp [~jk{Cx + Sy)]H=r)p(S)(P,O,l)exp[-jk(Cx

(19)(20)

where p(S) is the Fresnel reflection coefficient of the earth anddepends on the angle of elevation of the incident wave. Thereis also a wave transmitted into the earth, which in practice maybe regarded as travelling almost vertically downwards becausethe complex dielectric constant of the earth is usually quite large.The admittance looking downwards across the earth's surfaceis therefore t\'. That looking downwards in the incident waveis obtained by dividing the horizontal m agneticfieldstrength (18)by the horizontal electric field strength (17) and is rj/S. Hencethe Fresnel reflection coefficient of the earth is?(21)V

provided that the transmitted wave is travelling almost verticallydownwards. It will be convenient to write (21) as(22)

where (23)The reflection coefficient (22) vanishes when S = So, so thatd0 (sin 9 Q ~ SQ) corresponds to what is known as the Brewsterangle. With r/ small compared with rf, as it usually is inpractice, the Brewster angle of elevation is small. Strictly, theBrewster angle of elevation is the angle whose tangent [not sineas given by (23)] is 77/7/; but for small angles the difference isunimportant. We shall begin by considering an earth whosereflection coefficient is any function of the sine S of angle ofelevation, and subsequently pay special attention to the reflectioncoefficient (22), which covers most practical requirements inconnection with the earth.

As described in the Introduction, it will be convenient andadequate to consider two-dimensional radiation from a line

source parallel to the earth's surface. For a vertically polarizedhorizontal line-source we may use an infinitely long horizontalslot cut in an infinite vertical perfectly-conducting sheet, withan alternating voltage of amplitude V across the slot. Thissituation is depicted in Fig. 5, where the surface of the earth is

TzX

Fig. 5.Two-dimensional radiation from a horizontal slot parallel toa plane earth.the plane y = 0, and the slot T lies along the line x = 0, v ~ //.The image line of the slot in the earth is T', its location beingx = 0, y = h. We need consider radiation from the slotonly in the region x > 0.If the slot were along the z-axis, the aperture distribution inthe plane x = 0 would be (5) and the corresponding angularspectrum (6). In fact the slot is shifted along the aperture planeto x = 0, y = h, so that the aperture distribution is

By a well-known shift rule,8 it follows from (6) that the angularspectrum corresponding to the aperture distribution (24) isP(S) = Kexp (jkSh) (25)

For the angular spectrum into which the given primary sourcemay be analysed, (25) gives the amplitude and phase at theorigin of the plane wave whose direction of propagation is(C , S, 0) . It follows that the amplitude and phase of the planewave whose direction of propagation is (C, S, 0) are given by

P ( - S ) = V e x p ( ~ j k S h ) . . . . ( 2 6 )The corresponding wave reflected from the earth's surface inthe direction (C, S, 0) has amplitude and phase at the origingiven by P(- S)p(S) = Vp(S ) exp (~ jkSh) . . . (27)where p(S) is the Fresnel reflection coefficient of the earth. Itfollows that the angular spectrum of plane waves reflected fromthe earth as a result of the primary source at x = 0, v = // is (27).In order to discover what the earth re-radiates in the presenceof the primary source, the field corresponding to the angularspectrum (27) of reflected waves has to be evaluated.To discuss the angular spectrum (27) of reflected waves it isconvenient to shift the origin in Fig. 5 down through a distance //,so that it is on the image line T', as shown in Fig. 6. This wedo by means of the previously mentioned shift rule. The effectis to remove the factor exp ( jkSli) in (27), leaving the angularspectrum of reflected waves as

Vp(S) (28)This means that, in the absence of the earth, a two-dimensionalangular spectrum of plane waves, in which the wave travellingat an angle of elevation 6 (sin 6 = S) has amplitude and phasegiven by (28) at the image line T', reproduces the re-radiationof the earth in the presence of a horizontal slot of voltage Vabove the earth.As a simple example of (28), we may consider a perfectly



