272 Z RADIO WAVE PROPAGATION EXPERIMENTS 
These spherical vector waves will be denoted by m 
and n. They form complete orthogonal sets whose 
members are defined by the following equations: 
na OC) EEG cancth 
sin 0 
1 
— z,™ (kr) — sin ¢ is, 
Men = — ol zn™ (kr) P), (cos 6) sin ¢ is 
sin 6 
iP , 
—z,™ (kr) oe cos@is, (5) 
Non = n (n-+1) = oe eur (cos 0) sin ¢ i; 
Pp} 
+2 == = [rn 
é 
—_ () 1 
Pang BP o tn (kr) ] P23 (cos 6) cos ¢ is , 
— r) 
On =n (ji) 2 ———— P! (cos 8) cos ¢ it 
dP} 
ees (a) ett H 
fl = 2 ie: (kr)] Cos ¢ ig 
Ud: (a) v in gi 
GRTGL Gr [rzn™ (kr) ] P} (cos 6) sing@iz. (6) 
Here, P1(x) is the first associated Legendre poly- 
nomial of the first kind; i,, i,, and i, are unit vectors 
drawn in the increasing r, 0, and ¢ directions at the 
point (r,0,¢) on the sphere of radius r (Figure 3). 
The vectors i, and i, are tangent to the sphere along 
a meridian and a parallel circle respectively. The 
superscript a takes on two values. 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, 
2) (x) = (®/2x)*Tn44(x), 
Zz) = (w/2x)tH®, ,(2). @ 
In 44(2) is the Bessel function of the first kind and 
half integer order, while HO) (x) is the Hankel 
function of the second kind and half integer order. 
The expanded field strengths of the incident wave 
are then 
2n+1 
Bs = BoD (jp 21 (ea 4 jl, 
sea n(n-+1) . fae) 
8 
— jp 2+ F (mi? — jai). : 
n(n + 1) ne 
It is seen that the nth expansion coefficient of E, 
into the m waves is (—j)"[(2n-+1)/n(n-+ 1)], 
whereas the corresponding expansion coefficient into 
the n waves is(—)”- j"*1[(2n + 1)/n(n + 1)], ete. 
The radiation field induced by the incident radia- 
tion is composed of the transmitted radiation field 
(E., H:) and the scattered radiation (E,, H,) which, 
at large distances (r) from the scattering sphere, be- 
haves as a divergent spherical wave, whose amplitude 
vanishes as 1/r. 
The scattered and transmitted steady-state fields 
will now be expanded, in analogy with the incident 
field (Ei, H;). Thus, 
—~, . 2n+1 3) 1 ans (3) 
E, = E ear = 3 ‘on b;, De ? 
3 j) Mey aa (a, mon + jbn Den ) 
(9) 
Ey 2n+1 as (3) 
= 7 b, me; m *2on ] > 
ne BD n(n + Ge) Teeotte) 
valid at distances r>a, i.e., outside the sphere in 
medium 2. Clearly in equations (6) and (7), in the 
expressions of m and n, k, replaces.& according to 
equations (5) and (6). 
Inside the sphere (complex wave number k,, in- 
trinsic impedance 7,), the transmitted field is ex- 
panded in the following way: 
2 1 
E, = BA — rel Gt m® + 704, a), 
n(n + 1) (10) 
ntl (1) (1) 
H; saa taaa eS bh mex ‘on 
2 Sir wmap em —ja', a). 
The final determination of the scattered and trans- 
mitted fields is thus reduced to finding the coefficients 
(or amplitudes) a2, 6%, and a‘, bf. 
The preceding formulas permit one to write down 
rapidly the polar components of the different field 
strengths (E;, H;), (E, H.), and (E;, H;). The 
boundary conditions at the surface of the sphere de- 
mand the continuity of the tangential components of 
the total field outside the sphere and the transmitted 
field. If we denote these tangential components by 
subscripts @ or ¢, the boundary conditions take on 
the following form: 
E,+ £3 = Ej,Hj+ HH’ =Hj,r=a. (11) 
hese lead to the following systems of equations for the 
letermination of the coefficients (a°, b%,) and (a, b'): 
a’, 2 (Np) — ak 2 (pr) = 2 (p). — (12) 
mat oe [Ne20 (We)] — mes se? (o)] 
= ap 2 i (e)] ? 
p20}, 2 - p) — mds, 2© (p) = mize (0), 
bn [Np 2 (Np)] 
‘ain ) 
—N bs — soe? (p)] = sleet? (e)], (13) 
where N=4£,/k,, p=k,a and the z(a) and 
2® (x) are the spherical Bessel functions defined in 
