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282 Chapter 13 Backscattering at a Finite-Length Cylinder

and

Ex(1)right = E0x

a

 

eikR

2π

0sin ψ Fθ(1)(ψ , θ , φ) · θx

 

 

 

 

e−i2kl cos ϑ

 

 

2π

R

π

 

 

 

(1)

 

(1)

i2ka sin ϑ sin ψ

+ cos ϑ cos ψ [Gθ (ψ , θ ,

φ) · θx + Gφ (ψ , θ , φ) · φx ]1e

dψ ,

(13.60)

Ey(1,z)right = Hx(1)right = 0.

 

 

(13.61)

Equations (13.59) and (13.61) show that no cross-polarization takes place in this problem, due to its symmetry with respect to the plane x = 0. In the direction ϑ = π − 0, which is the focal line for EEWs, these equations predict the field

Ex(1)left = E0x

2

f (1)

 

2

,

2

,

2

− g(1)

2

,

2

,

2

 

 

R

e−i2kl ,

(13.62)

 

a

 

 

π

 

π

 

3π

 

π

 

 

π

 

3π

eikR

 

 

 

 

f (1)

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Ex(1)right = E0x

4

0, 0,

2

 

− g(1) 0, 0,

 

2

 

 

R

ei2kl .

(13.63)

 

a

 

 

 

 

 

3π

 

 

3π

 

 

 

eikR

 

 

 

 

The above equations allow the complete calculation of the total scattered field (13.45). They involve elementary functions, Bessel functions, and the one-dimensional integrals (13.58) and (13.60), which can be calculated numerically. However, one can avoid this direct integration by introducing approximations similar to (13.22), (13.23) and (13.26), (13.27).

The idea of these approximations is as follows. Away from the focal line (ka sin ϑ 1), the asymptotic evaluation of the integrals (13.58) and (13.60) leads to the following ray asymptotics:

Ex(1)left

a

1

 

 

[ f (1)(1)ei2ka sin ϑ −iπ/4

 

= E0x

 

 

 

 

√

 

 

 

 

2

 

 

 

π ka sin ϑ

 

 

+ f (1)(2)e−i2ka sin θ +iπ/4]

eikR

ei2kl cos ϑ

(13.64)

 

R

and

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

(1)right

= E0x

a

 

f (1)(3)

e−i2ka sin ϑ +iπ/4 eikR

e−i2kl cos ϑ .

 

Ex

 

 

√

 

 

 

 

(13.65)

2

 

R

 

π ka sin ϑ

 

These are the electromagnetic versions of the acoustic asymptotics (13.18) and (13.25). They reveal again the equivalence relationships existing between the acoustic and electromagnetic diffracted rays,

Ex(1) = us(1),

if u0 = E0x .

(13.66)

As shown above, the focal asymptotics (13.17) and (13.62) for acoustic and electromagnetic waves are different. However, they are small quantities of the order

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13.2 Electromagnetic Waves 283

(ka)−1, compared to the basic component E0(0x)disk scattered from the left base (disk). Therefore, the approximation (13.28) derived for acoustic waves can also be used for calculation of electromagnetic waves, but with the relative error of the order (ka)−1.

In this case, it is just sufficient to replace in Equation (13.28) the quantity u0 by E0x and us(1) by Ex(1).

13.2.2 H-Polarization

The incident wave

Hxinc = H0x eik(z cos γ +y sin γ ),

Hyinc,z = Exinc = 0

(13.67)

generates the uniform currents

 

 

jy(0)disk = −2H0x e−ikl cos γ eikρ sin γ sin ψ ,

jx(0)disk jz(0)disk = 0

(13.68)

on the left base of the cylinder (Fig. 13.1), and

 

 

jz(0)cyl = −2H0x sin ψ eik(z cos γ +a sin γ sin ψ ),

jx(0)cyl = jy(0)cyl = 0

(13.69)

on its cylindrical part (−l ≤ z ≤ l, π ≤ ψ ≤ 2π ). One can show that these currents radiate the field

Hx(0) = uh(0)

(13.70)

under the condition u0 = H0x , where function uh(0) is determined in Section 13.1.1 and represents the acoustic field scattered at a rigid cylinder. Therefore, the PO curves in Figures 13.5 and 13.7 for the backscattering of acoustic waves from a rigid cylinder also display the backscattering of electromagnetic waves from a perfectly

conducting cylinder.

