194 Chapter 7 Elementary Acoustic and Electromagnetic Edge Waves
Figure 7.8 Directivity patterns of elementary edge waves radiated by the nonuniform/fringe sources in the directions of the diffraction cone (ϑ = π − γ0, 0 ≤ ϕ ≤ α).
•For calculation of the field at the diffraction cone (ϑ = π − γ0), the following relationships are useful:
cos σ1 = − cos ϕ, |
cos σ2 = − cos(α − ϕ), |
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sin σ1 = | sin ϕ|, |
sin σ2 = | sin(α − ϕ)|. |
(7.126) |
They allow one to simplify functions Vt (σ1, ϕ0) sin ϕ and |
Vt (σ2, α − ϕ0) |
sin(α − ϕ), taking into account that |
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sin ϕ/ sin σ1 = sgn(sin ϕ) and sin(α − ϕ)/ sin σ2 = sgn(sin(α − ϕ)). (7.127)
•The numerical calculations confirm that in the region inside the wedge (315◦ ≤
ϕ ≤ 360◦) the total field Fs,h(t) = Fs,h(1) + Fs,h(0) equals zero in the directions of the diffraction cone (ϑ = π − γ0).
7.8ELECTROMAGNETIC ELEMENTARY EDGE WAVES
The theory of acoustic EEWs presented in Sections 7.1 to 7.7 was extended in the work of Butorin et al. (1987) and Ufimtsev (1991) for electromagnetic waves diffracted at
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7.8 Electromagnetic Elementary Edge Waves 195
perfectly conducting objects. The basic elements of this extended theory are similar to those in the case of acoustic waves.
The uniform current induced by the incident wave on strips 1 and 2 (Fig. 7.3) is determined according to PO as
j(0) |
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ε(ϕ0) n1 |
× |
Hinc |
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ε(α |
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ϕ0) n2 |
× |
Hinc |
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where nˆ1 and nˆ2 are |
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− j |
(0) |
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The nonuniform current j |
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the total surface current induced on the tangential perfectly conducting wedge. This current is determined by the exact solution of the wedge diffraction problem presented in Sections 2.1 to 2.4 and adapted for electromagnetic waves. The explicit expressions
(1)
for the current j induced on the elementary strips 1 and 2 (Fig. 7.3) are given below in the Problems 7.14 and 7.15. The field radiated by the nonuniform current is found by integration over the elementary strips 1 and 2. The basic integrals are the same as those for the case of acoustic waves. We omit all intermediate calculations and give here the final results.
It is supposed that the incident electromagnetic wave, |
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Einc = E0eikφi , |
Hinc = H0eikφi , |
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(7.129) |
propagates in the direction φ |
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perfectly conducting |
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and undergoes diffraction at a (1) |
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object with a curved edge (Fig. 8.1). The nonuniform current j |
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edge radiates the EEWs. Away from the diffraction point ζ at the edge (kR |
1), their |
high-frequency asymptotics are expressed as |
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dE(1) |
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(1)(ϑ , ϕ) |
eikR |
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dH(1) |
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R |
× |
dE(1) |
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/Z0. |
(7.130) |
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2π E |
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Here Z0 = √μ0/ε0 = 120π ohms is the vacuum impedance; the differential element of the edge is positive (dζ > 0) and is measured in the positive direction of the local polar axis zˆ = ˆt. The quantity
(1)(ϑ , ϕ) |
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E0t (ζ )F(1)(ϑ , ϕ) |
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Z0H0t |
E |
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is the directivity pattern of the EEWs.
Here, it is also necessary to repeat the note
(ζ )G(1)(ϑ , ϕ)]eikφi (ζ ) |
(7.131) |
from Section 7.2: |
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The local cylindrical coordinates r, ϕ, z and spherical coordinates R, ϑ , ϕ are introduced
ˆ × ˆ = ˆ ˆ × ˆ = ˆ according to the right-hand rule (with respect to their unit vectors, r ϕ z, R ϑ ϕ).
One should remember that we introduce these coordinates in such a way that the angle ϕ is measured from the illuminated face of the edge and the tangent ˆt to the edge is directed
along the local polar axis zˆ (ˆt = zˆ). When both faces are illuminated, one can measure the angle ϕ from any face, but in this case one should choose the correct direction of the polar axis z and the tangent ˆt (zˆ = ˆt = rˆ × ϕˆ). This note is important for correct applications of the theory of electromagnetic EEWs.
196 |
Chapter 7 |
Elementary Acoustic and Electromagnetic Edge Waves |
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(ζ )] and H0t (ζ ) exp[ikφ |
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are the components |
The quantities E0t (ζ ) exp[ikφ |
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(1) |
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of the incident field that are tangential to the edge. Vectors F |
and G |
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by their spherical components: |
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Fϑ(1)(ϑ , ϕ) = [U(σ1, ϕ0) + U(σ2, α − ϕ0)] sin ϑ , |
Fϕ(1)(ϑ , ϕ) = 0 |
(7.132) |
and |
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(ϑ , ϕ) = |
sin ϑ cos γ0 |
[ε(ϕ0) − ε(α − ϕ0)] |
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Gϑ |
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sin2 γ0 |
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+ (sin γ0 cos ϑ cos ϕ − cos γ0 sin ϑ cos σ1) V (σ1, ϕ0)
− [sin γ0 cos ϑ cos(α − ϕ) − cos γ0 sin ϑ cos σ2] V (σ2, α − ϕ0), (7.133)
Gϕ(1)(ϑ , ϕ) = −[V (σ1, ϕ0) sin ϕ + V (σ2, α − ϕ0) sin(α − ϕ)] sin γ0. |
(7.134) |
All functions and parameters in Equations (7.132) to (7.134) are the same as those introduced in the previous sections for acoustic waves.
