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

4.1 Integrals and Asymptotics 75

face ϕ = α):

One should note that the functions f (1) and g(1) are singular in the two special directions

which relate to the grazing reflections from the faces under the grazing incidence. This is a special case when the integrand in (4.9) cannot be expanded into the Taylor

Figure 4.1 Directivity patterns of edge waves radiated by nonuniform components of the surface sources. The function f (1)(g(1)) corresponds to the case of the acoustically soft (hard) wedge; they also describe the Ez (Hz ) component of the electromagnetic wave scattered at the perfectly conducting wedge. Reprinted from Ufimtsev (1957) with permission of Zhurnal Tekhnicheskoi Fiziki.

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4.2 Integral form of Functions f (1) and g(1) 77

According to Equations (2.61) and (2.63), the functions Js(1) and Jh(1) decrease as (kξ )−3/2 and (kξ )−1/2, respectively, with increasing distance ξ from the edge. At a certain distance ξ = ξeffective, these functions are sufficiently small and can be approx-

imated by zero for ξ ≥ ξeff . In the far zone, where r kξeff2 , the Hankel function in Equations (4.25) and (4.26) can be replaced by its asymptotics (2.29). This leads to

the asymptotic expressions

These expressions describe the field radiated from the face ϕ = 0. By the replacement of ϕ by α − ϕ and ϕ0 by α − ϕ0, one can find the field radiated from the face ϕ = α. The total field created by both faces must be equal to that of Equations (4.12) and (4.13). By equating them we obtain the useful relationships

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78 Chapter 4 Radiation by the Nonuniform Component of Surface Sources

These expressions show that the functions f (1) and g(1) can also be interpreted as the directivity patterns of elementary diffracted waves generated (in the plane normal to the edge) by the sources distributed at the wedge along the lines normal to the edge.

4.3 OBLIQUE INCIDENCE OF A PLANE WAVE AT A WEDGE

For an oblique incidence, the relationship us(t) exists for the diffracted rays generated by the total surface currents, j(t) = j(0) + j(1). However, us(0,1) is not equal to Ez(0,1),

because of the polarization coupling in the electromagnetic PO field.

The complete equivalence uh(0,1,t) = Hz(0,1,t) exists between the acoustic and electromagnetic diffracted rays.

Here, we extend the results of Section 4.1 to the general case when the incident wave propagates under the oblique direction to the edge (Fig. 4.3). It is given by the equation

with 0 < γ ≤ π/2. We use the “tilde-hat” for the angle α˜ to avoid possible confusion with the external angle α of the wedge.

The boundary conditions on the wedge faces are shown in Equations (2.2) and (2.3). To satisfy these conditions, the diffracted field must have the same dependence on the coordinate z as the incident wave (4.31):

The substitution of this function ud into the wave equation (2.4) leads to the equation for the function u(r, ϕ),

Figure 4.3 Oblique incidence of a plane wave at a wedge.

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4.3 Oblique Incidence of a Plane Wave at a Wedge 79

where the Laplacian operator is defined by Equation (2.5). It is also expedient to represent the incident wave (4.31) in the form of Equation (3.1):

Thus, we reduced the three-dimensional (3-D) diffraction problem for the oblique incidence to the 2-D problem for the normal incidence (γ = π/2) considered in Chapter 2. The solution for the oblique incidence can be automatically found by

simple replacements in the solution for the normal incidence. Namely, the quantity u0 should be replaced by u0eikz cos γ , the wave number k by k1 = k sin γ , and the angle

ϕ0 by ϕ0 = arctan(cos β/ cos α)˜ .

This rule has been established here for the exact solution of the wedge diffraction problem and for its asymptotics. One can show that it is also valid for the PO part of the field. First, by the substitution of the incident wave (4.34) into Equation (1.31)

Comparison with Equation (3.2) shows that the sources (4.37) and (4.38) satisfy the above rule for the transition from normal to oblique incidence. If the sources of the field satisfy this rule, one can expect the generated field does too. To verify that, we substitute Equations (4.37) and (4.38) into the original expressions (1.32) for the PO field:

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