Paper No. 1482
RADIO SECTION
621.396.11 : 538.566
SURFACE WAVES
By Prof. H. M. BARLOW, Ph.D., B.Sc.(Eng.), Member, and A. L. CULLEN, Ph.D., B.Sc.(Eng.), Associate Member.
{The paper was first received 29th October, 1952, and in revised form 5th January, 1953. Proofs were made available to the public \3th April, 1953,
and the paper was read before the RADIO SECTION 22nd April, 1953.)
SUMMARY
The paper is an attempt to present, in the simplest possible terms, a
unified picture of the theory of various forms of surface wave and a
clear physical interpretation of their behaviour.
The Zenneck wave, the radial cylindrical surface wave and the
Sommerfeld-Goubau or axial cylindrical surface wave are each discussed and shown to represent basically one and the same phenomenon.
The link with the Brewster angle, for which a wave incident on a
surface suffers no reflection, is clearly established. The transition
from a TEM wave supported by a parallel-strip transmission line,
with metal plates close together, to two Zenneck waves independently
supported by the plates, when these are separated by a very large
distance, is demonstrated.
The effect of bends in the supporting surface is considered, and
methods of reducing radiation are explained.
Finally the principles governing the launching of surface waves are
surveyed with particular reference to the Brewster-angle approach.
(1) INTRODUCTION
The type of electromagnetic wave with which the paper is
concerned may conveniently be described as one that propagates
without radiation along an interface between two different
media. If both media have finite losses, the main stream of
energy directed along the interface will be required to supply
these losses. This does not invalidate the description of the
surface wave if radiation is construed to mean that energy is
absorbed from the wave independently of the media supporting
it. The special interest centred in the surface-wave phenomenon
arises primarily from its unique non-radiating characteristic,
which enables high-frequency energy to be transferred intact
from one place to another, except in so far as demands are made
upon that energy to compensate for the losses in the two media.
In order to focus attention on a practical arrangement of particular importance, it is assumed that one of the media concerned
is a loss-free dielectric, such as air. This simplifies the discussion
and still enables essential principles to be established. In the
first place we observe that the interface must be straight in the
direction of propagation of the wave, although transversely it
can take a variety of forms. It will be convenient to refer to the
boundary of the medium surrounded by the loss-free dielectric
as the surface supporting the wave. Surfaces of transversely
cylindrical form are known to be suitable for the support of
surface waves, and flat surfaces should also be satisfactory. In
radio-wave propagation, as in optics, there is a particular angle
of incidence, known as the Brewster angle, for which no reflection
takes place. This is another way of stating that in such circumstances there is no outward radiation from the surface, and
it might therefore be confidently expected that the surface wave
is simply a wave of the required field configuration incident on
the surface at the Brewster angle. This proves to be the case,
and it can be readily established analytically for a flat surface.
Since the power flow is normal to the wavefront, defined as an
equiphase surface, it follows that the field distribution must be
evanescent over that surface, suffering a decay with increase of
Prof. Barlow and Dr. Cullen are at University College, London.
distance from the interface. We shall find that the waves of
interest are primarily E modes, having a component of the
electric field in the direction of propagation and a transverse
magnetic field. Mixed modes sometimes occur, but only in
special cases. For homogeneous media we are concerned with
a single interface, and there is then only one finite boundary
condition to satisfy so that the corresponding surface waves
exhibit no cut-off phenomenon.
It is proposed to concentrate attention on three distinctive
forms of the surface wave, namely:
(a) The Zenneck or inhomogeneous plane wave supported by a
flat surface.
(b) The radial cylindrical wave also supported by a flat surface.
(c) The Sommerfeld-Goubau or axial cylindrical wave, associated
with a transversely cylindrical surface (see Fig. 1).
Before entering on a more detailed discussion of these forms
of surface wave, it is relevant to recall that the Sommerfeld
theory of ground-wave propagation over a flat earth1 also
introduced a so-called "surface wave." This cannot be identified
in toto with the wave we are now considering, and the confusion
of names is unfortunate, particularly as Sommerfeld was
responsible for some of the early work on the true surface wave.
In his discussion of the problem of radiation from a vertical
dipole above a plane earth of finite conductivity, Sommerfeld
divided the ground wave into two parts which he called, respectively, a "space wave" and a "surface wave." The surface-wave
part is represented by one of the terms in the analysis of the total
field, and its particular feature is that it tends to predominate
near the earth's surface. Both parts are required simultaneously
to satisfy Maxwell's equations, and at long ranges, according to
Sommerfeld, the surface-wave part varies inversely as the square
of the range. In similar circumstances, a true surface wave
radiated from a vertical-line source over a horizontal surface
would be expected to decay exponentially with range, owing to
the losses, and at the same time to fall in amplitude inversely
as the square root of the range, owing to the expanding wavefront. In Sommerfeld's original paper a surface wave of this
type appeared in the expression for the total field, but the view
now seems to be generally accepted2 that this result arose from
an incorrect evaluation of the integral representing the field.
A correct evaluation of this integral does not yield the surfacewave term. Booker and Clemmow,3 in interpreting the Sommerfeld theory of propagation over a flat earth in the presence of
finite losses, invoked the aid of a Zenneck wave. They discussed
particularly the 2-dimensional case of a horizontal-line source
above the earth. In doing this they showed that a hypothetical
Zenneck wave, diffracted under a vertical-plane screen whose
lower edge coincided with the image in the earth of the line
source, made equivalent provision for the effect of the earth
losses. Booker and Clemmow were very careful to point out
that this was merely a convenient physical interpretation and
did not mean that a Zenneck wave actually existed above the
earth's surface, but it appears nevertheless that there has been
some misunderstanding which can, no doubt, be attributed to
the use of the term "surface wave" for two different things.
[329
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330
BARLOW AND CULLEN: SURFACE WAVES
Suppose the surface lies in the *z-plane at y — 0 and that the
media on each side of the interface are homogeneous.
Using the subscripts 1 and 2 to refer to the media below and
above the surface respectively, then for a wave travelling along
the interface in the positive x-direction with a propagation
coefficient
y = «+jp
(1)
Medium 2
/
/
h
the three components of the field required to satisfy the 2-dimensional wave equation are
(a) Below the surface.—In medium 1, having permeability
fx{ = (x0, permittivity /Cj and conductivity al, i.e. for y < 0.
/* V2^7
Reactive
plane surface
/
/V////////////////A
Hzl =
Medium 1
Ay
Medium2T *
:t
Elevation
'xl
y
(2)
*
////////?9///////A
where A is a constant and the propagation coefficient along the
.y-axis is
(3)
representing an attenuation av and phase change bv for a wave
travelling inwards from the surface.
(b) Above the surface.—In medium 2, assumed to be air for
which fx2 = /^o> K2 — Ko a n d °2 — 0, i.e. for y > 0,
Plan
Hzl =
Reactive
plane surface
JOJKQJ
Side elevation
End elevation
HNfiO
Medium
1
Here
r
CO
Fig. 1.—Forms of surface wave.
(a) Zenneck wave (inhomogeneous plane wave).
Medium I: m = no. «i, oi.
