P R O P A G A T I O N O F R A D I O - W A V E S IN S T R A T I F I E D M E D I A

M. V. T i n i n UDC 621.371.3

Cer ta in c a s e s of the propagat ion of waves along l a y e r s having a continuous squa re of the r e f r a c t i v e index that i n c r e a s e s monotonical ly f rom the homogeneous h a l f - s p a c e a r e in- inves t iga ted . It is shown that in such media the field of a point sou rce s i tuated in an in - homogeneous h a l f - s p a c e d e c r e a s e s in inverse propor t ion to the squa re of the d is tance at g r ea t d i s t ances along the l a y e r , r e g a r d l e s s of the p r e s e n c e of a shadow region. The s t rong effect of the smoothness of the p ro f i l e on the dependence of the field ampl i tude on f r e - quency is noted.

In inves t iga t ing the propagat ion of waves in l amina r ly - inhomogeneous media it is the usual p r a c t i c e to d i s t inguish between two c a s e s : waveguide propagat ion and propagat ion under condit ions of shadow zone fo rmat ion [1]. The shadow zone is evident ly formed for propagat ion of e l ec t romagne t i c waves in a ha l f - space which has a p e r m i t t i v i t y that i n c r e a s e s f rom the in ter face . This case and its acous t ic analog have been inves t iga ted in deta i l by var ious authors [2-6]. As is well known [6], the shadow zone is formed for ce r t a in cons t ra in t s which a r e imposed on the c h a r a c t e r of the growth of the pe rmi t t i v i t y .

The p r e s e n t pape r is devoted to a compara t ive ana lys i s of ce r t a in c a s e s of wave propagat ion along l a y e r s having monotonic p ro f i l e s of the squa re of the r e f r a c t i ve index which sa t i s fy and do not s a t i s fy the condit ions for the format ion of a shadow zone.

1. F i r s t of a l l l e t us c o n s i d e r the t h r e e - d i m e n s i o n a l p rob lem of the pene t ra t ion of the f ield into the shadow region on a weak in te r face . Assume that there is a l a y e r whose p e r m i t t i v i t y d e c r e a s e s mono- t on i ca l ly r ight up to the boundary z = 0 with a homogeneous ha l f - s pa c e on which the p e r m i t t i v i t y is a con- tinuous ftmction and its defi ivative undergoes a finite discont inui ty .

L e t the source be an e l e m e n t a r y v e r t i c a l magnet ic dipole having a t ime dependence e - iwt , which is s i tua ted in the lower ha l f - space (z -< 0) a t a d is tance Ihl f rom the in te r face between the two media. Let us d e t e r m i n e the field in the region z < 0 on the assumpt ion that the p e r m i t t i v i t y i n c r e a s e s l i n e a r l y with d e p a r t u r e f rom the z = 0 plane, while it is equal to the p e r m i t t i v i t y of the upper homogeneous medium a t z = 0. In the lower inhomogeneous h a l f - s p a c e a shadow zone is fo rmed [3] in which conventional geo - m e t r i c - o p t i c s methods of ca lcu la t ing the f ie lds turn out to be inappl icable .

The p r o b l e m s ta ted can be reduced to a solution of the Helmhol tz equation for the ve r t i ca l component IIm of the magnet ic Her tz vec tor :

arfm + k s n ~ (z) 1I m = - - Pm ~ (z - h) ~ (x) ~ (y) (1) I%

for the condition of f in i teness of the f ield eve rywhere except the point x = O, y = O, z = h (h < O) at which the sou rce is s i tua ted .

In Eq. (1) the square of the r e f r a c t i v e index is s t ipula ted by the re la t ionsh ip

n 2 ( z ) = { n~ = ~, z0 (z > 0) o 1 (z 0 - z ) (z ~ 0) (2)

I rku t sk State Univers i ty . T r a n s l a t e d f rom Izves t iya Vysshikh Uchebnykh Zavedeni i , Radiof iz ika, Vol. 16, No. 4, pp. 505-511, Apr i l , 1973. Original a r t i c l e submi t ted Sep tember 18, 1972

�9 1975 Plenum Publishing Corporation, 227 West 17th Street, New York, N. Y. 10011. No part o f this publication may be reproduced, stored in a retrieval system, or transmitted, in any form or by any means, electronic, mechanical, photocopying, microfilming, recording or otherwise, without written permission o f the publisher. A copy o f this article is available from the publisher for $15.00.

