154 



DIELECTRIC CONSTANT. ABSORPTION AND SCATTERING 



factor a 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 lie = 0.368, or the power per unit area (Poynting 

 vector) decreases in the same ratio over half that dis- 

 tance or l/3a meter. 



In the mks system, the attenuation factor is tlicii a 

 nepers per meter, whereas the power absorption co- 

 efficient is 20a (logidg) 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- 

 iiiiicli 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 

 diiferent partial attenuation factors due to different 

 physical phenomena. 



The particle attenuation factor will still be denoted 

 liy a. More appropriately we might call a the average 

 particle attenuation factor. It may be defined as 



a =-NQt neper per unit length, (31) 



where iV is the average number of water drops per 

 unit volume and Qt the total cross section of one 

 droplet. The absorption effect of one spherical water 

 drop is given by Qt which is the ratio of the power Pt 

 remo\'ed by the drop from the beam falling on it to 

 the incident power per unit area. Provided the effect 

 of all the dro]is be linearly additive, equation (31) 

 will express their average attenuation effect. The 

 incident power density is the complex Poynting vector 

 of the beam 



■in 2 

 Therefore, with ecjuations (3) and (26), 



(32) 



Qt = ~ (-Re) E (2«+l) « + K), (33)d 



where X = 2ir/k., is the wavelength of the radiation 

 in air or free space. Similarly the cross section for 

 scattering is, with equation (25), 



0.= ^ J.i2n+i)(w:,\^-+\Kr-). 



2ir 



(34) 



The differential cross section for back scattering (or 



''The 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 scatterinj^ 

 cross section Q, reduces in this case to the negative of the 

 total cross section Qt. Clearly Qs reduces to +Qt. 



radar cross section) is then, with equations (27) and 

 (32), 



= (:^YRe2 2 (-)''^'"(2,, + l)- 



(2«^ + l) [aX*+6:&:,*-2aX*]- (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 <^ (/, in which case for perfect reflection they 

 lead to a scattering cross section 27ra-, double of the 

 expected geometrical cross section. Since the modifica- 

 tions mentioned do not play any role for A > Sa/10, 

 which condition will always be satisfied in the present 

 report, they will not be discussed here. 



10.1.3 rpj^g Scattering Amplitudes' 



alandK 



The scattered fields (Eg, HJ outside the sphere 

 and the transmitted fields (E,, H,) inside are due to 

 forced oscillations of the sphere caused by the in- 

 cident field (E;, Hi). The fields (E^, HJ and (E^, 

 H,) given by equations (7) and (8) can be regarded 

 as due to electric and magnetic 2"-poles (/t = 1 cor- 

 responds to dipoles, n = 2 to quadrupoles, etc.) in- 

 duced in the substance of the spherical particle. In 

 the steady state these poles oscillate with the fre- 

 quency of the incident radiation field. When this fre- 

 quency approaches a characteristic frequency of the 

 free vibrations of the electric or magnetic 2''-poles of 

 the sphere, resonance will occur. It can, indeed, be 

 shown that the amplitudes a„. are associated with 

 \'ihra1ions of magnetic jioles and the ?)„'s with vibra- 

 tions of electric poles. The characteristic frequencies 

 of the free vibrations of magnetic poles of a sphere 

 are determined by a condition which annuls the de- 

 nominator of a,,, those of electric poles by a condition 

 which annuls the denominator of b,„ given by equation 

 (M)."*" The characteristic frequencies of the free 

 vibrations are, however, complex in contrast with the 



"Since henceforth we will deal only with the scattering co- 

 efficients a^, 6,^, we will omit the superscript s. 



