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

with n = 1 + (ω + )/π , where 0 ≤ ω ≤ π/2 and 0 ≤ ≤ π − ω.

In the limiting case when the spherical segment continuously transforms into the flat disk (ω → π/2 and l → 0), Equations (6.164), (6.165), (6.166) and (6.167), (6.168) are exactly reduced to Equations (6.150) and (6.151) and (6.152), respectively.

Numerical Analysis of Backscattering

In this section, we calculate the normalized scattering cross-section (6.112), taking into account Equations (6.110) and (6.111). According to the previous section, the following expressions for the ordinary scattering cross-section are valid. The PO approximation is given by

with n = 1 + (ω + )/π .

Two types of calculation are presented here.

•The first is the continuous transformation of the spherical segment into the flat disk in the limiting case ω → π/2. It is assumed that all transition surfaces are

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154 Chapter 6 Axially Symmetric Scattering of Acoustic Waves at Bodies of Revolution

Figure 6.18 Transformation of the spherical segment into a flat disk. According to Equation (6.166), the PO curve also represents the scattering of electromagnetic waves from a perfectly conducting spherical segment.

spherical with the curvature radius b = a/ cos ω. The initial object is given by the parameters ka = 3π , kl0 2.5π , ω0 = 10◦, and = 90◦. In terms of the wavelength, the base radius and the length of the spherical element are equal to a = 1.5λ and l0 1.26λ. The numerical results for the normalized scattering cross-section are plotted in Figure 6.18. It clearly shows the influence of the

Figure 6.19 Influence of the shadowed part of the spherical segment on backscattering. According to Equation (6.166), the PO curve also represents the scattering of electromagnetic waves from a perfectly conducting spherical segment.

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6.5 Axially Symmetric Bistatic Scattering 155

nonuniform component of the scattering sources ( js,h(1)) concentrated in the vicinity of the edge.

•In the next calculation we investigate the influence of the shadowed part of the

spherical segment on backscattering. The critical parameter of this part is the angle shown in Figure 6.17. It changes from zero to π − ω. In the limiting case, when = π − ω, the scattering object transforms into the perfectly

reflecting infinitely thin screen. The illuminated spherical part of the segment is determined by the parameters ω = 10◦, ka = 3π , kl 2.5π . In terms of the wavelength, a = 1.5λ and l 1.26λ. The results are plotted in Figure 6.19.

The PO approximation does not depend on the shape of the shadowed part and it is represented in Figure 6.19 by the horizontal straight line. However, according to PTD, the backscattering increases up to 10 dB for the acoustically soft object and up to 13.5 dB for the acoustically hard object.

6.5BODIES OF REVOLUTION WITH NONZERO GAUSSIAN CURVATURE: AXIALLY SYMMETRIC BISTATIC SCATTERING

The geometry of the problem is illustrated in Figure 6.20. The incident plane wave (6.1) propagates in the positive direction of the z-axis, which is the symmetry axis of a scattering body of revolution. The generatrix of the illuminated side of this body is given by the equation ρ = ρ(z), with 0 ≤ z ≤ l, under the condition d2ρ/dz2 =0. This condition ensures that the Gaussian curvature of this surface is not zero. The shadowed side is an arbitrary smooth surface with 0 ≤ ≤ π − ω. In the limiting case = π − ω, the scattering object is an infinitely thin perfectly reflecting screen with ρ = ρ(z) and 0 ≤ z ≤ l. The tangent to the generatrix forms the angle θ with the z-axis. At the edge points z = l − 0, this angle equals θ (l) = ω = tan−1[dρ(l)/dz]. The principal radii of curvature (R1, R2) are defined in Equations (6.121) and (6.122), and the Gaussian curvature is given by Equation (6.123). The unit normal nˆ to the illuminated surface is defined in Equation (6.128). Due to the axial symmetry of the

Figure 6.20 Generatrix of a body of revolution.

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156 Chapter 6 Axially Symmetric Scattering of Acoustic Waves at Bodies of Revolution

problem, it is sufficient to calculate the scattered field only in the meridian plane

ϕ = π/2.

6.5.1Ray Asymptotics for the PO Field

These asymptotics can be derived from the general integral expressions (6.132), (6.133) under the condition kρ sin ϑ 1. First we consider the observation points in the region π − ω < ϑ ≤ π , where the entire illuminated surface of the scattering object is visible (Fig. 6.21).

The integrals in Equations (6.132), (6.133) over the variable ψ are calculated by the stationary phase technique (Copson, 1965; Murray, 1984). The details of this method were briefly explained in Section 6.1.2. The phase function in these integrals has two stationary points: ψ1 = π/2 and ψ2 = 3π/2. Asymptotic evaluation of these integrals leads to the expressions

Figure 6.21 Cross-section of a body of revolution by the plane yoz.

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6.5 Axially Symmetric Bistatic Scattering 159

and

The functions us,h(0)(zst ) describe the ordinary ray reflected at the stationary point zst determined by Equations (6.179) and (6.180). The quantities R1 and R2 are the principal radii of curvature of the scattering surface. They are defined by Equations (6.121) and (6.122).

Expressions (6.188) and (6.189) determine the sum of two edge-diffracted rays diverging from the edge points 1 and 2 shown in Figure 6.21. The directivity patterns of these rays are defined by the functions

6.5.2 Bessel Interpolations for the PO Field in the Region π − ω ≤ ϑ ≤ π

With the application of relationships (6.45) and (6.46), the above ray asymptotics can be written in the form

+ a { f (0)(1)[J0(ka sin ϑ ) − iJ1(ka sin ϑ )]

2

eikR

+ f (0)(2)[J0(ka sin ϑ ) + iJ1(ka sin ϑ )]} eikl(1−cos ϑ ) (6.192)

R

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The functions f (1, 2) and g(1, 2) are defined by Equations (6.25), (6.26), and (6.30), and (6.31), where one should omit the last terms, which are exactly cancelled by the terms f (0)(1, 2) and g(0)(1, 2) during the summation of modified Equations (6.47) and (6.48) with Equations (6.193) and (6.194).

6.5.4 Asymptotics for the PTD Field in the Region 2ω < ϑ ≤ π − ω away from the GO Boundary ϑ = 2ω

In this observation region, the stationary edge point 2 (Fig. 6.21) is not visible (when< 2ω), and its first-order contribution to the scattered field equals zero. We also assume that the observation directions are far from the last GO ray (reflected at the point 1 and shown in Fig. 6.21), where functions f (1) and g(1) are singular. Under these conditions, one can use the obvious modifications of Equations (6.195) and (6.196) for the scattered field:

6.5.5 Uniform Approximations for the PO Field in the Ray Region 2ω ≤ ϑ ≤ π − ω Including the GO Boundary ϑ = 2ω

The above ray asymptotics (6.197), (6.198) are not applicable in the vicinity of the geometrical optics boundary ϑ = 2ω, where the wave field does not have a ray structure. In this so-called transition region, the process of the transverse diffusion of the

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