Appl. sci. ires.

Section 13, Vol. 10

THE TRANSIENT ELECTROMAGNETIC FIELD FROM AN ANTENNA NEAR THE PLANE B O U N D A R Y OF TWO DIELECTRIC HALFSPACES I. F I E L D

REPRESENTATIONS.

by N. J. VLAAR Vening Meinesz Institute for Geophysics and Geochemistry, Rijksuniversiteit Utrecht, Netherlands

Summary The properties of the transient field of a vertical electric a n t e n n a placed near or in the plane interface of two semi-infinite dielectric half spaces can be examined by means of the integral representations (5.4) for a transmitter action represented by the unit step function U(t). The expressions (5.4) are solutions of the wave equation for the medium for which they are derived and they satisfy the boundary conditions at the interface of the two media. The behaviour of the field depends on the location of the branch points of the denominator-radical in the relevant integrands, OI1 account of the approach by means of the theory of generalized functions, the location of these branch points relative to the path of integration is rigorously determined. The integral representation for the primary field, according to § 6, is the field t h a t results from a transmitter moment which varies in time like the Heaviside unit step function. I t is further demonstrated in sections 7 and 8 that the reflected and refracted fields vanish for t < 0. As a consequence of the foregoing we may conclude that the total field (5.4 a, b, c) constitutes a unique solution of the radiation problem treated in this paper. The exact determination of the above-mentioned branch points also provides a means for ascertaining the behaviour of the reflected field if n2 < h i , in which case a part of the wave front is of conical shape due to the appearance of critical reflection. For the refracted field in the lower medium it has been demonstrated that the wave front propagates according to Fermat's m i n i m u m time principle. All final integral representations of the field are integrals on a finite range. I t is also clear, t h a t the assumption of real refractive indices is necessary for the solution of the present problem. Non-real refractive indices, the

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353

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-

354

N . J . VLAAR

refractive indices depending on the circular frequency co, would not give rise to fixed branch points in the complex plane. Evidently the analysis in this case would have been impossible in the present paper.

§ 1. Introduction. Since S o m m e r f e l d 1) and W e y l 2) derived integral representations describing the electromagnetic field of a time harmonic dipole antenna, placed in the vicinity of the plane boundary of two halfspaces of different refractive indices, much attention has been given to associated problems concerning the propagation of radio waves. All work done on this subject was concerned with a periodically oscillating transmitter, until 1956, when V a n der Pol 3) examined the associated transient field. He considered the field emitted by an infinitesimal vertical electric dipole, whose moment varied in time like the Heaviside unit step function. Both the transmitter and the receiver were assumed to be placed in the interface of the two media, which, moreover, were supposed to be non-dissipative, i.e. to have zero conductivity. Van der Pol's paper stimulated the appearance of a series of papers, all of them treating the transient field from a vertical electric antenna situated in or near the interface of two loss-less halfspaces. P e k e r i s a n d A l t e r m a n 4) gave a detailed analysis of the field when the transmitter is in the interface and the receiver has an arbitrary location. De H o o p and F r a n k e n a 5) examined the reflected field of an elevated transmitter placed in the medium with the higher refractive index, which situation thus gave no rise to critical reflection. In a publication of V a n der Pol and L e v e l t 6), again an investigation is made of the field in the entire space for a transmitter located in the interface. Linking up with this publication, B r e m m e r 7) described the field of an elevated dipole in terms of contour integrals. The integrand of these integrals appears to be an algebraic function which solves the scalar wave equation for the medium under consideration, termed by B r e m m e r the "axially symmetric conical mode". In B r e m m e r ' s analysis, the field depends only on the singular points of the integrand and on the contours which are used. Consequently it is of crucial importance to determine exactly the position of these singular points. In certain cases lack of rigour leads to inconsistencies in the theory. In the present study

ELECTROMAGNETIC FIELD FROM AN ANTENNA. I

355

an attempt has been made to eliminate any flaws by a carefull analysis. As will be apparent from this short review the problem under discussion has been only partly examined. First, in the above-mentioned publications a special position of the transmitter or the receiver is assumed; secondly the field in the medium that does not contain the transmitter has not been investigated. Finally, no attention has been given to the reflected field in the rather complicated case of critical reflection. All methods mentioned above are based in some way or another on operational calculus, whereas our starting point is a Fouriersynthesis. Besides the above publications mention should be made of a paper by P o r i t z k y s) who expresses the primary field of the transmitter as a complex integral over plane waves. His approach is a generalization of the method of Weyl by assuming the time dependance of the transmitter moment to be arbitrary instead of periodic. However, this starting point does not lead to a proper analysis of the transient field. Instigated by Bremmer's publication, the object of the present study is to give a more rigorous and unified treatment, so as to arrive at a solution of the problem in its most general form. The field is determined for the case that the dipole moment is represented by an arbitrary, twice continuously differentiable function F(t), which is assumed to be zero for t <= 0. A special feature of this paper is the use of the theory of generalized functions in connection with Fourier-integrals, as has been advocated by T e m p l e 9) and L i g h t h i l l l ° ) . This approach, as will be proved, greatly improves the rigour of the analysis. The base of the analysis is to consider the Dirac delta function as a generalized function given by a Fourier-integral in which a parameter provides for the good behaviour of the generalized function concerned. Starting with the formal integral representations in connection with a time harmonic field, the final results are obtained straightforwardly by means of a Fourier synthesis. Thus field representations are found for the field in connection with a transmitter moment varying in time like the Heaviside unit step function. The field for an arbitrary F(t) can then be obtained through convolution. The value of the field representations, which are complex integrals,

356

N.j. VLAAR

depends, as in B r e m m e r ' s paper, on the location of the singularities of the integrand, which are now determined rigorously. Expressions are derived for the field in both media for an arbitrary position of the transmitter. The general validity of the results enables us to investigate the reflected field also in the case of critical reflection, and the refracted field in the medium not containing the transmitter. The only restriction limiting the applicability of the developed theory is that the assumption of real refractive indices has to be maintained.

