I I

CONDITIONS FOR THE SUPPORT OF SURFACE WAVES AT AN INTERFACE

BETWEEN TWO DIFFERENT HOMO­GENEOUS MEDIA

IN order to focus attention on a practical arrangement of particular importance we shall assume, to begin with, that one of the media con­cerned is a loss-free dielectric, such as air and that the interface is perfectly straight in the direction of propagation of the wave_ As long ago as 1899 Sommerfeld presented a theory of ground-wave propaga­tion over a flat earth excited from a vertical dipole and described, as part of the total field, what he called a 'surface wave' _ Although it is now generally agreed that his idea of such a wave differed radically from our present-day conception, there can be no doubt that Sommer­feld was the first to analyse the complete field existing in such circum­stances and to give a lead in defining the capabilities of the surface as a support for the true surface wave as we recognize it today. In fact it was Zenneck, one of Sommerfeld's students, who gave the appropriate solution of Maxwell's equations for the inhomogeneous plane wave over a flat surface with finite losses, and this is the simplest case of a surface wave.

2.1. Field components

2.1.1. The lnhomo�eneous plane wave, or Zenneck wave (sup­ported by a flat surface). This form of surface wave is shown in Fig. 2.1 (a). It is an inhomogeneous plane wave because the field decays (exponentially in this case) over the wavefront with increase of distance from the surface.

Suppose the surface lies in the xz plane at y = 0 and that the media on each side of the interface are homogeneous. Then, to describe the wave we require the three field components Ex, Ey, and Hz, both above and below the surface, so that Maxwell's equations reduce to

(2.1 )

Medium (1)

f12:fLO e! -Eo

O"l ::: 0

H

SECTIONAL ELEVATION

H ,I ?I t .. ,,-.,-: J H ... I I I I I Ii '"00 : 0( :� : :,_�.� 'I I I-� , I I I I I I:" ::-.�:� :-.� " , , , 1 , J' I f 1-: 1: O .I�: :.: I�I I 'or

II I I I I

z

1 1 " , , II 'I ' --•• � I ... I '---'--i I , - I I r I I I I ,'�._ I EO I I .. I ,,_ •• �I I I I I , 'I I..1-A/2.,

11 11.911 I I , : oE:'f'f .. � EO�

/I PL AN

Co)

F I at react Ive supporting surface

SECTIONAL ELEVATION

Medium (j)

Flat reactive support i ng

surfoce

u,

Medium (1)

fLt ::: fo t"2 '" 1':0 (T2 = 0

H H E E E

'':L--,-E E

S I DE

PLAN Cb)

FIG. 2.1. Typical surface wans:

II H

ELEVATION (e)

Cylindrical ,,-Isupparting

surface

, ,

, , ,

,,�- ---..,),.H E � , , , ,

END ELEVATION

{al Inhomogeneous plane wavE' (Zenncck wave). (b) Radial cylindrical surface wave. (c) Axial cylindrical surface wave.

8 CONDITIONS �'OR SURFACE WAVES 2.1

(2.2)

(2.3)

together with

The time variation of the electric and magnetic field components is taken as exp(jwt), so that differentiation with respect to t can be replaced by jw. From this point onwards, the time-dependent factor will be omitted.

oN. (" )E

cEil 1 82H, - � = a-r)WE 0;'11 or

�x ' ox {/ a+jwE" ox2

�11 "E 1 o�2H_ o - (+' )E U "x 0 � W = a JWE. .r.- or , oy (ly a+ jw£ ay2

and combining equations (2.4), (2.5), and (2.6), we find

(j2H,+

a2H, = _k2N. ax!! ay2 �,

with k2 = -jw!,( ,,+ jw£),

(2.4)

(2.5)

(2.6)

(2.7)

(2.8)

representing the intrinsic propagation coefficient for the medium. Now the solution to the wave equation (2.7) is of the form

H, = A, Fl(y)F2(x), (2.9)

where Al = constant and we know by analogy with the transmission line (supporting an outgoing wave only) that we can write

with F2(x) = e-Yx (2.10)

y = rx+jf3 = propagation coefficient along the interface. (2.11)

Using equations (2.9) and (2.10) in equation (2.7) gives

where

82F:('/) . I,. = _(k'+y2)Fl(Y) = u'F,(y), (2.12) oy (2.13)

Since we are assuming no discontinuity in the medium outside the surface, the solution to equation (2.12) is

Fl(y) = A,e-"", (2.14)

and the complete solution to equation (2.7) becomes

(2.15)

2.1 CONDITIONS FOR SURFACE WAVES

Hence, using equations (2.4), (2.5), and (2.6), and including the time

variation, we have for the three field components

E = y v u+jw£

(2.16)

(2.17)

(2.18)

where A is in amps per metre. Suppose we employ subscripts 1 and 2 for the various quantities

applied respectively inside and outside the surface. We have to deter­mine the appropriate values of u (the propagation coefficient in the y-direction) on each side of the interface, bearing in mind (i) that we require the field to decay with increasing distance from the surface on both sides of the interface, and (ii) that we mllst have a wave in the surrollnding air travelling towards the smfllce and then carrying on into the surface, so as to supply the losses in the medium below the surface.