Medium 2: pi2 = (lo, «2 = «o> 02 •= 0.
(b) Radial cylindrical surface wave.
(c) Sommerfeld-Goubau wave (axial cylindrical surface wave).
= a
2
(4)
(5)
- jb2
because the field not only decays at the rate a2 with increasing
distance from the surface but also suffers a progressive phase
change b2 for a wave travelling towards the surface. These
characteristics are in accordance with the specification of a
surface wave for which the power flow has two components, one
representing the main stream along the interface and subject to
the usual attenuation a with phase change j8, while the other,
generally a minor one, is directed into the surface to supply the
losses. No radiation therefore occurs.
On both sides of the interface we have
y 2 + M2 - K2 = ja)/x((T
and thus within the surface
+ j(DK)
y2 + u2 = K\ = jcn
•
• (6)
•
. (7)
•
• (8)
or in the air outside the surface
(2) FIELD COMPONENTS
(2.1) The Zenneck Wave
As long ago as 1907, Zenneck4 described, as a particular
solution of Maxwell's equations, a wave which travels without
change of pattern over a flat surface bounding two homogeneous
media of different conductivity and permittivity. This is a
form of surface wave, and the field distribution is shown in
Fig. \{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.
y2
+ «a = *3 =
— O)2IXQKQ
.
.
(2.2) The Radial Cylindrical Surface Wave
There is a radial form of surface wave which, supported by a
flat surface, is of particular interest, because unlike the planewave case the wavefront is here offiniteextent in the horizontal
direction. Fig. 1(6) shows the field distribution, and defining
this in cylindrical co-ordinates for a medium with constants
i".o» Ki> °\ within the surface, assumed to be surrounded by air,
we get the following field components:
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BARLOW AND CULLEN: SURFACE WAVES
surface. In general Zs is complex, having both resistive and
reactive components, so that
(a) Inside the surface when y < 0.
fK- jyr)
Z=Rs+jXs
^
= - A(
331
-jyr)
\ a , + ]o)Kx
(9)
with eqns. (3) and (7) as before.
(b) Outside the surface when y
-
JCOKQJ
jyr)
jyr)
(10)
with eqns. (5) and (8) as before.
Comparing eqns. (2) and (9) or (4) and (10) it will be seen
that the radial form of surface wave has precisely the same field
distribution in the ^-direction, normal to the surface, as the
corresponding Zenneck wave. Along the radial co-ordinate r
the field decays according to a Hankel function, which at large
distances becomes an exponential of the form e-vrl\/r.
(13)
For any medium constituting a surface of finite conductivity
and of thickness greater than the skin depth Rs and Xs cannot
be separated physically, since the existence of Rs necessarily
implies a corresponding Xs quantity arising from the penetration
of the field. In any good conductor, such as a metal, Rs is
slightly larger than Xs) although for most practical purposes
it is sufficiently accurate to regard the two quantities as equal.
With loss-free media it would be possible to have Xs without
Rs, and that condition is approached when using a polythenecoated smooth copper surface. It transpires that, for a thin
layer of dielectric over a flat surface of metal or one having a
radius large compared with the skin depth, the reactance due
to the dielectric coating may be added directly to the reactance
arising from the finite conductivity of the metal. As would
be expected, any increase of Rs increases the inclination of the
wavefront at the surface, measured from the normal, and
this in turn increases the phase velocity along the interface.
On the other hand, by analogy with electric circuits we might
reasonably anticipate that the corresponding phase velocity
would be reduced by an inductive surface reactance and increased
by a capacitive one.
(3.1) Zenneck and Radial Cylindrical Surface Waves
Looking into the surface supporting a Zenneck wave we have
(2.3) The Sommerfeld-Goubau or Axial Cylindrical Surface
Wave
7
_
^*2
(14)
Sommerfeld5 was the first to point out that a transversely
cylindrical surface would support a surface wave. Goubau6
subsequently developed the idea in its application to a waveguide or for the radial form of cylindrical surface wave
consisting of a metal wire having a dielectric-coated or corrugated
z--*a
surface. The field distribution is shown in Fig. l(c) and it will
(15)
5
be seen that, when the radius of the cylindrical surface is increased
u
to infinity, the Sommerfeld-Goubau wave becomes identical in
These values of Zs are independent of the distance from the
form with the Zenneck wave. For a surface with constants surface but are defined for y = 0.
K
H>o> i> °\ surrounded by air the field components in cylindrical
co-ordinates are
Thus
+ ja2) . . . (16)
Zs = KJOJKQ/
O)KQ
(a) Inside the surface when r <; s.
R =A
so that
(17)
*
(11)
*n =
A
and
X =
OJKQ
WKQ
(18)
It is therefore clear that for either of these forms of surface
wave the quantity a2, representing the rate of decay of the field
with distance from the surface, is directly proportional to the
surface reactance, and b2 (the corresponding phase factor)
depends only on the surface resistance.
^
with eqns. (3) and (7) as for the Zenneck wave.
(b) Outside the surface when r ^ s.
(3.1.1) The Particular Case of an Inhomogeneous Plane Wave
supported by a Dielectric-Coated Metal Surface.
Hat62 =
(12)
ju2
with eqns. (5) and (8) as for the Zenneck wave.
(3) SURFACE IMPEDANCE AND ITS EFFECT ON THE FIELD
DISTRIBUTION OUTSIDE THE SURFACE
The surface impedance Zs is defined as the ratio of the
tangential components of electric and magnetic field at the
In the case of an inhomogeneous plane wave supported by a
dielectric-coated metal surface, we have three media to consider
forming a stratified system, the arrangement representing a
development of the Zenneck wave which is analogous to the
Goubau modification of Sommerfeld's axial cylindrical surface
wave. It is assumed that the interface between the metal,
designated as medium 1 with constants /x0, K{, a,, and a loss-less
solid dielectric coating representing medium 2 with constants
/x0, K2 lies in the plane y = 0 (see Fig. 2). If the thickness of this
dielectric coating, assumed uniform, is /, the interface between it
and the surrounding air or medium 3 having constants [xQ, K0
lies in the plane y = I.
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BARLOW AND CULLEN: SURFACE WAVES
332
y
Medium 1
Fig. 2.—Dielectric-coated plane surface.
Medium 1 (metal): m «= no. *i> <*iMedium 2 (solid dielectric): (j.2 = 1*0. «2. 02 = 0.
Medium 3 (air): n 3 = p.o, K3 = * 0 , 03 = 0.
In the metal and in the air the field must decay exponentially
with distance from the corresponding solid dielectric boundary.
Thus using the appropriate suffices (in this case 1 and 3) we
have the same equations for the field in these media as given
by eqns. (2) and (4) supplemented by the usual equations (3) and
(7) for the metal with eqns. (5) and (8) for the air. Within the
solid dielectric of medium 2 we have a standing wave whose
magnetic field is given by
(19)
Hz2 ~ (A'2 cosh w2j> + A2
and in this medium
y2 + u\ = K\ = -
making b3 proportional to the 3/2 power of the frequency and
a3 approximately proportional to the square of the frequency.