(t) = (1) + (0)
The EEWs radiated by the total current j j j are determined by
dE(t) |
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dζ |
(t)(ϑ , ϕ) |
eikR |
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dH(t) |
= [ |
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dE(t) |
/Z0, |
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and |
R |
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2π E |
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(t)(ϑ , ϕ) |
= [ |
E0t (ζ )F(t)(ϑ , ϕ) |
+ |
Z0H0t (ζ )G(t)(ϑ , ϕ) |
eikφi (ζ ), |
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] |
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where |
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Fϑ(t)(ϑ , ϕ) = [Ut (σ1, ϕ0) + Ut (σ2, α − ϕ0)] sin ϑ , |
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Fϕ(t)(ϑ , ϕ) = 0, |
Gϑ(t)(ϑ , ϕ) = (sin γ0 cos ϑ cos ϕ − cos γ0 sin ϑ cos σ1)Vt (σ1, ϕ0) |
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− [sin γ0 cos ϑ cos(α − ϕ) − cos γ0 sin ϑ cos σ2]Vt (σ2, α − ϕ0),
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(7.138) |
and |
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Gϕ(t)(ϑ , ϕ) = −[Vt (σ1, ϕ0) sin ϕ + Vt (σ2, α − ϕ0) sin(α − ϕ)] sin γ0. |
(7.139) |
The difference |
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dE(0) = dE(t) − dE(1), |
dH(0) = dH(t) − dH(1) |
(7.140) |
(0)
is the electromagnetic field of EEWs generated by the uniform current j .
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(ϕ, ϕ , α)
0 eikR
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7.8 Electromagnetic Elementary Edge Waves 197 |
For the |
scattering directions ϑ |
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and G |
(1,t) |
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the functions F |
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reduce to |
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Fϑ(1)(π − γ0 |
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f (1)(ϕ, ϕ0, α), |
, ϕ) = − |
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sin γ0 |
Fϑ(t)(π − γ0 |
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f (ϕ, ϕ0 |
, α), |
sin γ0 |
Gϕ(1)(π − γ0 |
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g(1)(ϕ, ϕ0 |
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, ϕ) = |
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sin γ0 |
Gϕ(t)(π − γ0 |
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sin γ0 |
G(ϑ1)(π − γ0, ϕ) = [ε(ϕ0) − ε(α − ϕ0)] cot γ0,
and
G(ϑt)(π − γ0, ϕ) = 0.
(7.141)
(7.142)
(7.143)
(7.144)
(7.145)
(7.146)
The properties of functions U(σ , ψ ) and V (σ , ψ ) are described in Sections 7.5 and 7.6. According to these equations, one can represent the electromagnetic EEWs for
the directions ϑ = π − γ0 as
dEϑ(0) = Z0dHϕ(0) dEϑ(1) = Z0dHϕ(1) dEϑ(t) = Z0dHϕ(t)
dHϑ(0) = −Y0dEϕ(0) dHϑ(1) = −Y0dEϕ(1) dHϑ(t) = −Y0dEϕ(t)
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f (0)(ϕ, ϕ0, α) |
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Etinc(ζ ) |
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f (ϕ, ϕ0, α) |
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Htinc(ζ ) |
dζ |
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α) |
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g(ϕ, ϕ0, |
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where Y0 = 1/Z0 and ˆt is the unit vector tangent to the scattering edge at the diffraction point ζ . Notice also that the radial components of the far field are of the order 1/R2 and they are neglected here. Because of that, in general,
Eϑ = −Et / sin ϑ , Hϑ = −Ht / sin ϑ |
(7.149) |
and
dEϑ = −dEt / sin ϑ , dHϑ = −dHt / sin ϑ , |
(7.150) |
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198 Chapter 7 Elementary Acoustic and Electromagnetic Edge Waves
Taking into account these equations and the condition ϑ = π − γ0, one can rewrite the above asymptotics (7.147), (7.148) in the form
dEt(0) |
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ikR |
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Etinc(ζ ) |
f (1)(ϕ, ϕ0, α) |
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(7.151) |
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ikR |
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(7.152) |
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dH |
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Comparison of these equations with Equations (7.120), (7.121) allows one to establish the following relationships between the acoustic and electromagnetic EEWs for the directions belonging to the diffraction cone:
dEt = dus, |
if Etinc(ζ ) = uinc(ζ ), |
(7.153) |
dHt = duh, |
if Htinc(ζ ) = uinc(ζ ). |
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Notice also that the paper by Ufimtsev (1991) investigates in detail the ray, caustic, and focal asymptotics of electromagnetic EEWs, as well as their multiple and slope diffraction. The results of this investigation are presented below in Chapters 8 to 10.
7.9 IMPROVED THEORY OF ELEMENTARY EDGE WAVES: REMOVAL OF THE GRAZING SINGULARITY
The central idea of PTD is the separation of the surface scattering sources into the uniform and nonuniform components in such a way that they would be the most appropriate for calculation of the scattered field. In the case of scattering objects with edges, the nonuniform component is defined as that part of the field that concentrates near edges. Equations (7.7) and (7.8) determine it as the difference between the total field on the tangential wedge and its geometrical optics part. The latter is considered as the uniform component. As is shown in the present book and in other publications, the utilization of these components is really helpful for investigation of many scattering problems.
However, this is not the case for forward scattering in the directions grazing the edge faces (Fig. 7.9), where the above theory of elementary edge waves predicts infinite values for the functions f (1) = f − f (0) and g(1) = g − g(0). It turns out that for these directions, either function f or f (0) (either g or g(0)) becomes singular. Which of them becomes singular depends on how the grazing direction is approached: ϕ = 0 and then ϕ0 → π , or ϕ0 = π and then ϕ → 0. These singularities indicate that the
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