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

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

5.2.5Numerical Analysis of the Scattered Field

Here, the scattering cross-section σs(σh) for acoustic waves is equal to the scattering

cross-section for electromagnetic waves with the component Ez(Hz).

Numerical calculations were performed for the normalized scattering cross-section (5.20)

in the decibel scale for the equilateral cylinder with the parameter kl = 6π when l = 3λ. The results are plotted in Figures 5.11 and 5.12 for symmetric scattering (when the incident wave propagates in the direction parallel to the cylinder bisector; Fig. 5.9) and in Figures 5.13 and 5.14 for backscattering.

As is seen in Figures 5.11 and 5.12, PTD significantly improves the PO approximation at the minima of the scattering cross-section. The difference between the PTD and PO data is also appreciable at maxima. This difference is pronounced for the acoustically hard cylinder and it reaches about 6–9 db in the sector 160◦–180◦. A similar situation is observed for the backscattering in Figures 5.13 and 5.14. In particular,

Figure 5.11 Bistatic scattering at an acoustically soft cylinder (Ez -polarization of the electromagnetic wave).

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Figure 5.12 Bistatic scattering at an acoustically hard cylinder (Hz -polarization of the electromagnetic wave).

the difference between the PTD and PO curves for the acoustically hard cylinder is about 5–9 db in the directions 50◦–70◦ and 170◦–180◦.

Notice that more accurate PTD results for triangular cylinders are presented in the paper by Johanson (1996), where the second-order edge waves are partially taken into account.

Figure 5.13 Backscattering at an acoustically soft cylinder (Ez -polarization of the electromagnetic wave).

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

Figure 5.14 Backscattering at an acoustically hard cylinder (Hz -polarization of the electromagnetic wave).

PROBLEMS

5.1Derive the PO approximation for the field (in the far zone) scattered at a soft strip as shown in Figure 5.1. The incident wave is given by Equation (5.1). Start with the general expressions (1.32), use Equation (3.7) for the Hankel function, apply its asymptotic form (2.29). Represent the scattered field in terms of edge waves and compare the result with Equation (5.4). Verify that they are identical.

5.2Derive the PO approximation for the field (in the far zone) scattered at a hard strip as shown in Figure 5.1. The incident wave is given by Equation (5.1). Start with the general expressions (1.32), use Equation (3.7) for the Hankel function, apply its asymptotic form (2.29). Represent the scattered field in terms of edge waves and compare the result with Equation (5.5). Verify that they are identical.

5.3Derive the PO approximation for the field (in the far zone) scattered at a perfectly conducting strip as shown in Figure 5.1. The incident wave is given as

Ezinc = E0zeik(x cos ϕ0+y sin ϕ0)

Start with the general expressions (1.87) and (1.97). Use Equation (3.7) for the Hankel function, apply its asymptotic form (2.29). Represent the scattered field in terms of edge waves and compare the result with Equation (5.4). Formulate the relationship between acoustic and electromagnetic diffracted waves.

5.4Derive the PO approximation for the field (in the far zone) scattered at a perfectly conducting strip as shown in Figure 5.1. The incident wave is given as

Hzinc = H0zeik(x cos ϕ0+y sin ϕ0).

Start with the general expressions (1.88) and (1.97). Use Equation (3.7) for the Hankel function, apply its asymptotic form (2.29). Represent the scattered field in terms of edge

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Problems 113

waves and compare the result with Equation (5.5). Formulate the relationship between acoustic and electromagnetic diffracted waves.

5.5Use Equation (5.35). Prove Equation (5.39). Apply Equation (5.16) and calculate the total cross-section of a soft strip.

5.6Use Equation (5.37). Prove Equation (5.40). Apply Equation (5.16) and calculate the total cross-section of a hard strip.

5.7Investigate the diffraction of a cylindrical wave at a soft strip (Fig. 5.4). Related coordinates are shown in Figure 5.1. The incident wave is given by

Derive the PTD approximation for the scattered field as a sum of two separate components: the PO field us(0) and the fringe field us(1). Start the calculation of the PO field with the general expressions (1.32), use Equation (3.7) for the Hankel function, apply its asymptotic form (2.29). Represent the far field us(0) in integral form. Use Equation (4.12) and write the asymptotic expressions for the edge waves generated by the nonuniform/fringe sources js(1). Verify that these waves are finite at the shadow boundaries. Retrace the transition (in the field expressions) from the cylindrical wave excitation to the incident plane wave by setting

5.8Investigate the diffraction of a cylindrical wave at a hard strip (Fig. 5.4). Related coordinates are shown in Figure 5.1. The incident wave is given by

Derive the PTD approximation for the scattered field as a sum of two separate components: the PO field uh(0) and the fringe field uh(1). Start the calculation of the PO field with the general expressions (1.32), use Equation (3.7) for the Hankel function, apply its asymptotic form (2.29). Represent the far field uh(0) in integral form. Use Equation (4.13) and write the asymptotic expressions for the edge waves generated by the nonuniform/fringe sources jh(1). Verify that these waves are finite at the shadow boundaries. Retrace the transition (in the field expressions) from the cylindrical wave excitation to the incident plane wave by setting

5.9Investigate the diffraction of a cylindrical wave at a perfectly conducting strip (Fig. 5.4). Related coordinates are shown in Figure 5.1. The incident wave is given by

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

Derive the PTD approximation for the scattered field as a sum of two separate components: the PO field Ez(0) and the fringe field Ez(1) . Start the calculation of the PO field with the general expression (1.87), use Equation (3.7) for the Hankel function, apply its asymptotic form (2.29). Represent the far field Ez(0) in integral form. Use Equation (4.12) (adapted for electromagnetic waves) and write the asymptotic expressions for the field

Ez(1). Verify that these waves are finite at the shadow boundaries. Retrace the transition (in the field expressions) from the cylindrical wave excitation to the incident plane wave by setting

5.10Investigate the diffraction of a cylindrical wave at a perfectly conducting strip (Fig. 5.4). Related coordinates are shown in Figure 5.1. The incident wave is given by

Derive the PTD approximation for the scattered field as a sum of two separate components: the PO field Hz(0) and the fringe field Hz(1). Start the calculation of the PO field with the general expression (1.88), use Equation (3.7) for the Hankel function, apply its asymptotic form (2.29). Represent the far field Hz(0) in integral form. Use Equation (4.13) (adapted for electromagnetic waves) and write the asymptotic expressions for the field Hz(1). Verify that these waves are finite at the shadow boundaries. Retrace the transition (in the field expressions) from the cylindrical wave excitation to the incident plane wave by setting

eikr0

lim H0z √ = eikx . x0→−∞ kr0

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116 Chapter 6 Axially Symmetric Scattering of Acoustic Waves at Bodies of Revolution

Figure 6.1 A body of revolution (solid lines) with a circular edge and a conic surface (dashed lines) tangential to a body at the edge points.

from the illuminated side of the conic surface; and α − ϕ0 the angle of incidence measured from the shadowed side; the meaning of the angles ω and is clear; the edge points 1 and 2 are symmetrical.

Figure 6.2 Cross-section of the canonical conic surface.

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