As would be expected with a thin coating of loss-less solid
dielectric, the surface resistance Rs depends upon A and therefore
on the conductivity, au of the metal beneath.
The surface reactance Xs, on the other hand, is made up of two
parts added numerically together:
(a) The part arising from the surface resistance and with the
present degree of approximation equal to that resistance.
(b) The part for which the layer of solid dielectric over the metal
is responsible.
It will be observed that the two parts are of the same order of
magnitude when the thickness of the dielectric coating is about
equal to the skin depth.
A surface wave requires that the decay factor a3 in the air
surrounding the guide should have a positive finite value, and
for the Zenneck wave or this modified form of it we see from
eqn. (26) that Xs must also be positive and finite. It will be shown
in Section 3.2.1 that for the Sommerfeld-Goubau wave the
required conditions can be satisfied for positive, zero or negative
values of Xs. A perfect conductor, for which al = co, forming
a perfectly flat surface and covered with a layer of solid dielectric
is still capable of supporting a modified Zenneck wave, because
a finite surface reactance remains with a corresponding finite
value of a3.
In general, the higher the surface reactance and the higher the
frequency the greater the decay factor a3, so that the field
becomes concentrated more closely in the immediate vicinity of
the surface (see Fig. 3).
Energy flow into
and out of reactive
surface
Wet power
flow
(20)
Net power flow
Providing for the matching of the field components at the
boundaries between the different layers yields
CT) +
tanh u2l = —
/
"2*0
ja)Kx)\U3K2)
JU)K2 \ /«A
(21)
VCTj + j(xJKiJ\U2J _
and assuming that / is small, so that tanh u2l~u2l
we find
t\\\\\\\\\\\l
Slightly reactive lossless
surface
AAA
Highly reactive lossless
surface
(a)
and u3l < 1,
£ ] - , * } = a, -jb, . (22)
where the skin depth in the metal is
-VG
(23)
Dielectric coating of surface
to increase surface reactance
(b)
The impedance looking into the surface of the solid dielectric is
(24)
which corresponds to eqns. (13) and (14), so that
Corrugation of surface
to increase surface reactance
(25)
Fig. 3.—Methods of producing enhanced surface reactance.
and
=
——
= W^O
( —
(26)
(a) Effect of surface reactance on electric-field distribution.
(b) Dielectric-coated metal surface.
(c) Corrugated metal surface.
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333
BARLOW AND CULLEN: SURFACE WAVES
As an alternative to the dielectric coating of a surface to
produce an enhanced surface reactance, corrugations of the
metal may be used. In order to give the effect of a uniformly
distributed supplementary surface reactance, the pitch of the
corrugations must be smaller than the wavelength along the
interface. The behaviour of such a corrugated metal surface in
regard to surface reactance has been examined theoretically by
Walkinshaw,7 but no analysis was attempted by him of the
corresponding surface resistance. Morgan8 has made a calculation for the microwave power loss in a corrugated conducting
surface of defined contour, and some idea of surface resistances
can be obtained from this work. It has generally been assumed
that for small surface roughnesses Rs and Xs are increased in
the same proportion, but this can be only an approximation
and there is clearly a need for further investigation, particularly
of an experimental character.
xlO
20
(3.2) Sommerfeld-Goubau Wave
Provided that the cylindrical surface supporting the wave is
circularly symmetrical, it is unnecessary to define its constitution
for the purpose of calculating the surface impedance. Looking
into the surface, which we will suppose is medium 1, from the
surrounding air or medium 2, we have for r = s
10
.
H
•
rl
/
It
18
I!
16
/
14
i i
1
12
(iv)
y
• (27)
02
It is of interest to observe that when s->co we find
Zs = J{U2IOJKQ), which is eqn. (16) for the Zenneck wave
associated with a flat surface.
In many practical arrangements using the Sommerfeld-Goubau
wave supported by guides of small diameter, \ju2s\ < 0 - 0 5 , and
we can therefore apply the small-argument approximations for
the Hankel functions, giving
Zs = Rs + jXs
and remembering that
= -
OJK0
iog£ (O-89u2s) .
(28)
//:''A
/ . - • • '
V/
/
/
-2
(a)
(5)
u2 = a2 — jb2
we find
2
1
\(% - «2) tan- b: - 2a2b2\ogt [0'
COKQ I
Jiii)
3
Values of (a 2 s)
xlO
a2
. . . .
(29)
and
Fig. 4.—Curves showing the dependence of (ais) and (bis) on surface
resistance and reactance, for Sommerfeld-Goubau wave.
(«)
1
Xs = J—
ila2b2 tan" *?a + (b\ -aplog e [0-89V(«l + *
OJKQ y
2
. . . .
(30)
In order to examine the effect of Rs and Xs on the values of
a2 and b2 at a given frequency, it is convenient to plot, from
eqns. (29) and (30), curves of (COKOS)RS and (COKQS)XS against
the corresponding a2s and b2s quantities, thus making these
curves of general application at a given frequency.
It will be observed that, since (ajKQs)Rs — (2TTSIX0)(RSIZ0) and
(COKQS)XS — (27TSIX0)(XJZ0), where Z o is the free-space TEM wave
impedance and Ao is the free-space wavelength, the co-ordinates
of the curves become dimensionless. Fig. 4 shows families of
such curves representing (2TTSI\Q)(RSIZ0) and
(2TTSIA0)(XSIZ0)
plotted against a2s for a number of different values of b2s.
(3.2.1) Conditions for the Support of a Sommerfeld-Goubau Wave.
As previously mentioned, a smooth metal surface gives
Xs ~ Rs, representing a point of intersection of the corresponding
curves for any given value of b2s as indicated in Fig. 4. For
(2TTS/XO)«J/2TO for (b2s)
=
0.
(6) (.2nslXo)R,IZo for (b2s) = 10-*.
(c) (2ns/\o)R*IZo for (6 2i ) = 2 X 10" 4 .
(d) (2nsl\o)X,IZo for (b2s) = 0.
(e) (2nsl\o)XslZo for (b2s) = 10-«.
(/) (2nslX0)X,IZ0 for (b2s) = 2 x 10-*.
(i) Smooth metal surface with (b2s) = 10—'.
(ii) Region of added capacitive reactance,
(iii) Region of added inductive reactance,
(iv) Smooth metal surface with (b2s) = 2 x 10- 4 .
values of a2s larger than that of the smooth metal surface we
must have an enhanced inductive surface reactance, whereas for
smaller values of a2s capacitive reactance is required, and this
will tend, in the first place, to neutralize the inherent inductive
reactance arising from the surface resistance. It is clear from
Fig. 4 that the net surface reactance can be positive, zero or
negative whilst still providing for finite positive values of both
a2s and b2s necessary to the support of the wave. The contrast
between this result and that obtained for the Zenneck wave is
rather striking. If we use the asymptotic expansion for the
Hankel functions in eqn. (12) it is easy to show that at a large
radial distance from the x-axis the wave impedance looking
towards the wire for a given radial propagation coefficient u2 is
ju2jcoK0 whatever the radius of the wire. For wires of large
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BARLOW AND CULLEN: SURFACE WAVES
334
diameter the impedance at the surface is therefore the same as
that for a plane surface supporting a Zenneck wave with the
same value of u2 [see eqn. (16)].