Outside the Bu'face (y ?i' 0)

In this case:

so that

(2.19)

e-aou gi,-es an exponential decay of the field with positively increasing values of y (away from the surface) and e+;',y represents a wave travel­ling towards the surface. The field components in the air auove the surface are

and

with

R - A Y p' - 1IWe- YXe;wJ "JJh - -;-� � ) JWEO

-(y'+".ij = "i = w'/-'o<o,

(2.20)

(�.21 )

(2.22)

(2.23)

Ins-ide the smface (y .;; 0) Here we must remember that the values of yare negative, so that

to meet the requirements

( +'b) (�.24) " = - ", = - a, J,. - -

,\Ve then get ell11l = eflllleilJtll representing a wave travelling f>nto the

10 CONDITIO�S FOR SURFACE WAVES �.1

surface and attenuating as it progresses. The field components within the surface are (2.25)

u� e+1I1Ile-y:r:ejw{, u1+JW€1

(2.26)

E =A v. (2.27)

with (2.28)

At the surface (y = 0) When y = 0 the tangential components of elec(,ric field must be the

same on the two sides of the boundary so that EJ, = EJ• and conse­quently, from equations (2.21) and (2.26),

'U2 '1(1 ,iW€o at +JWE 1

•

The corresponding equality of H" aud H" is clearly satisfied.

(2.2H)

The foregoing analysis shows that Maxwell's equations can be satis­fied by a form of wave (having field components EJ' Ey, and HJ sup­ported by a flat surface and characterized by an exponential decay of the field on both sides of the interface.

Before discussing this case further it will be helpful to establish the possibility of other forms of wave having similar characteristics.

2.1.2. The radial cylindrical surface wave, or radial form of Zenneck wave (supported by a flat surface). To launch a plane wave discussed in Section 2.1.1 would clearly require an aperture of infinite extent in the transverse plane. It is possible to excite a very

similar forlll of wave over a flat surface from a vertical line source and giving a radial cylindrical field distrihution as shown in l<'ig. 2.1 (b) (W. M. G. Fernando and H. l\I. Barlow).

The corresponding field components would be expected to be E" Ev' and H� with Maxwell's equations in cylindrical coordinates

BE, BE" _ aH� ay 8r - -fL 8t '

BE" = 0 and o</>

(2.30)

(2.31 )

(2.32)

�.l CONDITIONS FOR SURFACE WAVES 11

Assuming as before) time, we find

sinusoidal variation of the field components in

8E. . H = -Jwp. -', or �

_ oH¢ = (a+jwe)E" By

� H-, + aH¢ = (a+jwe)Ey, r � or

with the resulting wave equation

_o'H+_i)'H¢+�H _IBH¢ =k'H, oy' or' r' ¢ r or +

where the value of k is given by equation (2.8). Now the solution to equation (2.36) may be written

ll¢ = A, F,(y)F,(r)

(2.33)

(2.34)

(2.35)

(2.36)

(2.37)

and substituting the corresponding values in equation (2.36) gives

iJ'}�(r)..L� of,(r) k'-� R(r) F,(r) o'F,(y) = 0 ar' 'r ar + r" + F,(y) By' ' (2.38)

We now seek a solution to equation (2.38) for an exponential decay of the field above and below the surface exactly as in the case of the plane wave.

Ou/side the surface (y ;;, 0)

Suppose F,(y) = cU." (2.39)

with '" given by equation (2.19), then equation (2.38) reduces to

a'F,(") � of,('') T'-� R(r) = 0 cr' + r er + r" '

,,�+lc� = -y' = T'. The general solution to equation (2.40) is

F,(r) = A, H[')(Tr)+A, H[')(Tr),

(HO)

(2.41)

(2.42)

but we require only a wave travelling towards the surface with the appropriate inclination of the equi-phase planes, and the conditions are satisfied when

where

F,(r) = A, H[')( -jyr) ,

T = -jy.

(2.43)

It is easy to show that this solntion meets the reqnirements by con­sidering the large argument approximation to the Hankel function.

12 CONDITIONS �'OH SURFACF: WAVES

Thus at large radii

Hi'l( -jyr) :;:::j J 2j e-Y'ejj·, 7Tyr

:!.l

(2.44)

and bearing in mind the value of y given by equation (2.11), we see

that the solntion provides for the right kind of field distribntion.

Moreover, for large va·lues of r the amplitude varies inversely as the square root of T, which is characteristic of this form of surface wave. Thus we have field components in air outside the surface

Here again A is in amps per metre.