§ 2. Fundamental equations. We are concerned with the problem of deriving the transient electromagnetic field emitted by a vertical infinitesimal electric dipole of finite (electric) moment. The transmitter is located at a distance h above the horizontal plane interface between two halfspaces filled with homogeneous isotropic dielectric media of different real refractive indices and so of zero conductivity. The three-dimensional space, x, y, z or r, % z, in respectively orthogonal cartesian- or cylinder coordinates, is divided in two halfspaces z > 0 and z < 0, separated by their common boundary z = 0; t represents the time; uz is the unit vector in the positive z-direction, (0, 0, h) the position of the dipole (h > 0), R0 = = [x 2 + y2 + (z -- h)2]½= [r 2 + (z -- h)2]~ the distance between an arbitrary point and the position (0, 0, h) of the dipole. Besides the usual notations for the field quantities, the following symbols will be used: nl and n2 are the refractive indices for the media concerned, c is the phase velocity of electromagnetic waves in vacuum, and F(t) the transient electric current of the dipole. For concreteness we assume that F is twice continuously differentiable for t > 0 and ---- 0 for t < 0. The "delta function" ~(x) should be taken in the sense that its integral on any real interval containing x = 0 equals unity. The electromagnetic field satisfies Maxwell's equations i.e. (in Gaussian units): 1 SH - - -- 0, c St

curl E + curlH

n~ SE _ c St

4~ c

dF(t) •

-

h ) . - - .

dt in the upper medium z > O,

ELECTROMAGNETIC FIELD FROM AN ANTENNA. I

357

and curl E +

1 8H c

-- 0,

8t 8E

curl H

c

--0

8t

in the lower medium z < 0. The horizontal components of E and/-/must be continuous at z = 0, and the field must satisfy radiation conditions, requiring a wave which is outgoing with respect to the transmitting dipole. The field quantities E and H are derivable from the Hertzian vector / / = H . Uz by means of the equations: E

n~ 8/-/

__

c

8t

~- c grad. d i v . / / ( x , y, z, T)

(1' --- 1,2)

(2.1)

and H = n~] curl

8/-/

Now H solves the scalar wave equations:

2

n~ 82 ) c2

8t 2

H -'

4x

-. d ( x ) . d ( y ) . d ( z - - h) F(t) cn~

V

n~ 8~ ) c2 8t 2 / 7 = 0 ,

z > 0, (2.2)

z<0.

The conditions imposed on E and H at z = 0, amount to the boundary conditions of continuity of n 2 H and OH/Sz at z -----0. The field which exists if nl = n2, (in which case the whole space is filled with the same dielectric) and which satisfies (2.2), can be verified to be given through: Ho =

F ( t - - Ronl/c) cn~Ro

,

Ro =/= O.

(2.3)

This field will be called the "primary field". If nl # n2, in order to satisfy the boundary conditions, we have besides the primary field H0 (2.3) in z > 0, a "reflected field" H I in z > 0, and a "refracted field" Ha in z < 0 instead of the field H0.

358

N. J. VLAAR

The field has to satisfy the boundary conditions: lim n~(IIo + HI) = lim n~II2, z--++ 0

~-->- 0

0

lim - - (/70 + II1) = lim z ~ + O OZ

z--+-O

OH2

(2.4)

~Z

§ 3. Preliminary theory on generalized/unctions. 3.1. The analysis of the problem under consideration will largely be based on the theory of generalized functions, especially in connection with Fourier analysis. For a complete understanding of the subject we m a y refer to a paper of T e m p l e 9) or to L i g h t h i l l ' s bookl0). Only the essentials of the theory are given here, in as far as it is necessary to this investigation. 3.2. The Dirac "delta function" is of particular importance for our analysis. This "function" is defined by the property +oo

f F(t -- r).d(r) dr = F(t),

(3.1a)

--oo

or +t~o

f F(-r) .6(t --r)d'r = F(t).

(3.1b)

--00

where F(t) is assumed to be a continuous real function of a real argument; in the present study we suppose this function to be zero for t < 0 and twice continuously differentiable for t > 0. The effect of the delta function combined with the integration performed on F(r) in (3. l b) is a linear operator on F(r) (a linear functional) which assigns to the function F(r) its value at ~- = t, namely F(t). Actually there is no function in the proper sense which has the properties required for the delta function. One way of overcoming this difficulty is to consider the Stieltjes-integral +00

f F(t -- "r) dU(r) = F(t), --oo

which leads to the same result if by U(t) we denote the unit step function, a discontinuous function which is zero for negative, and unity for positive t. 3.3. An alternative and more general approach is by means of

E L E C T R O M A G N E T I C F I E L D FROM AN A N T E N N A . I

359

generalized functions which are defined by sequences of arbitrarily differentiable functions. The generalized function 6(t) is then defined by all sequences of arbitrarily differentiable functions {bn(t)) which have the common property: +co

lim f F(t - r). n(r) dr ---- F(t). ~---~OO

--CO

Each of the sequences {6n(t)) with this property represents the generalized function ~(t). This definition enables us to differentiate 6(t) an arbitrary number of times. Hence, the k-th derivative of $(t), ~(~)(t), is defined by the sequences (6n(k)(t)} of which can be shown by repeated partial integration: +00

lim f F ( t - r ) . ~ ) ( . ) d r ~-->0o

=

--CO

+co

= (--1)~ lim f F(e)(t -- r) ~n(r) dr = (--1)k F(~)(t), U-+co

--CO

if at least F(t) is k times continuously differentiable. Likewise we have the k-fold iterated integral of 6(t), symbolically, The unit step function U(t) understood as a generalized function in the sense that +co

f F'(t -- r). U(r) dr = F(t) --oo

may now be differentiated, which results in 5(t) --

dU(t) dt

3.4. For our particular purpose we define the generalized delta function (~(t) by the sequence (~n(t)}"

(~n(t) = - i- Re ei~(~+i~") de). (Im to = 0) f

(3.2)

7~ 0

where (en} is a monotonously decreasing sequence converging to 0. For en we may take for example 1/n or 1In 2 with n as a positive integer.

360

N. j . V L A A R

Inserting 3n(t - - ~') defined by (3.2) into (3. lb) we arrive at +co

oo

dR) d T = 7~ --co

0 +co

+co

2~ --co

--co

from which it is proved that On(t) in (3.2) leads to the proper generalized delta function. We further remark that, by evaluating the integral (3.2) we obtain: 1

On(t) - -

Im - -

1

t + ien

--

1

en

2 ,

~ t 2 + en

(3.3)

which, in the limit as n -~ oo represents a familiar form of the delta function. 3.5. In order to make the theory outlined in the present section well applicable, we will in the following section first derive formal solutions of (2.2) in case F(t) = e ~'°t, subject to radiation conditions and the boundary conditions (2.4). These periodic field expressions (~o is the circular frequency) will be denoted b y / / 0 o , / / 1 ~ and//2o~. The field for a transient F(t) is then obtained through: +oo

//Fj = lim f F ( t - - "r).IIjo,(r, r, z) dl" n-+oo

--oo

in which co

II1o ~ - - -

Re

Ilto~(r + ien, r, z) d o

(i----0,1,2)-

(3.4)

7~ 0

B y / / j we denote the field associated with the unit step function U(t), also understood in the sense of a generalized function, namely: Hj = lim Hjn, n--~oo

where 0

//j,~ = ~/-Ijn.