For wires of small diameter, however, the curvature of the
equiphase surfaces near the wire, shown in Figs. 6(d) and 6{e),
has an important effect on the wave impedance. The curvature
of the equiphase surfaces is such as to retard the phase of &x
and advance the phase of HQ, so that the wave impedance may
change from being inductive at a great distance from the wire
to being capacitive near the wire. Thus, in general, with a
cylindrical surface the wave impedance becomes more inductive
at greater radial distances from the surface. It is for this reason
that a bare copper wire, which has a very small inductive component of impedance at its surface, is a practicable guide for the
Sommerfeld surface wave, whilst a smooth bare copper sheet is
certainly not a practicable guide for the Zenneck type of wave.
Referring to Fig. 4 it will be observed that at a point of intersection of any pair of curves representing (27TSIAQ)(RS/Z0) and
(27TSIXQ)(XSIZ0), we find a2slb2s ~ 2 • 25, which gives an angle
of about /66° for ju2 in accordance with Goubau's calculations
for a smooth metal surface. This angle varies slightly with s.
(4) ATTENUATION AND PHASE VELOCITY ALONG THE
SURFACE IN THE DIRECTION OF PROPAGATION
In the air outside the surface all three forms of wave comply
with the relationship
= K\ = u\\ =
. . .
(8)
surface, which is not always easy to achieve. With a surface
wave it means a large value of a2, and as shown by eqn. 33 this
increases a.
(4.1) Behaviour of the Zenneck and Radial Cylindrical Surface
Waves
Since the Zenneck and radial forms of surface value [see
Figs. l(a) and 1(6)] behave in exactly the same way, so far as
attenuation and phase velocity are concerned, the following
comments apply to both.
Equating the tangential components of the electric field at the
surface for media 1 and 2 and introducing eqns. (7) and (8) we
get
"i
K\
,2 —
(37)
K\
:
2
(38)
~ K\ K\
Eqns. (37) and (38) yield respectively the values of u{ and u2
in terms of the electrical constants of the two media, so that in
air surrounding a smooth metal surface for which ax > COK{ we
have
~\i * ~
i)lL
=s
2CT
I
\~J M +
J
L
T
2or
i
= a2-jb2
. (39)
Thus b2 is slightly greater than a2, and it is easy to show that
Using eqns. (1) and (5) we therefore have, in general,
(40
y = a + jp = {b\ - a\ + K\ + j2a2b2)±
Hence from eqn. (36) we find the phase velocity over a smooth
metal surface to be given by
so that
(41)
and
and with b2 > a2 we must have vp0 > v.
It has already been explained in Section 3.1.1 that the effect
Now when K\ > {b\ — a2.) which is substantially true for of coating the metal with a thin layer of solid dielectric is to add
substantially to the surface reactance and to cause a correspondunloaded surfaces supporting microwaves, a and jS reduce to
ing increase in a2. Alternatively a similar result can be obtained
by corrugating the metal surface. With both arrangements an
(33)
a ~ -(a2b2)
increase in b2 is also to be anticipated, but the net effect is
CO
probably to make a2 change more rapidly than b2 causing the
where
(34) wave to slow down. For an inductively loaded surface we can
v—
therefore have vp greater than, equal to or less than v according
to the amount of loading applied.
0=
r
7
or the phase velocity along the surface
.(32)
(4.2) Behaviour of the Sommerfeld-Goubau Wave
Attention has already been drawn to the fact that for the
CO
Sommerfeld-Goubau wave supported by a smooth metal surface
(36) (a slb s) ~ 2 • 25, and it follows from eqn. (36) that v < v.
«2
2
2
p
1 Inductive loading of the surface increases the ratio a2sjb2s be2co2 :« - 4) +"2o?rJ
yond the value of 2 • 25, and consequently slows the wave still
To reduce the attenuation of a waveguide system we require, further. With capacitive loading it should be possible to
as a rule, to keep as much of the field as possible out of the increase vp to v or even beyond that figure.
guide, which is inherently lossy, and accommodate it in the
Measurement of vp or the equivalent guide wavelength offers
surrounding air.
a prospect of estimating a2 for a highly reactive surface having
A high-conductivity metal guide allows very little field pene- a2 > b2.
From eqn. (36) we find vp = v when
tration, and Chandler9 has shown that a dielectric guide is
capable of supporting a form of surface wave (a dipole mode)
a2
with very small losses when the field is for the most part outside
or b2s — a2s
the surface. At the same time we require for practical purposes
to concentrate the field in the immediate vicinity of the guide
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335
BARLOW AND CULLEN: SURFACE WAVES
It is of interest to observe that for a given value of b2s the
surface reactance Xs with a2s = b2s is the same as the surface
resistance Rs with a2s = 0.
(5) THE COMPLEX BREWSTER ANGLE AND ITS RELATION
TO THE ZENNECK SURFACE WAVE
Let us consider first a homogeneous plane wave travelling in
the direction PQ and incident at an angle if, on the flat surface
The physical significance of this complex Brewster angle
becomes apparent when we replace ip by (if, — yX) in eqn. (42).
Since there is no reflected wave, the equation must give the total
magnetic field above the surface.
Thus
ff , = A€->at€~Po.y(sin <]) sinhx—7cosi>coshX)e — Pov(cosi|( sinhX+j sin^coshX)
. . . .
(46)
It will be seen that the field decays exponentially in amplitude
and suffers a progressive advance in phase with increasing distance above the surface. Furthermore, the wave is attenuated
in the x-direction whilst subject to a progressive lag in phase
along the interface.
These qualitative characteristics are those associated with the
Zenneck wave. Comparing eqn. (46) with eqn. (4) we see that
for the two waves to be identical we must have
Direction of
incident wave
u2 = j30(sin «/r sinh X — j cos ip cosh X)
= - yj80 cos (if, - yX)
y = j80(cos if, sinh X + j sin if, cosh X)
(47)
Using the relationship cos 2 z + sin2 z = 1, which is true for
complex as well as for real values of z, we find from eqn. (45) that
'//////////////////////
Medium 1
Direction of
transmitted
wave
cos (if, - jX) = 1/[1 + (/cr - jm)]*
sin (if, - jX) = (*r - ym)*/[l + (K; - jm)]l
(48)
and substituting in eqn. (47) we can derive expressions for u2
and y in terms of /cr and m.
When m > 1 we get
Fig. 5.—Plane wave incident on a flat surface at the Brewster angle.
Medium 1:
Medium 2:
"2 —
. «2 = KO> 0 2
of a loss-free medium, as shown in Fig. 5. The magnetic-field
component of such a wave above the surface is given by
#0
•
=
(42)
where
The two components of the electric field vary with x and y
in exactly the same way. In general a wave of this type will
give rise to both a reflected and a transmitted wave, but if the
angle of incidence has a certain critical value, known as the
Brewster angle, there will be no reflected wave.
