Appl. sci. Res.

Section B, Vol.

11

THE TRANSIENT ELECTROMAGNETIC F I E L D FROM AN ANTENNA NEAR THE PLANE BOUNDARY B E T W E E N TWO DIELECTRIC HALFSPACES II. & CLOSER INVESTIGATION OF THE FIELD b y N. J. V L A A R Vening Meinesz Institute for Geophysics and Geochemistry, Rijksuniversiteit Utrecht, The Netherlands Summary

In a preceding publication of the present author 5) expressions were derived to describe the pulse field from a vertical electric antenna placed near the plane boundary of two semi-infinite dielectrics. These expressions appeared to be integrals on a finite range in the complex plane and were derived for the field in both media. The analysis of the field also gave information about the time of arrival of the several pulse-fronts. In the present paper the field is studied in more detail, and with the aid of suitable approximations more specific results are obtained, such as logarithmic field singularities. Also a discussion of the pulse analogon to the Zenneck wave in the time harmonic case has been included.

§ 1. Introduction. T h i s is t h e second of two p a p e r s on the t r a n sient r a d i a t i o n f r o m an infinitesimal vertical electric dipole-antenna, which is placed in or n e a r t h e plane interface s e p a r a t i n g two semiinfinite dielectric m e d i a 5). I n the first p a p e r the field r e p r e s e n t a t i o n s for t h e p r i m a r y , reflected a n d r e f r a c t e d field are derived in the f o r m of c o m p l e x integrals. T h e object of the present investigation is to o b t a i n some specific i n f o r m a t i o n concerning these fields. I n § 2 of the present paper, a c o m p a r i s o n is given of our integral r e p r e s e n t a t i o n s a n d those derived b y B r e m m e r . B y m e a n s of a t r a n s f o r m a t i o n of the p a t h of integration, p e r f o r m e d in this section, --

49

--

50

N . J . VLAAR

we will examine in § 3 the field for the limiting case t -+ c~, which is to lead to the solution of an electrostatic problem. In § 4, closed expressions will be derived for the field at the interface, in the case that also the transmitter is placed in the interface, i.e. for z + h = 0. The results are in agreement with those obtained b y other authors, who used different methods. § 5 deals with the field in a point on the vertical through the point (0, 0, h), the location of the transmitter. In this case too, closed expressions are derived by simple means. § 6 and § 7 are dedicated to the study of the field, especially in the vicinity of the wave front of the reflected field, if h :fi 0, and of the field in the upper and lower medium if h ---- 0 respectively. Some remarkable features of the field, such as logarithmic field singularities, are shown. In § 8 we took into the possibility of the occurrence of the pulse-analogon of the Zenneck-wave that is encountered in time harmonic radiation problems. Finally, in § 9, some general remarks are made with regard to other problems in mathematical physics which are compatible with the problem discussed here.

§ 2. Alternative integral representations. 2.1. References to the first of the present two papers will be indicated b y the symbol I. B r e m m e r (I.7)) showed the equivalence of his field representations and those derived b y V a n d e r P o l and L e v e l t (I.6)) and De H o o p and F r a n k e n a (I.5)) respectively, at least for the special cases treated b y these authors (cf. 1.1). In the present section we will transform our field expressions into those obtained b y B r e m m e r . A discrepancy with B r e m m e r ' s work is met as regards the position with respect to the contour of integration of the real zeros of the denominator-radicals in (I. (5.4b)) Next, for completeness, the equivalent field expressions of the remaining above-mentioned authors will be briefly discussed. For these purposes we will confine ourselves to the treatment of the reflected field //1. (1.7.2a). The demonstration for the other field expressions is completely analogous. 2.2. Departing from HI--

1

lim Im

xec2nl ,n,o

1" V(cos 7) d cos y ----_--

J Dl(t + ~n,r,z, cos~) F

, z > 0, n2 > nl, (I.(5.4b))

ELECTROMAGNETIC

FIELD

FROM

ANTENNA

II

51

we note, t h a t for sn ~ + 0, the i n t e g r a n d assumes complex conjugate values for complex conjugate cos y.

