Vol.ll No.4 JOURNAL OF ELECTRONICS Oct. 1994

T H E E V I D E N C E OF L O S S Y W E D G E

D I F F R A C T I O N C O E F F I C I E N T IN T H E O R Y

Zhao Xiongwen(~]~5~:) Xie Yixi(~/~)

(China Research Institute of Redlowave Propagation, Xinxiang, Henan 453003)

A b s t r a c t Formulas of diffraction field of lossy wedges with less t han 180 ~ wedge angle are

derived on the basis of the Fresne]-Kirchhoff wave theory and thei r numerical results are compared

with those of the heuristic lossy wedge diffraction coefficient given by Luebbers (1984), showing

good agreement between the two types of numerical results wh ich have different bases in theory.

The agreement shows t ha t the lossy wedge diffraction coefficient as an extension of UTD is quite

reasonable.

K e y w o r d s Fresnel-Kirchhoff theory; Lossy wedge coefficient; Diffraction

I. I n t r o d u c t i o n

Since the heuristic coefficient involves such important parameters as wave polarization,

conductivity and permittivity, ridge profile and surface roughness, so it would be useful

for propagation path made of water surface, smooth earth and forested land etc. It has

given excellent results when applied to irregular terrain propagation and compared with

measurements [1-5]. More recently, a comparison between the heuristic lossy wedge coeffi-

cient and an exact Maliuzhinets diffraction coefficient for 180 ~ lossy wedge with different

wedge faces was reported, which showed good accuracy between them [3]. It must be pointed

out that lossy wedge coefficient as an extension of Kouyoumjian-Pathak UTD coefficient[ el

does not have rigorous bases in theory, its accuracy may be in doubt. So since it has been es-

tablished some people try to give its theoretical evidence, but, as far as we know, it seems that

little progress has been achieved. In the present paper, an evidence in validity of the heuris-

tic coefficient for a lossy wedge less than 180 ~ is given by means of Fresnel-Kirchhoff(F-K)

theory, and it can be seen that good agreements have been achieved between our numerical

results and lossy wedge field given by the heuristic coefficient.

I I . T h e H e u r i s t i c L o s s y W e d g e D i f f r a c t i o n C o e f f i c i e n t

If we use the coordinate system as shown in Fig.1 and assume incidence normal to the

340 JOURNAL OF ELECTRONICS Vol.11

edge, then the heuristic diffraction coefficient is[l]:

e x p ( - J 4 ) { ( ~ r + ( 2 ~ ~ D = �9 cot �9 F(kla+(~- ~')) 2nv/~-k

cot ( z r - (~ § ~') ~ . F(kla-(~ + ~')) § \ 2n /

(i)

r

Som'ce / / ~ Field point

/ 0 face n face

Fig.1 Geometry for lossy wedge diffraction coefficient

where F(x), l and a + were defined in Ref.[6], R0 and Rn are the reflection coefficients

respectively for the 0 face with incidence angle T' and the n face with the reflection angle

nlr'- ~. Here the usual Fresnel reflection coefficient of the plane wave is used for the reflective

surface with the finite conductivity and dielectric constant.

For vertical polarization

R0, Rn =

For horizontal polarization

sin r - X/~r - cos 2 r

sin r + x / ~ - cos 2 r

~ sin r - ~ / ~ - cos2 r R o , R n "~ ^

~ sin r + V/g~ - cos 2 r

where r is the incidence angle or reflection angle.

(2)

(3)

III. The Calculation o f L o s s y Wedge Field by F-K Theory

In Ref.[7], based on Helmholtz integral and considering the reflection of the flat-top, as

shown in Fig.2, the field relative to free space at point p is given as follows:

( k X j + l ~1/2 f ~ E(Xj+l , z j+l) = ~(x~ , zj)(e ~ + R ~ jk~ )dzj (4)

No.4 EVIDENCE OF LOSSY W E D G E DIFFRACTION C O E F F I C I E N T 341

P R

CJ r

f I

i ~zj

xj Xj+l

Fig.2 G e o m e t r y for f l a t - topped reflection

where

xj X J + l Z~ - - - - Z j + 1 (5 )

r - 2x jAx j X j + 1

Ax~+l = X~+l -- xj (6) i r ~ r + 2hlh2 (7)

Axj

h, = z j - ~ j (8)

h 2 = Z j + l - - ~ j + l (9)

j and j + 1 mean two neighboring knife-edges, which extend to infinite along the y axis; xj ,

zj and Xj+l, Zj+l are the coordinates of the integrated elemental surface over . the edge j

and those of the observation point p over the edge j + 1; ~j, ~j+l are the heights of the edges

relative to transmitt ing antenna; k is the wave number; RE is the reflection coefficient of

the flat-top.

