Page 348 - Electromagnetics

Figure 4.32: Geometry for scattering of a TM plane wave by a conducting half-plane.

symmetrically on the y-axis is equivalent for y > 0 to the problem of the line source
above a ground plane. The total field is the sum of the impressed and scattered fields:
∞+ j
˜
ω ˜µ 1 I(ω) e − jk y1 |y−h| − e − jk y1 (y+h)
˜ − jk x x
E z (x, y,ω) =− e dk x , y ≥ 0.
2π 2k y1
−∞+ j
We can write this in another form using the Hankel-function representation of the line
source (4.345):
ω ˜µ ω ˜µ
˜ ˜ (2) ˜ (2)
E z (x, y,ω) =− I(ω)H 0 (k|ρ − ˆ yh|) + I(ω)H 0 (k|ρ + ˆ yh|)
4 4

2
2
where |ρ ± ˆ yh|=|ρ ˆρ ± ˆ yh|= x + (y ± h) .
Interpreting the general case in terms of images is more difficult. Comparing (4.411)
and (4.412) with (4.410), we see that each spectral component of the field in region 1 has
the form of an image line source located at y =−h in region 2, but that the amplitude
˜
of the line source, R TM I, depends on k x . Similarly, the field in region 2 is composed of
spectral components that seem to originate from line sources with amplitudes −T TM I ˜
located at y = hk y1 /k y2 in region 1. In this case the amplitude and position of the image
line source producing a spectral component are both dependent on k x .

The field scattered by a half-plane. Consider a thin planar conductor that occupies
the half-plane y = 0, x > 0. We assume the half-plane lies within a slightly lossy medium
c
having parameters ˜µ(ω) and ˜ (ω), and may consider the case of free space as a lossless
limit. The half-plane is illuminated by an impressed uniform plane wave with a z-
directed electric field (Figure 4.32). The primary field inducesa secondary current on
the conductor and this in turn produces a secondary field. The total field must obey the
boundary conditions at y = 0.
Because the z-directed incident field induces a z-directed secondary current, the fields
may be described entirely in terms of a TM set. The impressed plane wave may be
written as
˜
˜ i
E (r,ω) = ˆ zE 0 (ω)e jk(x cos φ 0 +y sin φ 0 )

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