ATMOSPHERIC ABSORPTION AND SCATTERING 



169 



of p only since the coefficients of the successive powers 

 of p in the expansion (67) depend on the wavelength. 

 The comj)uted echo cross sections a-(7r) for spherical 

 water drops with diameters in the range 0.05 to 0.55 

 cm and the wavelength range 3 to 100 cm are given 

 in Table 1-1. ■ These cross sections reduce practically to 

 the Eayleigh 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 Eayleigh term predominates in ct{tt), with 

 the exception of the larger drops and smaller wave- 

 lengths, the trends of variation of cr(7r) with either 

 the diameter, at constant wavelength, or the wave- 

 length, at constant diameter, are similar to those of 

 Qs, the total scattering cross section. A graphical repre- 

 sentation of the data of Table 14 is thus of no particu- 

 lar interest; they aj)pear implicitly in Figures 15 

 and 16. 



In order to compute the radar attenuation factor a 

 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 h cm 

 diameter per cm^, 



a^,t = 2^k '^kM neper/cm, 

 and for a given distribution of particles 



(69)" 



an= 2!^ a^,k= ^Z^ Nk '^kM neper/cm. (70) 



i = A- = 



Using the radar cross section of Table 14 and the drop 

 size distributions in different rains as given in Table 



'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 Qs. 



"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.'* 



- -7 

 o 



10 20 30 40 60 60 70 80 90 lOi 



X IN CM 



Figure 17. Absorption coefficient, 2^a,r, due to back 

 scattering (echo) as a function of the wavelength in 

 different rains. The abscissa gives the wavelength, X, 

 in centimeters. The ordinate scale gives logio(2Q:n-), the 

 absorption coefficient 2a» being expressed in km~i. 

 The letters on the curves refer to the drop size distribu- 

 tions listed in Table 7. 



7, 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- 

 tion in different rains of given drop size distribution 

 and precipitation rate. As already emphasized in con- 

 nection with the study of the attenuation, these curves 

 are characteristic, probably, of those rains, but they 

 are not unique, since a given rain of known precipita- 

 tion rate might very likely be built up from a variety 

 of drojD size distributions. 



Since the absorption coefficient (3a^) for back scat- 

 tering represents also the fraction of the incident power 



Table 14. Back scattering cross section <r (ir) (cm^) of spherical water drops of D cm diameter. 