In Fig.3, assume that both the point source T and the observation point are close the

the lossy wedge and dl, d2 )~> H >> A, where A is the wavelength. Consider only the two

dimensional problem here and assume the cross section of the wedge is at x-z plane. Now

let point T be the origin of coordinate system, then, from Eq.(4), the received field is as

follows:

$ ( R ) = \ ~ i ] \ did2 E(Q)[exp(ikr) -~- RF2 exp(ikr )]dz (10)

where /~ represents the field which is relative to free space; Q is Fresnel source; r and r ' are

the distances of the direct ray and reflection ray from Q to receiver, RF1 and RE2 are the

342 JOURNAL OF ELECTRONICS Vol . l l

T

t t

J J

1

P i / ~ Q

Fig.3 Geometry of loss)" wedge diffraction by F-K theory

reflection coefficients for both wedge faces.

dl+d__.___2 [ dl ]2 r-- 2did2 z dl+d~(HR-HT)

[ ]" , d l + d 2 z + dl (I-IR+U-IT) + - - r -= 2did2 dl + d-----~ h= H - H T

2HRHT 2HRH dl+d2 d2

(11)

(12)

(13)

(14)

As a conductor wedge is concerned, the reflection coefficients of 0 face and n face

are constants regardless of polarization. However for a lossy wedge, its conductivi ty and

permit t ivi ty are finite, the reflection coefficient changes with the incidence or reflection angle,

which is related to the.height of the Fresnel source Q (varies with integral variable z), so RF1

and RF2 can not be taken as constants to put them outside from the integral in the Eq.(10).

In Eq.(1), when HT, HR, H, dl, d2 are definite, the incidence and reflection angles on both

sides are not changed with the change of Fresnel source Q, tha t is to say, in this circumstance,

R0 and Rn are taken as constants. It appears tha t there are some contradictions between

F-K theory results and the heuristic lossy wedge diffraction coefficient, but this is not imply

that the lossy wedge coefficient is not valid, because they are based on different theories, the

former on the wave theory, however, the lat ter as an expansion of the UTD {6] on geometry

optics.

There is a more accurate way to solve lossy wedge diffraction by making use of F-K

theory in the case of far field. Tha t is to solve Eq.(10) in different Fresnel regions. See Fig.4,

where Fi = v~F1, F1 is the first Fresnel radius. Assume that reflection coefficient does not

change in each Fresnel belt (in high frequency and far field approximation, this assumption

is accurate enough).

No.4 E V I D E N C E OF LOSSY W E D G E D I F F R A C T I O N C O E F F I C I E N T 343

T

- / i "-

F i g . 4 G e o m e t r y fo r t h e F r e s n e l b e l t s

From Fig.3, we have

HR . dl + HT " d2 H~ = H - d (15)

where d = dl + d2. It is necessary to compare H~ with F1, vf2F1, vf3F1 . . . , x/iF1 to

determine how many Fresnel belts are obst ructed by the wedge. In the diffraction region,

the contr ibution to the received field is mainly from first several Fresnel 's belts near to the

top of wedge. In illumination region, only a few Fresnel belts near the point p must be

considered in calculation of the field. For example, as to the diffraction field, if the i - th

Fresnel belt is not obstructed at the outset , so

\ did2 ] [JH--HT ~- JvqF1--HT "~-''"

+ Jv'7-~F1-HT J E,(Q)[exp(ikr) + RE2" exp(ik/)]dz (16)

In Eq.(14), it contains a series of integrals as follows:

( k )1/2 (dl~-d2)l /2~ E(Q)[exp( ikr) + RF~ . exp(ikr ' )]dz (17)

k(R) = ~ i \ did2 F1-HT

In each Fresnel belts, the reflection coefficient can be regarded as a constant, so the

received field expressed by Eq.(17) can be divided into four parts:

s E l ( R ) = C . exp( ikr )dz El --HT

E2(R) = RF1 �9 C. f ~ F1 --HT

P~3(R) = RF2. C. f ~ FI -- HT

/~4(R) = RF1Ry2" C . f~ i ~ F1 -- HT

E' (Q)exp( ikr )dz

exp( ikr ' )dz

E ' (Q) . exp(ikr ' )dz

(18)

(19)

(20)

(21)

344 JOURNAL OF ELECTRONICS Vol.l l

where

C = ~ \ d2d2 (22)

/~' (Q) = exp ~ ik 2(z -t- HT -- H)HT (2a) [ dl )

The physical meanings of the four rays are:

/~l(R)--diffracted field,

/~2 (R)--reflected~diffracted field,

E3 (R)--diffracted~reflected field,

/)4 (R)--reflected~diffracted~reflected field.

