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

292 Chapter 14 Bistatic Scattering at a Finite-Length Cylinder

Figure 14.5 Scattering at a hard cylinder. According to Equation (14.75), this figure also demonstrates the PO approximation for electromagnetic waves (with Hx -polarization) scattered from a perfectly conducting cylinder.

to the left and right edges, respectively. Only half of the right edge is illuminated by the incident wave (14.1), which is why the integration limits in Equation (14.13) are

π and 2π .

Functions Fh(1)left,right (ψ , θ , φ) are described by Equations (7.91) and (7.92). Section 13.2.1 shows how one defines the local angles γ0, θ , φ, and φ0 for the backscattering direction ϑ = π − γ . In the same way one can introduce these angles

for arbitrary scattering directions in the y0z plane. The relationships
		cos γ0 = sin γ cos ψ ,		cos θ = − sin cos ψ				(14.14)
are valid for both edges. For the left edge one should use the expressions
sin φ0 =		cos γ		cos φ0 =		sin γ sin ψ
			,				,	(14.15)
	'				'
		1 − sin2 γ cos2 ψ				1 − sin2 γ cos2 ψ

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14.1 Acoustic Waves 293

Figure 14.6 Shadow radiation as a part of the PO field.

sin φ = −

cos

cos φ = −

sin sin ψ

,

;

(14.16)

'

'

1 − sin2 cos2 ψ

1 − sin2 cos2 ψ

however, for the right edge one should use the definitions

sin φ0 = −

sin γ sin ψ

cos φ0 =

cos γ

,

(14.17)

'

'

1 − sin2 γ cos2 ψ

1 − sin2 γ cos2 ψ

and

sin φ =

sin sin ψ

cos φ = −

cos

,

.

(14.18)

'

'

1 − sin2 cos2 ψ

1 − sin2 cos2 ψ

The numerical results found with these expressions for the normalized scattering cross-section (13.14) are presented below for the two cylinders with parameters L = 3d = 3λ and L = 3d = 9λ. The direction of the incident wave (14.1) is given by the angle γ = 45◦. Figures 14.7 and 14.9 demonstrate the individual contributions

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294 Chapter 14 Bistatic Scattering at a Finite-Length Cylinder

Figure 14.7 Bistatic scattering at a hard cylinder. According to Equation (14.75), the PO curve here also demonstrates the scattering of electromagnetic waves (with Hx -polarization) from a perfectly conducting cylinder.

by the PO field and by the field generated by jh(1). The sum of these fields and its comparison with the PO field are shown in Figures 14.8 and 14.10.

These figures clearly show the influence of the field generated by the nonuniform/fringe scattering sources jh(1). In particular, this field fills in the deep minima in the PO field. More accurate PTD approximation can be obtained with calculation of the high-order edge waves. However, in contrast to thin dipoles, thick cylinders are not resonant bodies and all high-order edge waves can be neglected when the size of the cylinder exceeds 3–5 wavelengths. The larger the cylinders, the higher the accuracy of the PTD expressions (14.12) and (14.13).

The following comments explain some details of the numeric calculations:

•The functions Fh(1)left,right are determined using Equations (7.82), (7.87), (7.88), (7.92), and (7.94) through the functions Vt (σ1, φ0) and Vt (σ2, α − φ0), which

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14.1 Acoustic Waves 295

Figure 14.8 Scattering at a rigid cylinder. According to Equation (14.75), the PO curve here also demonstrates the scattering of electromagnetic waves (with Hx -polarization) from a perfectly conducting cylinder.

contain the factors 1/ sin σ1,2. These factors become singular when σ1.2 → 0 or σ1.2 → π . In the case σ1 → 0, the functions Vt remain finite. They can be transformed into the more convenient form

	4			cos				σ1								1
Vt (σ1, φ0) =					3														.	(14.19)
	9 sin2 γ	1		−	4		sin		2 σ1				cos		2φ0	− cos	2σ1
						3					3				3		3

The replacements of σ1 by σ2 and φ0 by α − φ0 in Equation (14.19) lead to the transformed expression for Vt (σ2, α − φ0).

•However, these expressions are still singular when σ1,2 → π . In this case one should calculate the products Vt (σ1, φ0) sin φ and Vt (σ2, α − φ0) sin(α − φ).

They remain finite when σ1,2 → π because the ratios sin φ/ sin σ1 and sin(α − φ)/ sin σ2 are equal to plus or minus unity.

