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

304 Chapter 14 Bistatic Scattering at a Finite-Length Cylinder

term shows the contribution by the sources js,h(1)sm caused by the smooth bending of the cylindrical surface.

Notice that the primary edge waves propagating along the cylinder base from point 1 (Fig. 14.1) to point 2 undergo edge diffraction and generate the secondary diffracted rays in the region 3π/2 < ≤ 2π . These secondary rays can be calculated by application of the theory developed in Chapter 10. Their contributions to the field in the specular direction are of the order (ka)−3/2 for the soft cylinder and of the order (ka)−1 for the hard cylinder. They are small compared to the field (14.59), (14.60) and can be neglected.

The counterpart of the present theory developed for electromagnetic waves scattered at a perfectly conducting cylinder of finite size has been published in the paper by Ufimtsev (1981) and is considered below in Section 14.2.3.

14.2ELECTROMAGNETIC WAVES

14.2.1E -Polarization

On the surface of a perfectly conducting cylinder (Fig. 14.1), the incident wave

generates the uniform currents given by Equations (13.43) and (13.44). The PO field radiated by these currents is calculated according to Equations (1.92) and (1.93). In the plane y0z (ϕ = π/2 and ϕ = 3π/2), this field is described by the expressions totally identical to Equations (14.3) and (14.4) derived above for acoustic waves:

In particular, this relationship means that the PO curves in Figures 14.2 and 14.3 for the acoustic waves scattered from a soft cylinder also demonstrate the electromagnetic waves with E-polarization scattered from a perfectly conducting cylinder of the same size.

(1)

The field radiated by the nonuniform current j is calculated according to Section 7.8 as

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14.2 Electromagnetic Waves 305

describes the field scattered from the left edge, and the function

× [G(θ1)right (ψ , θ , φ) · θx + G(φ1)right (ψ , θ , φ) · φx ]}eip sin ψ dψ (14.65)

represents the field scattered from the right edge. Here, p = ka(sin γ − sin ), q = kl(cos − cos γ ). In addition, Equations (14.15) to (14.18) define the local angles

are different.

Notice also that in the case of scattering at the disk, the expression similar to Equation (14.66) (but with the wedge angle equal to α = 2π ) exactly transforms into

are complete elliptic integrals (Gradshteyn and Ryzhik, 1994).

In order to simplify the comparative analysis of the electromagnetic and acoustic waves scattered at a finite cylinder, let us represent the total scattered field in the form

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306 Chapter 14 Bistatic Scattering at a Finite-Length Cylinder

similar to Equation (14.20) for acoustic waves. According to Equation (14.62), all of the beam and ray asymptotics of the electromagnetic field Ex(0) are identical to those presented in Section 14.1.4 for the acoustic field us(0):

The ray asymptotics of the field Ex(1) are also identical to those of the acoustic field us(1) shown in Section 14.1.4:

(ex1)( , γ ) = (s1)( , γ ). (14.72)

However, the beam asymptotics associated with the field generated by the nonuniform (fringe) components are different for electromagnetic and acoustic waves:

(ex1)beam( , γ ) = frs .beam( , γ ). (14.73)

14.2.2H -Polarization

On the surface of a perfectly conducting cylinder (Fig. 14.1), the incident wave

generates currents with the uniform components (13.68) and (13.69). The field radiated by these currents is calculated according to Equations (1.92) and (1.93). In the plane y0z (ϕ = π/2 and ϕ = 3π/2), it is described by the expressions identical to Equations (14.3) and (14.5) for acoustic waves:

In particular, this relationship means that the PO curves in Figures 14.4, 14.5, 14.7, 14.8, and 14.9 plotted for the acoustic waves scattered from a rigid cylinder also demonstrate the electromagnetic waves with Hx -polarization scattered from a perfectly conducting cylinder of the same size.

(1)

The field radiated by the nonuniform component of the current (j ) is calculated according to Section 7.8. The expression

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14.2 Electromagnetic Waves 307

represents the field scattered from the right edge. Here, p = ka(sin γ − sin ) and q = kl(cos − cos γ ). In addition, Equations (14.15) to (14.18) define the local

We emphasize again that the local angles φ, φ0 in (14.78) and (14.79) are different. Notice also that in the case of scattering at the disk, the expression similar to Equation (14.78) (but with the wedge angle equal to α = 2π ) exactly transforms into

the known function

derived earlier by Ufimtsev (1962) and Butorin et al. (1988). Here, functions E(π/2, x) and K(π/2, x) are the complete elliptic integrals (14.69).

To continue the comparative analysis of electromagnetic and acoustic waves scattered at a finite cylinder, it is convenient to represent the total scattered field in the form

similar to Equation (14.20) for acoustic waves. Due to Equation (14.75), all of the beam and ray asymptotics for the electromagnetic field Hx(0) are the same as those for the acoustic field uh(0) (presented in Section 14.1.4):

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308 Chapter 14 Bistatic Scattering at a Finite-Length Cylinder

It turns out that the ray asymptotics for the electromagnetic field Hx(1) and those (found in Section 14.1.4) for the acoustic field uh(1) are also identical:

However, the beam asymptotics for the fields Hx(1) and uh(1) generated by the nonuniform (fringe) components j(1) (in the scattering directions = γ , = π − γ , and = 2π − γ ) are different,

although they are of the same order of magnitude. Besides, the examination of Figures 14.7 and 14.9 reveals the following situations:

•Already in the case of the cylinder with diameter d = λ and length L = 3λ,

the quantity (h1)beam is about 18 dB less compared to the PO beams (h0)beam in the directions = γ , = π − γ , and about 25 dB less in the direction

= 2π − γ .

•In the case of the cylinder with diameter d = 3λ and length L = 9λ, the

quantity (h1)beam is about 28 dB less compared to the PO beams (h0)beam in the directions = γ , = π − γ , and about 35 dB less in the direction

= 2π − γ .

These observations clearly show that, for cylinders of such size and larger, the difference between acoustic and electromagnetic scattering (in the plane y0z (Fig. 14.1, and for electromagnetic waves with Ex - or Hx -polarization) is practically negligible.

14.2.3 Refined Asymptotics for the Specular Beam Reflected from the Lateral Surface

This section is the electromagnetic version of Section 14.1.5. It studies the scattered field in the vicinity of the specular direction = 2π − γ in the plane ϕ = 3π/2. This study is based on the paper by Ufimtsev (1981).

According to Sections 14.2.1 and 14.2.2, the following asymptotic relationships exist between the electromagnetic and acoustic scattered waves:

They are valid under the conditions ka sin γ 1 and kl 1. Here the fields with the superscript “0” are generated by the uniform components of the surface scattering sources ( j(0)) and represent the PO fields. The fields with the superscript “(1)fr” are generated by the nonuniform/fringe components of the surface scattering sources

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