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

294 Chapter 14 Bistatic Scattering at a Finite-Length Cylinder

Figure 14.7 Bistatic scattering at a hard cylinder. According to Equation (14.75), the PO curve here also demonstrates the scattering of electromagnetic waves (with Hx -polarization) from a perfectly conducting cylinder.

by the PO field and by the field generated by jh(1). The sum of these fields and its comparison with the PO field are shown in Figures 14.8 and 14.10.

These figures clearly show the influence of the field generated by the nonuniform/fringe scattering sources jh(1). In particular, this field fills in the deep minima in the PO field. More accurate PTD approximation can be obtained with calculation of the high-order edge waves. However, in contrast to thin dipoles, thick cylinders are not resonant bodies and all high-order edge waves can be neglected when the size of the cylinder exceeds 3–5 wavelengths. The larger the cylinders, the higher the accuracy of the PTD expressions (14.12) and (14.13).

The following comments explain some details of the numeric calculations:

•The functions Fh(1)left,right are determined using Equations (7.82), (7.87), (7.88), (7.92), and (7.94) through the functions Vt (σ1, φ0) and Vt (σ2, α − φ0), which

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296 Chapter 14 Bistatic Scattering at a Finite-Length Cylinder

Figure 14.9 Scattering at a rigid cylinder. According to Equation (14.75), the PO curve here also demonstrates the scattering of electromagnetic waves (with Hx -polarization) from a perfectly conducting cylinder.

• The function V (σ2, α − φ0) related to the left edge is singular at the points ψ = 0, π , 2π for the observation direction = γ . This is the grazing singularity mentioned in Equation (4.21). It is removed by the exclusion of a certain vicinity of the singular points from the integral (14.12). This exclusion is done only for that part of the integral (14.12) that contains the function V (σ2, α − φ0). The function V (σ1, ϕ0) is not singular and is integrated in Equation (14.12) over the entire region 0 ≤ ψ ≤ 2π . Notice that the theory of EEWs presented in Section 7.9 (which is free from the grazing singularity) cannot treat the above singularity in the direction = γ , because this theory is applicable only for objects with planar faces.

•Finally we note that in the case σ1 → φ0 or σ2 → α − φ0 one should use

Equation (7.107) for the functions V (φ0, φ0) and Equation (7.109) for the function V (α − φ0, α − φ0).

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14.1 Acoustic Waves 297

Figure 14.10 Scattering at a rigid cylinder. According to Equation (14.75), the PO curve here also demonstrates the scattering of electromagnetic waves (with Hx -polarization) from a perfectly conducting cylinder.

14.1.4Beams and Rays of the Scattered Field

The previous section provides a numerical investigation of the scattered field. Here we consider its physical structure and present simple high-frequency asymptotics for the directivity pattern ( , γ ) defined by the equation

From the physical point of view, the scattered field consists of the following basic components:

•The reflected beam in the vicinity of the direction = π − γ . This beam appears due to the transverse diffusion of the field in ordinary rays reflected from the left base of the cylinder (Fig. 14.1). It is described by the first terms in Equations (14.4) and (14.5), which contain the Bessel function J1( p). Exactly

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298 Chapter 14 Bistatic Scattering at a Finite-Length Cylinder

in this direction its value equals

•The reflected beam in the vicinity of the direction = 2π − γ . It appears due to the transverse diffusion of the field in ordinary rays reflected from the lower lateral part of the cylinder. This beam is described by the high-frequency asymptotics for the second terms in Equations (14.4) and (14.5):

•The beam of the shadow radiation in the vicinity of the direction = γ . It is described by both terms in Equations (14.4) and (14.5). Exactly in this direction its value equals

•The beam of edge-diffracted rays generated by the fringe scattering sources, which are located near the left edge. It propagates in the direction = π − γ and supplements the reflected beam (14.21). This beam is described by Equa-

tion (14.12). In the case of the soft cylinder, the corresponding expression

follows from Equation (14.12) with the obvious replacement of Fh(1) by Fs(1). Exactly in the direction = π − γ , it is determined by

•The beam of edge-diffracted rays generated by the fringe scattering sources, which are located near both edges. It propagates in the direction = γ and

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14.1 Acoustic Waves 299

supplements the shadow beam. This beam is described by Equation (14.11). Exactly in this direction it is described by

To avoid the grazing singularity, one should exclude a certain vicinity of the points ψ = 0, π , 2π in the integral over the left edge.

Away from these beams, the scattered field contains the three edge-diffracted rays generated by the total surface current js,h(0) + js,h(1). They can be determined by the

asymptotic estimation of the field us,h(0) + us,h(1) described by Equations (14.4), (14.5), and (14.11). However, a simpler way is to apply the modified asymptotics (8.12) and (8.13), where one should replace the functions f (1), g(1) with f , g. As a result, one obtains the following expressions for these rays.

propagates from the stationary point 1 (Fig. 14.1) and exists in the regions 0 ≤ < γ , γ < < π − γ , and π − γ < ≤ 3π/2. Factor exp(−iπ/4) is taken for positive values of p and factor exp(+iπ/4) is valid for the negative values of p.

• Ray 2:

propagates from the stationary point 2 and exists in the region π/2 ≤ < π − γ , π − γ < < 2π − γ , and 2π − γ < θ ≤ 2π . Factor exp(+iπ/4) is taken for positive values of p and factor exp(−iπ/4) is valid for negative values of p.

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