\

"\. \

'\ \ I

C OS~=I

.......

I

!

_.~7-.--.~

t

/

v

/i /

i, !

I i

.fl

!

I I

"/

/" .j/

i . . . . •~ ' ~ ' S ' ~

Fig. 1. The contour L. The position of cos ymax relative to L is visualized by indentations in L. Then, after having passed to the limit * n - + + 0, a n d b y extending the p a t h of integration F to a closed c o n t o u r L which is s y m m e t r i c a l with respect to the real axis, (I.(5.4b)) t r a n s f o r m s into

i 171 --

~ V(cos ~,) dcos7

2~c-2nl J Dl(t, r, z, cos y)

(2.1)

L

The expression (2. l) is used by B r e m m e r for studying the reflected field. However, there is an important discrepancy with B r e m m e r ' s publication as far as the location of the real zeros of D1 is concerned. F o r (nl/c) (z+h) < t < (nl/C)R, the real zero cos ymax ---- U l + V l (I, (7.1b)) should be considered to be twice inside L (fig. 1). If n~ < nz - which case was not t r e a t e d b y B r e m m e r - the b r a n c h

52

N.J. VLAAR

cut of cos y' is on the real axis. This circumstance (I. (7.2b)) leads to a reflected field with a partly conical wave front. In B r e m m e r ' s consideration the real zeros always have to be outside L, which assumption gives rise to inconsistencies. 2.3. If t > (nl/c)R1, when the complex conjugate zeros cos 7+ = = pl + iql, and cos y_ = P l - iql (I. (7.1a)) are in the 1st, respectively 4th quadrant, L can be replaced by a contour L' enclosing the relevant branch cut of D1. B y the substitutions

Dl(t, r, z, cos y) = (nl/c)R1 [(cos ? -- pl) 2 + q~]t

(2.2)

and cos ~ = Pl -- iql cos 1o, B r e m m e r transformed (2.1) into

,;

H I = ~c n~R~

V(pl -- iql cos t0) dr/

o

l

-- n~C ~R~

f V(pl

Re

-- if1 cos ~o) d~o.

0

The last expression is equivalent to the one used b y D e H o o p and F r a n k e n a for the reflected field. It should be remarked, however, that it is not possible to derive an analogous field representation for t h e lower medium in case hsa0. 2.4 Taking into account a contribution from the circle K (fig. 2) with the origin as center and a radius large enough to enclose all singularities of the integrand of (2.1), the integration on L' (or L) can be replaced by an integration on L", a contour which encloses the branch cut of cos y'. This deformation of the contour L' gives 1 n~ -- n~ H1 = cn~R1 n~ + n~

i ~ V(cos y) d cos y 2~c-2nl J Dl(t, r, z, cosy) "

(2.3)

L*

The first term in (2.3) arises from the integration on the circle K, which integration, b y letting the radius of K tend to infinity, leads to the closed term in (2.3).

53

ELECTROMAGNETIC FIELD FROM ANTENNA n

Considering t h a t V(cos 7) assumes inverse values at the opposite borders of the branch cut of cos 7', we arrive at

=

.'2 +

+ 1

+ --Im =c2nl

o

[" J

1/V (cos T)} d cos r Dl(t, r, z, cos 7)

{V(cos 7 ) -

o

(2.4)

--i(nz2/nl 2-1)~

This formula will be of use in the following section.

/ /

/

/

\'\K Cos~

,(~/~-,)~,,

X

i i I

't:' \

i(nz=/nZ-') v'

COS~. /"

%.

./I

(~,~//!/

/"

/ .]

Fig. 2. The contour L' may be replaced by L " + K .

B y introducing the new variable s ~ i nl cos 7, B r e m m e r dem o n s t r a t e d the equivalence of (2.3) and the expression for the field in the upper m e d i u m (by including the p r i m a r y H0), used b y V a n d e r P o l and L e v e l t , at least for the special case h ---- 0 examined b y the latter authors. It m a y be noted t h a t the obtained integral is an elliptic one. However, as the parameters of this elliptic integral are complex, no tables are available for numerical work. In the lower m e d i u m the expression analogous to (2.3) is 2

/-/2=

1 c(n~ + n~) [(z--h) 2 + r2] ~

in__ 2 [ {1 + V(cos 7)} d cos r 2~c~n~ ,I

D2(t, r, z, cos 7)

'

L"

which, for h = 0, also contains an elliptic integral.