(3.5)

ELECTROMAGNETIC FIELD FROM AN ANTENNA. I

361

The main object of the following sections is to derive the representations HI, since t h e y are better suitable for studying the field than are the Hjo. The field due to an arbitrary transmitter action F(t) is then obtained through: +co

HFj(t, r, z) = f F ' ( t -- r).IIj(r, r, z) dr.

(3.6)

--¢o

It should be noted, as will be shown later on, that the Hi(t, r, z) m a y have integrable discontinuities or discontinuous derivates with respect to t, r and z, which are associated with travelling wave fronts in the (r, z) space. As we have assumed that F(I) is twice continuously differentiable, the field expressions Hvj in (3.6) are twice continuously differentiable with respect to t. As far as differentiation with respect to the other variables is concerned, we remark that in virtue of the approach b y generalized functions, we can write (3.6) in the form: +co

/irrj = f F(a) (t -- r). Hi(-2) (r, r, z) dr, --oo

where Ha(-z) means twice iterated integration with respect to r. Hence (3.6) is twice continuously differentiable with respect to t, r and z and so constitutes a solution of our problem which leads to continuous field quantities E and H.

§ 4. The periodic/ield. 4.1. B y F1 we denote an oriented path in the complex plane (variable cos 7) from cos ~ = I along the real axis to cos y = 0, and subsequently from cos ~----0 along the imaginary axis to cos ~ ---- 0 -- i co (fig. 1). Let sin ~ = (1 -- cos27) ½be an analytic function with branch cut on [-- 1, + 1] of the real axis, where moreover Re sin ), > 0 in the lower half-plane (fig. 1). Let furthermore cos ~' = (1 -- nl/n 2 2 + nl[n2cos2~)½ 2 have a branch cut on [--(1 -- n~/n~)½, + (1 -- n~/n~)½] of the real axis if n2 <

nl, or on

[--i(n~/n~--1)½, + i ( n ~ / n ~ -

1)~] of the imaginary axis if n2 > nl, where in both cases Im cos ~,' < 0 in the third and fourth q u a d r a n t (fig. 2).

N. J. VLAAR

362

Re<

Re< o

o

Ira> o

Im
.........

+.;,~

Re > o

Re>o

Irn < o

Im > o

...=

_1o.~

Fig. I. T h e cos y-plane w i t h p a t h of integration F', b r a n c h cut of sin y, and the distribution of the signs of R e sin y and I m sin y.

Re > o Im>o

Re < o Im > o

7

+=

--1

|

b-I

Re < o Im
--1

Re>o Im
Fig. 2. B r a n c h cuts of cos y' for the cases n2 > nz and n2 < hi. Signs of Re cos ~' and I m cos y'.

E L E C T R O M A G N E T I C F I E L D FROM AN A N T E N N A . I

303

The cuts of sin y and cos ~' are situated in such a w a y as not to cross -P1. 4.2 a. Let co(o > 0, Im co = 0) denote the circular frequency. A formal solution of (2.2) in case F(t) = e i~t is given through the formulae

1

IIoo~- 2~c2n~

fdcos fd

St

Ft

0

•lc Iz--h[

• exp{icolt_

nl

cosT+

rsinycos~

,

(4.1a)

-nl - - r s i n T c o s ~ 0 1} ,

(4.1b)

6 2~

1 IIlo~- ~2~rc2n

afV(cosy) at

dcosyfd~o.

rz

•

exp{icolt_

-

0

nl ( z + h ) c o s y + -

c

c 2n

H2~-- 2:~c2n~

Ot [1 + V ( c o s ~ , ) ] d c o s ) , . /'z

o

• exp i~o t -- nl h cos )J + - - z cos c

d~0.

c

+--rsin~'cos~

,(4.1c)

c

where the suffixes 0,1 and 2 refer to the primary, respectively the reflected field in z > 0, and to the refracted field in z < 0. b. B y substituting in (4. I a) cos (9o -- v2) for cos ~0, and b y putting x = - - r cos ~0, y -~ - - r sin ~o, the exponent becomes: nl

t -- - - {x cos ~o sin ~, + y sin 9 sin )~ + ]z -- h] cos ~,}, c

(4.2)

which renders the integrand a plane wave solution of the wave equation for z > 0; (4.1 a) is then transformed into an integral over a continuous set of plane waves, which is essentially W e y l ' s ~ ) representation of the primary field caused b y a time harmonic point source. The form in curly brackets in (4.2) can be considered as the inner product (n. R0) of the vectors n --- (hi, n2, n3) (the w a v e normal) and R0 = (x, y, z -- h), where nl = cos ~ sin 7, ne = sin 9 sin ~, and n3 = cos ~.

364

N.J. VLAAR

The components of n m a y be complex. A rotation of the orthogonal Cartesian coordinate system with origin (0, 0, h) in the original coordinates, is a linear orthogonal transformatior~ given by a matrix with real coefficients. It is a well-known fact in linear algebra that under such a transformation the inner product (n. R0) is invariant, irrespective whether n is complex or not. For real vectors this result is trivial. On account of the foregoing, (4. l a) can be considered to be invariant under such a transformation also. Hence without loss of generality we then m a y put r = 0, [z -- h I ~ R0. The exponent in (4. l a), now being independent of ~, integrated with respect to 9 gives 2~. The integration on F1 amounts to the required result: exp(ho(t -- nlRo/c)}

IIo~o =

cn~Ro

(4.3)

Evidently, this field satisfies radiation conditions. c. The function V(cosT) in (4.1b) and (4.1c) is determined by the requirement that the field must satisfy the boundary conditions (2.4), which leads to n 2 COS 7 - - n l COS 7 '

V(cos 7) = n2 cos 7 + nl cos 7' '

(4.4)

where cos 7' is given by Snellius' relation nl sin 7 = n2 sin 7'. d. In virtue of I m cos 7 --< 0 and I m cos 7' --< 0 on /'1 the integrands of (4.1) are finite for [z[ = oo. Moreover, as the integrands are 0(e -i~c°~r) as cos 7 - + - ioo, where ~ is a positive quantitiy, the integrals (4. I b) and (4. lc) are convergent. R e m a r k s . (1) V (cos 7) appears to be the familiar Fresnel reflection coefficient encountered in the study of the reflection of plane waves at a plane interface. (2) By introduction of the integral formula for the Bessel function of order zero 2~

