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

of kl, this angle is in the interval 10.24◦ ≤ ω ≤ 14◦. For the angle (Fig. 6.13), we take its value as 90◦. The results of calculations are plotted in Figure 6.14. The difference between the soft and hard data approaches 16–19 dB. Figure 6.14 demonstrates the rough PO data, which do not depend at all on the boundary conditions and are totally incorrect in the vicinity of minima.

•The next topic is the transformation of the paraboloid into the disk. In this process, each intermediate shape between the initial parabolid and the final disk

is a paraboloid whose focal parameter p depends on its length l. It follows from the equation l = a2/2p that p = a2/2l. To find the angle ω (Fig. 6.13), we use the additional equation p = a tan ω and obtain tan ω = a/2l, or tan ω = ka/2kl. For the parameter ka we take the constant value ka = 3π , which does not depend

on the paraboloid length. In this case, the diameters of all the intermediate paraboloids and the final disk equal 2a = 3λ. For the initial paraboloid (which is transformed into the disk), we take kl = 6π , that is, l = 3λ. The results are plotted in Figure 6.15.

•Now we consider the influence of the paraboloid shadowed base on backscat-

tering. The illuminated part of the object under investigation is the paraboloid with parameters ka = 3π and kl = 6π , when the base diameter and the length of the paraboloid are equal to each other: 2a = l = 3λ. The critical param-

eter of the base, which affects the edge wave, is the angle . It changes from zero to = π − ω, where ω = tan−1(ka/2kl) 14◦. In the limiting

Figure 6.14 Backscattering at conformal paraboloids of different lengths (with constant focal parameter p). According to Equation (6.146), the PO curve also represents scattering of electromagnetic waves from perfectly conducting paraboloids.

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

Figure 6.15 Transformation of the paraboloid into a disk with continuous maintaining of the paraboloidial shape. According to Equation (6.146), the PO curve also represents scattering of electromagnetic waves from perfectly conducting paraboloids.

case = π − ω, the scattering object transforms into the perfectly reflecting infinitely thin screen. The results of this investigaton are shown in Figure 6.16.

The PO approximation does not depend on the shape of the shadowed part of the paraboloid. For the chosen parameter kl = 6π , it predicts the zero value for the scattered field. In the decibel scale it equals minus infinity and it is outside the figure area. As is seen in this figure, the backscattering from a soft paraboloid depends on the angle only a little (the change of scattering

Figure 6.16 Influence of the base shape on backscattering.

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6.4 Backscattered Focal Fields 151

cross-section is about 3 dB), although a strong dependence is observed for the hard paraboloid (about 20 dB).

6.4.4Backscattering from Spherical Segments

The PO approximations for acoustic and electromagnetic fields in this problem are

identical. Focal asymptotics for the total acoustic and electromagnetic fields are different.

Asymptotics for the Scattered Field

The illuminated surface of the scattering object is a spherical segment whose generatrix is shown in Figure 6.17 and given by the equation

The angle θ (z) in Equation (6.160) is displayed in Figure 6.13. At the point z = l, ρ = a, this angle equals θ (l) = ω. For the given quantities b and a, the angle ω and the segment length l are defined by equations

Figure 6.17 Generatrix of a spherical segment with an arbitrary shadowed base.

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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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