Z < 0.

(2.6)

54

N.J.

VLAAR

§ 3. The static state. If we let t t e n d to infinity, the integrals in (2.3) and (2.6) will vanish. B y adding the p r i m a r y field H0 to the remaining term in (2.3), we have, for t - + oo, in the upper medium

IL, = Flo + F11 --

-

1 cn~Ro -

-d-

(t -+ oo)

1

n~ -- n~

cn~R1 n~ + n~ '

z~O

and by means of (2.6) for the lower medium

//z =//2

(t --> c~) 1

1 c(n~ + n~) E(z--h) 2 + r 2 1 ~ '

z<0.

Hu and Hz constitute the solution of the potential problem connected with our wave problem, if, instead of a transient action of the transmitter, we consider the problem of the field from a vertical static point doublet of m o m e n t 1, placed at z = h, r = 0. § 4. Horizontal propagation /or h = 0. The field in the upper medium //u = / / o form

+//1

(I, (9.3)) for h = 0 can be put into the p

U(t -- (nl/c)R) Im 2 ( (1 + V (cos y)} d cos y ~c2nl ,] [ ( t - (nl/c)z cos ~)2 _ (n~/c2)r 2 sin 2 7]½"

Hu

~---/q

In the limit for z = + 0, R = r, the zero cos ~ - - - - - P l - iql (I, (7.1a))moves towards the negative imaginary axis, and hence, b y writing the integrand in a suitable form, the field representation for t > (nl/c)r becomes: 4n2 Hu

-=

2

2

(n2 -

nf)

0

I m f{n~ cos 2 y -- nl cos y (n~ - n~ + n~ cos2 y)~} d cos y {(n~ + n~) cos 2 y -- n~} Vt2 - (n~/c2)r 2 sin2 y]t ---/ql

ql = (c2t2 -- n~r2)/nlr.

55

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

By elementary though tedious calculations, the integration yields" 2n~

//u

I 1

nlnl2

cn~(n~ - n~) [ r 2n~ 1 --

H u - - c n~(n~ + n~)

1'

nl

n2(c2te- n~r2) ½

r
c

c

n2

--,

--r

r 1

r, '

(4.2) ~t,

c

1

n2

1

which results agree with those obtained by V a n d e r P o l and by Bremmer. The difference of the results in (4.2) for (nl/c)r < t < (n2/c)r and for t > (n2/c)r respectively, is due to the part of the integrand of (4.1) containing the square root in the numerator. This part of the integrand is purely imaginary on the branch cut of cos ?', and hence, its contribution to H u vanishes if ql < < (n~/n~ - - 1)~ i.e. if ct < n2r. If ct > n~r, its contribution to H u however, which is non-zero, is due to an integration between the limits -- iql and --i(n~./n~ - - 1)~. By virtue of the boundary condition of continuity of n ~ H at z = 0, the field Hi for z = --0, can be obtained from (4.2). It is obvious that the expressions in (4.2) then correspond with the expressions for Hz in the regions III and IV of the lower medium (I, fig. 10). § 5. Vertical propagation. The denominator-radical Dj in (I, (5.4))

is required to be + oo at cos~ ~ + oo. By putting r = 0, the radicand is purely quadratic. By maintaining the above requirement, the expressions (I, (5.4)) become: Ho -- - -

1

1"

d cos J (hi/C) [z--h[ cos y - - t - - ien '

lim I m

ac2nl ,~,o

F

H1

--

I ~ V (cos y) d cos 7 lim I m ~C2nl *.*0 j (r~l/C)[z + hi cos y t

-

-

-

-

-

isn

F

nl

(

{1 + V (cos ~)} d cos (n2/c)z cos y'

H2 - - ~c2n~ lim I m I (nl/c)h cos F

56

N.J.

