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

104 Chapter 5 First-Order Diffraction at Strips and Polygonal Cylinders

In the interval π/2 < φ < 5π/6, only two edge waves contribute to the total scattered field, and

with function f (0)(1) defined in Equation (5.92). The function f (0)(2) in this interval differs from Equation (5.93) and equals

because the face 2–3 is not illuminated by the incident wave and does not generate the scattered field (in the framework of the PO approximation). In the specular direction

φ= 2π/3, Equation (5.95) determines the value

which agrees with Equation (5.90).

In the interval 5π/6 < φ < π , again all three edges generate the scattered field, and the directivity pattern is determined by Equation (5.91), but with the different

However, the function f (0)(2) is still defined by Equation (5.96). The function (5.98) differs from (5.92) because in this interval of observation, two faces 1–2 and 1–3 are illuminated. The function (5.99) differs from Equation (5.94) because the different faces of edge 3 are illuminated in the intervals 0 ≤ φ < π/2 and 5π/6 < φ ≤ π .

5.2.3 Symmetric Scattering: First-Order

PTD Approximation

Here, the acoustic quantity s( h) is equivalent to the directivity pattern of the Ez(Hz)-

component of the electromagnetic field

To improve the PO approximation, we include into the scattered field the contributions generated by the nonuniform scattering sources js,h(1). In the first-order approximation, they have the form of edge waves (4.12) and (4.13). When these waves are added to

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the PO edge waves (3.53) and (3.54), one can see that (due to relationships (4.14) and (4.15)), the resulting edge waves are defined by the Sommerfeld asymptotics (2.61) and (2.63). Because the acoustically soft and hard cylinders are perfectly reflecting (nontransparent), only those edge waves contribute to the scattered field, which come from the edges visible from the observation point. We also note that in this section we consider the symmetric case when the incident wave is given by Equation (5.72) and propagates along the bisector of the cylinder (Fig. 5.9). The basic polar coordinates r, φ are used in the following for the description of the scattered field and the angle φ changes in the interval 0 ≤ φ ≤ π .

In accordance with these comments, the directivity patterns of the scattered field

where n = α/π = 5/3 (here α = 5π/3 is the external angle between the faces of the edge) and ψ2,3 is defined in Equation (5.76). The first terms in these functions are singular for the forward direction (φ = 0). However, the singularities of the functions related to the edges 2 and 3 cancel each other, resulting in the expressions

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106 Chapter 5 First-Order Diffraction at Strips and Polygonal Cylinders

Comparison of these equations with Equation (5.81) shows that the first term here relates to the PO field and the last two terms represent the contributions from the nonuniform component js,h(1).

In the interval π/6 < φ < π/2, three edge waves form the total scattered field:

There is the specular direction (φ = π/3) in this sector of observation. In this direction, the second terms in functions f (1), g(1) and f (2), g(2) are singular. However, these singular terms cancel each other, and the scattered field remains finite:

Due to the symmetry of the problem, these equations also describe the field in the direction of specular reflection from the face 1–3. Here the comments presented at the end of Section 5.2.1 are also pertinent. The power of the total reflected field is asymptotically equal to the power of the shadow radiation determined by the first term in Equations (5.104) and (5.105). It is also clear that the dominant term ±ikl/2 in Equations (5.109) and (5.110) relates to the PO contribution.

In the interval π/2 < φ < 5π/6, the edge 3 is invisible and the scattered field

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108 Chapter 5 First-Order Diffraction at Strips and Polygonal Cylinders

In the previous section it was explained that in the first-order PTD approximation, the field scattered by a triangular cylinder is a linear combination of the edge waves defined in general form by Equations (2.61) and (2.63). Now we apply this PTD theory for the investigation of the backscattering, assuming that the incident wave is given by Equation (5.84) with −π ≤ φ0 ≤ 0. The scattered field is found in the direction φ = π + φ0 in the sector 0 ≤ φ ≤ π . The geometry of the problem is shown in Figure 5.9. We omit the simple but tedious calculations of the individual edge waves in terms of the basic coordinates r, φ and present the final expressions for the directivity patterns of the total scattered field.

In the interval 0 ≤ φ ≤ π/6,

Here, the edge parameter equals n = α/π = 5/3, and the quantities ψ2,3 are defined in Equation (5.88). It follows from these equations that for the direction φ = 0,

where h = l cos(π/6).

In the interval π/6 < φ ≤ π/2, the additional wave appears incoming from