Inside Ihe surface (y � 0)

Here we have

where

yielding field components:

F1(y) = (�+III!I, '" = a,+jb,

H�, = Ae'''VHl'I{ -jyr)e;"",

•

EJl1 = A J� e'IiYII&2)(_jYI')ejwl, '" +JW€,

with equation (2.28) as before.

At ITte swiaee (y = 0)

(2.45)

(2.46)

(2.47)

(2.48)

(2.24)

(2.49)

(2.50)

(2.51)

Over this plane E" = E" and hence from equations (2.46) and (2.50) we get the same relationship as for the inhomogeneous plane wave, namely,

1£2 U1 -jWEo al +jWE:1

•

When 11 = 0 we also have H�, = H¢. as required.

(2.29)

2.1.3. The axial cylindrical surface wave (supported by a cylindrical surface). This is perhaps the most important form of surface wave from the point of view of applications, and consequently its performance in a variety of circumstances has been examined in more detail (G. Goubau (I) and (2); T. E. Roberts (1) and (2». As

2.1 CONDITIONS FOR SURFACE WAVES 13

will be seen from the analysis which follows, the limiting case of a sup­porting surface, whose radius is infinite, corresponds to that of the inhomogeneous plane wave and it is, in fact, quite enlightening to trace the changes in behaviour of the wave throughout that transition.

The axial cylindrical wave is shown in Fig. 2.1 (e) and it has field components He, E" and E, related by the Maxwell equations

with

af)� _ B!� = _I-' a!o, (2.52)

aHe = aE,+Ea�

, ox ot

.:..caE,c

x 0 d ,, = an

oe aE,

= O . b8

(2.53)

(2.54)

After applying the usual sinusoidal time variation we find

and writing

with

we get

BE, aEx . " - Or = -)Wl-'flO' ox U _ aHe

= (a+jwE)E" ax

�1I9+ 8�o = (a+jwE)E" r or

Ex = B,F,(r)Fix),

Fix) = e-YX, £-2F,(r) + 1 aF,(r) 'F, ( ) _ 0 - �'lt 1 r � or'.!. r or '

(2.55)

(2.56)

(2.57)

(2.58)

(2.10)

(2.59)

where ,,' = -(k'+y') as equation (2.13) and k is defined by equation (2.8). Equation (2.59) call be transformed into the standard form of Bessel's equation by putting

•

P = J"r and our wave equation then becomes

a'};(r) 1 a}�(r) F,() _ 0 • 0 +- + I r - . or l' 81'

O"taide the 81trface (r ;;,0 8)

(2.60)

(2.61)

The appropriate solution to equation (2.61) outside the surface is

F,(r) = B,H�'I(1') = B,ll�'I(j"2r) (2.62)

14 CONDITIONS FOR SURFACE WAVES 2.1

so that using equations (2.10) and (2.62) in (2.58). together with the time variation, we have

E" = Belw1e-yxH�')(juzr). (2.63) where U, = az-jb, (2.19) and B is in volts per metre. For the large argument approximation to the Hankel function in equation (2.63) we have

(2.64)

representing, as required, a wave travelling to'warda the surface and decaying in amplitude as r increases.

The other components of the field in the ail' outside the surface are

and

E" = B .Y e;wle-yxH\')(ju,r) JU,

Hoz = B WEo ejwfe-yxHp)(j1.l2r). 'Uz

Inside the surface (r � 8)

(2.65)

(2.66)

Since we are dealing with a cylindrical surface. forming a closed boundary of finite radius. there will be a standing Wave over the cross­section, just as in an ordinary cylindrical waveguide.

The Hankel function in equation (2.62). which is n. solution to the wave equation, can be rewritten in the form

H�\}(p) = Jo(p)+jYo(p) and within the surface only the Bessel function of the first kind can be retained. because when r = 0 then p = 0 and Yo(p) = -00.

Hence. instead of equation (2.62) we have in this Cllse F,(r) = B;Jo(p) = B;JoU", r) (2.67)

representing. as expected. a standing wave over the cross-section of the cylindrical guide.

Thus E = B' eiwte-Y"J, (J'u r) Zl 0 l ' where ", = a, +jb" and the other field components are

E = B' i' eiwle-yxJ (J'u r) 1, . 1 I • JU,

He = B' O',+jWE, eiwle-Y'J (jlt r) 1 JUt 1 1

with B' in volts per metre.

(2.68) (2.24)

(2.69)

(2.70)

2.1 CONDITIONS }'OR SeRFACE WAYES

At the 8"rface (r = 8)

Here we must have E.lt = Ex-I so that

BH�1)(j",8) = B'Jo(j",s) and He, = Ho. so that

B WEO /1(1) ( ' ) _ B' G,+jWE, T (' ) 1 )", s - . '" )", s .

1/.2 J'llt From equations (2.7 J) and (2.72) we therefore get

HUl( . ) . 1. ( . ) 'IlZ 0 JU28 JUt ' I) JUt8 - - .