VLAAR

The integrand of these expressions HI( ]. = 0, l, 2) has a simple pole at cos YvJ, where

C(t + ien) COS ~'io0 - -

nl[Z--h]

C(t -~ ien) , COS } ' p l - -

nl(z + h)

and

h(t + ien) c + z [c2(t + isn) 2 + (h 2 -- z 2) (n~ -- n~)]½ COS ~'p2 nl(h2

- - z 2)

If *n va 0, the poles are in the upper halfplane and no singularities will be passed if F is deformed into the traject (1, oo) of the real axis. If en ~ + 0 , the cos ~vJ are real. For t < ,j (*0= (nl/c) lz--hl, T1-~ (nl/c) (z + h), *2 ~- (nl/c)h--(n2/c)z), we have cos ~vt < 1, and hence, as the integrand for en --->+ 0 is real on (1, oo), it follows t h a t / / j = 0. For t > ,j, the only contribution t o / / j arises from a small semicircle around cos YvJ in the fourth quadrant. The ,integration on F then equals xi times the residue of the integrand at the pole cos ~vJThis leads to the following closed expressions for H j U(t //o -

U(t H1

II2=

---

*o)

cn~ [z -- hi '

V(cos 7vl) cn~(z + h) 71)

U(t--~'2) "cos Y'~2{1 -4- V (cos Yv2)} t

n2h cos Yv2 -- nlz cos Yv2

§ 6. A closer study o[ the reflected field i[ z + h ~ O. The field representations are integrals on a finite range, and therefore suitable for numerical computations. As the present study is mainly of theoretical interest, it does not seem useful to perform numerical work. It is possible, however, to show some remarkable features of the field b y elementary means. For that purpose we examine the behaviour of the refleced field especially in the vicinity of its wave fronts. 6.1. If n2 > nl, the branch cut of cos y' is on the imaginary axis. In this case the reflected field H I is given b y (I, (7.2)). We consider

ELECTROMAGNETIC

FIELD

FROM ANTENNA

II

57

the branch cut of D1 to be a straight segment between cos y+ = = Pl + iql and cos y - -----Pl -- iql (I. (7.1a)). The integration on C1 m a y then be replaced b y twice the integration on the right border of the branch cut, which g i v e s - i f we use for D1 the form (2.2) Pl

2

f

V(cosy) d c o s y

HI -- ~rc2nlR1 Im J [(cos }'_ pl) 2 q- q~j½

(6.1)

Pa --iql

The analytic function V(cos y) can be e x p a n d e d in a power series with argument (cos }' -- Pl). In the interior of the circle of convergence with centre cos }' = Pl and a radius equal to the distance from cos }' ---- Pl to one of the branch points of cos y', the series is uniformly convergent. For sufficiently small ql, the path of integration is in the interior of the circle of convergence. Hence it is justified to interchange the order of s u m m a t i o n and integration. B y substituting the power series co

V (cos},) :

1

Z

'--- V(") (pl) . (cos }' - P l ) "

n=o nT

(6.2)

and introducing the variable = arc sin

(cos ~, - Pi)

iql

(6.1) transforms - for sufficiently small ql - into 0

in+l r ~rc2nlR~ Im ~=o ~ ---n! q'~l V(n)(Pl) J s i n n ~p dy. 2

HI

-

-

oo

(6.3)

If the wave front is approached from within the disturbed region, that is if we let ql approach + 0 the field can be approximated b y the first term of (6.3), i.e. I / I '~'

V((z + h)/R1) c%1 R1

,

(6.4)

as

lim Pl = (z + h)/Ri. q~--~+ 0

6.2. If n2 < nl, when the branch cut of cos y' is on the real axis, we have to distinguish the regions I, II, and I I I of z > 0. (I, fig. 8).

