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

(2.33)

To clarify these expressions, we consider the solution of the wave equation (2.1) in the free infinite homogeneous medium, without the scattering wedge. For this problem, it is convenient to use the new polar coordinates (ρ, φ) with the origin at the radiating source (r0, ϕ0). It is clear that due to the azimuthal symmetry of the problem, its solution is a function of only one variable, u = u(ρ). In addition, as the wave equation is the differential equation of second order, its solution, in general, is the sum of two fundamental solutions of the related homogeneous equation:

because the second term, with H0(2)(kρ), does not satisfy the Sommerfeld radiation condition (2.12) and represents a nonphysical wave incoming from infinity. The constant c1 is found again with the Green theorem (2.15) applied to the circular region S of a small radius ε and with the center at ρ = 0 (Fig. 2.3).

For small values kρ 1, the function u(kρ) and its normal derivative du/dn = du/dρ (at the boundary of the region S) are described by the asymptotic approximations

By substitution of these quantities into the Green theorem and taking the limit with ε → 0, we find c1 = I0/i4 and

Figure 2.3 A circular region (0 ≤ ρ ≤ ε, 0 ≤ φ ≤ 2π ) with the radiating source at the center.

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40 Chapter 2 Wedge Diffraction: Exact Solution and Asymptotics

This solution explains the physical meaning of the factor in front of the series in Equations (2.32) and (2.33). It actually represents the field of the incident wave on the edge of the wedge

When r0 → ∞ and I0 → ∞, this field can be interpreted as the plane wave traveling to the wedge from the direction ϕ = ϕ0:

As a result, one can rewrite Equations (2.32) and (2.33) in the classical Sommerfeld form (Sommerfeld, 1935):

Equations (2.40) to (2.42) finally determine the total field generated by the incident plane wave in the presence of the perfectly reflecting wedge.

2.3 CONVERSION OF THE SERIES SOLUTION TO THE SOMMERFELD INTEGRALS

In his work Sommerfeld (1935) presented the solution of the wedge diffraction problem in integral form and then transformed it into infinite series. Here, we will perform a reverse procedure, and convert the infinite series (2.40) and (2.41) into integrals. To do this, we use the following expression of the Bessel function (Sommerfeld, 1935):

where the integration contour is shown in Figure 2.4. Then the function u(kr, ψ ) can be represented as

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2.3 Conversion of the Series Solution to the Sommerfeld Integrals 41

Here, the integration contour is shifted by −π compared to the contour shown in Figure 2.4. According to the difference inside the brackets, the function u(kr, ψ ) can be represented as the sum of two integrals. In the integral related to the first term inside the brackets, we replace β by β. In the integral related to the second term inside the brackets, we change β by −β. As a result, we arrive at the Sommerfeld integral (Sommerfeld, 1935)

The integration contour C, consisting of two branches, is shown in Figure 2.5. The integrand of u(kr, ψ ) possesses the first-order poles

Application of the Cauchy theorem to the integral over the closed contour C–D (Fig. 2.5) results in

Figure 2.4 Integration contour in Equation (2.43).

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42 Chapter 2 Wedge Diffraction: Exact Solution and Asymptotics

Figure 2.5 Integration contours in Equations (2.47) and (2.50).

where

The integration contour D consists of two branches (Fig. 2.5). In the integral over the left branch, we replace the variable β by ζ − π , and in the integral over the right branch we put β = ζ + π . Then the function v(kr, ψ ) transforms into the integral over the contour D0 (Fig. 2.6)

where n = α/π .

The physical interpretation of Equation (2.49) is the following. The function v(kr, ψ ) describes the diffracted part of the field, and the residues relate to the geometrical optics. This interpretation becomes clear if we consider functions u(kr, ϕ − ϕ0) and u(kr, ϕ + ϕ0).

In the case 0 < ϕ0 < α − π , when only one face (ϕ = 0) of the wedge is illuminated (Fig. 2.7), these functions are determined by

with 0 < ϕ < π − ϕ0

(2.52)

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