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and the following notation is introduced: P m is the dipole moment of the source;/~0 is the magnet ic p e r m e - abi l i ty of f r e e space (the Engineering s y s t e m of units is used); k = w / c is the wave number ; b is the Dirac del ta-funct ion, On the interface I Im sa t i s f i es the boundary conditions

Oil., = OII m I Hm it__0 = H= I,=+o, Oz =__o oz, I:=+0, (3)

Us ing a Four i e r - Bess el t r a n s f o r m by analogy with [1, 8] we for mulate the integral r ep resen ta t ion of the solution, which has the following fo rm in the domain h <- z --< 0:

c o

where It~l)(rl) is a Hankel ftmction of the f i r s t kind; Ai(rl) is an A i r y function;

2 " = d ' J

r = ~ x z + y2; 5 = (kd--~l)z/Z(z_z0 + sZ/~l ) ; ~h is the value of ~ for z = h.

T h e integrand express ion in (4) has the branching point s = :mr, and the integrat ion is p e r f o r m e d on that sheet of the Riemann s - s u r f a c e on which I m ~ - 0.

Using asympto t ic r ep re sen ta t i ons of the functions which enter into the integrand of (4), it is not difficult to show that in the deep shadow region the original integrat ion contour in (4) may be conver ted into contours around the s ingular i t ies of the coeff icient R(s) in the upper half-plane of s .

The integral around the cut I m ~ n ~ - s 2 = 0 may be reduced to an integral along the r ight bound:

where

" d o<.,= :, [ A, Eoxp( , + [ox, (2, t.o~ R e p l a c i n g the Ai ry and Hankel functions in (6) by the i r a sympto t ic r ep re sen ta t ions and calculat ing

the resu l t ing integral by the method of s t eepes t descent , we obtain

P,nn, exp {ikrn, + i 2 k~m[l hi~S' q-izl3/'] -t- i 3 } rl~ ) = C1

t ~o I zh I'" (kV~;) =/a t / ~ r [ r -- 2 I/Ih~-l-~o - 2 Vj-~0] ~'= ' (7)

where:

1 Ci = ~. 0.189.

Using power - law and asympto t ic r ep resen ta t ions of the Ai ry function [7], one may ver i fy the fact that the poles of R(s) a r e s i tuated fa r f rom the real axis; t he re fo re the contribution of the res idue ca l - culated at these poles a r e exponential ly smal l in compar i son with I I ~ ), and the field in the shadow region is comple te ly de te rmined by Eq. (7).

Thus, the field in the shadow region pene t ra tes via the l a t e ra l wave (7) which allows exact ly the s ame geomet r i c in terpre ta t ion of a ,wave glancing along the in ter face as does the solution of the s i m i l a r

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two-dimensional problem [5] bu td i f fe r s in the additional d!vergence of r -1 /2 and the l e s se r dependence of the amplitude on frequency (~-2/3 in compar ison with ~-7 /6 for the two-dimensiona, case [5]).

F r o m Eq. (7) it is evident that in the deep-shadow region where r >> 2C-f-ff~ 0 the f ie ld falls off ha inverse proportion to the squa re of the distance ra ther than according to an exponential law as in those cases of [2-4, 6] ha which an inhomogeneous hal f -space is bounded from above by an impedance plane or an ideally conducting plane z = 0.

2. Let us now go over to an analysis of the propagation of waves along a layer that is s imi lar to the one a l ready considered but does not form a shadow zone. For this purpose we consider a layer for which the square of the ref rac t ive index is determined by an analytic function ra ther than the piecewise- l inear function (2); however, the analytic function retains the proper t ies of the function (2) of a monotonic increase for z ~ - ~ and a tendency toward n} for z - - +~o. The ref rac t ive index satisfying these requ i re - ments may, for example,have the form

n ~ (z) = n~ + ~ e - ~ . (8)

The problem reduces to solving Eq. (1) for condition (8).

As in the previous case , one may construct the integral representat ion of the solution which has the following form for z ~- h:

c o

ik 2 P,~ ~ H(0 t) H (1) 1I m 4xI% ~ (ksr) ~ (%) J, ('c)sds, (9)

where T = (2k/3/n)e-1/2nz; 7 h is the value o f t for z = h; v = - i ( 2 k / ~ ) ~ n ~ = s 2 ; Jr(r) is a B'essel function. The integration (9) is ca r r i ed out on that sheet of the Riemann surface on which I m ~ n ~ - s 2 ~- 0.

