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

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

In the above expressions, functions f , g and f (1), g(1) are defined according to Chapters 2, 3, and 4:

The angles φ, φ0 for the functions f (1)left [φ (ψ ), φ0(ψ ), α], g(1)left [φ (ψ ), φ0(ψ ), α]

are determined by Equations (14.15) and (14.16), and for the functions f (1)right [φ (ψ ), φ0(ψ ), α], g(1)right [φ (ψ ), φ0(ψ ), α] they are found with Equations

(14.17), (14.18).

All diffracted rays undergo a phase shift equal to ±π/2 when they cross the focal lines = γ and = π − γ , which are the axes of the field beams.

14.1.5Refined Asymptotics for the Specular Beam

We refer again to Figure 14.1 and focus on the beam reflected from the lower lateral surface of the cylinder and propagating in the specular direction = 2π − γ . In the first-order approximation, this beam was evaluated in the previous section. Here we consider some fine features of the theory, which are beyond the first approximation. The final results of this section were published in the paper by Ufimtsev (1989).

According to PTD the scattered field is generated by the uniform j(0) and nonuniform j(1) components of the scattering sources induced by the incident wave on

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14.1 Acoustic Waves 301

the object. Up to now we have calculated the field radiated only by the basic part of the component j(1) that is caused by sharp bending (edges). This is the so-called fringe component j(1)fr . The other component j(1)sm is caused by the smooth bending of the scattering surface and it is asymptotically small compared to j(0) and j(1)fr . In particular on a circular cylinder, the ratio j(1)sm/j(0) is of the order 1/ka. That is why the component j(1)sm is usually neglected for thick cylinders. However, in our papers (Ufimtsev, 1979, 1981, 1989), it was shown that this small component distributed

in the beam field. This is the first fine feature of the theory.

It was also shown (Ufimtsev, 1979, 1981, 1989) that the second term of the asymptotic expansion for the PO field (in the specular direction) is also a quantity of the order (ka)−1/2 and it also should be included in the beam field. Usually, the high-order terms in the PO field are considered incorrect. However, in the framework of PTD, the PO field is the constituent part of the scattered field. Therefore one should incorporate into the field expression the high-order asymptotic terms of the PO field, which are of the same order of magnitude as those taken from the asymptotic expansion of the field generated by the nonuniform sources js,h(1). This is the second fine and important feature of PTD.

Here, these observations are demonstrated in the analytic form for the directivity pattern ( , γ ) introduced by Equation (14.20). The scattered field is evaluated in the vicinity of the specular direction = 2π − γ . The PO field generated by j(0) is described by Equations (14.4) and (14.5). The first term there relates to the field scattered by the left base (disk) of the cylinder. Its contribution to the field in the region 3π/2 < < 2π is created by the vicinity of point 2 (Fig. 14.1). By asymptotic evaluation of this contribution, the expressions (14.4) and (14.5) can be written as

where p = ka(sin γ − sin ) and q = kl(cos − cos γ ). Here, in accordance with the above discussion, we retain the two first terms in the asymptotic expansion for the integrals and obtain

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The nonuniform component j(1)sm caused by the smooth bending of the cylindrical surface is found by the extension of the results of the paper by Franz and Galle (1955) to the case of the oblique direction of the incident wave. The asymptotic approximations found in this way are

According to Equations (1.16), (1.17), and (14.20), the field radiated by these sources is described by

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