Completing some integral transformations and a series of simplification procedure, the

fields of the four rays can be expressed in very simple forms, which contain Fresnel integral,

as an example/~4 (R) is expressed as

E4(R) = RFI " RF2 " exp { - i k ( 2HRHd2

where

2HHT) } F(w4) (24) - - + d ~

1 F(~) = - -~ . exp ( - i~ /4 ) �9 {0.5 § C(~) § i[0.5 § S(~)l t

S(w), C(w) are the Fresnel integrals;

(25)

(26)

v/iF1 (27)

where Ao is the first fresnl radius and

A4 = --HT -- (Ha - HT) dl dl + d2 Then the total field is

4

= j = l

(28)

As concerns the far field, under the condition of dl, d2 >> H >> )~, because the incidence

and reflection angles are small and change only slightly with the height of Fresnel source Q,

so we can use R0 and R~ instead of RE1, RE2 in order to solve Eq.(10).

IV. C o n c l u s i o n

Lossy wedge diffraction field given by its heuristic coefficient has been compared to the

F-K wave theory for less than but near to 180 ~ wedge angle, as shown in Fig.5, Fig.6 and

Fig.7, Fig.8 with good agreement between them, and this perhaps implies tha t the heuristic

lossy coefficient as an extension of UTD is reasonable. But this conclusion may be valid

only under the condition of far field.

N o . 4 E V I D E N C E O F L O S S Y W E D G E D I F F R A C T I O N C O E F F I C I E N T 3 4 5

c

0

-10

-20

-30

-40

-50

-60

,M/" )

~r/if/ /

, . f i ; "

r, /

/ - - - F-K

o ~ o diffraction coefficient Lassy wedge

--r I 0 50

- I J ' - i ' - I " ' r " f " I 100 150 200 250

I~ Receiver height (m)

300 350 400

Fig.5 Lossy wedge dif f ract ion as s h o w n in Fig.3. HT ---- H ---- 200m, dl ~- d2 ---- 10km, ver t ica l po-

lar izat ion; 0 face ( lossy ear th) : s r : 8.0, a : 0.002 s / m ; n face ( a l u m i n u m ) : er ---- 1.0, a = 35 x 10 ?

s / m ; f requency: f ~ 1GHz

r o

0 - -

-10-:

-20

-30

-4O

-50

-60

l

/r

tl

J " F-K '/ , , o o Lossy wedge diffracgion coefllcie~t

/ ' I I

1 - -1 - - - [ - - -1 - J . . . . . . . f - i . . . . [ " - r - j . . . . w I F- I -!'

50 i00 150 200 250 300 3 5 0 , 400

Receiver height (m)

Fig.6 Lossy wedge diffraction. Hor izon ta l polar iza t ion; t h e p a r a m e t e r s are t h e s a m e as in Fig.5

3 4 6 J O U R N A L O F E L E C T R O N I C S V o l . l l

10 i

-10 '' : i : ! i : :

-70 0 50 100 150 200 250 300 350 400

Receiver height(m)

F i g . 7 L o s s y w e d g e d i f f r ac t i on . F r e q u e n c y : f = 1 0 G H z ; v e r t i c a l p o l a r i z a t i o n ; 0 face: e r = 15.0,

a ---- 0 .033 s / m ; rt face: e r = 30.0, a ---- 0 .15 s / m ; t h e o t h e r p a r a m e t e r s a r e t h e s a m e as in F i g . 5

10

ii!ili . -70 . . . . . . . . . i . . . . . . . . . . . . . . . . . . . �9 . . . . . . . .

0 50 100 150 200 250 300 350 40

Reoeiver height{m)

F i g . 8 L o s s y w e d g e d i f f r ac t i on . H o r i z o n t a l p o l a r i z a t i o n ; t h e p a r a m e t e r s a r e t h e s a m e a s i n F i g . 7

No.4 E V I D E N C E O F LOSSY W E D G E D I F F R A C T I O N C O E F F I C I E N T 347

R e f e r e n c e s

[1] R. J.. Luebbers, IEEE Trans. on AP, AP-32(1984)1, ?0-76.

[2] R. J. Luebbers, IEEE Trans. on AP, AP-37(1989)2, 206-211.

[3] R. J. Luebbers, IEEE Trans. on AP, AP-36(1988)7, 1031-1034.

[4] K. Chamberlin, J. Luebbers, IEEE Trans. on AES, AES-18(1982)1, 11-20.

[5] R. J. Luebbers, IEEE Trans. on AP, AP-32(1984)9, 951-955.

[6] It. G. Kouyoumjian, P. H. Pathak, [EEE Trans. on AP, AP-62(1974)11, 1448-1461.

[7] J. H. Whitterker, t~dio Sci., 25(1990)2, 831-851.

[8] ~ Y ( ; , ) ~ ~ , - ~ ~ ) ~ J ~ , 1989, ~ ~ .