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296 Chapter 14 Bistatic Scattering at a Finite-Length Cylinder

Figure 14.9 Scattering at a rigid cylinder. According to Equation (14.75), the PO curve here also demonstrates the scattering of electromagnetic waves (with Hx -polarization) from a perfectly conducting cylinder.

• The function V (σ2, α − φ0) related to the left edge is singular at the points ψ = 0, π , 2π for the observation direction = γ . This is the grazing singularity mentioned in Equation (4.21). It is removed by the exclusion of a certain vicinity of the singular points from the integral (14.12). This exclusion is done only for that part of the integral (14.12) that contains the function V (σ2, α − φ0). The function V (σ1, ϕ0) is not singular and is integrated in Equation (14.12) over the entire region 0 ≤ ψ ≤ 2π . Notice that the theory of EEWs presented in Section 7.9 (which is free from the grazing singularity) cannot treat the above singularity in the direction = γ , because this theory is applicable only for objects with planar faces.

•Finally we note that in the case σ1 → φ0 or σ2 → α − φ0 one should use

Equation (7.107) for the functions V (φ0, φ0) and Equation (7.109) for the function V (α − φ0, α − φ0).

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14.1 Acoustic Waves 297

Figure 14.10 Scattering at a rigid cylinder. According to Equation (14.75), the PO curve here also demonstrates the scattering of electromagnetic waves (with Hx -polarization) from a perfectly conducting cylinder.

14.1.4Beams and Rays of the Scattered Field

The previous section provides a numerical investigation of the scattered field. Here we consider its physical structure and present simple high-frequency asymptotics for the directivity pattern ( , γ ) defined by the equation

usc = u0 ( , γ )	eikR
	R .	(14.20)

From the physical point of view, the scattered field consists of the following basic components:

•The reflected beam in the vicinity of the direction = π − γ . This beam appears due to the transverse diffusion of the field in ordinary rays reflected from the left base of the cylinder (Fig. 14.1). It is described by the first terms in Equations (14.4) and (14.5), which contain the Bessel function J1( p). Exactly

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298 Chapter 14 Bistatic Scattering at a Finite-Length Cylinder

in this direction its value equals

					ika2
	beam 1	= −	beam 1	=			cos γ e−i2kl cos γ .	(14.21)
					2
	s		h

•The reflected beam in the vicinity of the direction = 2π − γ . It appears due to the transverse diffusion of the field in ordinary rays reflected from the lower lateral part of the cylinder. This beam is described by the high-frequency asymptotics for the second terms in Equations (14.4) and (14.5):

sbeam 2	= ikal sin γ	sin q			2				e−ipeiπ/4,
			q			π p
hbeam 2	= ikal sin		sin q		2				e−ipeiπ/4.
			q				π p

Exactly in the direction = 2π − γ its value equals

sbeam 2 = − hbeam 2 = l	ka sin γ	e−i2ka sin γ ei3π/4.
	π

We note that p = ka(sin γ − sin ) and q = kl(cos − cos γ ).

(14.22)

(14.23)

(14.24)

•The beam of the shadow radiation in the vicinity of the direction = γ . It is described by both terms in Equations (14.4) and (14.5). Exactly in this direction its value equals

sshad.beam = hshad.beam =	ika2	cos γ +	i2kal	sin γ .	(14.25)
	2		π

•The beam of edge-diffracted rays generated by the fringe scattering sources, which are located near the left edge. It propagates in the direction = π − γ and supplements the reflected beam (14.21). This beam is described by Equa-

tion (14.12). In the case of the soft cylinder, the corresponding expression

follows from Equation (14.12) with the obvious replacement of Fh(1) by Fs(1). Exactly in the direction = π − γ , it is determined by

			a	2π
fr.beam 1	=			e−i2kl cos γ	f (1)left (ψ , π	−	γ )dψ ,	(14.26)
s			2π	0
			a	2π
fr.beam 1	=			e−i2kl cos γ	g(1)left (ψ , π	−	γ )dψ .	(14.27)
h		2π		0

•The beam of edge-diffracted rays generated by the fringe scattering sources, which are located near both edges. It propagates in the direction = γ and

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14.1 Acoustic Waves 299

supplements the shadow beam. This beam is described by Equation (14.11). Exactly in this direction it is described by

	a		2π	2π	f (1)right (ψ , γ )dψ ,
sfr.shad.beam =				f (1)left (ψ , γ )dψ + π
	2π		0
					(14.28)
	a		2π	2π	g(1)right (ψ , γ )dψ .
hfr.shad.beam =				g(1)left (ψ , γ )dψ + π
	2π		0
					(14.29)

To avoid the grazing singularity, one should exclude a certain vicinity of the points ψ = 0, π , 2π in the integral over the left edge.