This critical angle is given by
=
tan iff = J—
V K
(43)
0
Eqn. (43) may be written in terms of the intrinsic wave
impedances of the media above and below the surface, i.e. Z o
and Zl respectively, thus
tan if, =
(44)
Now suppose that the medium within the surface has a finite
loss, so that yco/Cj is replaced by (al + JCOK^ in Maxwell's
equations, or instead of Kr we then write (/cr — jm), where
m = OIIOJKQ. Since the analysis leading to eqn. (43) is valid for
real or complex values of the wave impedances, we conclude that
a plane wave incident at a complex angle* (if, — jX) will not
give rise to any reflected wave, provided that
-yX) = f?
= y/(Kr-jm)
z
(45)
i
• It will be observed that if both media are lossy and the angles of the two wave
impedances are the same, the Brewster angle will be purely real.
which, apart from changes in notation, agrees with eqn. (39).
We also see that using the values of u2 and y obtained from
eqn. (47) eqn. (8) is satisfied.
Thus the values of u2 and y (which determine the form of the
field above the surface) given in Section 2.1 for the Zenneck
wave are identical to the corresponding quantities deduced from
the Brewster-angle analysis.* This provides a link with optics
which we shall find most useful in discussing the excitation of a
Zenneck wave. Moreover, the mathematical form of eqn. (46)
is convenient for obtaining surfaces of constant phase and
constant amplitude.
(6) EQUIPHASE AND EQUI-AMPLITUDE SURFACES
A simple plane wave propagated through a uniform lossy
medium of infinite extent suffers a reduction in amplitude and
retardation of phase as it progresses. The surfaces of constant
amplitude and constant phase are planes that coincide, and for
this reason such a wave is called "homogeneous."11 On the
other hand, waves exist for which equiphase and equi-amplitude
surfaces do not coincide, and such waves are called "inhomogeneous." The Zenneck wave and the Sommerfeld-Goubau
wave are both examples of inhomogeneous waves.
(6.1) Zenneck Wave
The magnetic field of the Zenneck wave given by eqn. (46) can
be expressed as follows:
H 2= AeJat€~$° sinh x(* cos ^+^ sin Wg-J'Po cosh X(x sinty-ycos ({/) # (46a)
* The complex Brewster angle, applicable to surface-wave theory, should not be
confused with what has been called the "pseudo Brewster angle." 10 That concerns
homogeneous plane waves only and is defined as the real angle for which the modulus
of the reflection coefficient at the surface is a minimum.
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BARLOW AND CULLEN: SURFACE WAVES
336
the field is to remain finite for all negative values of y, and this
medium may be regarded as providing a matched (reflectionless)
termination for the inhomogeneous wave incident upon it.
The amplitude is constant over planes defined by
x cos ifj + y sin ifj = Constant
and the phase is constant over planes defined by
x sin ifj — y cos iff — Constant
These surfaces are shown in Fig. 6(a).
y
Equiphase surfaces for
; Equi-amplitude surfaces
for
Guide surface
(a)
y
Equiphase surfaces y
fox H^ or ST
ii-amplitude
' surfaces for rLor#r
Guide
surface
Equiphase surfaces fbr#y
i-amplitude surfaces
Guide
surface
Ecpuiphase surfaces r.
for He or 67
]
cjui-amplitude
''surfaces for Hfior£.
^ '
Guide
surface,
Equiphase surfaces for # x
Equi-amplitude surfaces
7*7Guide
•^surface
(0
(6.2) Radial Cylindrical Surface Wave
The field components of the radial cylindrical surface wave
are given in eqns. (10). The field components H^2 and <?r2
have a radial variation which is different from that of the <py2component, so that we have two sets of equiphase and equiamplitude surfaces to consider. Let us first consider H^2:
- jx)r]
If the Hankel function is expressed in modulus and phaseangle form, surfaces of constant amplitude and phase can be
found, as in the preceding Section. However, tables of Hankel
functions for complex arguments are required, and these are not
complete. For small values of a, a reasonable approximation
for the modulus is |H(j2)(/3r)|e~ar, and the correction to the phase
angle is small, being of the order arctana//3. The result is
shown in Fig. 6(b).
Similar methods applied to the electric field <sv2 give the curves
shown in Fig. 6(c).
(6.3) Sommerfeld-Goubau Wave
The field components for the Sommerfeld-Goubau wave are
given by eqns. (12). The equiphase and equi-amplitude surfaces
are qualitatively similar to those of the preceding case, and are
shown in Figs. 6{d) and 6(e).
(7) EVANESCENT STRUCTURE OF THE FIELD OVER THE
WAVEFRONT OF SURFACE WAVES
Attention has already been drawn to the need for an evanescent
field distribution over the wavefront of surface waves as a
necessary condition in avoiding radiation. We shall now
examine the structure of a Zenneck wave in relation to the field
distribution between metal plates operated beyond cut-off.
Let us first express Hz2 as given by eqn. (46a) in terms of a
new co-ordinate system x', y', z, derived from the original one
x, y, z, by clockwise rotation about the z-axis through an angle
(TT/2 — iff), as shown in Fig. l(a). The co-ordinates of a point P
Fig. 6.—Equiphase and equi-amplitude surfaces.
(a) Zenneck wave.
(b) and (c) Radial cylindrical wave.
(cl) and (e) Sommerfeld-Goubau wave (axial cylindrical surface wave).
It follows that in loss-free media the planes of constant
amplitude and constant phase above the surface are orthogonal.
The direction of propagation of this inhomogeneous wave is
defined by the normal to the wavefront, and therefore makes
an angle I/J, representing the real part of the complex Brewster
angle, with the y-axis.
The physical behaviour of the Zenneck wave can be interpreted
very readily with the help of these results. Since there is no
reflected wave above the surface, the medium below the plane
y — 0 could be removed without affecting the field in the region
y > 0, and we need consider only the propagation of the incident
wave.
This wave travels without attenuation in a direction normal
to its wavefront. The decrease of amplitude with increase of x
can be interpreted as arising from the exponential variation of
amplitude across a wavefront as the wave sinks through the
plane y = 0. [See Fig. 7(a).]
The presence of the medium below the surface is necessary if
(50)
Net power flow
, Reactive and resistive surface
Fig. 7A.—Zenneck wave sinking into a lossy surface.
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BARLOW AND CULLEN: SURFACE WAVES
in the plane z = 0 are x, y in the old system, and x', y' in the
new one, where
x' = x sin «/r — y cos iff
(51)
y' = y sin tp + x cos
Substituting these values in eqn. (46a) we find
• (52)
s—/Bn*'coshX
This equation owes its simplicity to the orthogonal relationship
between the planes of constant phase and constant amplitude,
which we have already observed.
Now consider the magnetic field of an evanescent E01 mode
between two parallel perfectly conducting metal plates separated
337
wave complying with these conditions must inevitably have an
evanescent structure in the direction parallel to its equiphase
planes.
We have now established that the field pattern of the Zenneck
wave and of the evanescent E01 mode between perfectly conducting parallel plane sheets are very closely related, and can
be described respectively as a travelling inhomogeneous plane
wave and a standing inhomogeneous plane wave. A knowledge
of the behaviour of evanescent modes in waveguides can therefore be brought to bear on the physical behaviour of the Zenneck
wave. This is particularly valuable in considering the effect of
a small amount of curvature of the surface over which the wave
is propagated.
