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

11.2 Multiple Hard Diffraction 257

waves of the (m + 1) order. More precisely, u¯h(m) is the field of the m-order wave at the diffraction point ζ . One can show that

and

Here, the series is the geometric progression that can be converted to its sum. The physical meaning of Equation (11.7) is clear. The first term in the braces relates to the primary edge waves, the second to the secondary waves, and the third term represents the sum of all multiple edge waves of order 3 and higher.

The total scattered field on the focal line also includes the reflected rays (6.187) in front of the object (z < 0) and the shadow radiation (6.228) behind the object (z > 0). This approximation for the scattered field actually represents the incomplete asymptotic expansion, because it includes only the first term in the individual asymptotic expansion for each multiple edge wave. Also, Equation (6.187) is only the first term in the asymptotic expansion for the reflected field.

Expression (11.7) can be used to calculate the total scattering cross-section. In the case of the incident plane wave, uinc = eikz, this quantity is defined as

258 Chapter 11 Diffracted at a Convex Body of Revolution with a Flat Base

where uh,stot is the total field scattered in the forward direction (ψ = π/2). This field consists of the following components:

•The shadow radiation, which is equivalent to the PO field (for the forward direction) and determined by Equation (6.228), where one should set u0 = 1;

•The primary edge waves generated by the nonuniform/fringe scattering sources

jh,s(1) and determined by Equations (6.41), (6.42) or (11.2), where one should set u0 = 1 and u0 exp(ikφi) = 1, respectively;

•The sum of all multiple edge waves of order 2 and higher.

The substitution of this total field into Equation (11.8) results in the following asymptotic expression:

which is incomplete in the sense mentioned above. Notice that the series in Equation (11.9) equals zero, when 2ka − π/4 = lπ (l = 1, 2, 3, . . .). In this case, all corrections to the first term in Equation (11.9) are determined by the higher-order terms in the individual asymptotic expansions for each multiple edge wave.

11.3MULTIPLE SOFT DIFFRACTION

The primary edge wave excited by the incident wave (11.1) is determined by Equation (11.2). The higher-order edge waves arise due to the slope diffraction of waves running along the flat base of the scattering object. These higher-order edge waves are calculated on the basis of Equation (10.19). For the (m + 1)-order edge wave arriving at point P on the focal line, it can be written in the form

¯ (m)

Here, us is the amplitude factor of the wave, which is equivalent to the m-order edge

is calculated with application of Equations (10.18) and (10.20), where one should set γ0 = γ01 = π/2, R = 2a, and ρ = −a. These calculations result in

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11.3 Multiple Soft Diffraction 259

or

The series in Equation (11.15) is a geometrical progression.

Now we apply Equation (11.15) for calculation of the total scattering crosssection. In the case of the incident plane wave uinc = eikz, it is defined by

Equation (11.8), where

∞

Here, ush is the shadow radiation (6.228) (where one should set u0 = 1), the quantity

is the primary edge wave generated by the nonuniform scattering sources js(1), and the series represents the sum of all edge waves of order 2 and higher. Thus, the total

scattering cross-section of the acoustically soft object equals

+

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260 Chapter 11 Diffracted at a Convex Body of Revolution with a Flat Base

We emphasize again that approximations (11.15) and (11.18) are the incomplete asymptotic expansions in the sense discussed above in Section 11.2. The series in Equation (11.18) equals zero when 2ka + π/4 = lπ (l = 1, 2, 3, . . .). In this case, all corrections to the first term in (11.18) are determined by the higher-order terms in the individual asymptotic expansions for each multiple edge wave.

PROBLEMS

11.1Prove Equations (11.4) to (11.6) for the primary and multiple acoustic edge waves on a hard scattering object. Explain all of the details, including caustic parameters, phase shifts, directivity factors, and fractional coefficients.

11.2Prove Equations (11.11) to (11.13) for the primary and multiple acoustic edge waves on a soft scattering object. Explain all of the details, including caustic parameters, phase shifts, directivity factors, and fractional coefficients.

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