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

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

wave field happens to actually give birth to diffracted rays (Ufimtsev, 2003). The singularities of the functions f (1), g(1) and the singularity of the factor 1/ (l) in Equation (6.185) for the direction ϑ → 2ω (in this case, ξst → l and (l, P) → 0) are the mathematical evidence of the existence of this process. For the calculation of the field integral (6.184) in this region, one should apply a more accurate method of the stationary phase that allows the approach of the stationary phase to the end point (Felsen and Marcuvitz, 1972). In the following we present the basics of this technique.

First, we modify the canonical integral (6.184). For the sake of simplicity, the symbol P (related to coordinates of the observation point) is omitted. We notice that the edge point 2 (Fig. 6.21) is not visible and therefore its first-order contribution to the scattered field equals zero. In the integral (6.184), this point corresponds to the end point ξ = −l. To exclude its contribution, we set ξ = −∞ for the lower limit of integration. Then we introduce a new variable t with the equation

−ρ (ξst ) sin ϑ is positive and therefore the quantity (ξst ) is the minimum of function (ξ ). Taking this into account, we define the variable t as the continuous and differentiable function of the old variable ξ ,

where the radical is understood in the arithmetic sense. In the vicinity of the stationary point, where (ξst ) = 0, one can use the Taylor approximations

Now the canonical integral (6.184) can be represented in this form

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

the integral (1.71)) contains all reflected rays and an additional diffracted field that can be neglected in the shadow region, where the basic component of the scattered field is the shadow radiation (1.72). This observation leads directly to the following approximation

where the integration is performed over the illuminated side of the scattering object. This integral in general is also difficult to calculate. However, the Shadow Contour Theorem developed in Section 1.3.5 greatly simplifies the calculation. According to this theorem, the integral (6.225) is identical to the integral over the black disk located in the plane z = l and having the radius a.

The field scattered by the black disk can be represented in the form (1.73)

In the far-field approximation, the quantities us(0) and uh(0) have already been calculated in Section 6.2.1. Utilizing Equations (6.56) and (6.57), and taking into account the shift of the coordinates’ origin accepted in the present section, the field (6.226) can be written as

It is seen that this field really concentrates in the shadow region. It is zero in the backscattering direction (ϑ = π ), but it is large and exactly equal to the PO field generated by the perfectly reflecting disk in the shadow direction ϑ = 0:

The first-order PTD field in the shadow region can be found by the summation of the field (6.227) with the us,h(1) field generated by the edge sources js,h(1) and determined in Section 6.1.4. When utilizing the results of that section, one should include the additional factor exp[ikl(1 − cos ϑ )] into Equations (6.47) to (6.50) because of the above-mentioned origin shift.

This section completes the analysis of the field scattered by the bodies of revolution with nonzero Gaussian curvature.

PROBLEMS

6.1Verify the asymptotic approximation (6.17) for the integral from a function with two stationary points.

6.2Derive Equations (6.25) and (6.26) for functions f (1)(1), g(1)(1) related to the stationary point 1 (Fig. 6.3 in the canonical problem).

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Problems 167

6.3Derive Equations (6.30) and (6.31) for functions f (1)(2), g(1)(2) related to the stationary point 2 (Fig. 6.4 in the canonical problem).

6.4Verify Equations (6.34) and (6.35) for functions f (1)(1), g(1)(1) related to the specular direction ϑ = 2ω (in the canonical problem).

6.5Derive the PO approximation (6.64) for the field scattered from a perfectly conducting disk. Start with Equations (1.87) and (1.88), derive Equation (1.96), obtain Equations (1.98) and (1.99), and apply it for the disk problem.

6.6Derive the focal asymptotics (6.86) and (6.87) for the disk diffraction problem.

6.7Use Equations (6.104) and (6.105) for the focal field scattered by a cone and prove the limiting form (6.108) and (6.109) when a cone transforms into a disk.

6.8Equations (6.148) and (6.149) determine the field scattered by a paraboloid. Show that these expressions transform into Equations (6.151) and (6.152) when a paraboloid continuously transforms into a disk.

6.9Equations (6.169) and (6.170) determine the field scattered by a spherical segment. Show that these expressions transform into Equations (6.151) and (6.152) when a spherical segment continuously transforms into a disk.

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170 Chapter 7 Elementary Acoustic and Electromagnetic Edge Waves

(of the scattering surface sources) induced near the edge is asymptotically (with k → ∞) equivalent to the component jcan(1) induced on the canonical wedge tangent to the real edge. In order to understand what is the appropriate tangency point and what is the appropriate vicinity of this point responsible for the radiation of EEWs, one should appeal to the physical structure of edge waves diffracted at the canonical wedge.

7.1ELEMENTARY STRIPS ON A CANONICAL WEDGE

In Chapter 4 it was shown that under the oblique incidence on the wedge, the scattered edge waves have the form of conic waves, which can be interpreted as the edgediffracted rays distributed over the so-called diffraction cone (Fig. 4.4). Hence, the nonuniform component jcan(1) induced on the wedge faces is also the ray field running from the edge along the generatrix of this cone. Now it becomes clear how we should choose the appropriate tangency point (of the actual scattering edge L with edge E of the canonical wedge; Fig. 7.1) and how to determine its vicinity responsible for the radiation of the EEWs.

The appropriate tangency point must be the origin of the diffracted ray coming to the observation point on the tangent wedge. We emphasize that in general the tangency point is not the edge point nearest to the observation point! What concerns the appropriate vicinity of the tangency point, it must be an infinitely narrow (elementary) strip oriented along the diffracted ray. In Figure 7.1, that is the strip A.

This figure also helps to understand why any other elementary strip, for example strip B, is not acceptable. It is seen that the orientation of such a strip is not consistent with the localization principle. Indeed, the field on this strip does not depend on the local properties of the incident wave and the real scattering edge L at the tangency point. Instead, it consists of the spurious edge waves/rays coming from the fictitious scattering points on the auxiliary edge E, which do not belong to the real scattering edge L.

To complete the definition of the elementary strip, one should determine its length. Although the nonuniform/fringe sources j(1) concentrate near the edge, they

Figure 7.1 Here L is the edge of the actual scattering object and E is the edge of the canonical tangent wedge. The arrows show the edge-diffracted rays diverging from the edge E. The elementary strips A and B belong to the face of the canonical wedge.

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