IL NUOVO CIME.NTO

VOL. 60 B, N. 1

11 .Novembrc 1980

Propagation of Electromagnetic Waves in Two Media Separated by a Variable Interface ('). S . T . BISlIA~~ (*') a n d A. :E. 5[()H~,~I~ED

Department o/ Applied Mathematics, Faculty o/ Scie~2ce, Ain-Shams University Abbassia, Cairo, Egypt (ricevuto il 26 Novembre 1979; manoseritto revisionato rice,:uto 1'11 Giugno 1980)

Summary. - - The investigation of the propagation of electromagnetic waves in two media separated by a variable interface has required the use of the imaging space of a Hankel transformation. The reversal of such a transformation produces an integral representation, which ill this case has proved to be valid for all frequencies. 51oreovcr, no depolarization appears in the field as long as the variation in the interface remains small and is directly depeadent on the height of the second medium.

Introduction. KAHA~" a n d [EcKART

(l)

s t u d i e d t h e r e f l e c t i o n of a n e l e c t r o m a g n e t i c fiel4

o r i g i n a t i n g f r o m u dipole p l a c e d w i t h i n it u n i f o r m - h e i g h t d u c t . BECKER (2), On t h e o t h e r h a n d , i n v e s t i g a t e d t h e p r o b l e m i n t h e c~se of ~ dipole w i t h i n a d u c t of n o n u n i f o r m h e i g h t . I n t h i s work we arc c o n s i d e r i n g a dipole p l a c e d i n t h e i o n o s p h e r e , a b o v e d u c t w i t h a n o n u n i f o r m h e i g h t , t h e d u c t here, as i n t h e o t h e r ca.~es, b e i n g t h e m i d d l e m e d i u m of t h e t h r e e ( E a r t h , air, i o n o s p h e r e ) m e d i a . (*) To speed up publication, the authors of this paper have agreed to not receive the proofs for correction. (**) To whom correspondence is to be addressed. (l) T. KAHAN and G. ECKA1Vr: Z. Natur]orsch. Te.il A, 5, 334 (1950). (2) K. D. B~C~:~R: Die Ausb~'eitung elektromagnetischer Wellen in einem atmosphdrischen Bodenwellellleiter mit variabler oberer Grenze, A.E.~*., 23 (1969), Heft 2, p. 101.

46

P R O P A G A T I O N OF EL~.CTROMAGNETIC W A V E S I N TV~'O M E D I A S E P A R A T E D ETC.

~7

The emphasis in our work has b e e n on the s t u d y of the p r o p a g a t i o n of the electromagnetic waves f r o m a vertical m a g n e t i c dipole radiating a p u r e T E field. A previous s t u d y (5) was concerned with a similar a r r a n g e m e n t in the case of u n i f o r m - h e i g h t duct a n d the results obtained there shall be employed in the present work. T h r o u g h o u t this p a p e r the m a t h e m a t i c a l a r g u m e n t s depend~ however, m o r e or less entirely on the use of p e r t u r b a t i o n calculus.

1. - Structure o f the space under i n v e s t i g a t i o n .

I n the space under investigation the ionosphere (medium 1) is considered to be of infinite height a n d to h a v e a dielectric c o n s t a n t el. I m m e d i a t e l y below the ionosphere there exists a l a y e r of air, t e r m e d the d u c t ( m e d i u m 2). The duct is considered to h a v e a m e a n height d, as its u p p e r b o u n d a r y , separating it f r o m the ionosphere, is a s s u m e d to be c o n s t a n t l y varying. The duet is also assumed to haYe a dielectric c o n s t a n t e2. The E a r t h is the third m e d i u m a n d its surface (assumed to be uniform) represents the lower b o u n d a r y of t h e second m e d i u m (the duct). The :Earth, moreover, has b e e n t a k e n at b o u n d a r y conditions to be a good conducting medium.

reed~urn 1

h

rnecdum2 /

/

d

l

/ / / / / /

/ / / / / / / / / / / / / / / / / / . / / / /

Eaeth

Fig. 1. - The geometrical structure of the space under consideration. I n relation to the chosen space, the m a g n e t i c dipole (the source) is placed vertically in t h e first m e d i u m (the ionosphere) a t an a r b i t r a r y height (h) a b o v e t h e surface of the E a r t h , along t h e z-axis, i.e. the dipole m o m e n t is t a k e n to be perpendicular to the E a r t h ' s surface (fig. 1). I f cylindrical polar co-ordinates (r, ~o, z) are used to describe the geometrical structure of t h e described system, t h e n for the whole space 0~r~< co

and

0<~0<2~.