WEo Hl')(ju,.) G,+jWE, J,(j"I.j" 2.2. Surface impedance

15

(2.71)

(2.72)

(2.73)

Schelkunoff (1) has shown in a classic paper that electromagnetic, field problems can be considerably simplified by the introduction of the iue" of wave impedance. In dealing with surface waves it is often very convenient to specify the properties of the guiding structure in terllls of surface impedances, since this permit.a us to discuss the behaviour of the surface wave in t.he surrounding free space region with­out the need to know anything about the actual constitution of the supporting surface. The surface impedance, Z, = R,+jX" is defined as the ratio of one tangential component of the electric field to the tangent.ial component of magnetic field in the direction perpendicular to the chosen electric field. This mtio is evaluated at the surface which forms the boundary of the guiding structure. Obviously, for a given structure there are two possible values for this impedance, one depend, ing on the transverse electric field and the longitudinal magnetic field and the other on the transverse m agnetic field and the longitudinal electric field. :I<'or the particular surface waves which have been discussed so far, oilly the latter is needed, but in more general cases to be discussed in a later chapter, both impedances may be required.

The definition of surface impedance in terms of field component> makes it clear t.hat the value may depend not only on the physical properties of the guiding structure but aiso on the nature of the field which is being investigated. For example, some structures can support several surface waves of different types, and the values of the sUlfaee impedances may be different for each surface wave. I n many cases, however, it is possible to assume values for the impedances which are independent of the field patt.erns, and when this can be done very considerable simplifications can be effected in the calculations. This is particularly useful when we come to examine the properties of launch­ing systems, and most of the calculations in the later chapters will be

16 CONDITIONS FOR SURFACE WAVES • • -.-

carried out on the ""su.mption that the surface impedances can be regarded as parameters of the guiding structure only. It must be stressed, however, that this does involve an approximation and where appropriate the nature of the approximation will be indicated.

The use of the idea of a surface impedance can in principle be made completely rigorous: we will see later that any electromagnetic field can be synthesized from plane waves. The surface impedances can be determined uniquely for any specified plane wave, and we should there­fore regard the impedances for a given structure as being not only functions of frequency, but also of the direction and polarization of the particular plane wave considered. Unfortunately, the resulting com· plexity of the analysis makes it difficult to obtain solutions to the problems of interest. We therefore assume that the impedances are not strongly dependent on the plane· wave properties, an assumption which is justified for many cases of pract.ical interest. It may be noted that the idea of a surface impedance is widely used in calculating the attenua· tion in waveguides, the effect of the losses in the conducting walls being included by imposing an impedance boundary condition at the surface of each wall.

In dealing with surface waves we are mainly concerned with the reactance, since, as we shall see, this determines the rate of decay of the field outside the surface. The resistance is associated with the attenuat.ion of the snrface wave, and since we are naturaUy most in­terested in those surface waves which have low attenuation, we shall be seeking surfaces for which the surface resistance is low. In the examples which follow both the resistance and the reactance terms will be examined, but in later chapters we shall neglect the resistance term. The effect of this resistance can be included by a perturbation technique identical to that used for calculating the attenuation in waveguides.

The surface impedance of a structure has the properties which we normally associate with impedances in circuitry. For example, the resistance and reactance are associated respective1y with energy dissipa­tion and energy storago in the guiding structure. In particular, if we have an impedance Z" defined for tangential electric and magnetic field components, E" If, respectively, then the power dissipated in the guiding structure per unit surface area is �re Z.IE,12, and the difference between the stored electric and magnetic energies per unit surface area, IV. and Jv,,, respectively is given by:

IV,,,-w,; = -�illlZ,IE,I'. 2w

• • CONDITIONS FOR SURFACE WAVES 17 -.-

This result shows that the fields must penetrate a finite distance into the guiding structure, for we require a finite volume for the storage of energy. If the impedance is purely reactive, it must satisfy Foster's reactance theorem, i.e. the rate of change of reactance with frequency must be positive. All of these results are derived by the methods discussed in detail by Montgomery et al. for waveguides.

Once the surface impedance has been defined, the decay factor of the surface wave can be related to it. This relation will, of course, depend 011 the surface geometry, and in particular the effects of the curvature of the surface are of considerable importance. The impedance can be calculated from the properties of the guidiug structure, provided, as stressed above, that allowance is made for the field configuration involved.

2.2.1. The plane Zenneck wave and the radial cylindrical wave, supported by flat surfaces. Bearing in mind the definition of surface impedance and referring to Fig. 2.1 (a), we have for the plane Zenlleck wave

Z, = R.+jX, = E.. __ u, = 1 (b,+ja,). (2.74) H;;. 11==0 JWEO WEO

Similarly for the radial form of this wave, we find

Z, = R,+jX. = _ :.] = _ .u, = 1 (b,+ja,). (2.75)

�1 Jf=O Jweo WEO Thus for both of these waves

and

R, = b, (2.76) W'.