58

N . J . VLAAR

a. Region I. We have t > (hi~C) R1, (z + h)/R1 > cos~cr (=(l--n~/n~)½) and so Pl < cos 7cr. In this case formula (6.1) and the considerations made in 6.1 remain valid and consequently the field in I near the wave front RI -- ct/nl, can be approximated by (6.4). b. Region II. t > (nl/c) R1, (z + h)/R1 < cos),cr and so, as q l ~ + 0 we have Pl < cosycr. (6.1) holds if we take cosycr in stead of Pl as the upper limit of integration. The integral can then be considered to consist of two parts

HI

2 -

-

7ec2nlR1

Im (I~ + 12),

where f I1 =

V (cos $) d cos [(cos~ -- pl) 2 + q~l~

Pl --iql

and COS ~cr

f I2 =

V (cos ~) d cos ~, [(cosy -- Pl) 2 + q~]~

iOl

As ql -+ + 0, the contribution from I1 to HI can be approximated by Re {V(z + h/R1)/c2nlR1} and is finite. For sufficiently small ql, the series expansion of V (cos y) leads to (cos rcr-- p~)/qz

oo V(n)(pl) ( xn dx 12 = ~=0 X n ~ q7 j (x2 + 1)~ 0

in which all terms but the first are finite as ql ~ + 0. The first term contributes to the field (cos ycr--pz)/ql

2 /z+h\ Im V ( ~ ) l o g ~c2n l R1

{x + (2 2 + 1)½}

, (6.5)

#

0

on account of which HI becomes logarithmically infinite when the spherical surface R0 = ct/nl is approached from within region II. c. On the conical surface (z + h)/Rl=cos?/er (separating regions II and III) the field, which can be represented by (6.1) is finite.

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

59

d. Region III. (hi~C) (z + h) < t < (nl/c)R1. The field is given by COS "~or

H1 --

2 -

/' V(cos ?) d cos ? Im! ~ -~ U l + vl. (I, (7.1b)) m2nlR1 ,/[(cosy -- Ul) 2 -- vl]½' c°s 7max COS ~)max

If Vl ~ + 0, t h a t is, when the spherical surface R1 = ct/nl is approached from within I I I , the main contribution to H I becomes logarithmically infinite like (cos :~c~-ul)/vL

2

~c2nl R1

ImV(Z+h~log{x+

\

R1

/

(x 2 -

1)½} [ 1

and so likewise when R1 = ct/nl is approached from within II. W h e n the wave front t = (nl/c) (z + h) cos 7cr' + (nl/c)r sin ycr is approached from within II, t h a t is, when cos Ymax approaches cos ycr, the field H1 gradually vanishes. e. Summary. If n2 > nl, the reflected field/~1 will be zero up to the time of arrival of the wave front t = (nl/c)R1 which is m a r k e d b y a finite discontinuity. If n2 < nl we have to distinguish the regions of the upper medium (z + h)/R1 > cos ycr and (z + h)/R1 < cos ?or. In (z + h)/R1 > cos ?or H1 jumps from zero on a finite value at t = (nl/c)R1. In (z + h)/Rl < cos ?or HI is zero up to t=(nl/c)(z+h) cos ?cr + (nl/c)r sin 7or, when H1 starts from zero and increases gradually up to t = (nl/c)R1 which manifests itself as a logarithmic singularity of the field. For t > (nl/C)R1 the field again assumes finite values. 6.3. The case h = 0, when the t r a n s m i t t i n g dipole is situated in the interface z = 0, has been investigated in more or less detail b y P e k e r i s and A l t e r m a n n , V a n d e r P o l and L e v e l t and B r e m mer.

With the aid of the integral representations derived in I, § 9, several properties of the field can be derived b y methods as given in § 5. We shall only give the results for the case it is assumed t h a t n2 > nl (I, fig. I0). In z > 0 the total field Hu diverges from (0, 0) with a spherical wave front R = ct/nl which is a surface of finite discontinuity in

//u.

60

N.J. VLAAR

In z < 0 the field behaves differently in --z/R < cos y;r and --z/R > cos ~';r. In the first region the wave front R = ct/n~ marks a finite jump in Hi, in the latter region the field is zero until t = -- (n2/c)z cos 7'or + (nl/c)r sin y;r, when the point (r, z) is reached b y the conical part of the wavefront, and when the field begins to increase continuously up to t = (n2/c)R which is marked b y a logarithmic field singularity.