(8) S. T. BIStIAY and A. S. TADROS: %0 be published.

48

s . T . BIS~A~ and A. E. ~OHAM~D

I f t h e E a r t h ' s surface is chosen to be the plane z = 0, t h e n the E a r t h , the duct a n d the ionosphere will be as shown in t h e figure, where Az is the v a r i a t i o n of t h e u p p e r b o u n d a r y of t h e duct. These v a r i a t i o n s can be represented in the form

Az = ?]l(r) ~- ?~ f~(r) -~ ...,

(1)

where t h e functions fn(r), n ---- 1, 2, ..., are definite a n d continuous a n d ? is the disturbance p a r a m e t e r such t h a t

(2)

~: ly~L(r)l<< d.

W e shall limit ourselves to the first a p p r o x i m a t i o n of f~(r) a n d neglect the t e r m s of higher order O@ 2) and write ](r) for t h e function ]~(r). T h e unit n o r m a l to the variable interface b e t w e e n the two media is given b y n=

{n~,n~,n,}=

--?~

0,1

.

As the p r i m a r y fields ~ ( r , z), H~(r, z) are i n d e p e n d e n t of % we assume also t h a t the source function a n d all the field variables exhibit the c o m m o n t i m e dependence exp [-- io~t], where ~o is the fixed a n g u l a r frequency of the driving source. Thus Maxwell's equations (4) will t a k e t h e f o r m (3)

curl E = i w ~ H ,

curl H = - - ia)eE -{- I~

where E and H b o t h satisfy t h e w~ve equations [W + k ~ ] ~ = 0, k is called the p r o p a g a t i o n constant a n d is defined b y k2 =

eo-~ s / ~ .

The t o t a l fields Et(r , z), tIt(r , z) for the two m e d i a are

E~ = ~ ~- E~,

H~ : ~ ~- ~

for the first m e d i u m

and E~ = E~2 ,

H: =/~2

for the second m e d i u m ,

(4) _~k. SOMMERFELD**Partial Di]]erential .Equations in Physics (New York, N. Y., and London, 1949).

PROPAGATION

O F ~ E L E C T R O M A G N E T I C VJAV:ES I N T W O

MEDIA

SEPARATED

49

ETC.

where E , , H , are the secondary fields. These can be represented as

E. = E.o + ~E.,,

ft.-- ~ o + r ~ , ,

where E.o(r , z), H . o ( r , z ) are the secondary fields for the case of uniform interface, i.e. for the ease 7 - - 0, a n d E . l ( r , z),/'/.l(r, z) are the secondary fields due to the v a r i a t i o n of the interface between t h e two media. To obtain the value of the electromagnetic-field strength, Io is t a k e n as zero in eq. (3) a n d we apply, next, t h e following b o u n d a r y conditions: 1) A t t h e u p p e r b o u n d a r y (of the duct) the t a n g e n t i a l c o m p o n e n t s of the t o t a l electric field E~(r, z) a n d the magnetic field H t ( r , z) are continuous, i.e. ,, x [ ~ 4- ~'o + r ~ , - ~

(4)

- ;'~,] =

o,

and

n x E ~ + H:~ + ~H.',-- H:o-- ~H:,] = 0 ,

for all values of r and z =d-~

Az.

2) A t the lower b o u n d a r y (the surface of the E a r t h ) , since it is a good conductor, t h e n E t ( r , z) m u s t v a n i s h a t z = 0, i.e.

(5)

e~ X [E.2o + E.21] = 0

for all values of r a n d z - - 0,

where e: is t h e unit vector along t h e z-axis. The tields E~ + E.~o, H i -~ H,~o arc the solutions for the case 7 = 0. The b o u n d a r y conditions for the case 7 = 0 become

{ e.x[~ + ~ ' o - E:o]=0, (6) e~ x [H'~ + Hl.o --

H:o] = 0,

for all values of r

and

z =

d ,

and

(7)

ez x E.2o = 0

for all values of r and z = 0.