Jo(x) --- ~

exp {ix cos ~} d~o o

ELECTROMAGNETIC F I E L D FROM AN A N T E N N A . I

365

and substitution of the new variable ,~ = (nl/c) co sin 7, (4.1a) can be seen to transform into the well-known Sommerfeld expression for the primary field of a periodic point source. § 5. The transient field. 5.1. In the present section we will derive integral representations for the field when the transmitter moment is represented by the generalized unit step function U(t), by means of the theory of generalized functions outlined in § 3. Departing from the time harmonic field expressions (4.1), the successive steps to be taken will be performed for the primary field H0. As the analysis is completely analogous for HI and//2 their final integral representations will be given without proof. For transmitter moment F(t) subject to the conditions stated in § 2, the field can be obtained by means of the convolution integral (3.6). 5.2. Let the delta function d(t) be defined by the sequence of ~n(t) given by (3.2). The field associated with an(t) is obtained by means of (3.4), in which for Hlo~ we substitute the expression (4. l a) for Ho~. It then follows that 00

1 Refd

/r/0~ n

2~2c2nl

2~

.--0fdcosrfd,. St

0

• exp[i~o(t+ien

1~

nl [z

0

hlcos,+

nl r s i n y c o s 9 )/ . C

C

By interchanging the order of integration and differentiation, which is justified by the final result, the ~o-integration can be performed first, which in virtue of (3.3) leads to: i

0 2z~2c2nl S t " Im f d cos 7 • 1

Hoon--

P1 2~

f •

d~0 t + ien -- (nl/c) Iz -- hi cost + (nl/c) r sin• cos

0

The 9-integration amounts to evaluation of the integral 2~

I = f 0

d~ a + bcos~

. (5.1)

366

N.J. VLAAR

which gives"

(5.2)

I = 2 n / ( a 2 - - bZ) ½,

provided that a s - - b 2 is not negative-real, which condition, as en va 0, is satisfied on F1 in (5.1). Hence (5.1) transforms into" 1 Hoo~ =

+

- - -

0 _

_

.

nc~nl ~t

f

d cos y

[(t + ien -- (hi~C) Iz -- hl cos 7) - (n2/c2) r2 sin2 7] ½"

Im

(5.3)

Fx

The sign of the square-root in (5.2) is chosen negative while in (5.3) the denominator-radial is assumed to be + oo at cos 7 = + oo.

.... 9

..oo

i

J /

/

¢

/ / /"

/

/

./

Fig. 3. The ~ath of integration/' in the cos ~,-plane. Next we e x t e n d / ' 1 t o / ' (fig. 3) b y including a circular arc in the positive sense from - - i R to + R ( R > 0), where R is assumed to be sufficiently large to ensure that all singular points of the integrand are in the interior of the circle concerned. Then we let R tend to infinity. The contribution to the integral from the arc at infinity does not depend on t and consequently does not alter the value of HOOn, which is obtained b y differentiation with respect to t.

ELECTROMAGNETIC FIELD FROM ANTENNA. I

367

We now infer that, in virtue of (3.5), by leaving out the differentiation ~/~t the integral on/1 represents the field associated with U(t) ; a possible static term, not depending on t, must be taken into account. In the following, by verifying the formula for H0, it will be proved that this term equals zero, where, moreover, the choice of the sign of the denominator-radical in (5.3), and the interchanging of the order of integration, will be justified. The foregoing considerations now lead to the expression: Ho~

-

1 ( d cos y Im ~c2nl J E(t + ien(nl/c) [z - h] cos y) (n~/c 2) r 2 sin 2 7] ~'

-

P

associated with the transmitter moment Un(t), where

U(t) = lim Un(t). n---~O0

5.3. The total field for a unit step function pulse, where H1 and //2 are obtained in the same way as H0, can be demonstrated to be given through: Ho----

1

lim

~c2nl n--+oo

f • Im

d cos y I( t + ien (hi~C) ]z -- hi cos 7) -- (n~/c2) r2 sin2 Y]½'

(5.4a)

P

HI----

1

lim. aC2nl n~oo f V(cos 7) d cos Y (5.4b) • Im {It + ien (nl/c)(z + h) cos 7] 2 -- (n~/c s) r 2 sin2 y}~' nl

H2-

~C2rt~ n~oolim .

f

{1 + V(cos

d

• Im d{Et+ien(nl/c)h~os~-+~-~)~cosy~_~n~/c2)rg.sin2y}½. r

(5.4c)

The integrands of the expressions (5.4) satisfy the boundary conditions (2.4) and the scalar wave equations (2.2) for the media concerned.

368

N. J. VLAAR

If, moreover, we verify that (5.4a) equals U[t -- (nl/c) R0] cn~Ro

is.s)

and that HI and Ha vanish for t < 0, we are left with a unique solution of our problem. 5.4. If, in the expressions (5.4), F is deformed into the traject (1, oo) of the real axis, we have to take into account possible singularities of the integrands in the fourth quadrant. As for n -+ oO(~n -~ + 0), the integrands are real on (t, co), there is a zero contribution to the field from this traject of the real axis. Therefore the field will depend entirely on the above-mentioned singular points. If the fourth quadrant is free of singularities, the field will consequantly be zero. Hence it is of great importance to determine rigorously the position of these singular points during the limiting process n ~ oo. 5.5. The branch points of sin 7 (which are irrelevant because in 5.4 we have sin 2 7) and those of cos 7', do not enter into the discussion, as they will not be passed upon the above-mentioned deformation of F. The reflection coefficient V (cos ~) possibly has poles at the zeros of the denominator of (4.4), i.e., if n2 cos 7 + nl cos ~' = 0. This condition, however, because of the special choice of the branch cut of cos )/ and the distribution of the signs of I m cos ~,' and Re cos 7' in the complex plane (fig. 2), cannot be satisfied, and so, there are no poles of V(cos ~) to be taken into account. The only singular points that are significant for our analysis, prove to be the branch points of the denominator-radical in (5.4a, b, c). This denominator-radical, hence forward, will be denoted by Dj(t + ien, r, z, cos y) (1' ---- 0, l, 2 corresponding with the suffix

of Hi). R e m a r k . An alternative field representation can be arrived at by interchanging the order of integration in (5.1). Under the same conditions as made in this section, F1 can be extended to F, where moreover, the integrand is real on the traject (1, co) of the real axis as n ~ oo. Hence it is obvious that the resulting integration on

ELECTROMAGNETIC FIELD FROM AN ANTENNA. I

369

f ' equals the residue at the pole of the integrand in the fourth quadrant if n -+ co. The final field representation obtained in this way is an integral over % while the integrand is the residue at the above-mentioned pole. The difficulty in this case, however, results from the occurrence of cos 9 in (5.1), which makes the determination of the pole concerned troublesome. § 6. The primary ]ield. 6.1a. The branch points of Do if en --+ + 0 , i.e. the zeros of the quadratic form: Et -- (nl/c) [z - - hi cos y]2 _ (n~/c 2) r 2 sin 2 ),

