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

232 Chapter 9 Multiple Diffraction of Edge Waves

the next diffraction at edge L1. The unit vectors ˆt1 and ˆt2 are tangents to the edges L1 and L2. (Reprinted from Ufimtsev (1989) with the permission of the Journal of Acoustical Society of America).

The amplitude u02 of the equivalent canonical wave is defined by equating the normal derivatives of the real and equivalent incident waves at the diffraction point z1 = r1 = 0:

Now, we notice that the Helmholtz equation governing the wave field and its diffraction can be differentiated with respect to the parameter ϕ01 without changing the type of this equation. This means that the derivative (with respect to a free parameter) of any solution of the Helmholtz equation is also a solution of the same equation. The boundary conditions also admit differentiation with respect to ϕ01. In Chapter 7, we found the edge diffracted field generated by the incident plane wave (7.2). The incident

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9.2 Grazing Diffraction 233

wave (9.4) is the derivative of the wave (7.2). Therefore, the diffracted field generated

by the wave (9.4) can be found by the differentiation of the edge diffracted fields found in Chapter 7, if we replace uinc(ζ ) by u02 = w02 exp(ikR21). Before completing this

procedure, however, we make another observation.

The incident wave propagating in the grazing direction to the plate creates the identical scattering sources jh(0) = 2uinc on both sides of the plate. According to Equation (1.10), the field generated by the sources induced on one side of the plate is completely cancelled by the field generated by the identical sources induced on the opposite side of the same plate. Therefore, in this particular case of the grazing incidence, the field uh(0) equals zero and uh(1) ≡ uh(t).

In view of this observation and the one made in the previous paragraph, the elementary edge wave generated at edge L1 is found by the differentiation of Equation (7.97) with the simultaneous replacement of uinc(ζ ) by w02(ζ ) exp(ikR21):

The ray asymptotic of this field is found by application of the stationary-phase technique described in Section 8.1. We omit all intermediate details and present the final expression

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is the principal normal to edge L1 and a1 is the radius of curvature of this edge at the

ˆs

stationary point ζst . The unit vector k1 shows the directions of the diffracted rays (9.11) that form the diffraction cone. In accordance with Equation (8.7), this vector is defined by the equation

Asymptotic expression (9.11) can also be written in the form of Equation (8.13):

where (ζst ) = R21(ζst ) + R1(ζst ). Note that the quantity (ζ ) = d2 (ζ )/dζ 2 is easier to calculate than the caustic parameter ρ1(ζ ) in Equation (9.11).

The above approximations (9.11) and (9.15) are the nonuniform asymptotics. They are singular in the directions ϕ1 = 0, 2π , because

This singularity is a consequence of the singularity of function g(ϕ1, ϕ01, α1) in the directions ϕ1 = π ± ϕ0. It can be treated as shown in Section 7.9.

It is worth noting that, in view of Equation (9.8), the wave (9.11), (9.15) arising due to the grazing/slope diffraction is less in magnitude by a small factor of 1/kR21 compared to the wave (8.29) generated by the ordinary edge diffraction.

9.2.2Electromagnetic Waves

A similar problem for the grazing diffraction of electromagnetic waves was investigated in the paper by Ufimtsev (1991). Its solution is found in the same way as for

acoustic waves. Here, it is supposed that the edge wave traveling from edge L2 to edge

tot

L1 is polarized perpendicularly to the plate S1 (E2 S1) and its magnetic vector is parallel to this plate,

In the vicinity of edge L1, this wave is approximated by the equivalent wave (9.4),

eq = eq

where one should set u2 H2z . The quantity u02 in Equation (9.4) is defined by

1

Equation (9.6), and the equivalent wave is written as

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236 Chapter 9 Multiple Diffraction of Edge Waves

at the scattering edge L1

exists between the acoustic and electromagnetic waves arising due to the grazing diffraction.

9.3 SLOPE DIFFRACTION IN THE CONFIGURATION OF FIGURE 9.1

The following relationship exists between acoustic and electromagnetic waves arising due to the slope diffraction at edge L2 (Fig. 9.1):

uh = Ht , if ∂uinc = ∂Htinc

∂n ∂n

at the diffraction point on edge L2.

Here, ˆt is the tangent to the edge L2 and nˆ is the normal to the plate S1 (Fig. 9.1).

9.3.1 Acoustic Waves

Suppose that an external wave

with γ0 = π − γ01, cos γ01 = ˆt1 · φ0, and the angle γ01 is shown in Figure 9.2. This wave hits the edge L2 where it undergoes diffraction. That is why we denote it as u1inch . The function g(ϕ1, ϕ01, α1) is defined by Equation (2.64) and is equal to zero in the direction ϕ1 = π to edge L2.

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