The radiation conditions m u s t necessarily be satisfied for [r 2 ~- z2]~ -+ oo.

2.

-

Solution.

2"1. General solution. - As t h e dipole lies along the z-axis, the p r i m a r y field does not depend on the p a r a m e t e r T. I n this case, the p r i m a r y c o m p o n e n t s of the resulting T E field will be E 1 = {O,Z' 4 -

II

Nuovo

Cimento

B.

O}

1 H .1} , H ~ = { H o.,0,

s . T . B:SIIAY and A. ]~..~O::A.'~)IV.D

50

while the secondary-field components will be

E.~ = (0,

0),

E., = {0,

H.o---- {H o,, 0, H o,} ,

0)

H.: = {H.I,, 0, H , , } .

The representation of the electromagnetic field in terms of the Fitzgerald vector ~(r, z) (~) will be as follows: (8)

E~

i~o# curl g(r, z),

H~-- grad do)~(r, z) -~ k~(r, z),

such t h a t ~(r, z) satisfies the wave equations [V 2 + k-~

z) ~ 0

and, in terms of cylindrical polar co-ordinates, the latter equation becomes

(9)

L~rr2 q- r ~r q- ~-~ - k2 ~(r, z) -~ 0 .

Thus the components of the electromagnetic field in the case of the T E field are

(10)

E~--iw#

~-~,

- - c:rc~' ~ ~

H~--

~q-r~

"

The solution of these equations using the ]Iankel transformation of order v is oo

m(2, z) : ~f,{u(r, z)} -=;J,(2r)n(r,

(11)

Z)

rdr,

0

where the inverse IIankel function of order v is oo

n(r, z) = ~gf~-~{m()., r)} =;J,().r)m(~, z)).d2.

(12)

0

F r o m the wave equation (9) and from the differential equation of the Bessel function of order zero, n a m e l y

~

(13)

(s)

D. S. J o ~ s :

~- r

Jo()~r)= -- X'Jo().r) ,

The Theory o] Electromagnetism {Oxford, 1964).

PROPAGATION

OF

:ELECTROMAGNETIC

WAV:ES

IN

TWO

~IEDIA

S~EFAI~ATED

ETC.

51

we have

[~--~z2 + k~- 22]'m(2,z)----0.

(14)

The solution of this equation is

m(2, z) = A(2) exp [zV2u-c- k~] + B(2) exp [-- zV-~-- k~]

(15)

and thus eq. (12) becomes co

(16)

z) =j Jo(.)n(2) [e p

§

B(~)exp [-- z 2 ~ v ~ - -k2]] 2d),

--

2 .

0

with the condition lim

%/22--

k s :

~-..+ oo

2"2. The integral representation o/ the field due to variation o/ the inter/ace. - The secondary fields E ~ H I , n = 1, 2, for the two media written in terms of the integral representation are for medium 1

E.I,r = E '

=H'

alz

al~0

=0

oo

E1,1~= i~/~fJl(2r)Al(2) exp [-- iT, z] 2~d2, 0 oo

Hl.~r = f ivl J , ( / r ) A~(2) exp [-- iT, z] 2 3d 2 ,

(17)

0 oo

H'~,. = f Jo(ar)A,(2) exp [-- ivlz] 13 d2 o

and for medium 2

E~

E~

--~

0

oo

N.% = i~ogfJ,(2r) [A~(2) exp [ir2z] + B2(2) exp [iT~z]] 2' d 2 , o c~

(xs)

H,~,= - -

fJ,(2r) [A2(2) exp [iv2z] --B~(2) exp [-- iT2Z]]iT~2 ~d 2 , 0

oo

H 2,,, =fJo(2r) [A2(A) exp [iv2z] -4- B~(2) exp [-- iv~z]] 2 ~d2, 0

52

S . T. B I S t I A Y

and

A. E. M O H A M M E D

where k~ =

c:tts~,

~ = -- iv:): -- k~,

n = 1, 2,

i = v'~.