(6.1)

are conjugate complex: cosy+_=p0+iqo

Po-

ct Iz -- h I nlR~ ,

if

c2t 2 > n l2R 2o,

(6.2a)

r(c2t 2 -- n~R~)½ qo---nlR~

or real: COS ) ' m a x , m i n :

uo=Po--

U0 4 - V0

ct [z -- hi nlR2° ,

if

Vo=

C2~2 <

2 2 nlRo,

(6.2b)

r(n~R~ -- c2t2) ½ nlR~°

The square roots in qo and vo are assumed to be positive. b. If c2t2 > n l2R o2 the zeros Po 4-iqo are to the right of the imaginary axis if t > 0, to the left if t < 0. c. If c2t2 < n~R~ the situation is more complicated. As n -+ co ( t n - + + 0) the position of the real zeros u0 4-v0 m a y become critical with respect t o / ' . These zeros are on I--1, + I1, because on the remaining part of the real axis the form (6. l) is definitely positive. We note, moreover, that u0 + v0 > 0 if t > 0 and u0 -- v0 < 0 if t < 0. Their position relative t o / ' is determined b y the sign of the real quantity: d(u0 + vo) _ dt

c Iz -- h] nlR~ ~

rc2t n l R o ( n l2R 2 _ c2t2)~ ,

(6.3)

370

N.J. VLAAR

where the upper (lower) sign in the right- and lefthand member correspond. Since, by giving t a positive imaginary increment ien(en ~ 1), the sign of the corresponding imagiriary increment of uo ± vo is represented by the sign of the righthand member of (6.3). As we are interested in the real zeros on (0,1) which, for en ~ + 0 are below F, we investigate the conditions under which (6.3) is negative. If t < 0 this occurs for uo -- v0, and if t > 0 for uo + vo. For t < 0, uo -- vo is on (--1,0) and consequently this zero is not relevant to this investigation. COS ~ +

COS~max

COS~_

/i / / /

/" /

/

/.

Fig. 4. A b r a n c h c u t o n t h e real axis b e t w e e n cos 7maz a n d cos 7rain if (nl/c) ]z -- h[ < t < nlRo[c a n d a b r a n c h c u t b e t w e e n cos 7 - a n d cos 7+ if t > ~lRo/c. T h e b r a n c h c u t s do n o t cross t h e p a t h of i n t e g r a t i o n F. l d e n t a t i o n s in F d e m o n s t r a t e t h e p o s i t i o n of cos ~max a n d cos 7m*n r e l a t i v e t o F.

Proceeding with the remaining zero uo + vo, it follows that the

ELECTROMAGNETIC FIELD FROM AN ANTENNA. I

371

corresponding inequality

rc2t <0 ~ 2 _ c2t2)~ nlRo(nlRo

c [z -- h I

nlR~ is satisfied if

(nl/c) Iz

-

hi

<

t

(nl/c) Ro,

<

where uo + vo is on (0,1). d. The zeros Po + iqo and Po -- iqo if c2t2 > n~R~ or uo + vo and 2 o 2 are connected by a branch cut which we u0 -- v0 if c2t2 < nlR require not to cross the path of i n t e g r a t i o n / ' (fig. 4). According to § 5.2, Do is assumed to be + oo at cos y = + oo. 6.2. Next, the integration on _P is replaced by the integration on the traject (1, oo) of the real axis and on a possible loop Co which follows the borders of the relevant part of the branch cut of Do (fig. 5).

C

/cos ~o

c6s ~,~

'i-C t/

/ /

i/

/ /

~os V-

Fig. 5. The loop Co for the cases (nl/c) ]z -- h[ < t < n l R o / c and n l R o / c < t.

If en ~ + 0, when the integrand in (5.4a) is real on (1, oo), the only contribution to the field is due to Co. If t < (nl/c) lz -- hi no singular points will have to be taken into account and consequently H0 = 0.

372

N.J.

VLAAR

If (nl/c) [z -- hi < t < (nl/c) Ro, the loop Co will enclose the part of the branch cut below/~ between cos ~'max = U0 -~- V0 and cos ~---- 1. In this case also, the resulting integral is real, and hence, 17o = 0. If finally t > (nl/c) Ro, when Co is in the fourth quadrant and the integrand is complex, there will be a nonvanishing field Ho: 17o ---- U[t -- (nl[c) Ro] I m f . d cos ~ z~c2nl Do(t, r, z, cos ~) "

(6.4)

Co

By substituting Do

---- ( h i / c )

Ro. [(cos ~, -- po) 2 + q0~]½

the evaluation of (6.4) appears to be elementary, which leads to the required solution:

17o=

U[t -- (nl/c) RoT cn~Ro

This solution proves the correctness of interchanging the order of integration in § 5.2 and the choice of the sign of Do; furthermore it justifies the assumption of a zero static term mentioned in § 5.2. § 7. The reflected field. 7.1. By analogous arguments as for Do the zeros of D1 for en ~ + 0 (5.4b) can be localized. These zeros are:

cosv_,+=pl+iql

if

2 9. c2t 2 > nlR1,

where Pl-

nln~

,

ql =

nlR~

'

or COS ~,'max, m i n ~--- U l :ix Vl

if

cZt ~ < nlR~,

where Ul =

ct(z + h) nlR~ ,

vi =

r(n~R~ -- c2t2)~ nlR~

and R1 = [r ~ + (z + h)2]½

(7.1 b)

ELECTROMAGNETIC FIELD FROM AN ANTENNA. I

373

(nl/c)(z + h)

the fourth quadrant is free of singular points a n d / ' does not pass above one of the real zeros of D1. If (nl/c) (z + h) < t < (nl/c) R1, the real zero cos y m x = ul + Vl is below/1 on (0, 1). For t > (nl/c) R1 there is a complex zero Pl -- iql in the fourth quadrant. The zeros which are passed upon deformation o f / ' into the real traject (1, c~) give rise to branchline integrals on a loop C1. As in this case also the integrand of (5.4b) is real on (1, oo) the path of i n t e g r a t i o n / ' can be replaced by C1. If (nl/c)(z + h) < t < (nl/c) R1, the loop C1 encloses the part of the branch cut of D1 b e l o w / ' , between cos )"max ~ U l - - ~ Vl and cos y ---- l; if t > (nl/c) R1, C1 is in the fourth quadrant (fig. 5). 7.2a. n2 > nl. This corresponds with a higher phase velocity in the upper medium. The branch cut of cos y' is the segment If t <

i--i(n~/n~ -- 1)~], + i(n~/n~ -- 1)½] of the imaginary axis (cf. § 4.1). In this case the integrand of (5.4b) is real on C1 for (hi~C)(z + h) < < t < (hi~C) R1. As a consequence/71 ---- 0 if t < (nl/c) R1. If t > (nl/c) R1, (fig. 6), there is a non-vanishing reflected field:

II1 = U[t -- (nl/c) R1] Im f V(cos 7) d cos 7 ~cZnl D~(t, r, z, cos y)

(7.2)

U,

Contrary to the analogous expression for/-/0, (7.2) cannot be evaluated in a closed form on account of the occurrence of V(cos y) in the integrand. In the present case, the reflected field/I1 can be interpreted as a pulse with spherical wave front RI =: ct/nl, diverging outward from the image source (r = 0, z = --h) with wave front velocity

C/•I. b. n2 < nl. The transmitter is situated in the medium with the higher refractive index hi, so with the lower phase velocity c/nl. This circumstance, as will be demonstrated below, gives rise to the phenomenon of critical reflection. Mathematically this complication arises from the fact that the branch cut of cos y' in this case is on the segment [--(1 -- n~/n~)½, + (1 -- n~/n~)½] of the real axis, on which cos y' is purely imaginary.

374

N.J. VLAAR

The positive q u a n t i t y (1 -- n~/n~)½will be denoted in the following b y cos ~cr. It is obvious t h a t 7or = arc sin n2/nl is the angle of critical or total reflection for a plane wave incident on z = 0. Here a l s o / / 1 = 0 if t < (nl/C)(Z -~ h).

,+1 ----4

.....

-~

/

~os ~'_ / / / t' J

.j"

/

/

./

Fig. 6. The b r a n c h cut of cos 7' when n2 > n l a n d the c o n t o u r CI used to express the reflected field if t > nlRl/c.

If (nl/c)(z + h ) < t < (nl/c)R1 when C1 is on the real axis between cos ~max and cos ~ = 1 (fig. 7), we have to distinguish the two cases: (1) cos ~,r < cos ~max, when the integrand of the field expression for H I is real on C1 and hence H I = 0; (2) cos ~cr > cos ~max, when the non-zero contribution to H I arises from the part of C1 between cos ~max and cos ~,r b e l o w / ' , on which the integrand, containing cos ~', is complex.

ELECTROMAGNETIC FIELD FROM AN ANTENNA. 1

375

The condition of a nonvanishing H I if (nl/c)(z + h ) < t < < (nl/c) R1, i.e. cos 7or > cos 7max, amounts to the inequality:

r(n~R~ -- c2t2)~ < nlR~ cos 7er -- ct(z + h).

(7.3)

The lefthand member of (7.3) being positive, the inequality only holds if ct(z + h) < nln~ cos Fer (7.4) and if, moreover,

ct > ntr sin 7cr + nl(z + h) cos ?*r

(7.5a)

ct < --ntr sin 7or + nl(z + h) cos 7*r

(7.5b)

or

is satisfied. COS ~+

\ cos ~ max I

\

cos ~. 1

\__ ~---v" / /

-COS '~ cr

JJ f

I

COS ~ cr

// ¢os~_

Fig. 7. T h e b r a n c h c u t of cos 7 ' if n2 < nl. T h e c o n t o u r C1 o n t h e real axis if (z + h) nl/C < t < nlR1/c a n d in t h e f o u r t h q u a d r a n t if t > nlRl/c.

In virtue of t > (nl/c)(z + h), (7.5b) can be left out of consideration. From the remaining conditions (7.4) and (7.5a) we deduce the requirement for the existence of a non-zero HI if (nl/c) (z + h) < < t < (nl/c) RI: cos )1,r > (z + h)/R1

376

N.J.

VLAAR

R e s u m i n g we t h e n h a v e for the case n2 < n l :

(nl/c)(z + h) : HI = O,

(1)

t <

(2)

(nl/c)(z + h) < t < (nl/c) R1 : H I = 0 i n (z + h)/R1 > cosycr; :HI:#0

if t > r = ( n l / c ) ( z +

+ h) cos 7or + (hi/c) r sin 7or a n d

(z + h)/R1 < cos 7or;

in this c i r c u m s t a n c e we h a v e : 1 /I1

=

-

-

~c2nl

~ V(cos ~) d cos • Im

J Dl(t, r, z, cos 7) C~ CO S ~cr

_- -_

2 - ~c2nl -

Im .

f V(cos 7) d cos J Dl(t, r, z, cos 7)

COS

{7.6) )

max

. / t =n--1 c R1

, \

//

/

\~ z=--h

n2

Fig. 8. Wave fronts and different wave regions for the reflected field when critical reflexion occurs. Region I: t > nlR1/c, (z + h)/R1 > cos 7or. Region II: t > nlR1/c, (z + h)/R1 < cos 7or. Region I I I : t > ~ : (nl/c)(z + h) cos ~'cr + (nlr/c) sin 7cr, t < nlR1/c, (z + h)/R1 < c o s 7or.

T h e w a v e f r o n t t = T in the region (z + h)/R1 < cos ycr of z > 0 is a conical surface whose n o r m a l m a k e s the angle of critical reflection, 7,r, w i t h the z-axis.

ELECTROMAGNETIC FIELD FROM AN ANTENNA. I

(3)

377

t > (nl[c) R1 :IIl ~ 0 is represented by: H1 --

1

ac2nl

Im

( V(cos 7) d cos 7 J Dl(t, r, z, cos 7)

(7.7)

C1

The loop C1 in the fourth quadrant is associated with the complex zero Pl -- iql of D1. The situation of the wave fronts at t : T is depicted in fig. 8.

§ 8. The re/racted field. 8.1 For the study of the field in z < 0 we shall confine ourselves to the case n2 > nl. As the analysis for n2 < nl is nearly the same, it will not be dealt with in this paper. The case n2 > nl (the branch cut of cos 7' is on the imaginary axis) corresponds to the circumstance of a transmitter located in the air over a plane non-conducting earth. The main difficulty encountered in this section arises from the fact that the zeros of the radicand of D2 in (5.4c) cannot be determined in a simple way. Unlike Do and D1, because of the occurrence of cos 7' in the radicand, the branch points of D2 are not the roots of a simple quadratic form. 8.2a. The branch points of D2(en --> + 0), thus the zeros of I t - - (nl/c) hcosT + (n2/c) zcosT'l 2 -

(n~/c2) r2sin7

(8.1)

are the roots of:

t ---- (nl/c) h cos 7 -- (n2[c) z cos 7' + (nl/c) r sin 7

(8.2a)

t = (nl/c) h cos 7 -- (n2/c) z cos 7' -- (nl/c) r sin 7.

