274 RADIO WAVE PROPAGATION EXPERIMENTS 
kone 
We shall also need the fraction of the power scat- 
tered backwards by the sphere, i.e., in the direction 
6 =7, per unit solid angle. One thus obtains, with 
dw = sin 0d6d¢, 
a) E = = a 
‘) ==* Re > D(-) (ntl): 
(¢ O=4 8k, 1, 22 
(— Re) >, (2n +1) @ +8). (26) 
n=1 
P= 
(2m+1) (a—bn) (am*—bm*), (27) 
as a simple calculation shows, starting from equation 
(22). 
It has already been mentioned in connection with 
the definition of the complex wave number, equation 
(2), of a homogeneous and isotropic medium that its 
imaginary part is chosen to be negative. We shall 
write Fela a8. (28) 
where £ is the phase constant and @ the attenuation 
constant; both are real. The explicit expressions of 8 
and «@ in terms of the characteristic electromagnetic 
properties of the medium, namely, inductive capacity 
e, permeability » and conductivity o for the given 
frequency w/2z are the following :1% 
polit sty) @ 
ete yl @ 
With equation (28) the field strength, electric or 
magnetic, in a plane wave propagated in such a 
medium along, say, the z axis, is, omitting the time 
factor, of the form: 
F = Fo ¢-ié—az, 
F, being an amplitude vector directed along either one 
of the two remaining coordinate axes. The attenuation 
factor ~ simply means that in this medium an advance 
of the wave through a distance of 1/a meter is accom- 
panied by a decrease in the field strengths in the ratio 
of 1:e = 0.368, or the power per unit area (Poynting 
vector) decreases in the same ratio over half that dis- 
tance or 1/2a meter. 
In the mks system, the attenuation factor is then 
hepers per meter, whereas the power absorption co- 
efficient is 20a (log,,e) decibels per meter = 8.686a 
db per meter. 
Our problem is the study of propagation in a medi- 
um which is neither homogeneous nor isotropic, inas- 
much as it consists of a suspension of water droplets 
in the atmosphere. It can be proved that in such a 
medium the attenuation factor is the sum of all the 
different partial attenuation factors due to different 
physical phenomena. 
The particle attenuation factor will still be denoted 
by @. More appropriately we might call « the average 
particle attenuation factor. It may be defined as 
@ =5NQ: neper per unit length, (31) 
where N is the average number of water drops per 
unit volume and Q; the total cross section of one 
droplet. The absorption effect of one spherical water 
drop is given by Q which is the ratio of the power P; 
removed by the drop from the beam falling on it to 
the incident power per unit area. Provided the effect 
of all the drops be linearly additive, equation (31) 
will express their average attenuation effect. The 
incident power density is the complex Poynting vector 
of the beam 
pee (32) 
2n2 
Therefore, with equations (3) and (26), 
r? . ; 
Q.=5-(—Re) X (2nt1) (ah +5:), (83)! 
Ls n=1 
where A = 27/k, is the wavelength of the radiation 
in air or free space. Similarly the cross section for 
scattering is, with equation (25), 
Ie SS e124 152 
Q.= = DL Ant) (lawl*+/onF). 84) 
n=1 
The differential cross section for back scattering (or 
radar cross section) is then, with equations (27) and 
(32), 
(*)o— n= a(n) 
w (2)'Re > x (=) en+1) 
(2m+1) [a,a%,*+-b,b,*—2a,b,,*]. (35) 
These are the formulas on which the computations of 
attenuation have been based. They are certainly cor- 
rect in the wavelength region 1 cm to 100 cm with 
which the present study is mainly concerned, and they 
correctly take into account the linear dimensions of 
the scattering and absorbing particles. According to 
Brillouin? these formulas have to be modified in the 
limit A < a, in which case for perfect reflection they 
lead to a scattering cross section 2a, double of the 
expected geometrical cross section. Since the modifica- 
tions mentioned do not play any role for A > 3a/10, 
which condition will always be satisfied in the present 
report, they will not be discussed here. 
4The minus sign is missing in the presentation in reference 
18a; see formulas (26) and (29) on page 569. This leads to the 
incorrect result, for nonabsorbing spheres, that the scattering 
cross section Q, reduces in this case to the negative of the 
total cross section Q;. Clearly Q; reduces to +Q;. 
