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

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

Figure 6.8 Scattering at the acoustically hard disk. The curve “FRINGE” relates to the field generated by the nonuniform (“fringe”) scattering sources. According to Equations (1.100) and (1.101), the PO curve here also demonstrates the scattering of electromagnetic waves at a perfectly conducting disk.

6.3SCATTERING AT CONES: FOCAL FIELD

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

identical. Asymptotics for the total acoustic and electromagnetic fields are different.

6.3.1 Asymptotic Approximations for the Field

The geometry of the problem is shown in Figure 6.9. The incident plane wave is given by Equation (6.1). First, we calculate the focal field radiated by the uniform components js,h(0) of the induced scattering sources. They are defined by Equation (1.31) and determined on the cone surface by

The scattered field in the far zone is defined by Equations (1.16) and (1.17), where one should set

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for the backscattering direction ϑ = π . In these equations, n = 1 + (ω + )/π .

In contrast to the above noted equivalence of the PO acoustic and electromagnetic fields, the acoustic field us,h(1) generated by the nonuniform scattering sources is different from the electromagnetic field radiated by the nonuniform electric currents. The electromagnetic field is given by Equations (2.3.18) and (2.3.19) in Ufimtsev (2003) and is determined by a linear combination of functions f (1) and g(1), but in the acoustic case the field us(1) depends only on f (1) and the field uh(1) depends only on g(1).

The above equations describe the field us,h(1) generated only by the nonuniform scattering sources induced near the cone edge. There are two other types of nonuniform sources induced on the cone surface, which we have neglected here. The first is caused by the smooth bending of the surface, which is a small quantity inversely proportional to (kρ) at a distance far from the cone tip. Here ρ is a polar coordinate of the surface (ρ = a at the edge points). The second is the nonuniform component concentrated near the cone tip. It is caused by both the sharp tip and by the large curvature of the cone surface due to its smooth bending near the tip. At a far distance ξ from the tip, it is of the order 1/kξ . In the case of electromagnetic diffraction, it was shown (Ufimtsev, 2003) that in the backscattering direction (ϑ = π ) one can neglect the field radiated by these types of nonuniform scattering sources. The asymptotic analysis of the electromagnetic field scattered by a semi-infinite cone also confirms this observation (Felsen, 1955; Bowman et al., 1987).

Taking these comments into account, the first-order PTD approximation for the backscattered total field (at the focal line ϑ = π ) can be found by summation of

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

In the limiting case when ω → π/2 and the front part of the object (Fig. 6.9) transforms into the disk, the above equations for the backscattered field are reduced to

with n = (3/2) + ( /π ). Finally, with → π/2 these expressions transform into

which exactly coincide with Equations (6.86), (6.87) for the field scattered back from the disk.

6.3.2Numerical Analysis of Backscattering

For the field represented in the form

in the decibel scale, that is, the quantity 10 log(σs,hnorm). Note that this normalized scattering cross-section is different from the one used in Section 6.2.3 for the disk problem.

According to Section 6.3.1, the PO predicts the following approximation

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