Here V/t 2 - k ~ stands for the root of the positive real part of ) 2 _ k2. and

Jn(Ar) is a Bessel function of order n. The constant functions At, A2 and B2 are determined from the b o u n d a r y conditions (4) and (5). F r o m eqs. (5) and (7) we have the boundary conditions 2 r , 0)---- 0. El+(

(19)

F r o m the boundary conditions (4) and neglecting the higher-order terms of 0(72), we have

~Oq~2 c ~2EI,1~-- E~o+ -- ? E2,l~z=d+Az)

(

[ E pep 1 AE I --

x N [E~ +

{ ~' [~

.~,~-~.~o~

:

0

= o, z=a+Az

7H,i~-- H,o~ -- ?H,i,}z=d+Az=

~'~176162176

0

,

where E,~o, H~o are the secondary electric and magnetic fields in media i a n d 2, respectively, as i takes the value 1 or 2, in the uniform case (8). Applying Taylor's expansion for each function of the latter equations at Z ~ d~ we hsve

(20)

~/[E~ + E.o~--E~o+

= 0,

y ~-~L~,, -5 H,89 H,%,] -5 H~, + ?/~-~H,, + H.%, -~ X/~z

" -~

+ }'H.~r--H,0r-- Z/ ~Z H,or-- zH,Irj = 0. Z=d

Using the boundary conditions, eqs. (6) and (7), we get E:

__ E 1

--

(21) {H .I. -

H:..}.=~ = v ( r )

PKOPAGATION

O F :EL:ECTRO]~fAGN:ETIC W A V E S

IN TWO

~IEDIA SEPARATED

~TC.

53

where

1(

~

1

Es2 ] }

, z=d

(22)

__ 1 2 } [ ~t [H~ § H,o,-, ~ [H~, + H.o,-H,o,] V(r) = _ let H~o~] + ,fir) ez ~=~" B y substituting from eqs. (17), (18) in the b o u n d a r y condition (21) a n d in (19), we get A2 @ B~ = 0 , 1

A~ exp [iv2d] + B~ exp [-- iv2d] -- A~ exp [-- i~:~d] -- i o v ~ 5~f~{U(r)},

(23)

1 iv~A~ exp [-- iv~d] @ iv~ (A2 exp [iT2d]-- B2 exp [-- i~2d]) = -~ ~l'~ {V(r)} , ,r

from which we obtain the constant functions A~().), A~().) and B~(~) as A,(),)

=

= exp [iv~d][~o#(1--exp [ - - 2 i ~ f l ] ) ~ { V ( r ) } - - ~ 2 ( l + exp [ - - 2 i ~ f l ] ( ~ ( U(r)}]

(24) A2(~) = - - B~(~) = exp [-- iT&d] T , ~ { U(r)} + e3#5~f~{V(r)} io~/~(7:~ + ~)N(),) where N(),) = 1 + C, exp [-- 2ivzd],

Cr __

Tz - - r l T 2 ~- T 1

Now, using the above functions in eqs. (17) and (18), we get the secondary electromagnetic field in terms of the integral representation due to the variation of the interface between the two media. Finally, b y adding the p r i m a r y and secondary fields in the uniform case (3) to the secondary field in the n o n u n i f o r m ease, we can get the total electromagnetic field in terms of an integral representation.

3.

-

Conclusion.

As with other arrangements, no depolarization appears in the T E field originating f r o m a dipole placed in the ionosphere above a duet whose u p p e r b o u n d a r y is constantly varying.

5~

S.T.

BISItAY

and

A. E. 3[OHAMM]~D

A n expression for the resulting secondary electromagnetic field (reflection field) in integral form has been obtained. This was shown to hold for all frequencies, the one limitation imposed being t h a t the duct height varies only slightly from a mean average. I t is also possible to obtain the total electromagnetic field in terms of an integral representation b y combining results f r o m b o t h this and the previous s t u d y (~).

9

RIASSUNTO

(*)

Lo studio della propagazionc di onde elettromagnetiche in due mezzi separati da un'interfaeeia variabilc ha richiesto l'uso dello spazio delle immagini di una trasformazione di Hankel. L'inverso di tale trasformazione produce una rappresentazione integrale, ehe in questo easo si g dimostrata valida per tutte lc frequenze. Inoltre, nessuna depolarizzazione appare ncl campo fintanto chela variazione ncll'interfaccia rimane piccola e direttamentc dipendente dall'altezza del seeondo mezzo. (*)

T r a d u z i o n e a cura della R e d a z i o n e .

P e 3 I O M f i He HO.rlytleHO.