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22 BOOKER AND CLEMMOW: A RELATION BETWEEN THE SOMMERFELD THEORY OF RADIO PROPAGATIONconducting earth, for which p(S) is unity. The angular spec-trum (28) of reflected waves is then the same as the angularspectrum (6), so that the image consists of the aperture distribu-tion (5) with the origin on the image line. We thus have thewell-known result that the image of a slot above a perfectlyconducting earth is an identical slot along the image line.Now, for an imperfectly reflecting earth, the use of imagesnormally leads to the ray theory of reflection from the earth'ssurface, illustrated in Fig. 1. As explained in the Introduc tion,this image technique involves an approximation which is liableto break down when transmitter and receiver are sufficientlyclose to the earth's surface, and this is just the situation thatfrequently occurs in practice. From what has been said in thisSection, however, it is easy to arrive at an exact method wherebythe effect of an imperfectly reflecting earth may be representedby means of an image source. We shall now state what is theexact image for a line source parallel to an imperfectly reflectingearth, and then go on, in the next Section, to consider how thisexact image is related to the approximate image involved in theray theory, and when the approximate image fails adequately torepresent the exact image.

The exact image of a horizontal primary line-source in a flatimperfectly reflecting earth involves an aperture distribution overthe vertical plane through the image line and the p rimary source.This aperture distribution is calculated from the angular spectrumof reflected waves by substituting for P(S) in (3) the expres-sion (28). The aperture distribution forming the image sourceis thus the Fourier transform of the angular spectrum (28) ofreflected waves. An image consisting of this apertu re distribu-tion produces the angular spectrum (28j and therefore representsaccurately the re-radiation of the earth in the presence of theprimary source. Thus a horizontal slot of unit voltage placedabove a plane imperfectly reflecting earth has an exact imagein the form of a vertical aperture-distribution, given by theFou rier transform of the ear th's Fresnel reflection coefficient.A more general statement may be made as follows: If anyvertical aperture-distribution corresponding to a polar diagramP(iS), where S is the sine of the angle of elevation, is placed abovea plane earth whose Fresnel reflection coefficient is p(S), thenre-radiation by the earth is exactly represented by an image inthe form of a vertical aperture-distribution, given by the F ouriertransform of P(-S)p(S) (29)This more general result follows from consideration of the left-hand side of equation (27).(4) EVALUATION OF THE FIELD REFLECTED BY THE EARTHConsidering a transmitter T in the form of a horizontal slotof voltage V above a flat imperfectly reflecting earth, and takingthe r-axis along the image line as indicated in Fig. 6, let usexamine the field associated with the angular spectrum (28) ofreflected waves. Firs t we will consider the appro ximate evalua-tion of this field involved in the ray theory of reflection froman imperfectly reflecting earth. The approxim ation consistssimply in interpreting the angular spectrum (28) of reflectedwaves as a polar diagram.8 This means that we regard thewave re-radiated by the earth as a cylindrical wave emanatingfrom the image line and having a vertical polar diagram givenby the earth's Fresnel reflection coefficient. Com bination ofthis cylindrical wave from the image line T' with the cylindricalwave from the primary source T is equivalent to combinationof a direct ray from the primary source with a ray reflectedfrom the earth, allowing for the Fresnel reflection coefficient ofthe earth a t the app ropriate angle of elevation, as in Fig. 1.The assumption that the angular spectrum of reflected waves

Fig. 6.Diagram to illustrate the region (shaded) in which the raytheory is inapplicable, for an elevated transmitter.may be regarded simply as a polar diagram is not, however,satisfactory in all practical applications.8 Let us examine thisimportant point in detail by studying the nature of the angularspectrum of reflected waves, using the approximate expres-sion (22) for the Fresnel reflection coefficient of the earth . Sub-stituting from (22) into (28) we see that the angular spectrumof reflected waves becomes

V(S - S0)KS + So) (30)and this we may split into two angular spectra thus:

V-V2S0KS (3DNow, according to (5) and (6), the first of these angular spectracorresponds to a slot of voltage V along the z-axis in Fig. 6,i.e. to the image of the primary source in a perfectly conductingearth. The second term in (31) is therefore an angular spectrumwhich gives the correction to the field for a perfectly conductingearth, required for an imperfectly reflecting earth whose Fresnelreflection coefficient m ay be taken as (22). If we now concen-trate on the correction required to the field for a perfectly con-ducting earth, the angular spectrum that we have to study is

