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

9.3 Slope Diffraction in the Configuration of Figure 9.1 237

To calculate the slope diffraction of wave (9.28) at edge L2, we approximate it by the equivalent wave

The angle γ02 is shown in Figure 9.2. The amplitude of this wave is determined by the requirement

According to the idea demonstrated in Section 9.2, the elementary edge waves generated at edge L2 due to the slope diffraction are found by the differentiation of

Equations (7.90) and (7.97) with respect to ϕ02, and with the simultaneous replacement of uinc(ζ ) by (9.32):

( jh(1)) and the total ( jh(t) = jh(0) + jh(1)) scattering sources, respectively.

The diffracted waves diverging from the whole edge are determined respectively by the integrals:

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238 Chapter 9 Multiple Diffraction of Edge Waves

and

The ray asymptotics of these waves are found by the stationary-phase technique:

the first term in parentheses. This difference is a consequence of Equations (8.16) and (8.17), which were used in the derivations of both Equations (9.12) and (9.40). According to Equations (8.16) and (8.17), these terms appear due to the differentiation

shows the direction of the ray coming to edge L1 (L2) from edge L2 (L1), and the angles γ01 and γ02 are shown in Figure 9.2.

As well as in the case of grazing diffraction as considered in Section 9.2, the waves (9.38) and (9.39) arising due to the slope diffraction are also less in magnitude by a factor of 1/kR21 compared to the waves (8.13) and (8.29) generated by the ordinary edge diffraction. This result is not surprising, because the intensity of the incident field hitting the edge in the case of the slope diffraction is significantly less.

9.3.2Electromagnetic Waves

Here, the above theory is extended for electromagnetic waves (Ufimtsev, 1991). The edge wave traveling from edge L1 to edge L2 can be considered as the sum of two

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9.3 Slope Diffraction in the Configuration of Figure 9.1 239

with the function v1 shown in Equation (9.28) and equal to zero in direction to edge L2. It is clear that the diffraction of this wave at edge L2 can be investigated in the same way as the diffraction of the acoustic wave (9.27). One approximates the incident wave by the equivalent wave

with w01 as defined in Equation (9.33). The EEWs are found by the differentiation of Equations (7.135) and (7.136) (with respect to the angle ϕ02), where one should set E0t = 0 and

H0t exp(ikφi) = w01 exp(ikR21).

Then, for the total edge wave arising at wedge L2, one obtains

which completely agrees with Equation (9.39) and allows the formulation of the relationship

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9.4 Slope Diffraction: General Case 241

Figure 9.3 The plane P contains the tangent ˆt to the edge L, the incident ray Qζ , and the point q (which is the projection of the point Q on the perpendicular rq to the tangent ˆt). The vector nˆ is the unit normal to the plane P. (Reprinted from Ufimtsev and Rahmat-Samii (1995) with the permission of

Annales des Telecommunications.)

Figure 9.4 Here W is the tangential wedge to the scattering edge L, the plane T is the face of wedge W , the vector τ is perpendicular to the tangent t and belongs to the plane T . The angle γ0 indicates the direction of the incident ray. The point P(R, ϑ , ϕ) is the observation point. (Reprinted from Ufimtsev and Rahmat-Samii (1995) with the permission of Annales des Telecommunications.)

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