 

 

 

 

 

 

 

 

 

 

 

The nonuniform

 

currents induced

in the vicinity

of the left (z

= −

l) and

 

 

 

 

(1)

(1)left

(1)right

 

 

right (z = l) edges radiate the field Hx

= Hx

+ Hx

 

, which is determined

according to Section 7.8 as

 

 

 

 

 

 

 

a eikR

 

 

 

 

 

 

Hx(1)left = H0x

 

 

 

ei2kl cos ϑ

 

 

 

 

 

 

2

R

 

 

 

 

 

 

 

2π

{sin ψ [Gθ(1)(ψ , θ , φ) · φx − Gφ(1)(ψ , θ , φ) · θx ]

 

 

×

0

 

 

 

− cos ϑ cos ψ Fθ(1)(ψ , θ , φ) · φx }ei2ka sin ϑ sin ψ dψ ,

(13.71)

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284 Chapter 13

Backscattering at a Finite-Length Cylinder

 

and

 

 

 

 

 

 

 

 

Hx(1)right

 

 

a eikR

 

 

= H0x

 

 

 

e−i2kl cos ϑ

 

 

2

R

 

 

 

 

2π

5 sin ψ [Gθ(1)(ψ , θ , φ) · φx − Gφ(1)(ψ , θ , φ) · θx ]

 

 

×

π

 

 

− cos ϑ cos ψ Fθ(1)(ψ , θ , φ) · φx

6ei2ka sin ϑ sin ψ dψ .

(13.72)

Due to the symmetry of this problem, the cross-polarized component of the backscattered field equals zero, Hy(,1z) = 0. The local spherical coordinates θ , φ of EEWs involved in these integrals are defined in Section 13.2.1.

For the observation point on the focal line (ϑ = π − 0 ), these integrals are

calculated in the closed form:

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

a

g(1)

π

 

π

 

3π

 

− f (1)

π

 

π

 

3π

 

eikR

 

Hx(1)left = H0x

 

 

,

 

,

 

 

 

,

 

,

 

 

 

e−i2kl

(13.73)

2

2

2

2

2

2

2

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

R

 

and

a

g(1) 0, 0,

3π

0, 0,

3π

 

Hx(1)right = H0x

 

 

− f (1)

 

 

 

4

2

2

The ray asymptotics of the field Hx(1) are given by

 

 

 

a

1

 

 

 

 

)

Hx(1)left = H0x 2 √ g(1)(1)ei2ka sin

π ka sin ϑ

eikR

ei2kl . (13.74)

R

ϑ −iπ/4

 

 

 

 

+ g(1)(2)e−i2ka sin ϑ +iπ/4*

ikR

 

 

 

 

 

 

 

 

e

 

ei2kl cos ϑ

(13.75)

 

 

R

 

and

 

 

 

 

 

 

 

 

 

Hx(1)right = H0x

a

1

 

 

eikR

 

 

 

√

 

g(1)(3)e−i2ka sin ϑ +iπ/4

 

 

e−i2kl cos ϑ .

(13.76)

2

 

R

 

π ka sin ϑ

 

These asymptotics are identical to those found for the acoustic waves uh(1) in Section 13.1.2.

For electromagnetic waves with this polarization, one can also suggest the approximation similar to Equation (13.29), which follows from Equation (13.29) with the replacements u0 → H0x and uh(1) → Hx(1). The expression obtained in this way exactly transforms into the ray aymptotics (13.75) and (13.76), but it leads to the approximations for the total scattered field, which possess the relative error of the order (ka)−1 in the vicinity of the focal line.

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(0)disk mation E

Problems 285

PROBLEMS

13.1Use Equations (13.20) and (13.21) and verify the transition from the focal (13.17) and ray (13.18), (13.19) asymptotics to the uniform approximations (13.22) and (13.23).

13.2Verify Equation (13.43), use Equations (1.92) and (1.93), and derive the PO approxi-

for the field backscattered from the left base (disk) of a finite cylinder. Prove that Ey(0,z)disk = 0. Compare the results with the acoustic field us(0)disk scat-

tered from a soft cylinder (see Section 13.1.1). Establish the equivalence relationships between acoustic and electromagnetic diffracted waves.

13.3Verify that the approximations (13.22) and (13.23) exactly transform into the focal asymptotics (13.17) and asymptotically reduce to the ray asymptotics (13.18), (13.19).

13.4Verify the approximations (13.47) for the distance between the diffraction point on the edge and the observation point in the far zone.

13.5Use Equation (13.50) and verify the expressions (13.51) and (13.52) for the unit vectors

ˆ, ˆ .

