154 Chapter 6 Axially Symmetric Scattering of Acoustic Waves at Bodies of Revolution
Figure 6.18 Transformation of the spherical segment into a flat disk. According to Equation (6.166), the PO curve also represents the scattering of electromagnetic waves from a perfectly conducting spherical segment.
spherical with the curvature radius b = a/ cos ω. The initial object is given by the parameters ka = 3π , kl0 2.5π , ω0 = 10◦, and = 90◦. In terms of the wavelength, the base radius and the length of the spherical element are equal to a = 1.5λ and l0 1.26λ. The numerical results for the normalized scattering cross-section are plotted in Figure 6.18. It clearly shows the influence of the
Figure 6.19 Influence of the shadowed part of the spherical segment on backscattering. According to Equation (6.166), the PO curve also represents the scattering of electromagnetic waves from a perfectly conducting spherical segment.
6.5 Axially Symmetric Bistatic Scattering 155
nonuniform component of the scattering sources ( js,h(1)) concentrated in the vicinity of the edge.
•In the next calculation we investigate the influence of the shadowed part of the
spherical segment on backscattering. The critical parameter of this part is the angle shown in Figure 6.17. It changes from zero to π − ω. In the limiting case, when = π − ω, the scattering object transforms into the perfectly
reflecting infinitely thin screen. The illuminated spherical part of the segment is determined by the parameters ω = 10◦, ka = 3π , kl 2.5π . In terms of the wavelength, a = 1.5λ and l 1.26λ. The results are plotted in Figure 6.19.
The PO approximation does not depend on the shape of the shadowed part and it is represented in Figure 6.19 by the horizontal straight line. However, according to PTD, the backscattering increases up to 10 dB for the acoustically soft object and up to 13.5 dB for the acoustically hard object.
6.5BODIES OF REVOLUTION WITH NONZERO GAUSSIAN CURVATURE: AXIALLY SYMMETRIC BISTATIC SCATTERING
The geometry of the problem is illustrated in Figure 6.20. The incident plane wave (6.1) propagates in the positive direction of the z-axis, which is the symmetry axis of a scattering body of revolution. The generatrix of the illuminated side of this body is given by the equation ρ = ρ(z), with 0 ≤ z ≤ l, under the condition d2ρ/dz2 =0. This condition ensures that the Gaussian curvature of this surface is not zero. The shadowed side is an arbitrary smooth surface with 0 ≤ ≤ π − ω. In the limiting case = π − ω, the scattering object is an infinitely thin perfectly reflecting screen with ρ = ρ(z) and 0 ≤ z ≤ l. The tangent to the generatrix forms the angle θ with the z-axis. At the edge points z = l − 0, this angle equals θ (l) = ω = tan−1[dρ(l)/dz]. The principal radii of curvature (R1, R2) are defined in Equations (6.121) and (6.122), and the Gaussian curvature is given by Equation (6.123). The unit normal nˆ to the illuminated surface is defined in Equation (6.128). Due to the axial symmetry of the
Figure 6.20 Generatrix of a body of revolution.
156 Chapter 6 Axially Symmetric Scattering of Acoustic Waves at Bodies of Revolution
problem, it is sufficient to calculate the scattered field only in the meridian plane
ϕ = π/2.
6.5.1Ray Asymptotics for the PO Field
These asymptotics can be derived from the general integral expressions (6.132), (6.133) under the condition kρ sin ϑ 1. First we consider the observation points in the region π − ω < ϑ ≤ π , where the entire illuminated surface of the scattering object is visible (Fig. 6.21).
The integrals in Equations (6.132), (6.133) over the variable ψ are calculated by the stationary phase technique (Copson, 1965; Murray, 1984). The details of this method were briefly explained in Section 6.1.2. The phase function in these integrals has two stationary points: ψ1 = π/2 and ψ2 = 3π/2. Asymptotic evaluation of these integrals leads to the expressions
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2π k sin ϑ |
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Figure 6.21 Cross-section of a body of revolution by the plane yoz.
