168 



DIELECTRIC CONSTANT. ABSORPTION AND SCATTERING 



AecurdJiig to equatiou {22), the power seattered by 

 a spherical particle per unit solid angle at a point 

 {r,e,cl>) is 



(^) = ±.[\El\^-+\E-l\"-]r\ (64) 



Using equations (16) and (17), we obtain, remember- 

 ing that the incident power per unit area is (1/2770) i?o, 

 the following expression for the differential scatter- 

 ing cross section: 



E Z 



+ bJ4 



(2*( + l)(2w + l) 



'1 ,;!fi «(»+!) m{m+l) 



, ^ . dPl dP}„ . „ 



cos- d) + fsin- 



de dd 



P^ P dPl dPl 



sin- e de dd 



/ P^ dP^ pi dP^ 



+ 2a „ 5* ( -^ ^^ cos2 4,+ ±^'^ sin^ <^ I cm^ 

 \sine dd sine d6l 



(65)*= 

 Or, limiting ourselves to the approximation where only 

 the electric dipole (hi), electric quadrupole {bz), and 

 magnetic dipole (aj are efEective, we find, using the 

 explicit expressions of the associated Legcndre poly- 

 nomials, 



■(/ft 



r/co 



(fj 



Re 9! 61 1 - (sin^0 -I- cos- 6 cos' 4>) + 9\ «il'-(cos-0 



^ +cos^9sin-<^) 



-f 25| 60.1 2 (cos2 6 sin2 <t> + cos- (2 d) cos- 4>) 

 + 18ai5i*cos6i -|- 3O6162* cos9(sin-<^ + cos(2e)cos=<^) 

 -f ;30ai52'''(cos-e sin-</) + cos (26) cos-4>) cm-. (66) 



Hero the first term inside the brackets represeids 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 = ttD/x, i.e., the nearer the wavelength 

 is to the diameter of the scattering sphere, the more 



''With 8 = IT this reduces to equation (27) of the radar 

 cross section. 



Ilie radiation is scattered forward than backward. A 

 study of equation ((J6) for water drops of 1-cm diam- 

 eter shows that for spheres of this size it is only wdien 

 X> 15 cm or p < 0.2 that the back-scattered intensity 

 is about the same as the forward-scattered intensity. 

 For such p values only the dipole term in equation 

 (C6) remains of practical importance. 



Supjiose 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 

 cm. 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 ca.ses 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 tlie 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 difEerential cross section for back scattering re- 

 sults from equation (66) by taking = w there. Using 

 the explicit expressions (13) of the amplitudes a^, h^, 

 and Ik, one obtains for this back scattering (or radar 

 cross section) 



+ ---)cm% (67) 

 with the following ct»ellicienls A", using the notation 

 defined by equations (45), (16), and (59) : 



A. = 18 [/3if''>ft(6'ft<3'^i<"' - ai(«/3i(« - ai'^^iCS)] 



- 30 [ft('^'ft(=' + ^i('^'^2'«] , 



A, = 18 [ft'-^'ft*"' + ^/-^'ft""] , (68) 



^4=9 [[a/'^'P + W[^ - 18 [ai<^'/^i(=> + «i<«ft(^>] 



- 30 [iSi'^-'ft^"* + A<"'/32*-^' - q:i'"/32<« 

 -ai(5'ft(«] +25!/32'')p, 



A, = 18 [/3i<^>ft(« + ft<5*ft<« - ai(«/3it« - a/«^i(6)] 

 -30[i3i<''W')-F^i<«W"], 



A6 = 9|/3i<"|l 



The I'adar cross-section formula (67) is the same 

 as that given by Ryde.^' Again <j{j) is not a function 



