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Chapter 9

Multiple Diffraction of Edge

Waves: Grazing Incidence

and Slope Diffraction

9.1 STATEMENT OF THE PROBLEM AND RELATED REFERENCES

Clearly, the theory developed in the previous chapter can be applied to the investigation of multiple diffraction at edges that are spaced apart. Only two special cases need a individual investigation.

The first case is a grazing incidence of edge waves on acoustically hard planar plates. In the above asymptotic theory, the incident wave is approximated by an equivalent plane wave. However, a plane wave does not undergo diffraction at an infinitely thin plate under grazing incidence for the following reason. When this wave propagates in the direction parallel to the plate, its wave and amplitude fronts are perpendicular to the plate. As this incident field is constant in the direction normal to the plate, it automatically satisfies the boundary condition du/dn = 0 on the plate. Such a wave does not “see” the plate and propagates as if a free space is in its path. Because of this, the above theory predicts a zero diffracted field in this case. However, in the process of multiple diffraction, every diffracted wave is not plane. If its normal derivative du/dn is not zero on the plate, it undergoes diffraction. Such grazing diffraction is studied in Section 9.2.

The second case that also needs special treatment occurs when the scattering edge is located in the zero of the incident wave. This is the case of so-called slope diffraction. One distinguishes a slope diffraction of different orders, depending on the zero orders. Here we consider the most important one to be the slope diffraction of the first order, when the first derivative of the incident wave is not equal to zero. Such a situation occurs, for example, in reflector antennas, when one tries to decrease side lobes, and in the process of multiple diffraction between several scatterers, or between different parts of the same scatterer.

Fundamentals of the Physical Theory of Diffraction. By Pyotr Ya. Ufimtsev

Copyright © 2007 John Wiley & Sons, Inc.

229

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230 Chapter 9 Multiple Diffraction of Edge Waves

Many authors have studied the phenomenon of slope diffraction. Ufimtsev (1958a,b,c, 1962) suggested uniform asymptotics for the secondary edge waves arising due to the slope diffraction on plane screens (strip, disk). Karp and Keller (1961) derived nonuniform ray asymptotics for the same edge waves. Mentzer et al. (1975) published uniform asymptotics similar to those found by Ufimtsev (1958a,b,c). The spectral theory of diffraction (Rahmat-Samii and Mittra, 1978) also enables one to investigate the slope diffraction. In the particular case of the half-plane diffraction problem, Boersma and Rahmat-Samii (1980) analyzed this phenomenon in the framework of the ray-based theories (uniform asymptotic theory (UAT) and uniform theory of diffraction (UTD)). Pathak (1988) constructed the general UTD for the slope diffraction at the wedge. The general PTD for the slope diffraction of electromagnetic waves (based on the concept of elementary edge waves) was elaborated in the papers by Ufimtsev (1991) and Ufimtsev and Rahmat-Samii (1995). A similar theory for both the grazing diffraction and the slope diffraction of acoustic waves was developed earlier in the work of Ufimtsev (1989, 1991). The theory presented below is based on the papers by Ufimtsev (1989, 1991) and Ufimtsev and Rahmat-Samii (1995).

9.2GRAZING DIFFRACTION

The following relationship exists between the acoustic and electromagnetic diffracted rays arising due to the grazing diffraction at the plate S1 (Fig. 9.1):

at the scattering edge L1. Here, ˆt is the tangent to the edge L1 and nˆ is the normal to the plate S1 at the diffraction point ζ .

9.2.1 Acoustic Waves

Figure 9.1 shows the configuration appropriate to studying both the grazing diffraction and the slope diffraction. There are two acoustically hard scattering objects with edges L1 and L2. One of them is a planar plate S1. The boundary conditions du/dn = 0 are imposed on the surfaces S1 and S2. Edge L2 is located in the plane containing the plate S1. No more than one edge diffracted ray comes to every point on edge L1 (L2) from edge L2 (L1). The wave initially diffracted at edge L2 propagates to edge L1 and undergoes grazing diffraction at plate S1. This problem is investigated in the present section. The wave field diffracted at edge L1 equals zero in the direction to edge L2, where it undergoes slope diffraction. This problem is considered in the next section.

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9.2 Grazing Diffraction 231

Figure 9.1 The problem of multiple diffraction. The edge L2 is in the plane containing plate S1 with edge L1. Edge L1 is perpendicular to the figure plane at the intersection point. Edge L2 intersects the figure plane at an oblique angle. The line R21 shows the diffracted ray coming from L2 to L1. (Reprinted from Ufimtsev (1989) with the permission of the Journal of Acoustical Society of America.)

Suppose that the wave

propagates from edge L2 and undergoes grazing diffraction at plate S1. This is a wave with a ray structure of the type of Equation (8.29). Consider the two first terms of its Taylor expansion in the vicinity of the grazing direction ϕ2 = ϕ02:

Here, the first term represents the wave that does not undergo diffraction at plate S1, because its normal derivative on the plate equals zero [∂v2(R2, ϕ02, ϕ02)/∂ϕ2 = ∂(const)/∂ϕ2 = 0]. Therefore, that part of the incident wave that experiences diffraction at the plate can be approximated by the wave

which has the zero field in the grazing direction. Thus we see that the grazing diffraction of the wave (9.1) actually represents itself a particular case of the slope diffraction.

We approximate the wave (9.3) by the equivalent canonical wave

obtained by the differentiation of the plane wave. The quantities r1, ϕ1, z1 are local polar coordinates with the origin at the diffraction point on edge L1 (Fig. 9.1), and

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