§ 7. The Zenneck-pulse. So far no mention has been made of the pulse-analogon to the Zenneck-wave encountered in the study of periodic radiation from an antenna over a flat earth (S o m m e r f el d). According to B a r l o w and B r o w n l), the Zenneck-wave is a surface wave, "a wave which propagates without radiation along the interface between two different media". Such a wave pattern can be obtained by a plane periodic wave incident on z -----0 under such an angle that no accompanying reflected wave occurs. That is, when V (cos 7) = 0. The angle of incidence determined b y this relation is called "the Brewster angle". If the refraction indices of both media are real, the Brewster angle = arctg (n2/nl), is real. In conducting media, when one or both refractive indices are complex, ~p is a complex angle which causes the field to be attenuated with increasing distance from the interface. The phase-velocity of the Zenneck-wave is c/nl2, where n12 is given b y n~-~ = n~-~ + n~ ~, and hence is a quantity which is symmetrically dependent on the refractive indices of the two media. In radiation from an antenna in the neighbourhood of a plane interface, the Zenneck-wave contribution to the field appears to be associated with a pole of the integrand in the formal integral-representation. This pole is the zero of the denominator of V (cos ~). It then becomes obvious that the Zenneck-pulse has not yet made its appearance in the present paper, since, b y the specific definition of cos y' in the complex plane, V (cos y) hitherto had no poles. 7.2. We will study only the case n2 > nl because the essential features of the Zenneck-pulse are similar for n2 < nl. Contrary to the definition of cos ~' b y means of a branchcut connecting its zeros, we here consider the analytic function cos ~,' having branch-

ELECTROMAGNETIC

FIELD

FROM

ANTENNA

II

61

cuts between

<-', )'

+ikn~

-- 1

and

+ioo,

--

-- 1

and

- - i o o respectively, and by imposing

Im cos r' < 0 in the fourth quadrant (fig. 3).

Jm>o

Jm
c9~z A!

............ ** t

,

.d- "-r

,.T

it

t4/ c

\

I Jm:~o

Jm
.2,',,

,s

./"

/-

/.

tl'

i

B

Fig. 3. Branch cuts of cos ~', signs of Im cos ~', the loop CI, the pole cos ?z and the path of integration/". As a consequence of Re cos 7' > 0 in the entire complex plane, we have Im cos y' < 0 in the second, and Im cos ~,' > 0 in the first and third quadrant. B y the new choice of the branch cuts, the signs of Re cos ~' and Im cos ~' in the fourth quadrant are not altered, and so, the field is determined by the same singularities of the integrand in the fourth quadrant, giving rise to the same loops C as in (I, 7.2). In the lefthand halfplane, however, the situation has changed, which causes the appearance of a pole of V (cos y). This pole, cos 7~, is determined by n2 cos ~ + nl cos ~' = 0 and is on the negative real axis at cos ~z = -- nl2/nl. 7.3. The reflected field HI which is zero up to t = (nl/c)R1 is determined for t > (nl/c)R1 by singularities in the fourth quadrant, and so, as there are no singular points in the third quadrant, we are allowed to replace the path of integration F by F" (fig. 3).

62

N.J. VLAAR We now are able to write:

,

n ~ ' + n~

1

171 = cn~(n~ -- n~) -

R1

2n~'nl2[{nlnl2 c.~ ~-~) ~ +

(z + h)

.... ,,~

}2

]-½

- n~r~

+

D

+

1 Im ~" J

{V(cos~)-~/V(cos~)}dcos~

(7.1)

1~'

The first term in (7.1) arises from integration on the arcs AB and DE, where it has to be taken into account that lim V (cos y) -Ioo~~,~ -

-

n~ + ~ ~-

~

ng + n~

in the third quadrant in the fourth quadrant

according to the definition of cos y'. As the integrand is real on the real axis, the only contribution to the integral from it is due to the pole cos Yz, which gives rise to the second term in (7.1). (Since cos Yz is on the real axis to the left of the branch cut of the denominator-radical, and in order to let the radical in the second term take its arithmetic value, the sign of this term has been reversed). The remaining integral in (7.1) is obtained by considering that by crossing the branch cut of cos y', V (cos ~) changes into its inverse. A remarkable result is the equivalence of (4.2) and (7.1) if in the latter formula we put R1 = r, z + h = 0, and add the primary field H0. 7.4. The integral on FD in (7.1) may be replaced by the sum of the integrals on FO, the positive real axis in the positive sense, the loop C1, and ED taken clockwise. The only contribution from the real axis is due to the pole of 1/V (cos y) at cos y = +n12/n2. It m a y be noted, moreover, that the integral on FO equals the