The integral (9) may be calculated by the method of s teepest descent af ter f i rs t replacing the cy - l indrical functions by their asymptot ic representa t ions , the saddle points s s being found f rom the equa- tion for the rays [8]. An analysis of the equations for the ray

2 S$ $ - - e l l 2 xh S - - 1 r ~--- -- elfZ xz arccos q- arcccs �9 x V ' s s _ n ~ ~

shows that in the case given the shadow zone is not formed (i.e., one may always find a ray that penetrates a rb i t r a r i l y deep into the layer) .

Le t us go over to an analysis of the field far f rom the source while applying a method for analyzing the solution (9) which is analogous to that used in Section 1. Using the asymptotic representat ions of cy- l indrical functions [9], it may be shown that in the region of int.._ c res t to us which is far away along the layer Eq. (9) may be reduced to an integral around the cut Im~n12-s 2 = 0 in the upper half-plane; with allowance for the bypass relat ionship for cyl indrical functions this integral may be reduced to the integral for the r ight bound of the cut:

l i r a - - k"Pm4xl~o ~ sin ~r,I-l(o 1) (ksr) tt~ 1> (~) H! 1) (%) e l~ sds. (10)

nt

Since in the previous ease the shadow zone was formed in the lower hal f -space , we shall likewise investigate (10) in the lower ha l f -space T ->- M >> 1. Therefore the Hankel functions in (10) may be replaced by the asymptot ic representa t ions , and (10) may be calculated by the method of steepest descent:

Pm n, exP [ 4 (z + h ) ]

llm~_... { 2~e__,/,e~ ~ 2~ ,,2~,~1 [ 2n, e'/2"~) V/2 e x p i k n , r q - . + - - e . (11) 2Xl~o[~ V' r �9 r -]- ~ (el/2 ~ q- / x

F r o m (ii) it is evident that, just as in the case of a p iecewise- l inear layer , the field along the layer ' (8) at a grea t distance f rom the source falls off in inverse proportion to the square of the distance, but unlike (7) its amplitude is no longer dependent on frequency.

3. In the previous sect ions we considered piecewise- l inear and analytic (exponential) profiles of the square of the ref rac t ive index. Now let us turn to the intermediate case in which there is an interface on

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which continuity of the N-th der ivat ive of the prof i le is violated - i .e. ,

n~(z ) _- { n; (z > o) (12) n~ + ~ ( - z ) ~ (z < 0)'

where N is an a r b i t r a r y posit ive number , the shadow region being formed only for N < 2 [6].

The integral representa t ion of the cons idered prob lem (1), (12), (3) takes on the following form (for s implici ty we shall r e s t r i c t ourselves to the domain h -< z -< 0):

II~ ") = k~ P" H(~ (ksr) Ul ") (h) [Ul")(z) 4n1% W(t~} Jr R# (s) U~ tO (z)] sds, (13)

where

RN(s ) = _ dz ] ik V ~ i - s ' u'# ) ' a b e " ) ' dz z =o

W(m dU~ N) U(~N) dV~#) U~.)

dz dz

u(N)(z s) a re the solutions of the differential equations while uIN)(z, s) and z ,

dz ~'q' 2 jr k= [n~ - - s ' -k a s ( - - z) s] U (N),. 2 = O,

which have an asymptot ic representa t ion for z ~ - ~ in the form

(14)

(15)

z

u~ m (z, s) ~ A,(s) exp (!k ~ V ~ ( z ' ) - s 'd(_

~/n'(Z) - s~ " (16) z

exp (-- ik ~ W n~ (z') - s ~ dz" U~ N) (z, s) ~ A~ (s) ,-4 , ~ .... �9

V n ~ ( z ) - s

The integration in (13), jus t as in the preceding cases , is ca r r i ed out on the sheet of the Riemann s - s u r - face on which I m ~ n ' ~ F - > - 0.

h t h e problem cons idered in Sections 1 and 2 the deformation of the original integration contour in the integral representa t ions of the solutions (4) and (9) in the contours around the s ingulari t ies of the integrand express ion was based on a knowledge of the asymptot ic (in s) representa t ions of the functions that a re included in the integrand express ions of (4) and (9). In view of the absence of analogous asymptotic r e p r e - sentations for u~N)(z," s) and --'u(2N~(z, s) for a r b i t r a r y N, it remains for us to assume that in this in ter - mediate case it is also impossible to r ep re sen t the field in a region that is fa i r ly distant along the l aye r in the form of an integral around the,out Imf~n~--~ = O, which is not difficult to t r ans fo rm into an integral along the r ight bound of the out:

I I ~ = i o o

~k~p~ uV ) V ~ 2~0,: S I-I(~ (ksr) Ui N) (z, s) (h, s)

In o rde r to de te rmine Uz(N)(z, s) and du~2N)(z, s ) / d z we use the fact that at a great distance f rom the source the main contribution to (17) is made by the integration segment in the neighborhood of the point s = n i. For s = n~ the solution of Eq. (15) is expressed in t e rms of cyl indrical functions - in par t i cu la r ,

u ~ ( z , al) -- V---z Ht,(~+2)(')[ 2 ~ k ~ : ( _ z)(N+~}/2],

where a r g ( - z ) (N +2)/2 = 0 for z < 0.

(18)

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Substituting (18) into (17) a re using the well-known asymptot ic and power- law expansions of cyl in- dr ica l functions, the integral (17) may be calculated for r ~ oo by the method of steepest descent:

I I~ ) -~ (19)

2t~0(N +2)u/(N+2)[F(I - i 2 _ __ N + 2 ) ] V ~ ~ ( k V % , :~( N + ') l zh lJV /4r2

Equation (19), a par t icu lar case of which for r ~oo, N = 1 will be (7), is the expression for the la te ra l wave propagating along the interface on which continuity of the N-th derivat ive is violated.*

Comparing the derived expressions (7), (11), and (19), one may note the following.

In all of the cases considered, the field at a great distance from the source falls off along the l ayer in inverse proport ion to the square root of the distance, regard less of whether a shadow zone is formed or not. The p resence of homogeneous sec tors in the monotonic profUe which have a minimal ref rac t ive index for the given layer and are situated at a finite or infinite (in an exponential layer) distance from the source and f rom the observer unite all of the cases considered. Therefore , one may conclude that for p ropaga- tion along such layers the presence of a shadow region does not determine the charac te r of the field decay.

R is also n e c e s s a r y to indicate s t rong effect of the degree of smoothness of the profile on the f r e - quency dependence of the field amplitude. F r o m (19) it is evident that to the extent that the smoothness of the profile increases the dependence of the field amplitude and frequency decreases (see also [11]); under these conditions the diffractional cha rac te r of the field is, as it were, lost, and in the case when the p r o - file is an analytic function (8) the field amplitude (11) becomes independent o f frequency.

The author thanks G. I. Makarov, V. S. Buldyrev, and N. V. Tsepelev for discussing the resul ts of the presen t paper.

LITERATURE CITED

I. L.M. Brekhovskikh, Waves in Layered Media, Academic Press (1960). 2. C.L. Pekeris, J. Acoust. Soc. Amer., 18, No. 2, 295 (1946). 3. L. Levey and L. B. Felsen, Inst. Maths. Appl., 3, No. i, 76 (1967). 4. D.S. Jones, Phil. Trans., 255A, 1058, 341 (196~. 5. N.V. Tsepelev, in: Mathematical Problems in Wave Propagation Theory, Part If, Seminars in

Mathematics of the V. A. Steklov Mathematical Institute, Vol. 25, Consultants Bureau (1971). 6. N.V. Tsepelev, in: Mathematical Problems in Wave Propagation Theory, Part III, Seminars in

Mathematics of the V. A. Steklov Mathematical Institute, Vol. 17, Consultants Bureau (1972). 7. I~. T. Copson, Asymptotic Expansions, Cambridge University Press. 8. G.I. Makarov, in: Problems in the Diffraction and Propagation of Waves [in Russian], Vol. 2 (1952),

p. 81. 9. V.M. Babich, V. S. Buldyrev, and I. A. Molotkov, First All-Union School-Seminar on Diffraction

and Propagation of Waves (Palanga), Texts of Lectures [in Russian], Vol. 3, Moscow-Khar'kov (1968).

10. L .B. Felsen, Electromagnetic Wave Theory, Vol. 11, Pergamon Press, Oxford (1967). 11. Y. Nakamura, J. Geophys. Res., 69, No. 20, 4349 (1964).

* Felsen [10] mentioned the necess i ty of deriving such an equation.

385