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

where, as is shown in Figure 3.2, the integration contour F skirts above the pole

ξ = π − ϕ0.

In Figures 3.3 and 3.4, we introduce two additional contours G1 and G2, related to the cases ϕ < π and ϕ > π , respectively. In the dashed regions of these figures, the relations Im cos(ξ − ϕ) > 0 and Im cos(ξ + ϕ) > 0 are valid, which guarantees the convergence of integrals Is and Ih. We then apply the Cauchy residue theorem

Figure 3.3 Integration contours F and G1 for the case ϕ < π . In the shaded regions, Im cos(ξ − ϕ) > 0.

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64 Chapter 3 Wedge Diffraction: The Physical Optics Field

Figure 3.4 Integration contours F and G2 for the case ϕ > π . In the shaded regions, Im cos(ξ + ϕ) > 0.

to the integrals over the closed contours F − G1,2 (they are closed at the infinity, ξ = ±∞) and obtain

(3.25)

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3.2 Conversion of the PO Integrals to the Canonical Form 65

In these equations, we change with integration variable ξ by ζ + ϕ in the region 0 ≤ ϕ < π , and by ζ + 2π − ϕ in the region π < ϕ < 2π . As a result, the integrals in Equations (3.25) to (3.28) are transformed into the integrals over the contour D0 shown in Figure 2.6:

We omit similar transformations for integral Ih and present only their results:

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As is seen in Equations (3.41) to (3.48), the PO scattered field consists of the reflected plane waves and the diffracted part described by functions vs± and vh±. Simple asymptotic expressions for the functions vs,h± and for the diffracted field are derived in the next section.

3.3 RAY ASYMPTOTICS FOR THE PO DIFFRACTED FIELD

The relationships us = Ez and uh = Hz exist between the acoustic and electromagnetic

fields studied in this section.

Again, as in √Section 2.4, we introduce in the integrals (3.39) and (3.40) the new variable s = 2eiπ/4 sin ζ2 and transform functions vs,h± as

These asymptotics are valid under the conditions kr| cos(ϕ ± ϕ0)/2| 1, that is, away from the geometrical optics boundaries, where the diffracted field has a ray structure and can be interpreted in terms of edge diffracted rays.

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68 Chapter 3 Wedge Diffraction: The Physical Optics Field

In view of Equations (3.51) and (3.52), the ray asymptotics for the diffracted part of the fields (3.41) to (3.44) and (3.45) to (3.48) can be written in the following form:

The directivity patterns of these diffracted rays are different for the situations with one or two illuminated faces. In the case 0 < ϕ0 < α − π , when only the face ϕ = 0 is illuminated by the incident wave,

It is seen that these functions are singular at the boundaries of reflected plane waves, where ϕ = π ± ϕ0 or ϕ = 2α − π − ϕ0. The asymptotics of the PO diffracted field, which are uniformly valid around the wedge, can be easily derived by the application of the Pauli technique demonstrated in Sections 2.5 and 2.6. We do not consider them here, because our main purpose in the canonical wedge diffraction problem is the calculation of the field generated by the nonuniform component of the surface sources.

PROBLEMS

3.1 Analyze the accuracy of the PO approximation. Compute and plot the relative error

δf (0)(ϕ, ϕ0, α) = |F(ϕ, ϕ0, α) − 1| · 100%,

where

F(ϕ, ϕ0, α) = f (0)(ϕ, ϕ0)/f (ϕ, ϕ0, α).

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