Page 323 - Electromagnetics

Anyattempt to satisfythe boundaryconditions byusing a single nonuniform wave
fails. This is because the sinusoidal dependence on φ of each individual nonuniform wave
cannot match the more complicated dependence of the impressed field (4.356). Since the
sinusoids are complete, an infinite series of the functions (4.353) can be used to represent
the scattered field. So we have internal to the cylinder
∞
#
˜ s
E (r,ω) = [A n (ω) sin nφ + B n (ω) cos nφ] J n (kρ)
z
n=0
c 1/2
where k = ω( ˜µ˜ ) . External to the cylinder we have free space and thus
∞
#
˜ s (2)
E (r,ω) = [C n (ω) sin nφ + D n (ω) cos nφ] H n (k 0 ρ).
z
n=0
Equations (4.349) and (4.350) yield the magnetic field internal to the cylinder:
∞ jn 1
˜ s #
H = [A n (ω) cos nφ − B n (ω) sin nφ] J n (kρ),
ρ Z TM k ρ
n=0
∞ j
#
˜ s
H =− [A n (ω) sin nφ + B n (ω) cos nφ] J (kρ),
φ n
Z TM
n=0
where Z TM = ω ˜µ/k. Outside the cylinder
∞
# jn 1
˜ s (2)
H = [C n (ω) cos nφ − D n (ω) sin nφ] H (k 0 ρ),
ρ η 0 k 0 ρ n
n=0
∞ j
˜ s # (2)
H =− [C n (ω) sin nφ + D n (ω) cos nφ] H (k 0 ρ),
φ n
η 0
n=0
where J (z) = dJ n (z)/dz and H (2) (z) = dH (2) (z)/dz.

n n n
We have two sets of unknown spectral amplitudes (A n , B n ) and (C n , D n ). These can
be determined byapplying the boundaryconditions at the interface. Since the total field
outside the cylinder is the sum of the impressed and scattered terms, an application of
continuityof the tangential electric field at ρ = a gives us
∞
#
[A n sin nφ + B n cos nφ] J n (ka) =
n=0
∞
# (2) − jk 0 a cos φ
˜
[C n sin nφ + D n cos nφ] H (k 0 a) + E 0 e ,
n
n=0
which must hold for all −π ≤ φ ≤ π. To remove the coefficients from the sum we apply
orthogonality. Multiplying both sides by sin mφ, integrating over [−π, π], and using the
orthogonalityconditions (A.129)–(A.131) we obtain
π
π A m J m (ka) − πC m H (2) ˜ sin mφe − jk 0 a cos φ dφ = 0. (4.357)
m (k 0 a) = E 0
−π
Multiplying by cos mφ and integrating, we find that
π
2π B m J m (ka) − 2π D m H (2) ˜ cos mφe − jk 0 a cos φ dφ
m (k 0 a) = E 0 m
−π
˜
= 2π E 0 m j −m J m (k 0 a) (4.358)
© 2001 by CRC Press LLC