-2VS0l(S + S0) (32)This expression gives the amplitude and phase at the image lineof a plane wave whose angle of elevation is 6 (S = sin 0) ,#o ($o = s u l ^o) being t n e earth's Brewster angle of elevation.We immediately recognize that the angular spectrum (32) isof the same type as (12). It follows that the angular spectrum (32)corresponds to diffraction under the image line T', in Fig. 6,of a plane wave of the type represented by (9) and (10). Accord-ing to (23) this wave is travelling in a direction correspondingto the Brewster angle of elevation for the earth, and is thereforethe Zenneck wave mentioned in the Introdu ction. The ampli-tude and phase of the wave at the image line, obtained by com-paring (32) with (12), are represented by the modulus andargument of the complex amplitude

A = - 2jkVS0IC 0 (33)Substituting this value of A into (9) and (10), we see that thefield of the Zenneck wave before diffraction is%> = _ 2jkV(S 0/C 0) (S o, C o , 0) exp [ - y*(C ox - Soy)] (34)H = - ijkrjViSolCQXO, 0, 1) exp [ - jk(C ox - SQy) ] . (35)the z-axis being along the image line. Thus the correction



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OVER A FLAT EARTH AND THE THEORY OF DIFFRACTION AT A STRAIGHT EDGE 2J-required to the field for a perfectly conducting earth is the fieldproduced by diffraction under the image line of the Zenneckwave (34) and (35).

We are now in a position to see clearly why the ray theoryof reflection from the earth's surface has its limitations, andwhat must be done to remove them. The complete image of ahorizontal line-source in a flat imperfectly reflecting earthconsists of the image appropriate to a perfectly conductingearth, together with an aperture distribution extending from theimage line vertically downwards and given by (11), with thevalue (33) for A. The exact image of a source in an imperfectlyreflecting earth is thus infinitely wide, vertically. Great care istherefore necessary in interpreting the angular spectrum as apolar diagram,8 and it is precisely this approximation that ismade in the ray theory. No difficulty occurs for a perfectlyconducting earth, because then the characteristic admittance ofthe earth is infinite, making the sine of the Brewster angle, So,zero by (23), and this makes the amplitude of the Zenneckwave (34) and (35) zero. For an imperfectly reflecting earth,however, the amplitude of the Zenneck wave to be diffractedunder the image line is not zero, and the image involves anaperture distribution, infinitely wide, extending indefinitelydownwards from the image line.

Fig. 6 illustrates diffraction of the Zenneck wave (34) and (35)under the image line. The associated shadow edge slopes down-wards from the image line at the Brewster angle, and the wholeof the region above the earth's surface, where the diffracted fieldmust be calculated, is within the shadow. It is clear that inmuch of this region the edge-wave approximation to diffractionof the Zenneck wave under the image line may be used. Butthe edge-wave approximation is one in which the angularspectrum (32;, or (12) with the value (33) for A, is merely inter-preted as a polar diagram. This is simply equivalent to inter-preting (31), and therefore (30), as a polar diagram (since theangular spectrum V of reflected waves for a perfectly conductingearth may always be interpreted as a polar diagram). Thususe of the edge-wave approximation for diffraction of the Zenneckwave (34) and (35) under the image line is merely equivalent tointerpretation of the angular spectrum of reflected waves as apolar diagram, and this, as we saw at the beginning of thisSection, is what leads to the ray theory. It follows that theray theory of reflection from the earth is unsatisfactory underthose conditions in which the edge-wave approximation is notan adequate description of diffraction of the Zenneck wave (34)and (35) under the image line.

Now, we know the conditions under which the edge-waveapproximation may be used.8 We draw, in Fig. 6, a parabolawith focus on the image line, with axis along the shadow edgecast by diffraction of the Zenneck wave under the image line,and with semi-latus-rectum A/277. It is the region inside thisparabola in which the edge-wave approximation is unsatisfactory.If, therefore, the parabola extends above the surface of the earth,there is a region above the surface and within the parabolawhere the edge-wave approximation, and consequently the raytheory, is unsatisfactory. In such a region, shown shaded inFig. 6, diffraction of the Zenneck wave under the image linemust be calculated using the formulae (14) and (15) involvingthe Fresnel integral (13), with the value (33) for A. Thus,radiation from a horizontal slot of voltage V above a flat imper-fectly reflecting earth is obtained by adding to the field for aperfectly conducting earth the field

5 - - V(2)jkV(S0/C0)F[(S +( - S, C, 0) exp { - j[k(CQx - Soy) - TT/4]} . (36)

H = - V(2)jkr,V(SQIC0)F[(S + S0W(.7Tr/X)](0, 0, 1) exp {- j[k(Cox - Soy) - TT/4]} . (37)

where the axes are as indicated in Fig. 6, and the Fresnelintegral F{u) is given by (13). But above the parabola drawnin Fig. 6 the argument of the Fresnel integral in (36) and (37)becomes large, and the field then becomes substantially identicalwith that given by the ray theory.