(8) TRANSMISSION-LINE MODES AND THEIR RELATION
TO SURFACE WAVES
The parallel-plate transmission line is an important form of
waveguide, and in examining its behaviour it is usual to assume
that the plates are perfectly conducting and separated by a
distance small in comparison with the wavelength. In such
circumstances the only propagating mode is the simple TEM
wave.
With plates of finite conductivity, a longitudinal component of
the electric field must exist, as shown in Fig. 8(a), and the mode
is no longer strictly TEM. We observe, in the first place, that
the wave of interest is one for which the current flow is equal
and opposite in the two conducting plates, so that the magnetic
-I
f-f
Fig. 7B.—Evanescent structure of Zenneck wave over the wavefront.
by a distance d and of infinite extent in the planes x' = ± d\2,
as shown in Fig. lib). In this case we have
'k
o
T x
6
•
Medium 1
i
• (53)
I
+1
Medium 2
Medium 1
where
(54)
Eqn. (53) describes the evanescent EOi mode in terms of a
standing-wave pattern formed by a pair of inhomogeneous plane
waves travelling back and forth along the x'-axis between the
guiding plates, without making any net forward progress. (If
the separation between the plates is increased so that the mode
is no longer evanescent, we get the usual zig-zag picture of
waveguide propagation.)
Comparison of eqn. (52) with the first term in the brackets
of eqn. (53), representing the forward-travelling component wave,
shows that these are identical when
Tt\d = j80 cosh X
and
<xc = j80 sinh X .
Medium 2
(55)
(56)
Eqn. (55) will be satisfied if the plate separation d is suitably
chosen, and eqn. (56) then follows from eqns. (54) and (55),
since cosh 2 X — sinh 2 X — 1.
Thus the Zenneck wave is identical to the correspondingly
directed component wave of the evanescent E 01 mode between
parallel perfectly conducting plates.
The plate separation d given by eqn. (55) is necessarily less
than Ao/2, since p0 = 2-TTIXQ and cosh X > 1. It follows that in
the Zenneck wave the separation between the equiphase planes
representing the half wavelength in the ^'-direction for which
the phase difference is v must also be less than AQ/2. Any plane
VOL. 100, PART III.
Medium 1
z
Medium 1
Fig. 8.—Parallel-plate transmission-line wave modes.
Medium 1:
|/,o, i, a;.
;
Medium 2:
no, *2 — KQ, 02 •= 0.
(a) Plates close together.
(6) Plates separated by large distance.
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23
BARLOW AND CULLEN: SURFACE WAVES
338
field has the same direction and magnitude at the surface of each, conducting material, i.e. from a lossy dielectric, so that a{ <
giving even symmetry about the plane y = 0. Thus, in the space binomial expansion of eqn. (62) gives
between the plates, i.e. between the planes y = ± d, we have
Hz2 =
=
-v* cosh
A(-^2-
\jo>i€-v* sinh utf
\J(X)KQ/
.
> . . .
.
OJKU
. (63)
(57)
Transmission-line analysis again leads to the same result, the
first equation (63) corresponding to the term GZ0/2 of eqn. (60).
So far we have assumed that \u2d\ is a small quantity and
together with eqn. (8).
tanh u2d ~ u2d.
If d is increased indefinitely this approximation
For the limiting case in which a{ = oo, u2 is zero and y = yj80, is no longer valid, and for sufficiently large plate separations the
so that eqns. (57) reduce to the well-known expressions of the approximation tanh u2d ~ 1 becomes appropriate. As the plate
TEM wave. It is reasonable to assume, therefore, that if ax is separation, Id, tends to infinity it transpires that the propagation
large \u2\ is small and y~yjS 0 . Provided that u2d is small coefficient, y, tends towards the value we have found for the
enough to make the approximation tanh u2d ~ u2d applicable, Zenneck wave. The field components are then strong near the
we find by matching thefieldsin the two media at the surface of plates and resemble the Zenneck wave very closely. The electricthe plates that
field pattern under these conditions is as shown qualitatively in
Fig. 8(6). In the central overlap region there is some interaction
.
2 2
. (58) between the individual Zenneck-type fields associated with the
y two plates, and it is the magnitude and phase of this interaction
which governs the extent to which the propagation coefficient y
Putting y = a + yj8 = y'j80 + 8 and treating 8 as a small of the system departs from that for a pure Zenneck wave.
(complex) quantity, which implies that a{ > o>/c0, we find
We can correlate these results by considering how the TEM
type of wave between plates of finite conductivity varies with
the separation between the plates. Three main types of propa. . . . (59) gation can be distinguished, depending on the plate separation
Id in relation to the two characteristic lengths, A3/A§ and Ag/A.
(a) If the separation is exceedingly small, so that Id < A3/AQ'
the wave is essentially of the TEM pattern within the metal, and it
Thus j8 > jS0 and the phase velocity is less than . . „ „ „ .
is, of course, very rapidly attenuated, falling to 1/e of its initial
The corresponding approximate transmission-line theory yields
value in a longitudinal distance A measured in the direction of
increasing x and with very small phase velocity. Practically no
travels in the narrow air-gap between the plates.
« = £- + ^r
(60) energy
(b) If A3/AJ) < Id <Ao/A we have the usual TEM type of wave
normally assumed in transmission-line analysis. The attenuation is
where R = Loop resistance/unit length (both "go" and "return")
generally small, and the phase velocity is slightly less than that of
of the plates.
light. Almost all the energy travels in the region between the plates.
G = Dielectric conductance per unit length.
(c) If 2d > Ao/A, the wave tends to separate into two isolated
Zo = Characteristic line impedance.
Zenneck waves, one associated with the top plate and the other with
the bottom plate. The attenuation is very small and the phase
In the present case,
velocity slightly greater than that of light. Most of the energy
travels in the medium between the plates and close to the plates
(within a distance of about Ao/A from each plate).
JU)KQJ
V Kn
The three modes of propagation described in (a), (b) and (c)
are not, in fact, sharply defined, and in principle we can pass
continuously from one to another by varying the plate separation.
In particular, the Zenneck wave becomes the limiting case of
and
the field distribution near one plate of a parallel-plate transwhere A is the skin depth in the metal plates and is defined by mission line when the other plate is removed to a very great
eqn. (23). Substituting from eqn. (61) in eqn. (60) we find distance.
A similar analysis can be carried out for a parallel-wire transagreement with eqn. (59).
mission line of finite conductivity, which shows, as might be
The change in phase velocity due to finite conductivity, which expected from the foregoing results, that when the wire separation
is predicted by eqn. (59), also follows from the usual trans- is increased to a large value the TEM type of wave mode passes
mission-line analysis if we take into account the internal re- over into two axial cylindrical surface wave modes, one on each
actance of the conductors. The resistance of the conductors, wire. Thus surface waves can be regarded as a limiting case of
as such, has only a third-order effect on the phase velocity.
