DIELECTRIC CONSTANT, ABSORPTION AND SCATTERING 287 
is about the same as the forward-scattered intensity. 
For such p values only the dipole term in equation 
(6€) remains of practical importance. 
Suppose that we adopt a p value of 0.2 as a rough 
indication of what happens in the case of actual rain- 
drops, the diameter of which is less than about 0.55 
em. It is then seen that for radar purposes the use of 
longer waves is favored, as far as the amount of back- 
scattered power is concerned, viz., in those cases where 
the greatest amount of back scattering from water 
drops is of operational importance. This will clearly 
occur in radar meteorology. However, when it is 
desirable to limit as much as possible the back scatter- 
ing from rain or rainclouds, one might make use of 
this forward-backward scattering dissymmetry, which 
is the more pronounced the shorter the wavelength as 
compared with the diameter of the raindrops. This 
dissymmetry might, however, be counterbalanced by a 
rapid increase in’ the attenuation as well as a general 
decrease in the intensity of scattering. 
The differential cross section for back scattering re- 
sults from equation (66) by taking 6 = 7 there. Using 
the explicit expressions (43) of the amplitudes a,, b,, 
and 6,, one obtains for this back scattering (or radar 
cross section) 
Xr 
o(r) = G)e (Ap+Asp?+Asp?+ Asp! A5p>+ Agp® 
+++)em?, (67) 
with the following coefficients A", using the notation 
defined by equations (45), (46), and (59) : 
A, = 9|6:|?, 
As = 18 [8,B,B, 8, — 0,8, — a, B,] 
— 30 [6.6 + BB] , 
A; = 18 [8:, + BiB, ] 5 (68) 
Az = 9 [la®|? + [B,|2] — 18 [a B, + a,8,] 
— 30 [6,6 + B,B. — Bo 
= a Bo] + 25|B.|2 , 
As = 18 [8,B, + BB, — a6, — a8, ©] 
— 30 [AB + BB] , 
Ag = 9161. 
The radar cross-section formula (67) is the same 
as that given by Ryde.’” Again o(7) is not a function 
of p only since the coefficients of the successive powers 
of p in the expansion (67) depend on the wavelength. 
The computed echo cross sections o(7) for spherical 
water drops with diameters in the range 0.05 to 0.55 
em and the wavelength range 3 to 100 cm are given 
in Table 14.’ These cross sections reduce practically to 
the Rayleigh type, i.e., the series (67) reduces to its 
first term for the smaller drops at any wavelength and 
for any drops for wavelengths larger than about 15 cm. 
Since the Rayleigh term predominates in o(7), with 
the exception of the larger drops and smaller wave- 
lengths, the trends of variation of o(7) with either 
the diameter, at constant wavelength, or the wave- 
length, at constant diameter, are similar to those of 
Q,, the total scattering cross section. A graphical repre- 
sentation of the data of Table 14 is thus of no particu- 
lar interest; they appear implicitly in Figures 15 
and 16. 
In order to compute the radar attenuation factor « 
associated with echo phenomena occurring with rain 
of known drop size distribution, we have but to use 
equation (31) and hence obtain for N;, drops of & em 
diameter per cm'®, 
On = 5M %,(3) neper/em, (69)™ 
and for a given distribution of particles 
a, = » nn 5 >» Nx %(ar) neper/em. (70) 
k=0 k=0 
Using the radar cross section of Table 14 and the drop 
size distributions in different rains as given in Table 
%, we have computed a,, the attenuation factor due to 
back scattering in the wavelength range 3 to 100 cm. 
The results of these calculations are included in Table 
15 and in Figure 17. The variation of a, is represented 
as a function of the wavelength of the incident radia- 
For the shorter waves and large drops the cross sections 
given are merely orders of magnitude, as the convergence of 
equation (67) is too slow in that case; in fact, it is even 
slower than the expression for Q,. 
=™The coherent portion of the scattering is neglected here 
on account of the assumed random distribution of the scatter- 
ers. See, nevertheless, a recent note by F. Hoyle.*4 
Tas.E 14. Back scattering cross section o (7) (em?) of spherical water drops of D cm diameter. 
, em 
D, em 3 5 8 10 15 20 30 50 75 100 
0.05 4.25 10-9 5.551019 86310- 3.501072 6.96 10-12 2.1810-!2 4.321078 5.601074 1.1110°! 3.50 10% 
0.10 2.6410-7 3.521078 5.4710-9 2.241079 4.441079 1.401079 2.77107! 3.5910°2 7.091018 2.24 10718 
0.15 2.88 10-6 3.9710-7 6.28108 2.54108 5.1010-9 1.601079 3.18107 41210 8.14107 2.57 10-2 
0.20 1.48 10-5 2.1510-§ 3.451077 1.421077 2.8410-§ 8.9410°9 1.7710°9 2.291079 4.52104 1.43 1071! 
0.25 5.02 10-5 7.4210-§ 1.3010-6 5.3410-7 1.0710-7 3.42108 6.7310°9 8.72107 1.721019 5.45 10°" 
0.30 1.34104 2.2510-5 3.8010-6 1.5710-§ 3.191077 1.201077 2.0210°§ 2.6210°9 5.171071 1.64 10-10 
0.35 2.48 10-4 5.4010-5 9.371076 3.9110-6 8.011077 2.581077 5.041078 6.5310°9 1.2910°9 4.08 10-1 
0.40 5.0410-4 1.121074 2.031075 8.5510-§ 1.7710-§ 5.751077 1.13107 1.461078 2.881079 9.13 10-10 
0.45 7.76 10-4 2.12104 3.99105 1.70105 3.5510-6 1.1610-§ 2.321077 3.00108 5.921079 1.87 10-9 
0.50 9.91 10-4 3.6510-4 7.3010-5 3.1410-5 66310-§ 2.1810°§ 4321077 56010°§ 1.11108 3.5010 
0.55 5.95 10-4 5.8210-4 1.2410-4 5.4410 1.1610 3.87106 7.701077 9.98108 1.97108 6.24109 