(8.2b)

and (we adopt for the signs of the real and imaginary parts of sin 7 and cog 7' the conventions of § 4.1.) For some fixed t, the roots of (8.2a) and (8.2b) are conjugate complex. It will thus be sufficient to investigate only equation (8.2a). By squaring twice, (8.2) can be demonstrated to lead to an equation of the fourth degree in cos 7, which equation consequently has four roots. It m a y be, however, that some or all of these roots do not satisfy (8.2). We are therefore forced to determine the possible location and prove the existence of these roots. Hence-forward, a possible root of (8.2a) will be denoted by cos 7o and corresponding with it, we have cos 7'0 and sin 7o.

378

N. ]. VLAAR

Separation of (8.2a) in its real and imaginary part gives: t = (nl/c) h. Re cos 7o -- (n2/c) z. Re cos 7~ + (nl/c)r. Re sin 7o (8.3a) and (nl/c) h Im cos 70-- (n2/c) z. Im cos 7~ + (nl/c) r. Im sin 70 : 0.

(8.33)

If there is a root in the upper half plane, that is, if Im cos 70 > o, we have also Im cos 7~ > 0 and Re sin 7o > 0. Then it follows from (8.3b) that Im sin 70 < 0 and hence that cos 7o must be in the second quadrant. In the second quadrant we have Re cos 70 < 0 and Re cos 70 < 0 and therefore we deduce from (8.3a) that t < 0 . A second zero of (8.1) tor the same t is then the complex conjugate of cos To. Similarly it can be shown that for a possible cos 7o in the fourth quadrant, the condition t > 0 must be satisfied. If Re cos 7o :- 0 we have Im sin 7o : 0. As Im cos 7o and Im cos 7~ cannot vanish simultanelously, (8.3b) will not be satisfied on the imaginary axis. As a consequence cos 7o cannot be located on the imaginary axis. If Im cos 7o : 0 and so Im cos 7~ = 0, (8.3b) can be satisfied only on [-- 1, + 11 of the real axis. Consequently real roots cos 7o are possible only on [-- 1, + 11. b. For a fixed point in the (r, z) space, the value of the integral (5.4c) depends on the zeros of (8.1). These zeros give rise to a nonz e r o / I 2 if they are in the fourth quadrant or on the real interval (0,1) below F and hence below the branch cut of sin 7 in (8.2a). The existence of these zeros, the roots of (8.2a), has not yet been proved. If n -+ oo, (sn -+ + 0), analogous to Do in (6.3), the position of a real root cos 7o relative to F, is determined b y the sign of: t' -- dt(cos 70) d cos 7o

(8.4)

I f 0 < cos 7o < 1 and t'(cos 7o) < 0, the real zero will be below F and hence has to be taken into account. The function t(cos 7o) given b y (8.2a)is an analytic function of its argument. Then it follows that cos 7o determined b y . t'(cos 7o) = nl h c must be a saddlepoint.

n~ n2. c

z

cos 7o

nl

cos 7~

c

r

cos 7o sin 7o

: 0 (8.5)

ELECTROMAGNETIC FIELD FROM AN ANTENNA. I

379

This specific value of cos 7o, say cos 75, corresponding with a certain t = T, m a y be interpreted as the cosine of the angle of incidence of a plane wave travelling along a minimum time path from P(0, h) in the upper, to Q(r, z) in the lower medium (fig. 9). The corresponding value of t is given b y ~- = (nl/c)(rl~ + h2) } -~-

(n2/c)(r~ + z2) ½,

(8.6)

where rl -~ h tg 70, r2 ---- --z tg yb and rl + r2 ---- r.

/

/

/ / I P (o,h)

Z >0

/

I

/

z

./7 /

/

/

Q

/

(r,Z)

/ /

7

/ /

Fig. 9. M i n i m u m t i m e p a t h P R Q for a plane wave. y* and ~*' are t h e angles of incidence and refraction corresponding to the v a l u e t ---- . given b y dt -

-

--

0.

d cos Y0 *

¥

As 7~ is real and 0 < 70 < ~/2, the value of cos 70 is between 0 and 1. On (0,1) of the real axis t(cos 7o ) is real and t'(cos 70 ) is positive for 0 < cos 7 < cos 70 and negative for cos 70 < cos 7 < 1. Hence ~- is a m a x i m u m of t(cos 7o) on (0,1) of the real axis. In virtue of cos 7o being a saddlepoint, the curves Im t ---- 0 intersect each other at right angles in cos 70. For t > ~, Im t ~ 0 is a curve in the fourth quadrant, and hence, for t > ~-, cos 7o is a complex root of (8.2a) in the fourth quadrant.

380

N. J, VLAAR

A further consequence of t'(cos 70) < 0 on (cos 7o, 1) is that possible real zeros below F must be situated on this interval of the real axis. If cos 70 = 1 we have t = (nl/c) h -- (n2/c) z, which implies that real zeros below F only exist if (nl/c) h -

(n2/c) z < t <

~.

c. (1) If t < (hi/C) h -- (n2/c) z, no singular points of D2(t, r, z, cos 7) are passed upon deformation of F i n t o the real interval (1, co). As the integrand in 5.4c) is real on (1, co) as n -+ co, it follows that //2=0. (2) If (nl/c) h -- (n2/c) z < t < "r, the real zero below F will give rise to a loop C2 enclosing the real segment [cos 7~, 1]. The integrand of (5.4c) being real on C2 in this case, we have also/7o = 0. (3) If t > T, there are two complex conjugate zeros of (8.1) in the righthand halfplane. The zeros, cos 7+,-, are connected by a branch cut. The path of integration F can then be replaced by a loop C2 following the borders of the part of this cut in the fourth quadrant. As the integrand of (5.4c) is complex on Ca it follows: nl Ha -- ~~can

~ {1 + V(cos 7)} d cos 7 (8.7) U(t -- ~.). Im o D~(t, r, z, cos 7) ' Us

where T is given by (8.6). Physically, (8.7) signifies that for a point (r, z) in z < 0, the field is zero up to the time t = r when the wavefront reaches (r, z) following Fermat's principle of minimum time. For increasing t the wavefront is a downward travelling surface of revolution with the z-axis as axis of symmetry. The surface is determined by t = ~ and is concave downward. § 9. The transmitter in the interlace. 9.1. When the transmitter is in the interface z = 0, that is, if h = 0, the analysis of the field is less complicated than if h :fi 0. In the upper medium we have:

H~, = Ho + H~

(h = 0)

z > 0,

(9.1)

(h = 0)

z < o.

