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4 ANALYTIC WAVE THEORY
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Mixture of Two Constituents
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The parameters are Eb, fb' Es, and fs with fb + J.s = 1. The quantity Eg is first determined by using (4.3.20). We use the simple correlation functions
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Taking averages, (e) = fs(; + fb~l and (1~12) = fs l~sl2 (4.3.9) we have ~s = 3r sEg/E o, ~b = 3rbEg/Eo and
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Eb -
+ 2Eg
Using (4.3.36), (4.3.44), and (4.3.45) gives
Eelf
= = =
Eg [1
. 3 2 2 + z2kga3 Usr s + fb 7'b)]
(4.3.52) (4.3.53) (4.3.54)
(Jhh
31 kg l a fslrsl 31 kg l a
4 3[ 2 + fblrbl 2] IK l2 IXOliXlOd 24K:~ krz 1
(Jvv
4 3[ 2 + fb hi 2] IKizl2 lYoli YlOil 24K:~ krz 1 J.slrsl
In Figs. 4.3.1 and 4.3.2 the real and imaginary parts of Eg/E o and Eelf/Eo are plotted as a function of the fractional volume of scatterers. The limit of fs = 1 corresponds to the entire volume being occupied by scatterers. This is not attainable for particles of the ~;ame size and is purely of theoretical interest. In Fig. 4.3.3 the backscattering coefficients are plotted as a function of incidence angle (}i at a frequency of 8.6 GHz and for a half-space with a fractional volume of scatterers equal to 30%. The calculated effective permittivity is (1.47 + iO.663 x 1O-3)Eo The results of the figure indicate that the polarization difference is small.
Mixture of Three Constituents
For three constituents we have a background medium with permittivity Eb, two kinds of scatterers with permittivities Esl and Es 2, and fractional volumes fsl and fs2. The scatterers are assumed to be spherical with radii al and
3.5 Results of Effective Permittivity and BistHtic Coefficients
3.5r-----,---r---r--..,-----,--,---,------,---r---,
t:: 25
"iii
f.~ff/f.(J
f~J/fo
101o<::.---,'-,...--...I0.-2--0J...3--0"'.4---0L.5--0.J...6--...IO.-7--0J....a--0"'.9----l
Fractional volume of scatterers
Figure 4.3.1 Real part of fg/f o and
for Eb = f o , Es = 3.2(1
+ iO.001)f o , frequency
felf I f o
as a function of fractional volume of scatterers 8.6 GHz, and a = 0.5 mm.
3 X 10. 3.5,.-----.-----.--,------,-----,--..,---,-------r---,----,
~ ~ ~ ~ ~
<:tr/fo
~ ~ ~
.- .-
.- ..0.5
.- ,.'
f'r;/ o
o"""=-------'-__--'-__-'----_--'__-'-__-'-__L - _ - - , ' -_ _...L.-_--'
0.1 0.2
0.3 0.4 0.5 0.6 0.7 0.8
Fractional volume of scatterers
Figure 4.3.2 Imaginary part of fg/f o and fefflf o as a function of fractional volume of scatterers for fb = f o ' f s = 3.2(1 + 'iO.001)f o , frequency = 8.6 GHz, and a = 0.5 mlll.
4 ANALYTIC WAVE THEORY
-8,---,-------,----,----,-----,------.----,
-10r--
:!3. E -12 Q)
G vv
0- 14
<II CO
_20'----L-----'-----l.----'-----'-------'----' o 10 20 30 40 50 60 70
Incidence angle (deg)
Figure 4.3.3 BackHcattering coefficientH aH a function of incidence angle (}i for Eb = Eo, E = 3.2(1 + iO.OOl)E o , f.. = 30%, a = 0.5 nUll, and frequency = 8.G GHz. Calculated value 8 of felf is (1.47 + iO.663 x lO-:l)E o .
a2. We have Ib + lsI + Is2 = 1. The quantity Eg is first calculated by using (4,3,20), Assuming, without loss of generality, that a2 ~ aI, the correlation functions are
(4.3.55)
for 0 ~ T ~ aj (4.3.56) for al < T ~ a2 otherwi8e Using (4.3,35) and (4.3.43) to calculate U and W respectively, we have
_ 3E~ (J.sI1,s1 U - 2 {,2
2 3 : + Is2 1.2 + Ib T b)al + 1.2 1s2(1 + Is2)(a2 3 :al) } - 3 82 82
(4.3.57)
3.5 Results of Effective Permittivity 'IDd Bistatic Coefficients
1.7.----,----,---,---,----,---,---.----,---,----,
0. 1.3
"iii Q)
t:;'ff / f o
('t,/f o
1'----'---"-----"-----'------"---'-----''----'----'----'
Fractional volume of scatterer 2
Figure 4.3.4 Real part of f y / f
<a' <sl
3.2(1
+ ill,OOl)eo,
and (eff / f a aH a function of fractional volume of scatterel' 2 fb = SO'Yc" (1.l = 0.5 mm, (/,2 = 0.5 Illlll, and frequency =
17 GHz.
007,---,,_____----.---,------,---,---,----,-----,----,--,
0. 004
'g> 003 ,g
O"------'~--'----'------'-----'----'-----L----',_____--'-,_____----,'
Fractional volume of scatterer 2
Figure 4.3.5 Imagiuary part of eg/<o and eeH)<o as a fUllction of fractiolla\ volume of scatterer 2 for <b = <0, <"I = 3.2(1 + iO.llOl)(o, !b = Sll%, (l1 = 0.5 Illln, (/,2 = 0.5 mm, and frequency = 17 GHz.
4 ANALYTIC WAVE THEORY
-4,--r---r----.------,----,--.,----,----,---,-----r--,
~-10
:2 \
:ai -14 o
C -12 Q)
01-16
s:::
8, = 50
:m- 18
~ -20
<II U
al -22
-24 -28
- - - __
Hi --- --- 70 --=
--- --0.09
-28 '----_'--_'--_'--_-'--_-'--_--'--_--'--_-L-_---'-_----'---l
Fractional volume of scatterer 2
(Jhh at ()i = 50 and ()i = 70 as a function of fractional volulIle of ,;catterer 2 for fb = f o , f 81 = 3.2(1 + i.O.OOl )f o , !b = 80%, al = 0.5 mill, (t2 = 0.5 mlll, and frequency = 17 GHz. Comparison with experimental data.