(2.77)

It will be observed that the rate of decay of the field with distance from the surface, namely a., depends only upon X, whilst R, deter­mines the phase·change coefficient b.. Moreover Z, is independent of the distance y from the surface.

When the medium below the surface is homogeneous and of infinite extent, the reactance X, arises from the resistauce R, associated with the finite conductivity u of the medium. Thus for a plane wave incident on such a surface at right angles to it, we have

X,.= wl4'{'J(w'.;+un-w·,l! (2.78) 2{w'.�+uU

aud R. = WI4,U(w'.;+ull+w·,l t. (2.79) 2{w2.l+ui} 47111.12 o

18 CO�DITIONS FOR SURFACE WAVES 2.2

As would be expected, R, is always a little greater than X,. By differentiating X, and R, with regard to wand equating in each cnse to zero, it is easy to show that Xs is a maximum when Rs is a minimum (for given electrical constants of the medium) and that this OCOlUS for

u = "J3WE. (2.80) In general an increase in X, will reduce the effective spread of the surface wave field outside the sluface, and at the same t,ime reduce its phase velocity vp along the interface. On the other hand, any inc.rease in R, tends to tilt the wavefront (represented by an equi-phase plane outside the surface) forward, so that a larger proportion of the power is directed into the snrface. This results in increasing the value of vp as the component of the wave velocity resolved in the direction of the interface.

Although the behavionr of the wave outside the surface is determined by the value of the surface impedance, it is immaterial to that part of the wave how the surface impedance is produced. A simple calcula­tion is sufficient to show that when the surface has a homogeneous medium of infinite extent beneath it, olily small values of X, are possible. On the other hand, a dielectric coated or corrugated metal surface can have a much larger reactance and for flat surfaces this is the only means of increasing the rate of decay of the field in the sur­rounding air (A. E. Karbowiak (3».

2.2.1.1. Dielectric coat.ed 1IIetal ... <rjace. Referring to Fig. 2.2 we have in this case three different media to consider. In thc metal represented by medium 1 and in th� air (medium 3) we can write (as in Section 2.1.1) for the tangential components of the magnetic field II,

and

where

and

H = A eU11fe-)lX %, 1 , H = A e-tlJVe-YX ;:::, 3 •

u, = a, +.ib" U3 = a3-jb3·

(2.81) , (2.82) (2.83) (2.84)

If medium 2 is assumed to be a solid dielectric with negligible losses, then we would expect to have a standing wave within it such that

H" = [A;coshu2y+A;sinh",yJe�Yx, (2.85) where - (y'+uiJ = k; = w'P.o E,. (2.86) Matching the tangential electric and magnetic fidd components at the boundaries between the different media gives

tanhu,l = !':.! ) . U 2 (2.87)

00 -. - CONDITIONS FOR SURFACE WAVES

When l is small so that tanh".l � u2l and u31 < 1 (2.87) reduces to yield

, [E,-EO I t A} . .I A U3 = w-J-Lo EO E2

+:f.:..l -J;1L.lo J

where the skin depth in the metal

y

-Ll.--J 2 . - --wfLo G1

I o

Mtdlum Q) (Air)

- -_ . ., .r

FIn. 2.2. Dielectric-coated met,a] surface.

19

then equation

(2.88)

(2.89)

The corresponding surface impedance, looking into the solid dielectric coating the metal, is

Z - E +:X _ Ex. "3 8 - '8 J 8 - ' = - ' , H::'3 JWEU and therefore we find

with

where

E .• = ba = !W�.Ll., WEo

(2.90)

(2.91)

(2.92)

The contributioll to X, made by the penetration of the field into the metal is generally negligible.

2.2.1.2. Corrugated ",etalsurface. The enhanced reactance of a corrn­gated surface (Fig. 2.3) relies upon the setting lip of .tanding waves in the grooves, much in the same way as the dielectric coated surface

20 CONDITIONS FOR SURFACE WAVES 2.2

employs the layer of solid dielectric. Since the corrugations form a

periodic structure, arrangements must be made to ensure that the

surface impedance behaves as if it were uniformly distributed, and this

requires at least three complete corrugations within the wavelength along the surface, together with a depth of groove which is preferably

not less than its width. Each groove short·circuited at the bottom will

FlG. 2.3. Corrugakd metal surfaco.

-- � T

then support a T.E.M. wave and the impedance looking iuto the mouth will be Z",oove = Zw tanhy'h, (2.93)

where the characteristic wave impedance for the groove is Z", = 377 Q when air-filled and y'

is the propagation coefficient. Neglecting losses and the resistive component of Zgroove equation

(2.93) reduces to .J h Z ·X ·Z t _7T

groove = J groove = J w an .\ • (2.94)

The metal ridges have relatively negligible surfacc impedance, so that

approximately we can write

Z d Z ·Z d 27Th Il = D groove = J w D

tan -X- . (2.95)

2.2.2. The axial cylindrical wave. In this case, Fig. 2.1 (0), we have

Z = R +:X _ [Ex,] _ U2] [H�I)(jU2 S)] , , J , - H - EIUI( . ) 81. r=s W£O 1 JU28 (2.96)

and the surface impedance becomes a function of the radius 8 of the surface as well as of 'lt2, We note, moreover, that when 8 -7 00, the large argument approximations to the Hankel functions are applicable,

in equation (2.96), which then simplifies to the corresponding expression

for the plane Zenneck wave, namely

Z . U2 s = J .