θφ

13.6Derive Equations (13.54) and (13.55) for local coordinates φ, φ0 of EEWs diverging from the left edge (z = −l).

13.7Derive Equations (13.56) and (13.57) for local coordinates φ, φ0 of EEWs diverging from the right edge (z = l).

13.8Use the general Equations (7.130) and (7.131) for EEWs and derive Equation (13.58) for the field scattered by the left edge.

13.9Use the general Equations (7.130) and (7.131) for EEWs and derive Equation (13.60) for the field scattered by the right edge.

13.10 Use the definitions of the quantities F(1)

, G(1)

and prove that

θ

θ ,φ

 

(1) − = (1) (1) − = − (1) (1) − = (1)

Fθ (π ψ ) Fθ (ψ ), Gθ (π ψ ) Gθ (ψ ), Gφ (π ψ ) Gφ (ψ ).

Then, use these relations and prove Equation (13.59), which indicates the absence of cross-polarization in the field backscattered from a finite cylinder.

13.11Apply the stationary-phase method to the integrals (13.58) and (13.60) and obtain the ray asymptotics (13.64) and (13.65) for the waves diffracted at the left and right edges. Compare them with similar expressions for acoustic waves (see Section 13.1.2). Establish the relationships between the acoustic and electromagnetic edge-diffracted rays.

13.12Compute and graph the normalized backscattering cross-section for a soft cylinder with parameters d = 2a = 2λ, L = 2l = 4λ. Set γ = 45◦. Follow Section 13.1.3.

13.13Verify Equation (13.68), use Equations (1.92) and (1.93), and derive the field Hx(0)disk backscattered from the left base of a cylinder. Compare the result with the acoustic field (13.3). Formulate the relationships between acoustic and electromagnetic waves.

13.14Verify Equation (13.69), use Equations (1.92) and (1.93), and derive the integral

(0)cyl (0)cyl = expressions for the field H backscattered from a cylinder. Prove that Hy,z 0.

Compare Hx(0)cyl with the acoustic field (13.4). Formulate the relationships between acoustic and electromagnetic waves.

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286 Chapter 13 Backscattering at a Finite-Length Cylinder

13.15Use the general Equations (7.130) and (7.131) for EEWs and derive Equation (13.71) for the field scattered by the left edge.

13.16Use the general Equations (7.130) and (7.131) for EEWs and derive Equation (13.72) for the field scattered by the right edge.

13.17Use Equations (7.130) and (7.131) and derive the integral expressions for the field components Hy(,1z)left similar to Equation (13.71). Prove that Hy(,1z)left = 0 in the backscattering direction.

13.18Use the general Equations (13.71) and (13.72) and derive the focal asymptotics (13.73) and (13.74).

13.19Apply the stationary-phase method to the integral (13.71) and derive the ray asymptotic (13.75). Compare it with Equation (13.19) and formulate the equivalence relationships between acoustic and electromagnetic edge-diffracted rays.

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Chapter 14

Bistatic Scattering at a

Finite-Length Cylinder

14.1 ACOUSTIC WAVES

The geometry of the problem is shown in Figure 14.1. The diameter of the cylinder is d = 2a, and its length is L = 2l. The incident wave is given by

uinc = u0eik(y sin γ +z cos γ ),

0 < γ < π/2.

(14.1)

The scattered field is evaluated in the plane y0z (ϕ = π/2 and ϕ = 3π/2). It is convenient to indicate the scattering direction by the angle (0 ≤ ≤ 2π ),

ϑ ,

 

ϑ ,

if ϕ

=

π/2

(14.2)

= 2π

−

if ϕ

3π/2,

 

 

 

=

 

 

where ϑ (0 ≤ ϑ ≤ π ) is the ordinary spherical coordinate of the field point (R, ϑ , ϕ). One should not confuse this angle with the local angle θ used for description of an EEW diverging from the diffraction point ζ at the edge. The relevant local coordinates r, θ , φ were introduced above in Section 13.2.1.

14.1.1PO Approximation

The incident wave (14.1) generates the uniform component js,h(0) of the scattering sources only on the left base (disk) and on the lower lateral part (π ≤ ψ ≤ 2π ) of the cylindrical surface. The scattered field is determined by Equation (1.32). We omit all intermediate calculations and obtain the final expressions for this field

us,h(0) = u0 s,h(0)( , γ )

eikR

,

(14.3)

R

Fundamentals of the Physical Theory of Diffraction. By Pyotr Ya. Ufimtsev

Copyright © 2007 John Wiley & Sons, Inc.

287

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