(9.2)

and in the lower medium: //z = / / 2

ELECTROMAGNETIC F I E L D FROM AN A N T E N N A . I

381

The field expressions (9.1) and (9.2) are assumed to be the limiting forms for n -+ oo. (sn -+ + 0). Without loss of generality we suppose that the upper medium has the higher phase-velocity, i.e., that n2 > hi. This implies that the branch cut of cos ?" is on the imaginary axis of the cos ?'-plane. The upper medium m a y then be considered to be the air, while the lower is the dielectric earth. 9.2. The field//~ is the superposition of the primary field/7o and the reflected field/71 in the limit for h = 0: II~ =

{

U(t -- (ni/c) R) cnl

if

1

+ ~-c

V(cos ?') d cos ?' }(9.3) [(t -- (nl/c) z cos ?')2 _ (n~/c2) r 2 sin 2 ?'It

C:

where R = Ir2 + z2]½ and C1 the loop used for the analysis of/71 in (7.2) but now also taken for h ---- 0. For some point (r, z) in the air the disturbance arrives at the time t = nj/c and the wave front for increasing t is a sphere diverging from (0, 0). 9.3. In order to determine the field/71 it m a y be useful to proceed with the analysis in the cos ?,'-plane 7). In virtue of cos ~ =

El - n~ln~ +

(n~l~)cos2

?"1½

there is a branch cut on the real axis of the cos ?"-plane for -(:

-

,,~In~)~ < cos ?" < +

(: - ,~I,~)~.

The sign of the real and imaginary part of cos y are settled according to the corresponding signs of cos ? / i n the cos ?'-plane. The path of integration F is mapped into a similar path N', running from cos ?" = 1 to cos ?" -= 0 on the real axis and subsequently from cos 7i' = 0 to cos ?" = -- i oo in the cos ??'-plane. By means of the substitutions cos ?' 4 cos ?' = (n~/n~) cos ?" d cos ?" and sin 71 = (n2[nl) sin ?",

382

lq. j. VLAAR

we can transform (9.2) into n2 f [1 -- V (cos y)] d cos y' (9.4) TI, -- z~dn~ " I m ~ [(t + (n2/c) z cos y,)2 _ (n~/c 2) r 2 sin~ ?'lt " I"

Hz can be seen to consist of two parts:

//11

--

n2 ~C2n~

"

Im

//z = / / 1 1 + / / i 2 .

f

d cos y' E(t+(n2/c)zcos~----(n~/c)rg.

(9.5) sin2y,3 ½,

F"

//12

nz f V(cos y) d cos y (9.6) :rc2n21 . I m [(t + (n2]c) z cosy') 2 -- (n~/c 2) r 2 sin 2 y']½" /,,

H n is easily d e m o n s t r a t e d to be: //n =

U(t - n2/c R) cn~R

On (1, oo) of the real cos ?'-axis the integrand of (9.6) is real. W h e n F ' is deformed into this traject, the following zeros of the denominator-radical in (9.6) will be passed: (1) T h e real zero --ctz r(c2t 2 -- n~R2) ~ COS ? ' m a x = - -{nlR 2 nlR z if

--(ng[c) z < t < (n21c) R

or

(2)

The complex zero --az nlR2

cos Y'

if

ir(n~R 2 -- c2tz) ~ nlR2 t > (n2/c) R.

As the branch cut of cos y is on the real axis, these zeros will lead to a non-zero field//19 (in analogon with § 7.2b): •08

_

(1)

/71~=

7'er

2n_____2_2 [ V(co~y) ~ dcos?' e r c 2 n ~ ' I m-L j [ ( t + ( n ~ / c ) z c-----~, -s-v,'~~,~- r- ~ ' s i n ~ ? , COS

~'nmz

if

-zlR

< cos r'c, [=(1 - ~/n~)~]

, ~½ (9.4)

ELECTROMAGNETIC FIELD FROM ANTENNA. I

383

and -- (n2/c) z cos 7'or + (n2/c) r sin 7'or < t < (n2/c) R.

(2)

$*2

Lr12 =

~c2n21

f

V(cos 7) d cos ~' [(t + (n2/c) z cos ~,)2 _ (n~/c2) r 2 sin 2 7,]t '

• Im

(9.5)

C'

if

t > (n2/c) R.

areeos. ~

Fig. 10. Wave fronts and wave regions on t = ~ of the field in both media if h = 0. Points in the regions I l i and IV m a y be reached along the refracted rays OPQ b y the disturbance travelling in the upper medium with velocity c/n1 and on the leg PQ with velocity c/ng. The regions I I and I I I are passed b y the direct disturbance having a spherical wave front and travelling with velocity c/n2. The conical wave front t = 7 is determined b y -t ~ -- (n2z/c) cos ~'cr + (n2r/c) sin 7'er.

C' is the loop which encloses the branch cut of the denominatorradical between cos 7-' and some point on (1, oo) of the real axis of the cos 7' plane.

384

ELECTROMAGNETIC FIELD FROM AN ANTENNA. I

9.4 We thus have derived the field in the lower medium: //z =

1Ill

+//12

where//11 is a contribution to the total field which propagates with the spherical wave front R ---- ct/n2. H12 has a spherical wave front R = ct/n2 if - - z/R > cos 7'cr, and a conical wave front t = --(n2/c) z cos 7'cr + (n2/c) r sin 7'or if --z/R < cos ~'cr. In the latter case H12 can be considered to be a diifracted field: points in the region --z/R < cos ~'cr are reached by the wave front at a time equal to the minimum time for a plane w3ve in the upper medium, -leaving the transmitter at an angle of grazing incidence with z ---- 0, to reach the point concerned. Received 19th October, 1962. REFERENCES 1) 2) 3) 4) 5) 6) 7)

S o m m e r f e l d , A., Ann der Phys. 28 (1909) 665. Weyl, H., Ann. der Phys. 60 (1919) 481. Pol, B. v a n der, Trans. Inst. Radio Engrs. A P - 4 (1956) 288. P e k e r i s , C.L. andZ. A l t e r m a n , J. Appl. Phys. RS(1957) 1317. De Hoop, A. T. and H. J. F r a n k e n a , Appl. so. Res. B8 (1960) 369. Pol, 13. v a n d e r and A. H. M. L e v e l t , Ned. Akad. Wetensch. Proc. A63 254. B r e m m e r , H. Electromagnetic Waves, Ed. Rudolph E. Langer, The University of Wisconsin Press, 1962. 8) P o r t z k y , H., Brit. J. Appl. Phys. 6 (1955) 421. 9) T e m p l e , G., Proc. Roy. Soc. A 2 2 8 (19551 175. 10) L i g h t h i l l M. J., Fourier Analysis and Generalized Functions, Cambridge University Press, 1958.