6.5 Axially Symmetric Bistatic Scattering 157
and
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uh(0) = −u0 |
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0 eik 1(z)'ρ(z)[sin ϑ − ρ |
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[sin ϑ + ρ (z) cos ϑ ]dz |
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0 eik 2(z)' |
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1(z) = z(1 − cos ϑ ) − ρ(z) sin ϑ |
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2(z) = z(1 − cos ϑ ) + ρ(z) sin ϑ . |
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Here the integrals with the factor exp[ 1(z)] represent the field generated by the vicinity of the stationary line ψ = π/2, 0 < z ≤ l, and the integrals with exp[ 2(z)] describe the field generated by the vicinity of the stationary line ψ = 3π/2, 0 < z ≤ l (Fig. 6.21).
Now we check the functions 1(z) and 2(z) for the presence of stationary points zst . It follows from the equation
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ρ (zst ) = dρ/dz = tan θ (zst ) = tan(ϑ/2) |
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θ (zst ) = ϑ/2. |
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This equation determines the reflection point zst on the scattering surface shown in Figure 6.20. At this point, the tangent to the generatrix ρ(z) forms the angle θ = ϑ/2 with the z-axis, which agrees with the reflection law.
We then check the function 2(z) for the stationary point. It follows from the equation
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ρ (zst ) = dρ/dz = tan θ (zst ) = − tan(ϑ/2) |
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θ (zst ) = −ϑ/2, |
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(6.183) |
TEAM LinG
158 Chapter 6 Axially Symmetric Scattering of Acoustic Waves at Bodies of Revolution
This stationary point relates to the reflected ray in the meridian plane ϕ = 3π/2. For the same value zst in Equations (6.180) and (6.183), this ray is exactly symmetrical to the reflected ray shown in Figure 6.20. As we consider the scattered field only in the meridian plane ϕ = π/2, the function 2(z) does not have any stationary points for the scattering directions in this plane.
By introducing into Equations (6.174) and (6.175) a new integration variable ξ = z for the integrals with function 1(z), and ξ = −z for the integrals with function2(z), one can represent their sum as
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−l
where symbol P denotes the location of the observation point. For a high frequency of the field (when k 1), the factor exp[ (ξ , P)], being a function of the integration variable ξ , undergoes fast oscillations. Because of this, most differential contributions F(ξ , P)eik (ξ ,P)dξ to the integral I(P) asymptotically cancel each other. Only those that are in the vicinity of the stationary point ξst and in the vicinity of the end points ξ = −l and ξ = l provide substantial contributions to I(P). The contribution of the stationary point is calculated by the stationary phase technique, and the contributions by the end points are found by integrating by parts (Copson, 1965; Murray, 1984). The resulting asymptotic approximation for I(P) is given by
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F(ξst , P)eik (ξst ,P)+iπ/4 |
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The outlined procedure was used to represent the scattered field us,h(0) in the form of three contributions:
us,h(0) = us,h(0)(zst ) + us,h(0)(1) + us,h(0)(2), |
(6.186) |
where
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6.5 Axially Symmetric Bistatic Scattering 159
and
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The functions us,h(0)(zst ) describe the ordinary ray reflected at the stationary point zst determined by Equations (6.179) and (6.180). The quantities R1 and R2 are the principal radii of curvature of the scattering surface. They are defined by Equations (6.121) and (6.122).
Expressions (6.188) and (6.189) determine the sum of two edge-diffracted rays diverging from the edge points 1 and 2 shown in Figure 6.21. The directivity patterns of these rays are defined by the functions
sin ω
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cos ω − cos(ω − ϑ )
sin(ω − ϑ )
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cos ω − cos(ω − ϑ )
sin ω
(6.190)
cos ω − cos(ω + ϑ )
sin(ω + ϑ )
. (6.191)
cos ω − cos(ω + ϑ )
6.5.2 Bessel Interpolations for the PO Field in the Region π − ω ≤ ϑ ≤ π
With the application of relationships (6.45) and (6.46), the above ray asymptotics can be written in the form
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− 2 'R1(zst )R2(zst )eik 1(zst ) |
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+ a { f (0)(1)[J0(ka sin ϑ ) − iJ1(ka sin ϑ )]
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eikR
+ f (0)(2)[J0(ka sin ϑ ) + iJ1(ka sin ϑ )]} eikl(1−cos ϑ ) (6.192)
R
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