It is the necessity, in the shaded region of Fig. 6, of calcu-lating the field produced by diffraction of the Zenneck waveunder the image line by using the complete expressions (36)and (37) involving the Fresnel integral, instead of by means ofthe edge-wave approximation, that constitutes, from the pointof view adopted in this paper, the Sommerfeld theory of propa-gation over a flat, imperfectly reflecting earth, as distinct fromthe ray theory.(5) TRANSMITTER AND RECEIVER AT THE EARTH'SSURFACELet us examine the implications of (36) and (37) for the special

case treated originally by Sommerfeld,1 namely that for whichtransmitter and receiver are at the earth 's surface. Both theprimary line-source and the image line then coincide at theearth's surface; both lie along the z-axis at the earth's surface,and Fig. 6 is replaced by Fig. 7. Fo r a receiver close to the

Fig. 7.Diagram to illustrate the region (shaded) in which the raytheory is inapplicable, when the transmitter height is zero.earth's surface the ray theory is inapplicable for all ranges upto a certain characteristic range rQ, at which the parabola (focusat 0, axis along shadow edge formed by diffraction of the Zenneckwave under 0, semi-latus-rectum A/(2TT) intersects the earth'ssurface. Beyond the range r0 the field begins to resemble thatappropriate to the ray theory in a way that we shall investigate.

With transmitter and receiver at the earth's surface, we putx = r, y - 0 (38)

(39)in (36) and (37). The electric field (36) is then vertical, andits value is$ = - V(2)jkV(S0IC0)F[S0V(.rrr/X)] exp [ - j(/


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24 BOOKER AND CLEMMOW: A RELATION BETWEEN THE SOMMERFELD THEORY OF RADIO PROPAGATIONvertical electric field strength near the earth's surface, at range r,due to diffraction of the Zenneck wave under 0, is thereforer = - V(2)JkVS 0F[S 0VMX)]

exp[;S 2(7rr/A )]exp[- X*r -7r/4)] (42)To this must be added thefieldapprop riate to a slot of voltage V,at 0, in the presence of a perfectly reflecting ear th. This isdouble the field in the absence of the earth, and is therefore,from (7), , - 2 K e x p [ - ^ r - 7 r / 4 ) ] / V ( r A ) (43)Adding (42) and (43) we see that the complete vertical electricfield near the earth's surface at range / is

(44)'6 = {2Kexp [ - j{kr -{1 - jrrS0V{2rlX)F[S0V(7TrlX)) exp [JS^nr/X)]}Dividing (44) by (43), we obtain the ratio of the vertical electricfield, at range r, near the surface of an imperfectly reflectingearth, to that which would exist for a perfectly conducting earth,namely"/'a8 = l-yTr5oV(2r/A)F[1SoV(w/A)]exp[;S2(7rr/A)] (45)

In this form there is no need to retain the specialization of aline source along the earth's surface. By an application of themethod of steepest descent9'10 we may extend the validityof (45) to almost any practical transmitter close to the earth'ssurface. This means that, if we substitute for cf in (45) thefield at range r (appreciably greater than A) near a perfectlyreflecting earth for the transmitter in which we are interested,then % will be the field for the same transmitter at the samerange near the surface of an imperfectly reflecting earth, thesine of whose (small) Brewster angle of elevation is SQ> givenby (23).The parabola in Fig. 7 is the curve on which the argumentof the Fresnel integral in (45) is unity, so that the range r0 atwhich the parabola crosses the earth's surface is given by

WCw oM) = 1 (46)Hence r0 = XI(TTS$ (47)