63

ELECTROMAGNETIC FIELD FROM ANTENNA II

opposite of the integral occurring in (1.4), and so is equal to 1

n~

-

nl~

-

/71.

cn~R1 n~ + n~

Including the contributions from C1 and ED we arrive at: 2/I1

n~)

2(n~ -]-

-

1

2n~'n12

~-

~t

cHR~

•

a+

nl'n12 (z + h) ~

--

{(

ct

cnl(n~-

nl'nm (z + h) n2

1 --Im

f

)' )'

~)

}-, },]

-- n~2 r~

--n~

--

r2

+

{V ( c o s y ) - 1/V(cosy)} dcosy . . . .

-,J[{t--(nl/c)(z+h)cos~'}2--(n2/cgr2sinb' ?. (7.2)

+ 2~c2nl

C1

This result resembles an expression derived by B r e m m e r for the field in the lower medium for the special case h = 0. He arrived at this result by factorizing V (cos y) in the form

n~ + n I

2n~.n212

V(cosy)-- n~--n~ +

n~--n I COS2 Y

2

n2 cos y • cos y'

2n~ "nl

n~

"

(7.3)

In (7.2), the terms not involving integrals, arise from the first two factors in (7.3), yielding elementary integrals, whereas the elliptic integral in (7.2) is due to the last factor containing cos y'. 7.5. In z < 0 an expression analogous to (7.1) can be derived. For t > T, the time of arrival of the disturbance, we then have

/h =

2n~ 1 c(n~ -- n~) [(z + h) ~"+ r2] ~ -

--

~

-

~h

r~

+

nl f {1 -+- V (cosy)} d c o s y + ~c2n-~ Im [ ( t - ( n l / c ) h c o s r + (n2/c)zcos~/)2--(n~/cgr2sin27]~" BFD

(7.4)

64

N.J. VLAAR

B y the occurrence of cos 7' in the denominator-radical it is not possible, however, to perform similar transformations and simplifications as has been done in the section 7.3 and 7.4, except for the case h = 0. For h ---- 0 B r e m m e r derived expressions b y means of elementary integration. 7.6. The second terms in (7.1) and (7.4) i.e.

2n~'n12

ct +

(z + h)

--n~2rZ

, ( z > 0 ) , (7.5a)

and

• ( 2-- 1)[[

nl'nl2h+ n2"n12 ] 2 _ ~ 2]-~ n2 nl zf nl~r ] ,(z<0),(7.5b)

constitute a pair of solutions of the scalar wave equations which satisfy the boundary conditions at z = 0. For the existence of a field (7.5) no radiation is involved. This means, that a field of this form is self-supporting, and that no source or sink of energy is needed to govern the field. We note, moreover, that this field satisfies the two-dimensional wave equation in the interface: +

r Or

cz

at z

. //,j=0,

(/=

1 2),z=0.

where the phase-velocity c/nl2 along the interface exceeds those of the bordering media. The field is confined to the boundary z ---- 0 and is attenuated with increasing distance from this boundary. B y virtue of the foregoing, we are inclined to consider (7.5) as a surface-pulse, which, as it is associated with the Zenneck-pole, could be called Zenneck-pulse. In the form (7.5) it m a y indeed be considered to be a free mode of the interface. In the present study, however, the Zenneck-pulse appears to be a part of the total field, and is obtained b y means of mathematical manipulations. As the Zenneck-pulse does not appear before the arrival of the main wave, there is no travelling Zenneck-pulse ahead of the main wave front, and so, it will not be recognized as a true surface-wave; hence no physical significance must be attached to its appearance in the formulae (7.1) and (7.4).

