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

44 Chapter 2 Wedge Diffraction: Exact Solution and Asymptotics

Figure 2.8 The incident plane wave illuminates both faces of the wedge. The line ϕ = π − ϕ0 is the boundary of the wave reflected from the face ϕ = 0, and the line ϕ = 2α − π − ϕ0 is the boundary of the wave reflected from the face ϕ = α.

Figure 2.8 illustrates the situation when both faces of the wedge are illuminated. In this case α − π < ϕ0 < π , and functions u(kr, ϕ − ϕ0), u(kr, ϕ + ϕ0) are determined by

u(kr, ϕ − ϕ0) = v(kr, ϕ − ϕ0) + e−ikr cos(ϕ−ϕ0) u(kr, ϕ + ϕ0) = v(kr, ϕ + ϕ0) + e−ikr cos(ϕ+ϕ0)

(2.57) The term e−ikr cos(2α−ϕ−ϕ0) describes the plane wave reflected from the face ϕ = α.

2.4 THE SOMMERFELD RAY ASYMPTOTICS

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

edge-diffracted rays.

A simple asymptotic expression for the function v(kr, ψ ) with kr 1 can be found by the steepest descent method (Copson, 1965; Murray, 1984). With this purpose we replace the integration variable in Equation (2.51) by

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46 Chapter 2 Wedge Diffraction: Exact Solution and Asymptotics

Equations (2.41), (2.49), and (2.60) determine the wave arising at the edge of the acoustically hard wedge

Functions f and g describe the directivity patterns of the edge waves. The asymptotic expressions (2.60) to (2.64) were introduced by Sommerfeld (1935) and are well known. It is easy to verify that they satisfy the boundary conditions (2.2) and (2.3).

One should mention that the two-dimensional edge waves (2.61) and (2.63) can be interpreted as continuous sets of edge diffracted rays. They arise due to diffraction, but propagate from the edge in the first asymptotic approximation as ordinary rays in accordance with geometrical optics laws for 2-D fields. As shown in the work of Pelosi et al. (1998), the edge diffracted rays had already been visually observed already by Newton, although he did not use such a terminology. The term “diffracted ray” was introduced by Kalashnikov (1912), who was also the first to present an objective experimental proof of the existence of edge diffracted rays by recording them on a photographic plate. Theoretically, their existence was established first by Rubinowicz (1924) and later on by many other researchers. Keller (1962) formulated the concept of diffracted rays in a general form.

Here it is also pertinent to remind one about Sommerfeld’s warning against the too formal ray interpretation of diffraction phenomena. He wrote that shining diffraction points on edges do not exist in reality and they are just optical illusions: “Das ist naturlich eine optische Tauschung” (Sommerfeld, 1896, p. 369). He explained that such seemingly shining edge points are the result of our perception, or in Sommerfeld’s words, the result of “analytical continuation” of diffracted rays by our eyes.

Because of the ray structure of edge waves, the asymptotic expressions (2.61) to (2.64) derived above can be called the ray asymptotics, as emphasized in the title of the present section. These asymptotics have an essential drawback. They are not valid near the shadow boundary (ϕ ≈ π + ϕ0) and near the boundaries of reflected plane waves ( ϕ ≈ π − ϕ0, ϕ ≈ 2α − π − ϕ0) . The mathematical reason for this drawback

ψ = ϕ ± ϕ0 → π and ψ = ϕ + ϕ0 → 2α − π . In this case, the Taylor expansion

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2.5 The Pauli Asymptotics 47

for the integrand becomes meaningless because its terms tend to infinity. There is a physical background behind this mathematics. The vicinity of boundaries of incident and reflected waves is the region of the effective transverse diffusion where the field cannot be described in terms of diffracted rays and has a more complex structure. This phenomenon is considered in detail in Section 5.5 of Ufimtsev (2003).

2.5 THE PAULI ASYMPTOTICS

In 1938, Pauli suggested the asymptotic expansion for the function v(kr, ψ ) that is valid at the geometrical optics boundaries ϕ = π ± ϕ0 and transforms to the Sommerfeld asymptotics away from these boundaries (Pauli, 1938). In this section we provide the derivation for the first term of the Pauli expansion. Usually, for engineering analysis, only the first terms in asymptotic expansions are of practical value. Higher-order terms commonly are not utilized, because they are smaller in magnitude, and are quite complicated to evaluate. Besides, the high-order terms can occur beyond the frames of validity of idealized mathematical models used for description of real physical phenomena. That is why we focus here on the first asymptotic term.

According to Equation (2.59),

does not have a pole at the saddle point s = 0 (ζ = 0) when ψ = ϕ ± ϕ0 → π . Therefore it can be expanded into a regular Taylor series. By integrating this series

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