WE, (2.74)

CONDITiONS FOR SUItI"ACE WAVES

In an ordinary application of the axial cylindrical wave ]ju,sl � 1 (2.97)

and consequently the slllall argument approximations arc valid for the Hankel functions in equation (2.96). PuUing

we can therefore write

•

1', = J'1I-,8

o 0 H�l)(P,) = Jo(p,)+jY.(p,);:::j 1 +j':: (In 41'2+0'577) ;:::j 1+j':lnO'89p,

and

Thus

� � (2.99)

(2.100)

(2.101)

The numerat.or of this expression can be converted t.o a different form as follows:

q 9 ? l+j':ln O·80jn,8 = 1+j':[lnj+ln O'89",8] = j':: In O'S9U,8

� � rr

and substituting in equation (2.101) gives H�1)ti"28) � -fu sln(0'801l 8) liil)(ju2 8) � 2 2 ,

so that from equation (2.06) we get . ,

Z, = R,+jX. = _JU'Sln(O·8911,.,), WE.

(2.102)

(2.103)

since lu2.1 < 1 and hence In(0'8971,8) is negative. Remembering that "., = a,-jb, equation (2.103) gives

R, = 8 (bl - "�)tan-l b'-2a,b2In{0'89s�(bH"1)} (2.104) WEo a2

and x. = 8 [2G, 6, tan-1 � + (bi-a!)ln{0'898J(bHam] . WEO a'Z

Alt t· I ' 'of = _ 1 = _ I , . ernalve v, BIIICe A " "j(JL. '0) '0 Z.

we can rewrite equations (2.104) and (2.105) as

( )R 2"s R, W£OS S = A Z • •

(2.105)

(2.106)

{(b,s)2-(a, 8)')tan-' �:: - 2(a,8)(b, 8)ln{0·89�[(a28)'+ (b,s)'])] (2.107)

and (W<os)X, =

CONDITIONS FOR HUR�'ACE WAVES

21T8 X.�

"0 Zo

00 - .-

[2(a, 8)(b, 8)tan-1 b," + ((b, 8)2_ (a, s)2}ln(O·89J[(a, 8)'+ (b, 8)'ll . aos •

20 X 10-7 r---�---�--�------,------, 11 •• �(1E')' �

-o o

., 16 � 10

., 1 � I 10

., H 110

o

" '__ _ __ _ o &110 t- ----. .

c(i�

-o . ., " �110 , -o >

., hl0

o

., - 2110

Ao Z"

��. (7:) �: /1 • (2"�') x .. 1/ A" Z"

. I ! I I I

-� 110 ·'L _______________________ .L ______ .1 ______ �

(2.108)

FIG. 2.4. The dependence of surface resistance and reactance on tho vslues of (aI8) and (baS) for an axial cylindrical surface wavo.

Equations (2.107) and (2.108) are dimensionless, and if the values of (27TS/Ao)(R,/Zo) and (27TS/Ao)(X,/Zo) are plotted against corresponding values of (a,s) for given values of (b,s) the curves shown in Fig. 2.4 are obtained.

CONDITIONS FOR RURFACE WAVES 23 It will be observed that over a limited range, negative values of X,/Z.

representing a capacit.i"e surface are suit,able for the support of the wave, and this has been confirmed experimentally. Another interesting feature is that for a given vallie of (b,.), the curves of (2rrs/A.)(R,/Z.) and (�rrs/).o)(X,IZ,) intersect at the point where X, and R, are equal, corresponding very nearly to the case of a bare metal guide with a smooth surface for which (n,s) � 2'25(b,s) (H. 111. Barlow and A. L. CuUen).

2.3. Propagation coefficients and phase velocity outside the surface For all three forms of surface wave discussed, when the medillln Ollt�

side the snrface is air, we have

-(y'+7Ii) = ki = W'fLo€., witl! y = a+j{3,

and y., = a, -jb,.

Combining equations (�.109), (2.110), and (2.111) gives

'" = W,,(,ll'+N')+Jf}]!,

and f3 = ['lU(M'+N')-lIf}]f,

where 111 = (bi-ai-Q),

(2.109)

(2.110)

(2.111)

(2.112)

(2.ll3) (2.114)

with N = 2a,b,. (2.115)

Equations (2.112) and (t.ll :1) are exact, but in some cases calculations can be simplified without introducing serious errors, by making approxi­mations.

