286 RADIO WAVE PROPAGATION EXPERIMENTS 
In Table 13 are given the probabilities @, in the 
drop diameter range 0.05 to 0.55 cm and wavelength 
range 3 to 100 cm. A glance at this table shows that 
(2n + 1) (2m + 1) : 
n(n+ 1) m(m + 1) 
he 
n=1 m=1 
with the exception of the shortest wavelengths and co oN 
+) : : — Pe adP;, dP 
largest drops the probability of the waves being truly ana* 
absorbed is always mnch larger than that of their being 29. dé i 
scattered. The smaller the drops the greater the chance Pip) .. dP} dP} Ba 
; : 3 3 +b, b¥( —2=™ gin? ¢ + —2 52 
of absorption, since, according to the cross-section n sin? 6 Wael ia BON 
formulas, for small drops Qs is proportional to D°/A* : f 
: l é 1 
(Rayleigh’s law) whereas Vt ~ Qavs is proportional Aiton ae P, dP; Esaneen + <8 Pn aP,, sin? )] ed 
to D®/\ and in our case the drop diameter D is always sin@ do in 6 do 
smaller than the wavelength A of the radiation. (65)* 
Taste 13. Probability of scattering @, by spherical water drops of D cm diameter. 
A, cm 
D, em 3 5 8 10 20 30 50 75 100 
0.05 3.94 10-3 1.6410-3 6.631074 4.251074 1.941074 1.091074 49210-5 1.7410% 7.7410-6 4.33 10-6 
0.10 1.5410-2 1.091072 4.891073 3.22107 1.53108 860104 3.911074 1.401074 6.331075 3.47 10-5 
0.15 2.1110-2 2.9010-2 1.4710 9.961073 5.0010°§ 2.8710°8 1.321073 4.731074 2.111074 1.17 1074 
0.20 2.86 10-2 5.15 10-2 2.9110-2 2.1010-2 1.11107? 6.511075 3.0510-§ 1.101073 3.95104 2.76 10-4 
0.25 3.72 10-2 7.6010-2 4.7510-2 3.161072 1.9710-2 1.2110? 5.811078 2.141078 9.661074 5.37 10-4 
0.30 5.31 10-2 1.03107! 6.9110 5.331072 3.121072 1.75107? 9.771073 3.661078 1.61107 9.32 10-4 
0.35 8.06 10-2 1.29107! 9.061072 7.2810-2 4501072 2.96102 1.45102 5.76108 2.6010°3 1.47 10-8 
0.40 1.21101 1.54107! 1.13107! 9.311072 6.06107? 4.11107? 2.17102 8.421078 3.881078 2.18 10-3 
0.45 1.95 10-1 1.84107! 1.835107! 1.141071 7.7910-2 5.471072 2.9610-2 1.191072 5.5010-3 3.11 1078 
0.50 2.96 10-1 2.1710-1 1.58101 1.341071 9.631072 6.881072 3.901072 1.5910-2 7.4310-8 4,23 10-8 
0.55 482.107! 2.49107! 1.80107! 1.56107! 1.15107! 8.44107? 4951072 2.081072 9.821078 5.61 10-8 
Back Scattering (Echoes) 
Whereas the attenuation of microwaves is of inter- 
est to both communication and radar, back scattering 
is of importance to radar only. The importance of the 
echo phenomena is twofold. On the one hand, it is 
of operational interest to distinguish between atmos- 
pheric echoes of the waves and their reflection from 
other targets in the atmosphere. On the other hand, 
the observation of these phenomena has led to the rec- 
ognition of its meteorological value in helping to map 
the storm topography of the atmosphere (storm detec- 
tion) around the position of the observer and at 
ranges limited only by the characteristics of the radar 
set used.**5*88 
The echo intensities may be computed from for- 
mula (35) for the differential cross section of drops 
a(7) for back scattering (scattering angle 7). 
According to equation (22), the power scattered by 
a spherical particle per unit solid angle at a point 
(7,0,) is 
aP, = 1 2 S12] 2 
(2) = Uimete 
Using equations (16) and (17), we obtain, remember- 
ing that the incident power per unit area is (1/22) 2, 
the following expression for the differential scatter- 
ing cross section: 
Or, limiting ourselves to the approximation where only 
the electric dipole (b,), electric quadrupole (6,), and 
magnetic dipole (a,) are effective, we find, using the 
explicit expressions of the associated Legendre poly- 
o(6, 4) 
nomials, 
a) 
dw / %¢ 
e 
4a. 
Re [stl (sin? ¢ + cos? @ cos? d) + 9] a:|2(cos? ¢ 
+ cos? 6 sin? ¢) 
+ 25|bs|2 (cos? @ sin? ¢ + cos? (2.6) cos? ) 
+ 18a1b:* cos@ + 30b:b.* cos@(sin? ¢ + cos (26) cos?¢) 
+ 30a1b.* (cos?6 sin?¢ + cos (26) cos?¢) | em. (66) 
Here the first term inside the brackets represents the 
contribution of the electric dipole, the second is the 
magnetic dipole term, the third is the electric quad- 
rupole term, and the three others correspond to inter- 
ference terms between these three poles. 
In the optical case it is known that the larger the 
parameter p = D/A, i.e., the nearer the wavelength 
is to the diameter of the scattering sphere, the more 
the radiation is scattered forward than backward. A 
study of equation (66) for water drops of 1-cm diam- 
eter shows that for spheres of this size it is only when 
A> 15 cm or p < 0.2 that the back-scattered intensity 
kWith 9 = 7m this reduces to equation (27) of the radar 
cross section. 
