DIELECTRIC CONSTANT, ABSORPTION AND SCATTERING 271 
with the wavelength, scems to suggest that the rain 
attenuation might level off or even decrease for waves 
shorter than 1 em. However, without a closer inves- 
tigation of the raindrop absorption in this wavelength 
region, no precise statement can be made on this 
subject. 
As regards the normal atmospheric absorption of 
microwaves, it may be mentioned that the oxygen ab- 
sorption is due to the paramagnetic character of this 
gas. It is through the interaction of the magnetic field 
strength with the magnetic dipole moment of the oxy- 
gen molecule that microwaves are absorbed by this 
gas. In the microwave region the oxygen molecule has 
a resonance line at A = 0.25 em and a band near 0.5 
em, while water vapor seems to have a resonance line 
around 1.25 em and interacts with the radiation field. 
through its electric dipole moment. The whole subject 
has been discussed exhaustively.» 
The study of the scattering of microwaves by rain- 
drops shows that the radar observations of rainclouds 
can be explained satisfactorily if the scattering is at- 
tributed to spherical particles of dimensions similar 
to those of raindrops, even though no rain reaches the 
ground. Recent experimental work**’ has helped 
considerably in clearing up the apparent inconsist- 
ency which previously existed in this subject. 
On the whole, taking into consideration the irregu- 
larities of the precipitation forms in space, it may be 
said that theory provides a fairly good picture of mi- 
crowave propagation through a cloudy, foggy, or rainy 
atmosphere. 
The major object of the present paper is to report 
the theoretical and experimental work done on atten- 
uation of microwaves by liquid or solid water particles 
falling through the atmosphere, as well as clouds and 
fog, which are water and ice particles in suspension 
Theoretical work has thus far been concerned with 
the problem of a plane electromagnetic wave scattered 
and absorbed by a.single spherical or spheroidal par- 
ticle, first studied in detail by Mie* for other purposes. 
The application of the results of Mie to very short 
radio wayes propagated through rain, clouds, and fog, 
i.e., through a swarm of spherical water droplets, was 
made by Ryde.'#1? The present report is, in part, an 
extension of his work using more detailed meteorologi- 
eal data on rains. 
A compact and elegant presentation of the problem 
of absorption and scattering of a plane wave by a 
sphere is given by Stratton.* The method followed 
by him was first used by Lord Rayleigh.*® In the follow- 
ing section a brief review of this method will be given. 
Scattering and Absorption of Radio 
Waves by Spherical Particles 
Let the center of a sphere of radius a be the origin 
of a rectangular coordinate system and suppose a 
plane wave to be propagated along the positive z axis 
and to fall on the sphere (Figure 3). The sphere of 
permeability », (henrys per meter) and complex in- 
ductive capacity «, (farads per meter) is em- 
bedded in a medium of permeability », and inductive 
capacity e,. The plane wave is supposed to be polarized 
parallel to the « axis. 
Vicure 3. Spherical coordinates. 
The electric and magnetic field strengths E;, H, 
of the incident wave (subscript 1) are expanded into 
spherical wave functions.*** The reason for this expan- 
sion lies in the boundary conditions and will appear 
clearly below. 
E; = E, = a, Eye ~# et ot , 
1 
H; = H, = n 4 Eo e ~ihz + iat (1) 
where : 
k = (uew? — jucw)? (2) 
is the complex wave number of the medium (here 
ky = 2a/X, d being the wavelength referred to air or 
free space), the square root is so taken that the imag- 
inary part is negative, and 7 is the intrinsic impedance 
of the medium, or 
1= ee = _ = 377 ohms. (3) 
€ 
2 
The conductivity o is expressed in mhos per meter; 
the frequency o, in radians per second; a, ay, and as 
are unit vectors pointing in the positive z, y, and 2 
directions, respectively. 
The plane wave vectors® 
will now be expanded into spherical wave vectors 
at a point of spherical polar coordinates (7,0,¢). It is 
readily recalled that 
a,e*F and ayes 
18a 
x =rsin 6 cos q, 
y = rsin sin ¢, 
z =r cos 0, (4) 
with 
0<0<7and0<¢S 27. 
°Henceforth the time factor e+! will be omitted, as it 
does not play a direct. role in what follows. 