Thus, when q p (a�- bl), as for example in the case of an unloaded surface consisting of a smooth homogeneous medium, and frequencies in the V.H.1<'. band upwal'us, or for some loaued surfaces operated in the microwave band where a, p b" we have from equations (2.112) and (2.113), after expansion by the binomial theorem and neglecting higher-

order terms,

and

v '" = - (n, b,) ,

w

f3 - W [1 1 v '(b' ') ' 1 'v'a,b, 2] - - - - - z-a2 T- , v 2 w 2 w2

or the phase velocity along the interface

where

w V vJ• = fJ = 1-(v'l�w')(bi-ail+·(v'aibiI2w·)·

wAg v = . �tr

(2.116)

(2.117)

(2.11S)

(2.119)

Putt.ing

CONDITIONS I··on SUln'A8Jo: WAVES

• • 'b' _ V

( '_ b

')+

V a, , T- -a2 2 -

w2 w4

we can therefore rea.rrange equation (2.118) to give

v _ >'0 _ J >'n 2+.,-- 1+27" or , vp - Ao - AO-'\q T

and since .,- <{ 2 hence

>'0->'0 Ll>' 1 - = - =·.�-T. >'0 >'0 -

2.3

(2.120)

(2.121)

(2.122)

Thus from equation (2.122) the fractional change in wavelength along thc surface of the guide can be determined in terms of a, and b,.

metal plot.s terml.r.orioq cylindrical qulde

-" . __

, lCYli'�riCOI quide for

5uppor", of wav.

FIG. 2.5. Axial cylindrica.l surface wave resonator.

Using an axial cylindrical surface wave resonator, Fig. 2.5, the number of half guide wavelengths within a given length of resonator can easily be determined at a number of different resonant frequencies. Alterna­

tively, a lossy dielectric rod, e.g. Perspex, can be used as a gu.ide, and arrangements made to rednce its diameter step by step, so that the surface reactance changes from an inductive to a capacitive one. In this way the conclusion drawn from equation (2.108) and Fig. 2.4, that .. capacitive surface is capable of supporting over a limited range an axial cylindrical wave, has been verified (H. M. Barlow and A. E. Karbowiak (3». Fig. 2.6 shows the results obtained in a given case, and it will be seen that in these circumstances we can have vp = v when

Ll>' = 0, or even v" > v until quite suddenly a point is reached at which the surface wave breaks up altogether.

With the ordinary type of unloaded smooth metal wire guide sup­porting an axial cylindrical wave, we have seen from Section 2.2.2 that a, � 2·25b,. Thus, referring to equation (2.118) it is clear that for such a guide vp < v.

2,3 CONDITJONS .FOB. SUltF;\f'E WAVES

1 o·.�---+- I

;;.. 0·. o 5? •

"I ' ., "

I

I �. 0 --\;::---+:,.,. ""-l'lA cm : H :}} ,

.l_-L'_� Surface wove could no't be supported bt!yond t�IS pOinT

Perspe:c rod gUide of radius O·Q7t!11 em

FlO. 2.6. Percentago decrease in guide wavelengt·h 118 a funct.iotl of free SpRCO wavelength for a Perspex roll of l'/l.diu$ 0'9781 cm.

0' .-, - '

2.3.1. Waves supported by flat surfaces. Plane Zenneck wave and radial form of Zenneck wave. By matching t.he tangential com­ponents of the field at the interface we obtained for both of these waves the relationship

U2 111

or

Now and

. = - . ) JW<o 0", +Jw<l

111 Ul +jW€l kr �

= - jw£o = -X1'

-(Y'+"il = k� = -jwPO(O"I +jw<,)

(2.29)

(U�3)

(2.124)

(2.125)

Thus, assuming a smooth unloaded metal surface, for which 0", » w<, we can combine equations (2.123), (2.124), and (2.125) to give

= -'b � w'Po<. (1-wh+<o)\_ '(1 ' w(€,+€O») '" u, J,

� ..)(2 ) ., J J T o ' Wilo U1 _U1 ""U1

(2.126) It follows from equation (2.126) that b, > u" and consequently

equation (2.118), yields VI' > V. By loading the supporting surface to give it an enhanced inductive reactance vp is reduced in relation to v. It will be observed that v,. � v, when b, � u,.

2.3.2. The axial cylindrical wave. Goubau (1) has made detailed calculations for this type of wave using as the guide a smooth metal wire of circular cross�section.

By matching the field components at the surface, we found '" Hb')UU,8) .i'" ';;lei"18)

H(1)( . ) = . J ( .

) ' W£O 1 J'lL28 Ul+JlU£l 1 JU18 (2.73)

C O N D I T I O N S F O B. 8 11 J{. «,..\l' .E \V .-\ \·}�K

and since Ij-U 2 8 1 <� 1 , we showed in Sect,ion 2.2.2 that

Jim( jl(. s) .

Iiill'. 2 � -J", 8 In( fl' HHu,8) .

