ATMOSPHERIC. ABSORPTION AND SCATTERING 



151 



free space), tlie square root is so taken that tlie imag- 

 inary part is negative, and ij is tlie intrinsic impedance 

 of the medium, or 



,,= ^, 77,= jH' = 377 ohms. (3) 



The conductivity o- is expressed in mhos per meter; 

 the frequency a>, in radians per second; ax, ay, and Ej 

 are unit vectors pointing in the positive a', y, and z 

 directions, respectively. 



The plane wave ^■ectors'' a^e''''^ and a^e'^^^ 

 will now be expanded into spherical wave vectors'^'' 

 at a point of spherical polar coordinates {r,6,<l>). It is 

 readily recalled that 



X = r sin 6 cos <^, 



y = r sin 6 sin <p, 



z = r cos 6, f^\ 



with 



<d <ir andO < <t><2Tr . 



These sijherical vector waves will be denoted by m 

 and n. They form complete orthogonal sets whose 

 members are defined by the following equations : 



mon 



(a) = . 



sin 6 



zj"^ (kr) Pi (cos ff) cos 4> I2 

 dPl 



z J"' (kr) -— sin 4> 13, 

 ad 



m,„'«' = - ^-.Sn'"' (kr) Pi (cose) sin</) 12 

 sin 6 



dP^ 



-.'J"' (fcr)-fcos0i3, (5) 



no/"' = n (n + l) -^— Pi (cos 9) sin <^ ii 



ki- 



, 1 ^ r (-,.1/7 \ T dPn . 



+ 7~ 7" C''^" ^^"'■> ] "irsin <^ 12 



+ 



1 



At sinS 5r 



[r3„(<'' (A;r) ] P,^ (cos 6) cos <^ 13 , 



i") = 



n{n+l) 



M 



(kr) „, 



kr 



Pn (cos e) COS <j> h 



1 a ^ ,,,, ,,dPl 

 + ^, ^[-"^"'(fcr)]-cos^x. 



1 



fcr sin 5 dr 



[?-2;„("> (fcr) ] P,i (cos 6) sin <^ 13 . (6) 



"Henceforth the time factor e+-''"' will be omitted, as it 

 does not play a direct role in what follows. 



Here, P,|(.t) is the first associated Legendre poly- 

 nomial of the first kind; ij, i,, and I3 are unit vectors 

 drawn in the increasing r, 6, and <j> directions at the 

 point {r,6,(f>) on the si^here of radius r (Figure 3). 

 The vectors ij and £3 are tangent to the sphere along 



FiciURE 3. Spherical coordinates. 



a meridian and a parallel i-ircle respectively. The 

 superscript a takes on two \alues. In the expressions 

 for the incident wave and the transmitted wave inside 

 the scattering sphere, it has the value 1, while its 

 value is 3 in the expressions for the scattered wave. 

 Explicitly, 



z^!\x) = (^/2.r)V„+5(.r), 



(7) 



zf\x) = {W2xy^Hl~iiix). 



t^n+jX''') is the Bessel function of the first kind and 

 half integer order, while Hf>^,{.i-) is the Hankel 

 function of the second kind and half integer order. 



The expanded field strengths of the incident wave 

 are then 



„=i n(n+l) 



Hi 



E, 



E (-./)"■ 



V2 „ = 1 



2n + 1 

 ?i(n + 1) 



(8) 





(1)^ 



It is seen that the nth expansion coefficient of E; 

 into the m waves is ( — j)"[{.2ii -\- l)/n{n -\- 1)], 

 wliereas the corresponding expansion coefficient into 

 the n waves is(— )"• j" + ''[(2>i + l)/n{n-}- 1)], etc. 



The radiation field induced by the incident radia- 

 tion is composed of the transmitted radiation field 

 (E,. H,) and the scattered radiation (Eg, H5) which, 

 at large distances (r) from tire scattering sphere, be- 

 haves as a divergent spherical wave, whose amplitude 

 vanishes as 1/r. 



The scattered and transmitted steady-state fields 