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

65

Because of the initial conditions of a vanishing total field for t < 0, the solution of our radiation problem is uniquely determined. If no initial conditions had been imposed, it would have been admissable to add a field of the form (7.5), on account of which the solution would lose its uniqueness.

§ 8. Concluding remarks. 8.1a. In the equations Dj(t, r, z, cos 7) = 0 (Dj is the denominator-radical in the integrands of (I. (5.4a, b, c) for en ~ 0, /" = 0, l, 2), tj can be made explicit, and b y suitable branch cuts, tj = tj (r, z, cos 7) can be made an analytic function of cos 7. The time corresponding with dO(r, z, cos 7) -- o, (8.1) d cos y say tj = ,j, can be demonstrated to be the time of arrival of the main wave front, i.e., T1 = (nl/c)R1 for the reflected field and T2 (I. (8.6)) for the refracted field. The Tj mark a discontinuity of the field. It m a y further be noticed that the cos 7J determined b y (8.1), say cos 7~, is a saddle point of the function tj(r, z, cos 7). b. It is well known that in the time harmonic analogon to the present pulse problem, approximations to the field expressions can be found b y means of the method of steepest descent 3). This method is based on a deformation of the path of integration into a path which passes through the saddle point of a function equal to tj ~- tl(r, z, cos 7) mentioned above. The main contribution to the field - at least for large circular frequency co or large distance from the source - comes from the part of the path of integration situated in the close vicinity of the saddle point., c. There is a relation between the behaviour of the field near the wave front in the present pulse problem and the approximation arrived at by the method of steepest descent in the corresponding time harmonic case. This is due to the fact that in the Fouriersynthesis of the pulse problem, the wave front discontinuity is mainly governed b y the part of the spectrum with large co, while the wave front is determined b y the above mentioned saddle point. It may be noted, moreover, that the approximation to the field for example (6.4) - apart from a periodic factor - is the first term of the asymptotic approximation b y the method of steepest descent ill the time harmonic case.

66

ELECTROMAGNETIC F I E L D FROM A N T E N N A II

The approximation (6.4) is valid for large distances from the source and equals the geometrical optics approximation to the field. 8.2a. A remarkable feature of the reflected pulse if n2 < nl is the logarithmic field singularity at R1 = ct/nl. This singularity does not have its counterpart in the time harmonic case. In the latter case, however, there is a caustic, which for large ~o coincides with the cone (z+h)/R1 = cos 7cr, while for moderate o~ it is shifted somewhat into the region (z+h)/R1 < cos Ycr 3). This caustic is the effect of interference of waves in the spectrum (which represents the spherical wave) with angle of incidence close to the angle of critical reflection. As has been mentioned in 6.2c, in the present case the reflected field is finite on or in the vicinity of the cone (z + h)/R1 : cos 7cr. A singular behaviour of the reflected field comparable with that in the present study, has been found b y A r o n s and Y e n n i e 1), who examined the behaviour of the reflected field in the acoustic case. Their investigation dealt with an acoustic plane pulse incident beyond the angle of critical reflection; the wave front in that case was also marked b y a logarithmic singularity. This result - in view of the present study - is not very surprising, as the approximations (6.4) and (6.5) to the reflected field represent the geometric optics approximation, which, for plane waves, is rigorously valid. It should be kept in mind, however, that already in 1939, Cagni a r d a), b y investigating the reflection and refraction of a spherical seismic pulse, noticed a similar singular behaviour. Received 7th March, 1963 REFERENCES 1) A r o n s , A. B. and D. R. Y e n n i e , J. Acoust. Soc. Amer., 20 (1950) 231-237. 2) B a r l o w , H. M. and J. B r o w n , Radio Surface Waves, Clarendon Press, Oxfiord, 1962 3) B r e k h o v s k i k h , L. M., Waves in Layered Media (Engl. Transl.), Aead. Press, New York and London, 1960. 4) C a g n i a r d , L., Reflection and Refraction of progressive seismic waves (Engl. Transl.) McGraw Hill, New York and London, 1962. 5) V l a a r , N. J., Appl. sci. Res. B. 10 (1964) 353.