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

1.3 Physical Optics 19

Figure 1.9 Illustration of the equivalency theorem applied to the reflected field. Here, as in Figure 1.8, SR is a circular plate on the plane z = 0 with radius R, and HR s a hemisphere with the same radius R.

To clarify the physical meaning of function us,2sc , we introduce new denotations into Equation (1.58),

It is supposed that the image source, that is, the source of the reflected field, is inside the volume V .

According to the equivalency theorem,

Thus, the function (1.58) represents the reflected field in the region z > 0.

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20 Chapter 1 Basic Notions in Acoustic and Electromagnetic Diffraction Problems

The field (1.54) scattered by the hard infinite plane S also can be represented as the sum of the reflected field and the shadow radiation:

The shadow radiation does not depend on the boundary conditions and is the same both for the soft and hard planes. It is defined by Equation (1.63).

The above definitions of the reflected and shadow radiations are applicable to the PO field scattered by arbitrary soft and hard objects. In this case, however, the integration surface in (1.63), (1.66), and in (1.68) must be specified as the illuminated side (Sil) of the object (Fig. 1.4). Thus, in general,

where

and us,hrefl, ∂us,hrefl/∂n are defined by Equations (1.64) and (1.69). Equation (1.72) can be interpreted as the generalization of the Kirchhoff definition for the black bodies

suggested earlier by Ufimtsev (1968, 2003). It is clear that the PO formulation in the form of Equation (1.70) is valid for electromagnetic waves as well.

One should also notice another interesting relationship between the PO field and the shadow radiation. According to the definitions (1.31), (1.32) and (1.72) for these quantities, the shadow radiation can be represented in the form

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If we now take into account Equation (1.37) and substitute it into (1.73), we immediately come to the fundamental conclusion that the shadow radiation exactly equals zero in the direction to the source of the incident wave. In other words, the black bodies (as they are defined above) do not generate the backscattering.

In contrast to the infinite plane problem, where the reflected field and shadow radiations exist in the separated half-spaces, the fields (1.71) and (1.72) caused by diffraction at the finite size objects exist in the whole surrounding space. However, their spatial distributions are different. The reflected field dominates in the ray region, and the shadow radiation concentrates at the shadow region and in its vicinity (Ufimtsev, 1968, 1990, 1996, 2003). Well-known manifestations of the shadow radiation are the phenomena Fresnel diffraction and forward scattering (Glaser, 1985; Ufimtsev, 1996; Willis, 1991).

Indeed, the diffraction bands bordering the geometrical optics shadow of opaque objects observed in Fresnel diffraction are nothing but the result of interference of an incident wave with shadow radiation. It is interesting that the history of diffraction as a science started in the seventeenth century with investigation of just this phenomenon (Grimaldi, Newton,Young, Fresnel). The forward scattering is the enhancement of the scattered field in the directions approaching the shadow boundary behind the object. It was extensively investigated both experimentally (Willis, 1991) and theoretically [Bowman et al. (1987)]. The numerical data for the field scattered by acoustically soft and hard objects, as well as perfectly conducting objects, presented in the work of Bowman et al. (1987), clearly illustrate the existence of this phenomenon. Our present analysis of the PO approximation reveals the nature of this phenomenon, which is inherent for scattering at any large opaque objects.

One should note that, due to transverse diffusion, the shadow radiation can penetrate far from the shadow region (Ufimtsev, 1990); see also Section 14.2 in the present book. Also, it gives origin to the edge waves, creeping waves, and surface diffracted rays (Ufimtsev, 1996, 2003).

1.3.5 Shadow Contour Theorem and the

Total Scattering Cross-Section

Among the properties of shadow radiation, the most significant are the Shadow Contour Theorem and the Total Power of Shadow Radiation. They have already been established for electromagnetic waves (Ufimtsev, 1968, 1990, 1996, 2003) and will now be verified for acoustic waves.

Let us compare the shadow radiation generated by two scattering objects with different shapes, but with the same shadow contour (Fig. 1.10). Their illuminated sides are S1 and S2.

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