= (X/rrXy'h)2 . . . . ( 4 8 )from (23). Jn terms of r0 we may rewrite (45) asW


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OVER A FLAT EARTH AND THE THEORY OF DIFFRACTION AT A STRAIGHT EDGE 25and for an earth acting mainly as an imperfect conductor wehave

0?7T?)2 = - JOIK0O) (CO < CT/K) . . . (57)Substituting from (56) and (57) into (48) we have for dielectricbehaviour of the earth (co > O/K)

r0 = (A/TT)(IC/ICO) (58)and for conducting behaviour (co < O/K)

ro=-j\ro\ (59)where |ro| = (A/T7)[CT/0


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26 BOOKER AND CLEMM OW : A RELATION BETWEEN THE SOMMERFELD THEORY OF RADIO PROPAGATIONsufficiently accurate for most practical purposes, and it simplifiesthe discussion. But the methods, in fact, apply to the accurateform of the Fresnel reflection coefficient. The accurate formcontains a simple pole of the type (16) which corresponds todiffraction of the Zenneck wave under the image line. Onseparating the pole from the remainder of the Fresnel reflectioncoefficient [cf. equation (31)], we find that the remainder doesnot correspond simply to the image in a perfectly conductingearth, as was the case for (22). The corresponding angularspectrum may, however, be satisfactorily interpreted as a polardiagram, and this is all that is necessary for arriving at a usefulanswer.(e ) Production of curves exhibiting propagation over a flatimperfectly reflecting earth involves tabulation of the Fresnelintegral (13). Fo r a dielectric earth we merely require valuesof the Fresnel integral for positive real values of the argument u,and these are completely given by the Cornu spiral. To dealwith earth of any conductivity, however, we require values ofthe Fresnel integral throughou t the region 0 < arg u < i?r inthe complex plane. Tables of the Fresnel integral in this regionhave been prepared for values of |M| between 0 and 1. No rton 'scurves of propagation over a flat imperfectly reflecting earth18constitute a way of plotting the Fresnel integral convenient forthe problem concerned. In particular his curves for propaga-tion over a flat, purely dielectric earth are simply a re-plot ofthe Cornu spiral, well known in optics.(S) PROPAGATION PATHS PARTLY OVER LAND AND

PARTLY OVER SEAThe foregoing investigation of the Sommerfeld theory wasundertaken with a view to finding a solution of the problemof propagation along a path lying partly over one type of surfaceof the earth and partly over another, land and sea being theobvious practical examples. This apparen tly simple problemhas so far eluded satisfactory solution;19-2 plausible answersthat have been suggested often fail to satisfy even approximatelythe law of reciprocity, which states that the answer must beunaffected if transm itter and receiver are interchanged. It isnot intended to describe in this paper the solution of the problemthat has been obtained, but sufficient will be said in this Sectionto indicate the advantage of the treatment of the Sommerfeldtheory given in the preceding Sections.We suppose, for simplicity, that the land and sea are at thesame level (though there is no difficulty in dealing with a cliffedge if desired). This is indicated in Fig. 8, the coast line C

f

T' j r"Land

C

Sea

Fig. 8.Radiation from a transmitter over land and sea.being straight and normal to the diagram. Let T be a trans-mitter which we suppose is located over the land, but whichmight equally well be over the sea. Let us suppose for sim-plicity that T is a horizontal line-source parallel to the coastline; there is no particular difficulty in converting to a pointsource. T', in Fig . 8, is the image line of T in the surface ofthe earth.The first point to notice about the problem depicted in Fig. 8