TEM waves on 2-conductor systems when the conductors are
Returning to the general equation (58) we observe that for separated by an infinitely large distance. We have therefore
sufficiently small values of d (small in comparison with A), we found another link between surface waves and better known
have
modes of propagation.
K
(61)
y = V t M ^ o f a i + J«> \)]
. . . .
(62)
This is simply the intrinsic propagation coefficient for a plane
wave within the plates, and it is clear that the infinitesimal air-gap
between them has a negligible effect.
If we now assume that the plates are made from a very poor
(9) LAUNCHING
The waves, whose characteristics have been discussed in the
preceding Sections, are known to satisfy Maxwell's equations.
Although this is a necessary condition for a wave to occur in
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BARLOW AND CULLEN: SURFACE WAVES
339
practice, it is not a sufficient condition. For example, a homogeneous plane wave (the simplest wave solution of Maxwell's
equations) is not physically realizable, because it entails a source
Zenneck/^ I
of infinite extent and an infinite amount of power. At a sufficient
wave / 1^A.
distance from any finite aperture the field produced in free space
Region of diffracted
must take the form of an outward-travelling spherical wave,
field only
known as the "radiation" field. This field represents leakage
of the energy which we wish to guide by means of a surface
. ^ v l 7 Region of
(a)
wave, and therefore an important aspect of the problem of
^ . ' T direct field
excitation is to make the radiation field from the launching
rM 01 \
aperture as small as possible. Unfortunately, the theory of
excitation of surface waves is not yet fully established, and we
shall be restricted to a qualitative discussion of a particular
2-dimensional case which has been worked out in detail by
Macfarlane.12
It is convenient to adopt the Brewster-angle approach to the
theory of the Zenneck wave, and to start with a semi-infinite
loss-free dielectric medium below the surface. We know that a
homogeneous plane wave incident on such a surface at the
Brewster angle does not give rise to any reflected wave. Let
us consider the arrangement shown in Fig. 9(a) employing a
limited aperture of height h from the surface, illuminated by a
plane wave incident from the left.
If the wavelength is small in comparison with the height of
the aperture, optical-ray theory will give a first approximation
to the effect of the screen, namely that below the "shadow line"
AB there will be a plane wave travelling down into the dielectric,
whilst above AB and to the right of B there will be no field at all.
In the second approximation the finite wavelength produces
an edge effect, well known in optics and in the theory of radio
aerials. Thus the sharp shadow line of ray theory is blurred
by the effect of diffraction to an extent which increases with
increasing wavelength. Moreover, for a given wavelength, the
blurring becomes more pronounced with increasing distance from
the aperture. This is shown qualitatively in Fig. 9(b). A more
detailed discussion of this phenomenon is given by Slater and
Frank.13
To an observer situated on the surface and moving from O to
B the field is substantially constant except in the vicinity of B,
where it begins to show variations of amplitude with position
as indicated in Fig. 9(c). Beyond B, the field, which is then
plane wave launched at the Brewster angle
solely caused by diffraction, falls off very rapidly at first with Fig. 9.—Inhomogeneous
from a limited aperture.
2 2
increasing distance, eventually settling down to vary as x~ l at
(a) Illumination of surface from a limited aperture.
sufficiently large distances, where it represents the 2-dimensional
radiation field.
Let us now suppose that the medium below the surface is
(<?) Field distribution in the horizontal plane for lossy surface.
slightly lossy (e.g. the earth). The Brewster angle then becomes
complex, so that the incident wave must have an exponential greater than its height, so that only the finite dimension normal
taper of amplitude across a wavefront. Qualitatively, the to the surface is significant in determining the range of the
argument is unchanged.
surface wave. At distances large in comparison with the aperture
If the appropriate Zenneck wave is incident on the left-hand width, thefieldmust become a spherical wave as already indicated.
side of the aperture the field on the right will again be modified
by diffraction, as shown in Figs. 9(d), and an observer on the
ground would find a field variation substantially as shown in
(10) THE EFFECTS OF BENDS
Fig. 9(e). The exponential decrease of amplitude with distance
Practical
application
of surface waveguides must be restricted
between O and B at a rate characteristic of the appropriate
unless
arrangements
can
be made to "turn corners" without
Zenneck wave may be regarded as being due to the exponential
serious
loss
of
energy,
and
it is the purpose of this Section to
amplitude taper across a wavefront as the wave sinks down into
the lossy medium below the surface. Beyond B, the field dis- explain the physical phenomena involved.
A gradual change in boundary conditions usually results in
tribution does not resemble the Zenneck wave at all.
We therefore conclude that an aperture of finite height h, over a correspondingly slow transition from one wave mode to
which the field distribution characteristic of the Zenneck wave another. We may therefore reasonably expect to find that a
is maintained, will give rise to a wave approximating to the solution of Maxwell's equations exists which represents a wave
Zenneck form at ground level out to a distance h cot if* from circulating around a cylinder of large radius as in Fig. 10, and
which degenerates into the Zenneck wave as the radius tends to
the aperture, where </» is the real part of the Brewster angle.
It is assumed that the width of the aperture is very much infinity.
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BARLOW AND CULLEN: SURFACE WAVES
340
and for the first term of the series to give a useful approximation
we must have
-2n
/Wave/ front
Let us consider a circulating wave whose wavelength measured
circumferentially at the surface is approximately equal to AQ, the
free-space TEM wavelength.
Since v is nearly real if the circumferential attenuation is small,
it follows that for such a wave
v90 ~
where
2TT
r
o
or
Fig. 10.—Effect of curved surface on propagation of an inhomogeneous
plane wave.
In cylindrical co-ordinates, r, 9, z, the 2-dimensional wave
equation is
(64)
= o
<)02
A circulating wave will be described by a factor c-M, where,
in general, v is a complex number. (If v is real, the wave
circulates without attenuation.)
with the notation indicated in Fig. 10.
Thus the asymptotic expansion cannot be valid near the
cylindrical surface where p ~ p0, because p0 > 1 and the
condition 8p > (4J>2 — 1) is not satisfied.
This leads to a difficulty in the mathematical analysis of the
behaviour of this form of wave. We are interested in large
radii, say 1 m or more, and comparatively small wavelengths,
say A ~ 10 cm. For these figures v — 60, and no tables of
Bessel functions for such large arguments are available. However, the general behaviour of these functions is well known, and
is illustrated qualitatively in Fig. 11.
Therefore, above the surface, Hz must satisfy the equation
= 0.
. (65)
to which the solution is
€-J^
. (66)
At a sufficiently large distance, the first terms of the asymptotic
expansions of the Hankel functions may be used, and we have
^y-J^-i^-^ V>
. . . .
where
2TJT
.
.
.
(67)
. (68)
It will be observed that, for sufficiently large values of r, the
radial variation of the field becomes independent of v and is
represented simply by an outgoing and an incoming wave.
There is therefore no possibility of an evanescent decrease of
amplitude with increasing distance in the radial direction at
indefinitely large distances, whatever the boundary conditions
may be.
At great distances the field is composed of travelling waves
only, and on physical grounds we can reject the inward travelling
wave except in so far as it is required to supply losses in the
surface. However, the asymptotic expansion of the Hankel
functions is valid only if p > v.
