ATMOSPHERIC ABSORPTION AND SCATTERING 



153 



Since the resultant field at any point outside the 

 si^here is obtained by superposition of the incident 

 and scattered or reflected fields, one has 



E = E; + E^; H = Hi + H, . (18) 



In view of equation (16), the complex Poynting vector 

 associated witli this resultant field is radial, so that 



Sc=HEeH^*- E^He*), (19) 



where an asterisk denotes the complex conjugate. 

 Using equation (18) one gets 



S, = i {E'o Hy- - E'^ Hi*} + i (E'e H^* - E^ HI*) 



+ h{KH\* + We H'^* - E\ HI* - E% H'^*). ' (20) 



The first term on the right-hand side is the rate of 

 flow of energy in the incident wave and the second 

 term is the rate of flow in the scattered wa\'e. The 

 total scattered power is then 



P, = i Re f" f {El HI* - El HI*) r^ sin ededcj,, 

 Jo Jo (21) 



where Ee denotes "Eeal part of . . ." and the integral 

 is extended over the surface of a large sphere of radius 

 r. In our case, using equations (16) and (17), one 

 gets 



P, = ^Re f" f\\El\^+\El\ 2) r^sin dddd<j>. 



2j72 Jo Jo ^22) 



In the case of an absorbing sphere, the net flow of 

 energy across a closed surface around the sphere is 

 absorbed energy flow, and it is directed inward. One 

 may thus write tliat this absorbed energy, which dis- 

 appears in the form of heat, is 



/2ir r IT 

 / {-S,)rHmeddd4>. (23) 



Since the integral of the incident flow across a closed 

 surface is zero, equation (23) in connection with 

 equation (31) leads to the definition of the rate of 

 flow of total energy or the power subtracted from 

 the beam, i.e., (Pab + Ps), as an integral over a closed 

 surface of the third term on the right-hand side of the 

 radial eompojient Sc of the Poynting vector, equation 

 (19). Thus 



Pt = Pa6 + P. = I (- Re) 



/:/: 



{E\H%* + EIH'^ 



E'^ HI* - E\ HI*) y^ sin ededcl>. 



(24) 

 Substituting equations (16) and (17) into equation 

 (24) and remembering that the ^ integration leads 



to a factor ir and that the integrals over products of 

 the associated Legendre polynomials P,,'(^) ^^^ dif- 

 ferent from zero only in the following combination of 

 these products appearing in eciuatious (22) and (24), 



E 

 P. = ^ (- Re) Z i2n + 1) («■; + K) . (26) 



AVe shall also need the fraction of the power scat- 

 tered backwards by the sphere, i.e., in the direction 

 6 = IT, per unit solid angle. One thus obtains, with 

 d(o = sin dddd(j), 



/ , ^^ \ CO CO 



fdPA E' "v x^ "+"' 



, {2m+l) iai-K) ia:*-b:*), (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 



jk = a+ ji3, (28) 



where j8 is the phase constant and « the attenuation 

 constant; both are real. The explicit expressions of ^ 

 and a in terms of the characteristic electromagnetic 

 properties of the medium, namely, inductive capacity 

 e, permeability ju, and conductivity <t for the given 

 frequency m/'Ztt are the following :i®" 



(29) 



(30) 



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 e-'"^-°% 

 F(, being an amplitude vector directed along either one 

 of the two remaining coordinate axes. The attenuation 