is that refraction below the earth's surface at the interfacebetween land and sea is unlikely to be of importance in practice:the refracted waves in both land and sea are travelling almostvertically downw ards and are rapidly attenuated. It is obviouslyreasonable to assume that electrons vibrating near the surfaceof the land are doing so in substantially the same way as ifthe land extended indefinitely in a horizontal direction . Thesame assumption may (with less certainty) be made about thesea. This is equivalent to saying that the wave reflected fromthe earth is a combination of:(a) A wave from the image in the land (as though the landextended indefinitely) diffracted over the coast line.(b) A wave from the image in the sea (as though the seaextended indefinitely) diffracted under the coast line.These two waves, added to the direct wave from the primarysource, should give a useful first approx imation to the fieldexisting above b oth land and sea.If we can assume that ray theory is applicable to reflectionboth from the land and from the sea, application of the abovemethod is straightforward . But the cases of greatest interestare those in which the ray theory is inapplicable and theSommerfeld theory must be used. We need to know, in fact,the exact images in both land and sea, not the approximate oneson which the ray theory is based. These exact images areprecisely what have been calculated in Section 3 . It followsthat the field produced above the surface of the earth in Fig. 8is, to a first approximation, a combination of:(i) The field for a perfectly conducting earth of infinite extent.(ii) The field formed by the Zenneck wave appropriate to theland, diffracted under, the image line T' a nd over the coas tline C.(iii) The field formed by the Zenneck wave appropriate to thesea, diffracted unde r the image line T' and under the coast line C.The amplitudes of the two Zenneck waves are determinedby (33), using the appropriate Brewster angle of elevation ineach case.It is apparen t tha t each Zenneck wave is diffracted by twostraight edges in succession. This involves certain difficultiesinto which we shall not enter.It may be noticed that the method outlined above for handlingpropagation across a coast line is quite different from thatsuggested by Ratcliffe.21 Ratcliffe points out that for a low-level transmitter and receiver above a dielectric earth, equa-tions (51) and (52) imply a phase slip of \n in going from zerorange to a range appreciably greater than r0, while for a con-ducting earth the phase slip is TT in accordance with (61) and (62).He interprets this as a small but significant reduction in phasevelocity, and then goes on to regard coastal phenomena asordinary optical refraction using different phase velocities overland and sea. In particular, for oblique propagation across thecoast line, he interprets the phenomenon of so-called "coastalrefraction"22 '23 as arising from the difference of phase velocityon the two sides of the coast, m aking the assumption that, thephase velocity along the coast must be the same on either side.But, according to the view outlined above, coastal phenomenahave nothing to do with refraction a nd are, in fact, diffractionphenomena. A field approp riate to the land fades over con-tinuously to one appropriate to the sea as the shadow edge,formed by T'C produced (Fig. 8), is crossed. There is no reasonwhy the component of phase velocity parallel to the coast shouldnot change smoothly from one value to another as this shadowedge is crossed, and this seems to destroy the basis of therefraction argum ent. It appears more likely that so-calledcoastal refraction is in reality a diffraction phen ome non , con-fined mainly to a region within A/(2?7) of the coast, and arisingin the manner suggested by Barfield.24



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OVER A FLAT EARTH AND THE THEORY OF DIFFRACTION AT A STRAIGHT EDGE 27(9) REFERENCES(1) SOMMERFELD, A.: Annalen der Physik, 1909, 28 , p. 665.(2) WEYL, H.: ibid., 1919, 60, p. 481.(3) VAN DER POL, B., and NIESSEN, K. F.: ibid., 1930, 6,p. 273.(4) NIESSEN, K. F. : ibid., 1933, 18 , p. 893.(5) VAN DER POL, B .: Physica, 1935, 2, p . 843.(6) ZENNECK, J. A.: Annalen der Physik, 1907, 23 , p. 846.(7) BOOKER, H. G.: Journal I.E.E., 1947, 94, P art III, p . 181.(8) BOOKER, H. G., and CLEMMOW, P. C : See page 11 .(9) JEFFREYS, H. and B. S.: "Methods of Mathematical Physics"(Cambridge University Press, 1946), p. 472.(10) WATSON, G. N.: "Theory of Bessel Functions" (CambridgeUniversity P ress, 1944), p . 235.(11) JAHNKE, E., and EMDE, F.: "Funktionentafeln" (B. G.Teubner, Leipzig, 1938,) p. 35.

(12) SOMMERFELD, A.: Annalen der Physik, 1926, 81, p. 1135.(13) NORTON, K. A.: Nature, 1935, 135, p. 954.(14) BURROWS, C. R.: ibid., 1936, 138, p. 284.(15) BURROWS, C. R.: Proceedings of the Institute of RadioEngineers, 1937, 25 , p. 219.(16) WISE,W.H. : Bell System Technical Journal, 1937,16 , p. 101.(17) ROLF, B .: Proceedings of the Institute of Radio Engineers,1930, 18 , p. 391.(18) NORTON, K. A.: ibid., 1941, 29, p. 623.(19) GRUNBERG, G.: Physical Review, 1943, 63, p. 185.(20) MILLINGTON, G.: Proceedings I.E.E., 1949, 96 , p. 53.(21) RATCLIFFE, J. A.: Proceedings of the Institute of RadioEngineers, 1947, 35, p. 938.(22) ECKERSLEY, T. L.: Radio Review, 1920, 1, p. 421.(23) SMITH-ROSE, R. L.: Nature, 1925, 116, p. 426.(24) BARFIELD, R. H.: ibid., 1925, 116, p. 498.