In the present case this condition is not met near the cylindrical surface, and, in fact, the field there is quasi-evanescent, as
we shall see.
More precisely, the asymptotic expansion for the second kind
of Hankel function is
(69)
Fig. 11.—Curves of Bessel functions.
For values of p much less than v, the Hankel function
Hv2)(p) = Jv(p) — jYv(/>) is almost purely imaginary, and in
eqn. (66) describes a reactive storage field of a quasi-evanescent
type. For p much greater than v this Hankel function is
oscillatory and represents an outward-bound wave.
A simple physical interpretation of this result can be given.
Curvature of the surface may be regarded as bringing about an
increasing separation between adjacent equiphase planes* with
increasing distance from the surface, as shown in Fig. 10. Near
the surface, the separation is slightly less than A0/2, but as the
radial distance increases a point will be reached at which the
separation becomes greater than AQ/2.
The evanescent character of the field then disappears, and the
energy is no longer trapped in a surface-wave mode but is partly
radiated outward.
* To a first approximation it is sufficient to neglect possible curvature of the equiphase surfaces.
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BARLOW AND CULLEN: SURFACE WAVES
For surface waves over a plane surface we have given an
analogy with the E01 mode between parallel plates. In the
corresponding analogy for the present case with a curved surface
the plates must diverge at an angle of Ao/2ro radians. A pair of
diverging plates will support a propagating mode where the
plate separation exceeds A0/2, but the mode will be evanescent
where the plate separation is less than AQ/2.
An estimate of the permissible curvature can be made as
follows. If the phase velocity at the surface is vp, the separation
between two adjacent equiphase planes is (XQI2)(VPIV). At a
height h above the surface the separation increases to
(X0vpl2v)(l + hlr0). The critical height hc is reached when the
separation is equal to AQ/2, and so
evanescent than before, so that no radiation will occur in that
direction.
(11) ACKNOWLEDGMENT
The authors wish to express their thanks to the many friends
and colleagues with whom they have had helpful discussions on
this subject. In particular, Dr. G. G. Macfarlane, Mr. A. E.
Karbowiak and Mr. J. C. Parr have, in this way, made a real
contribution to the substance of the paper.
(1)
(70)
(2)
If this height is much greater than the height at which the
field falls to an insignificant value for the flat surface, i.e. if
hc > l/fl2 the loss of power by radiation will be small. Using
the relationship
(3)
K = rJ —
(4)
(8)
and writing y—jfi with u2 — a2, which is reasonable if the
attenuation coefficient a is small, then
«2 = V(|82 - ©
•»-
£)-'
(5)
(6)
(7)
The condition hc > lja2 is then seen to be equivalent to
-3/2
2W2W,
(8)
/
For example, if vp\v = 0-995, we find r0 > 1 000 AQ/TT, whilst
if vplv = 0-98 we find r0 > 125^/77. It is obvious that the permissible curvature depends very markedly on the amount by
which vplv differs from unity, and a reduction in phase velocity
to a comparatively low value by increasing the surface reactance
will help to cut down radiation loss at bends. To calculate the
power radiated at bends is a much more difficult matter, and is
beyond the scope of the paper. The same physical ideas can
be applied to the cylindrical surface wave supported by a guide
with longitudinal curvature, but in this case the equiphase
surfaces will be cones coaxial with the wire. Energy will be
radiated outward from a bend, away from the centre of curvature.
On the inside of the bend, the field will be more strongly
341
(9)
(10)
(11)
(12) REFERENCES
A.: "Uber die Ausbreitung der Wellen in der
drahtlosen Telegraphic," Annalen der Physik, 1909, 28,
p. 665.
BOUWKAMP, C. J.: "On Sommerfeld's Surface Wave,"
Physical Review, 1950, 80, p. 294.
BOOKER, H. G., and CLEMMOW, P. C.: "A Relation between
the Sommerfeld Theory of Radio Propagation over a
Flat Earth and the Theory of Diffraction at a Straight
Edge," Proceedings I.E.E., 1950, 97, Part III, p. 18.
ZENNECK, J.: "Uber die Fortpflanzung ebener elektromagnetischer Wellen langs einer ebener Leiterflache und
ihre Beziehung zur drahtlosen Telegraphie," Annalen der
Physik, 1907, 23, p. 846.
SOMMERFELD, A.: "Uber die Fortpflanzung electrodynamischer Wellen langs eines Drahtes," Annalen der Physik
und Chemie, 1899, 67, p. 233.
GOUBAU, G.: "Surface Waves and their Application to
Transmission Lines," Journal of Applied Physics, 1950,
21, p. 1119.
WALKTNSHAW, W.: "Theoretical Design of Linear Accelerator for Electrons," Proceedings of the Physical Society,
1948, 61, p. 246.
MORGAN, S. P.: "Effect of Surface Roughness on Eddy
Current Losses at Microwave Frequencies," Journal of
Applied Physics, 1949, 20, p. 352.
CHANDLER, C. H.: "An Investigation of Dielectric Rod as
Waveguide," ibid., p. 1188.
BURROWS, C. R.: "Radio Propagation over Plane Earth.
Field Strength Curves," Bell System Technical Journal,
1937, 16, p. 45.
STRATTON, J. A.: "Electromagnetic Theory" (McGraw-Hill,
New York, 1941), p. 516.
SOMMERFELD,
(12) MACFARLANE, G. G.: "Notes on the Launching of a
Zenneck Surface Wave over the Earth," T.R.E. Memorandum No. 472. (By courtesy of the Ministry of
Supply.)
(13) SLATER, J. C , and FRANK, N. H.: "Electromagnetism"
(McGraw-Hill, New York, 1947), p. 182.
DISCUSSION BEFORE THE RADIO SECTION, 22ND APRIL, 1953
Dr. G. G. Macfarlane: The authors have described the properties of surface waves as if a surface wave can exist as a
complete field all by itself, just like an H01 mode in a perfectly
conducting waveguide. From the behaviour of the surfacewave field at large distances from the surface, it is immediately
clear, however, that this cannot be so, and that the surface
wave is a pseudo-mode and not a pure one. The surface-wave
field is an incoming progressive wave, whereas the field at large
distances from the surface must be an outgoing progressive wave.
There must therefore always be a radiation field associated with
the surface-wave field. The only exception could be if the field
were produced by an aperture of infinite extent. The qualitative
discussion of Section 9 of the paper by Prof. Barlow and Mr.
Karbowiak illustrates this point.
This leads one to consider how the contribution of the surface
wave to the total field can be enhanced. This is indeed a big
question, on which I hope the authors will have something
further to say. There is one facet of it, however, on which I
would like to remark. It concerns the ease with which a surfacewave-like field can be launched along a thin resistive wire, but
the difficulty in launching it along a planar surface of the same
material. The problem is to understand the changes in the field
as the radius of the wire is increased indefinitely. In order to
find out the answer one can write down the expression for the
field due to a ring source of dipoles coaxial with the wire (Fig. A).
This can be expressed as an integral with respect to the propagation coefficient along the path of integration shown in Fig. B.
The integrand has a pole at the value of y for which the
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