, Un, s)

" 3 - .

(2. 103)

Now within the surface of t.hc cylindrical guide, assuming t.his to be metal or a medium of similar charact.eristics, ,ve know that [jU1 8 i ?> I and therefore we can apply the large argument approximations to the Bessel functions on the right-hand side of equation (�. 73) above.

Thus JoCi", s) � .

J,(jll, s) � �J' (2. 1 27)

Substituting from equations (2.127) and (2. 1 02) in equation (2.73) gives

(2 . 128)

where

In order to determine the value of ", from equation (2 .1 2R) it is convenient to let

so that and then to put

g = (0·89",8)2,

In g = 2 In(0'891I, s)

(2. 1 29)

'I) = g hl g = 2(0·S91l, 8)2hl(IJ·89!l,s). (2 . 130) Substituting the value of In(0·89u,s) = -(wEoll,linisa,) as given by

equation (2. 128) in equation (2 . 130), yields

'I) =

j2WEO(0'S9)'u, s. (2 . 1 3 1 )

a, Moreover, equations (2 . 124) and (2. 1 25) are applicable with the addi­tiona.! conditions a1 /> WEI and aI }> WE"o so that

'I' "12 - 1,2 k' � y'w" a ,- 2 - h2- 1 """" ,....-0 I ' (2. 1 :l2) Now the medium outside the metal guide is supposed to be air and it follows therefore that lu, l ;> lu, � . Equation (2 . 1 :12) thus reduces to

•

or JU, =

2 . 1./,1 ::::::: JWfLo VI

Using this result in equation (2. 1 3 1 ) gives

'I) = [2(0'89)'(27T)!Eof!l-'�a,!8leji" =

24·6 X 1 0-" Hence 1 '1) 1 = f's .

-Ja, Suppose g = Ig lei8,

171 1 ,,)171" 1 - / 1 ' •

(2. 1 33)

(2. 1 34)

(2.135)

(2. 136)

CONJ)ITIONS FOR S U R FACE WAVES

where then from equations (2.130) and (2. 1 34) we get � = 1 � lrijrr = t ln g

27

(2.1 37)

= It le-Jllrr+f)[ln( lt le-iU rr+,I)}] = It le-JUrr+fl[ln It l -j(!7T+�)]. (2. 138) Multiplying botb sides of eqllation (2 .138) by eil. and putting ei• = - \

• gives - 1 '71 = Ig le-jf[ln l t l -j(l7T+�)] = It l [cos �-j sin�][ln lg l -j(!7T+�)].

(2. 139) The imaginary part of the expression on the right.hand side of equation (2.139) must be zero, wWlst the real part is - I � I . Hence

or

(cos � )(t7T+�) = -(Bin � Hln 1m. tan � = _ [t7T+ �

In Ig I ' and remembering that (l+tan2�) = l/cos2�

_ I 1 = It lln lg l . � COB �

(2.140)

(2.141)

Applying equation (2.135) to a copper wire guide of radius 8 = 1 em and a. = 6 X 10' then at a frequency f = 10.0 cis we get

I � I = 3·17 X 10-4. Thus in general for unloaded metal guides � < 4° and cos � � I with tan � � �, giving as good approximations to equations (2 .140) and (2.141 ) 7T �

= - 4(lnlg l+I) '

with I S = -(!7T+�)

= -17T I -In lg l + I '

and - I� I = I t lln lt l · }'rom equations (2.129) and (2.136) we have

or

gl l -l23 It l ' iI8 u.z = = e , 0·898 8

·b 1 . 123Ig l' [ •• + . . 1.] 1lz = a'l,-J t = 8 cos "2;O J 8m 2 °

(2. 142)

(2.143)

(2.144)

(2. 145)

•

CONl.l I T l O � i:l F O .H. S U H F A CE W A V E S �.3

:1<'01' OUl' metal guide 4> � 3°, so tha.t from equa.tion (2. 146) we find

lL;! _ COS �H77+<p) _ O) . () ;-- - - - _.) b, sin�U""+4»

as outa.iued previously. We can also calculate phase coefficients for the wave along the surface .

the attenuation and Thus, from equation

(2. 125) and since 1", 1 ' � Ik,I' = W'fLo 'o = (w/v)' we have

[ ' ] . 'U2 Y � J k.,+ l ' � . ) . .. '2 so that using equations (2 .129) and (2.136) we find

or

and

_ " I . ·Q _ . l. + !� Icoso j ltlsin 0 y - -, J,.. - J , 2(O'SO)'k,s'+ 2(O'89)'k,s'

[ v 1g 1Sin O ] . [w v lt lcos o = - 2(U'89)'W8' + J -;;-+ 2(O'89)'W8' '

[ v lt isin o ] "

= - 2(U·g0j2ws' ' f3 = W + vltlcos o

. v �(O'89)'wS2

(2. 147)

(2.148)

(2 . 149) N wllerical examples of the application of these expressions will be given later .