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

16 Chapter 1 Basic Notions in Acoustic and Electromagnetic Diffraction Problems

1.3.4Introducing the Notion of “Shadow Radiation”

This notion was introduced for electromagnetic waves by Ufimtsev (1968). It was additionally investigated in Ufimtsev (1990) and discussed in Ufimtsev (1996). A significant part of the results relating to electromagnetic shadow radiation was included in Theory of Edge Diffraction in Electromagnetics (Ufimtsev, 2003). In the present section, the notion of shadow radiation is introduced in conjunction with scalar waves. Consider again the reflection from an infinite perfectly reflecting plane (Fig. 1.6) located in an homogeneous medium. The scattered field in the region z > 0 is determined by the Helmholtz integral expression (Bakker and Copson, 1939)

where utot = uinc + usc is the total field, ds = dxdy is a differential area of the infinite plane S (z = 0), and r is the distance between the integration and observation points.

Let the reflecting plane be acoustically soft. Then, on its surface,

ussc = −uinc,

but

Figure 1.6 Reflection of waves from an infinite plane S in an homogeneous medium. The source is in the region z > 0. The total field in the shadow region (z < 0) equals zero.

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1.3 Physical Optics 17

and

To evaluate the integral in Equation (1.57), we utilize the Helmholtz equivalency theorem (Bakker and Copson, 1939). According to this theorem, the field u created by the acoustic source at the point P in an homogeneous medium (Fig. 1.7) can be represented as the radiation generated by the equivalent sources uinc and ∂uinc/∂N distributed over the closed imaginary surface of the volume V :

(1.59)

Here it is supposed that the source of the incident wave is located inside the volume V . One should emphasize the following wonderful property of this theorem. The field at the point P inside V or outside V does not depend on the shape of its surface . One

can deform this surface in any way, but the result of the integration in Equation (1.59) will be the same: u(P) = 0 if P V , and u(P) = uinc(P) if P / V .

In order to evaluate integral (1.57), we first apply the equivalency theorem (1.59) to the closed surface = SR + HR (Fig. 1.8). Here SR is a circular plate with a radius R, which is a part of the infinite plane S shown in Figure 1.6, and HR is a hemisphere with the same radius R. It is supposed that the source of the incident wave is inside the volume V .

(1.60)

Figure 1.7 Illustration of the equivalency principle. is an arbitrary imaginary surface covering

ˆ ˆ

a volume V of a free homogeneous medium, n and N are respectively the inward and outward unit vectors normal to , and a source of the incident wave is inside V .

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