276 RADIO WAVE PROPAGATION EXPERIMENTS 
as to have the real and imaginary parts of the ampli- 
tudes separated easily. The refractive index N of the 
spheres is connected with their complex dielectric 
constant by 
ee = N? Cr le Jéiy (40) 
or with 
N =n (1— jx), (41) 
the complex index of refraction, one has 
e, = n? (1—x?); 6; = Qnx. (42) 
Using equation (40) in the amplitudes (38) one 
gets 
1 i 
Oh Fe [—e¢—jle—1)]p%, 
— 2; 2 
f= Geeayereat need 
[ (e + 2) (Ze, — 10) + 7e,?] had: 
Geach 9 
(e-—1)?(- +2)? +€2[2(c-— 1) (-+2)—9] +e , 
[( +2)? +e]? ieee 
Imp, = —2@ = VG@+2) +e? , 2 
BR Gao © B 
(e-— 1) (e-—2) (e--+2)?+ €,?7[2(€-+1)?— (3e-+20)]+-€,4 ; 
[(e + 2)? + €:7] ? 
id. 8« L(- — 1) (+2) +67] -*, 
3 [ (e- + 2)? + e,?]? 
pp = ELST LG/S) [= V) er +38) + 24) 
3 (2e, + 3)? + 4e2 ; 
(43) 
These amplitudes are the same as those found by 
Ryde. They allow the computation of attenuation 
and back scattering with a certain approximation. 
The results thus obtained are the more accurate, the 
smaller the parameter p = 27a/X. 
In the computation of the amplitudes a, and 6, 
we have used the same values of the real and imag- 
inary parts of the dielectric constant of water «- and 
e, as the ones used by Ryde. These were obtained by 
using the Clarendon Laboratory values for «, and ¢ 
for waves of 1.26-cm wavelength?’ and determining 
with them the transition wavelength A, in the Debye’® 
formulas 
iy PN 
= fo toe my e = > (@ = €or); (44) 
r 
ed, = 1.33, ¢, = 81, do= 1.59 cm. 
£In his first report Ryde!” gave incorrectly the coefficients 
of p® in both the real and imaginary parts of b;. The coefficient 
of p® in the real part was corrected in the second report. In 
comparing the b,’s with the amplitudes given by Ryde, the 
relations (38) have to be taken into account. 
Tasie 1. Values of the dielectric constant of water at 
t — 18C, used in this work.* 
A, cm €r 6 = 60cr o mhos/m 
1 24.2 35.6 5.93 x 10 
1.26 32.5 38.6 5.11 x 10 
1.62 43.3 39.5 4.13 x 10 
2 50.6 38.5 3.20 x 10 
2.5 58.2 35.9 2.39 x 10 
3.0 63.6 32.7 1.81 x 10 
4.0 70.3 27.2 1.13 x 10 
5 73.8 22.7 7.56 
6 76.0 19.3 5.36 
8 78.0 15.1 3.15 
10 79.0 12.3 2.05 
15 81 8.40 9.33 x 107! 
20° 81 6.30 5.25 x 1071 
30 81 4,20 2.33 X 107! 
50 81 2.52 8.40 x 107? 
75 81 1.68 3.73 X 1072 
100 81 1.26 2.10 x 107? 
*The computations of the attenuation and scattering effects are all 
based on this table and refer therefore always to temperatures of about 
18C, unless stated otherwise. 
Taste 2. Temperature variation of the dielectric 
constant of water (K band). 
Degrees C €r Ci 
Water 3 27 27 
25 35 23 
60 44 14 
Iee —15 3.3 0.011 
The values of ¢, computed with these formulas happen 
to be in fair agreement with the experimental values 
obtained by a large group of independent workers.*”?? 
Yhere seems to be a regrettable situation concerning 
the values of ¢;, and no serious studies have been made 
on the temperature and frequency variation of this 
quantity, so fundamental for the microwave region. 
A beginning in this direction has been undertaken by 
the Radiation Laboratory. In Figure 4 we have 
drawn the curves ¢,(A) and (A) in the range 1 to 11 
em, and Table 1 gives the values of the dielectric con- 
stant used in this work in the wavelength interval 1 
to 100 cm. 
It is interesting to consider here the temperature 
variation of ¢,. Recent measurements made in the 
Radiation Laboratory in the K band” gave the results 
shown in Table 2. 
Figure 4. Dielectric constant of water (t—18C) e,= 
Jet 