SOUTH MIDLAND RADIO GROUP: CHAIRMAN'S ADDRESSBy K. R. STURLEY, Ph.D., B.Sc, Member.

"PROBLEMS IN BROADCAST TRANSMISSION AND RECEPTION"(ABSTRACT of Address delivered at BIRMINGHAM, 26th September,1949.)

Before considering the problems encountered in broadcastingit is necessary to have a clear picture of the chain of processes.The transmitting chain involves the studio, microphone, audio-frequency amplifiers and programme-distributing system, therecording apparatus, Post Office lines, transmitter and aerial.The receiving chain consists of the aerial, r.f. amplifier, frequencychanger, i.f. amplifier, detector, a.f. amplifier and loudspeakerAt the present time the weakest links are at the end of eachchain, namely the studio and the loudspeaker.When a programme originates from an enclosed space, thesound received by the microphone is modified by reflectionsfrom the boundaries of that space and from any large objectsin it. In the early days, attempts were made to suppress studioacoustics by heavy draping, but the result was unsatisfactoryfor such studios had a large output at low frequencies due to theselective absorption of the draping. The present-day trend is totry to control studio acoustics so that artistic presentation ofthe programme is improved. Reverberation time has not provedan entirely satisfactory measure of studio performance, and it isbeing supplemented by examination with pulsed tone. Examplesof acoustic effects are easily obtained by recording speech in theopen air (little reverberation), in a studio (more or less controlledreverberation) and in a reverberant room (uncontrolledreverberation).The microphone can introduce all forms of distortion. Theearly carbon types were lacking in bass response and hadresonances in the middle-frequency range; a later development,the Reis7, was a great improvement but had a low signal/noiseratio and tended to emphasize sibilants. The moving-coil andribbon microphones now used in broadcasting represent a con-siderable advance. The ribbon has a good frequency responseand has the advantage, exploited in drama, of an approach to afigure-of-eight directional diagram with very m uch reducedresponse at the sides. The moving-coil microph one is practically

Dr. Sturley is with th e British Broadcasting Corporation.

omni-directional. Developments are taking place towards theproduction of a unidirectional condenser microphone, whichhas advantages when outside broadcasts from theatres arerequired.The chief problem in the a.f. amplifiers and programme dis-tribution system is to preserve good signal/noise ratio and toreduce attenuation and intermodulation distortion to negligiblepropo rtions. The majority of broadcast recordings are carriedon discs, and one of the main troubles is to preserve the h.f.response as the centre of the disc is approached. Increasedloading at the recording cutter point and failure of the recordinghead to track the reduced wavelength cause appreciable h.f.loss, which is only partially counteracted by equalization circuitsin the recorder.At the transmitter, constancy of carrier frequency to betterthan 1 part in 106 presents no difficulty. Over-m odulation mustnot be allowed to occur because, apart from possible damage totransmitter apparatus, it causes harmonic sidebands to be pro-duced and these spread over into adjacent channels. When acommon aerial is used for two different programmes, verycareful filtering is necessary to prevent one modulated carrierfrom entering the r.f. stages of the other transmitter, where cross-modulation can take place.At the receiver, signal/noise ratio is an important factor, anda satisfactory value can generally be achieved by correct sitingof the aerial and by including a r.f. amplifier before the frequencychanger. In detection, the circuit constants (C and R) must beso proportioned that the voltage across the detector loadresistance is able to follow the r.f. peaks of modulation when ther.f. envelope voltage is falling. A wrong value of a.c.-to-d.c.load can also cause distortion due to the d.c. charge on thecoupling capacitor between detector load resistance and a.f.stage. Proba bly one of the weakest links in the chain of receptionis the loudspeaker, which is rarely able to reproduce faithfullytransient electric signals applied to its input.

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