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

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

rays incoming from the stationary points 1 and 2 at the circular edge. As is also seen there, the ray from point 2 undergoes the additional phase shift equal −π/2 while crossing the focal line along the z-axis.

The above approximations were derived for the regions 0 < ϑ ≤ and π − ω ≤ ϑ < π when both stationary points are visible from the observation point. The same technique is used for the calculation of the diffracted field in the region≤ ϑ ≤ π − ω, when the stationary point 2 is not visible and therefore does not contribute to the first-order edge waves. In this case, only the vicinity of the stationary point 1 participates in the calculation that results in

Functions f (1) and g(1) are defined by Equations (4.14) and (4.15) with Equations (3.55) to (3.57) and (2.62) and (2.64). According to these equations,

These expressions for functions f (1) and g(1) are valid in the entire region 0 ≤ ϑ ≤ π , although we have two different expressions for functions f (1)(2) and g(1)(2). For the

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6.1 Diffraction at a Canonical Conic Surface 123

Some terms in functions f (1) and g(1) are singular in certain directions ϑ ; however, such singularities cancel each other and these functions are always finite. For the forward direction ϑ = 0 (that is, the shadow boundary behind the body),

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6.1 Diffraction at a Canonical Conic Surface 125

The upper (lower) sign here relates to ϑ = 0 (ϑ = π ). In terms of the local coordinates they look the same for both directions:

with ϕ1 = ω(ϕ1 = π + ω) when ϑ = π (ϑ = 0). Then we use Equations (4.29) and (4.30) and find the following expressions for the focal field:

It is seen that the focal field is ka times higher in magnitude than the ray fields (6.21) and (6.22). In conjunction with Equations (6.41) and (6.42), we recall relationships (6.36), which are valid for the focal points.

6.1.4Bessel Interpolations for the Field us,h(1)

In the previous sections, we derived the ray and focal asymptotic expressions for the field us,h(1). The ray asymptotics (6.21) to (6.24) are valid under the condition ka sin ϑ 1 (i.e., away from the focal line), and the asymptotic expressions (6.41) and (6.42) determine the field directly on the focal line (where sin ϑ = 0). Now our goal is to construct such approximations, that would continuously describe the diffracted field both in the ray and focal regions. This can be done utilizing the Bessel functions J0(ka sin ϑ ) and J1(ka sin ϑ ).

For the large argument (ka sin ϑ 1), they can be approximated by (Gradshteyn and Ryzhik, 1994)

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

It follows from these expressions that

Now we analytically continue these expressions into the focal region. If we take into account Equation (6.36), as well as the relationships J0(0) = 1 and J1(0) = 0, one can see that expressions (6.47) and (6.48) exactly transform into the focal asymptotics (6.41) and (6.42), when ϑ → 0 and ϑ → π . This means that the above formulas (6.47) and (6.48) can be considered as the appropriate approximations for the diffracted field in the regions 0 ≤ ϑ ≤ and π − ω ≤ ϑ ≤ π .

In the region ≤ ϑ ≤ π − ω, where the stationary point 2 is not visible, the modified versions of Equations (6.47) and (6.48),

represent the analytical continuation of the ray asymptotics (6.23) and (6.24) into the entire region ≤ ϑ ≤ π − ω.

In the next sections, the above results found for the canonical problem are applied for calculation of the field scattered at certain bodies of revolution.

6.2SCATTERING AT A DISK

The geometry of the problem is shown in Figure 6.6. The incident plane wave is given by Equation (6.1) and propagates in the positive direction of the z-axis. Because of

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6.2 Scattering at a Disk 127

Figure 6.6 An incident wave propagates in the direction along the z-axis and undergoes diffraction at a circular disk of radius a. Edge points 1 and 2 are in the plane x = 0.

that, and due to the axial symmetry of the disk, the scattered field is the same in all meridian planes ϕ = const. (0 ≤ ϕ ≤ 2π ). In addition, the scattered field is symmetric with respect to the disk plane: us(−z) = us(z), uh(−z) = −uh(z). Therefore, it is sufficient to investigate the field only in the plane ϕ = π/2 for the directions 0 ≤

ϑ ≤ π/2.

6.2.1Physical Optics Approximation

The relationships us(0) = Eϕ(0), uh(0) = Hϕ(0) are valid for the disk diffraction problem

under the conditions usinc = Eϕinc, uh(0) = Hϕ(0).

Acoustic Waves

In this approximation, the scattered field is considered as the radiation generated by the uniform component (1.31) of the scattering sources, which are induced on the illuminated side (z = −0) of the disk. This field is determined by Equations (1.16)

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