Ufimtsev P. Fundamentals of the physical theory of diffraction (Wiley 2007)(348s) PEo

192 Chapter 7 Elementary Acoustic and Electromagnetic Edge Waves

This means that 0◦ ≤ θ ≤ 180◦ when ϕ = α/2 and 180◦ ≤ θ ≤ 360◦ when ϕ = α/2 + 180◦.

Figures 7.6 and 7.7 show the elementary edge waves both outside and inside the wedge. The presence of elementary waves inside the wedge does not contradict the fact that the wedge is nontransparent. The PTD is based on the Helmholtz equivalency principle (Equations (1.10), (1.54), and (1.59)). According to this principle, a real perfectly reflecting object is replaced by the equivalent scattering sources distributed (in free space (!)) over the geometrical surface conformal to the actual scattering surface. These sources generate the field everywhere, including the region inside the object, where this field completely cancels the incident field and ensures the zero total field there. The nonzero elementary edge waves inside the wedge also cancel each other.

Figure 7.8 demonstrates the elementary edge waves propagating in the directions along the diffraction cone.

The following comments are pertinent regarding these calculations:

•As ϕ0 = 45◦, only the wedge face ϕ = 0 is illuminated. Because of this, only the functions Ut (σ1, ϕ0), Vt (σ1, ϕ0), U0(β1, ϕ0), and V0(β1, ϕ0) are singular,

Figure 7.6 Directivity patterns of elementary edge waves (radiated by the nonuniform/fringe sources js,h(1)) in the plane ϑ = 90◦. The interval 315◦ ≤ ϕ ≤ 360◦ relates to the region inside the wedge.

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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

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.

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

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.

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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)

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

− [sin γ0 cos ϑ cos(α − ϕ) − cos γ0 sin ϑ cos σ2]Vt (σ2, α − ϕ0),

(0)

is the electromagnetic field of EEWs generated by the uniform current j .

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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

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,

and

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200 Chapter 7 Elementary Acoustic and Electromagnetic Edge Waves

The last terms in these equations (together with the factor 2 in front of the brackets) relate to the sum of the incident and reflected plane waves. When ϕ0 → π , these equations transform into

where jh(0) is actually the grazing incident wave, and js(0) represents the edge wave. The quantities jh,s(0) on strip 2 (of the half-plane ϕ = α) can be found from Equations (7.157) and (7.158) with the replacement of ϕ0 by α − ϕ0 and ξ1 by ξ2.

The new nonuniform components jh,s(1) are defined as the difference between the total scattering source jh,s (induced on the wedge faces) and the uniform component jh,s(0) ≡ jh,shp (induced on the tangential half-planes). These nonuniform components are determined by Equations (7.12) and (7.13), where one should replace

The field generated by the new component jh,s(1) is found with a procedure completely similar to that described in Sections 7.3 to 7.5. The associated field integrals are defined by Equations (7.44) and (7.45), where one should apply the replacement (7.159). It is worth noting that now the contribution to these integrals caused by the residues at the poles η = ±ϕ0 is equal to zero. The field of EEWs found in this way can be represented again in the form of Equations (7.89) and (7.90), with the new functions Fh,s(1):

+ [−Ut (σ2, α − ϕ0) + ε(α − ϕ0)Uthp(σ2, α − ϕ0)]} sin2 γ0. (7.164)

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