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

248 Chapter 10 Diffraction Interaction of Neighboring Edges on a Ruled Surface

Figure 10.1 Element of a ruled surface S with two edges (L1 and L). The unit vectors ˆt1, ˆt are tangential to the edges. The unit vectors τˆ1, τˆ are tangential to the surface S and perpendicular to ˆt1, ˆt, respectively. The quantities r1, ϕ1, z1 and r, ϕ, z are local polar coordinates. The unit vector nˆ is normal to the plane tangential to S and containing the generatrix R10 as well as the tangents ˆt1, ˆt and τˆ1, τˆ. (Reprinted from Ufimtsev (1989) with the permission of the Journal of Acoustical Society of America.)

10.1 DIFFRACTION AT AN ACOUSTICALLY HARD SURFACE

Suppose that the edge wave propagating from the edge L1 has a ray structure and can be represented in the form of Equation (8.29) as

where the function g(ϕ1, ϕ01, α1) is defined in Equation (2.64). In the vicinity of the edge L, this wave can be approximated by the two merging plane waves

The first term here plays the role of the incident wave in the canonical wedge diffraction problem utilized in Chapter 7 to derive the asymptotic approximation (8.3), (8.4). Therefore, replacing the quantity u0(ζ ) exp(ikφi) in Equation (8.4) by (1/2)u1h(R10, 0), we obtain the asymptotic expression

for the edge wave generated by the total scattering source jh(t) = jh(1) + jh(0). One should note that in this particular case, jh(0) = u1h(R1, 0).

The ray asymptotic of this wave can be found by the stationary-phase technique. However, it can be obtained directly from Equation (8.29) if we replace there uinc(ζst ) by (1/2)u1h(ζst ) and set ϕ0 = 0:

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10.1 Diffraction at an Acoustically Hard Surface 249

where

The letters α1, α denote the external angles of edges L1, L (π ≤ α1 ≤ 2π , π ≤ α ≤ 2π ), and the caustic parameters ρ1, ρ are defined according to Equation (8.23).

The ray asymptotic (10.4) is not applicable near the shadow boundary ϕ = π , where it becomes singular. Instead, one can suggest the following heuristic approximation of Equation (10.3) valid in all directions along the diffraction cone (ϑ = π − γ0, 0 ≤ ϕ ≤ α):

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250 Chapter 10 Diffraction Interaction of Neighboring Edges on a Ruled Surface

the diffracted field u(t) is discontinuous at the shadow boundary (ϕ1 = 0, ϕ = π ):

However, the sum of the diffracted and incident fields (uh(t) + u1h) is continuous there. One can also show that the normal derivative of this sum is also continuous at the shadow boundary.

Utilizing the asymptotic approximation

10.2 DIFFRACTION AT AN ACOUSTICALLY SOFT SURFACE

The geometry of the problem is shown in Figure 10.1. Suppose that the edge wave

propagates from the edge L1 along the face S and, due to the boundary condition, equals zero in the direction ϕ1 = 0 to the edge L. We note that f (0, ϕ01, α1) = 0 in accordance with the definition given by Equation (2.62). Thus, the diffraction of the wave (10.14) at the edge L is a particular case of the slope diffraction, and it can be treated, as shown below, with a little modification of the technique developed in Chapter 9.

First, we notice that an appropriate wave (equivalent to the incident wave (10.14) in the vicinity of edge L) can be constructed from the combination of the incident and reflected plane waves

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10.2 Diffraction at an Acoustically Soft Surface 251

and equals

According to the idea introduced in Chapter 9, the edge wave arising due to the slope diffraction of the equivalent wave (10.16) is determined by the derivative of (8.4) with the simultaneous replacement of u0(ζ ) exp(ikφi) by u0eq:

The ray asymptotic of the diffracted field (10.19) can be found by the stationary-

phase technique or directly by the differentiation of Equation (8.29) with the replacement of u0(ζ ) exp(ikφi) by u0eq:

This function is singular in the direction of the shadow boundary (ϕ = π ). Instead of it one can suggest the following heuristic approximation of Equation (10.19) valid in all directions along the diffraction cone (ϑ = π − γ0, 0 ≤ ϕ ≤ α):

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