Away from these beams, the scattered field contains the three edge-diffracted rays generated by the total surface current js,h(0) + js,h(1). They can be determined by the

asymptotic estimation of the field us,h(0) + us,h(1) described by Equations (14.4), (14.5), and (14.11). However, a simpler way is to apply the modified asymptotics (8.12) and (8.13), where one should replace the functions f (1), g(1) with f , g. As a result, one obtains the following expressions for these rays.

• Ray 1:

sray	(1) =		a							f (1)ei(p+q)e iπ/4
		√2π			p
				|			|
and
hray	(1) =		a							g(1)ei(p+q)e iπ/4
		√2π				p
				|				|

(14.30)

(14.31)

propagates from the stationary point 1 (Fig. 14.1) and exists in the regions 0 ≤ < γ , γ < < π − γ , and π − γ < ≤ 3π/2. Factor exp(−iπ/4) is taken for positive values of p and factor exp(+iπ/4) is valid for the negative values of p.

• Ray 2:

sray	(2) =	√		a			f (2)ei(q−p)e±iπ/4	(14.32)
			2π		p	|
	|
and
hray	(2) =	√		a			g(2)ei(q−p)e±iπ/4	(14.33)
				2π	p	|
	|

propagates from the stationary point 2 and exists in the region π/2 ≤ < π − γ , π − γ < < 2π − γ , and 2π − γ < θ ≤ 2π . Factor exp(+iπ/4) is taken for positive values of p and factor exp(−iπ/4) is valid for negative values of p.

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300	Chapter 14 Bistatic Scattering at a Finite-Length Cylinder
•	Ray 3:
	sray(3) =	√		a			f (3)e−i(q+p)e±iπ/4	(14.34)
			2π		p	|
	and	|
	hray(3) =	√		a			g(3)e−i(q+p)e±iπ/4	(14.35)
				2π	p	|
		|

propagates from the stationary point 3 (Fig. 14.1) and exists in the regions 0 ≤ < γ , γ < ≤ π/2, π ≤ ≤ 2π − γ , and 2π − γ < ≤ 2π . Factor exp(+iπ/4) is taken for positive values of p and factor exp(−iπ/4) is valid for negative values of p.

In the above expressions, functions f , g and f (1), g(1) are defined according to Chapters 2, 3, and 4:

f (m) = f (φm, φ0m, α), g(m) = g(φm, φ0m, α),			(14.36)
f (1)(m) = f (1)(φm, φ0m, α),		g(1)(m) = g(1)(φm, φ0m, α),	(14.37)
with m = 1, 2, 3. Here, α = 3π/2 and
φ1	= 3π/2 − ,	φ01 = π/2 − γ ,	(14.38)
φ2	= − π/2,	φ02 = π/2 + γ ,	(14.39)
φ3	= − π ,	if π ≤ ≤ 2π ,	(14.40)
φ3	= π + ,	if 0 ≤ ≤ π/2,	(14.41)
φ03	= γ .		(14.42)

The angles φ, φ0 for the functions f (1)left [φ (ψ ), φ0(ψ ), α], g(1)left [φ (ψ ), φ0(ψ ), α]

are determined by Equations (14.15) and (14.16), and for the functions f (1)right [φ (ψ ), φ0(ψ ), α], g(1)right [φ (ψ ), φ0(ψ ), α] they are found with Equations

(14.17), (14.18).

All diffracted rays undergo a phase shift equal to ±π/2 when they cross the focal lines = γ and = π − γ , which are the axes of the field beams.

14.1.5Refined Asymptotics for the Specular Beam

We refer again to Figure 14.1 and focus on the beam reflected from the lower lateral surface of the cylinder and propagating in the specular direction = 2π − γ . In the first-order approximation, this beam was evaluated in the previous section. Here we consider some fine features of the theory, which are beyond the first approximation. The final results of this section were published in the paper by Ufimtsev (1989).

According to PTD the scattered field is generated by the uniform j(0) and nonuniform j(1) components of the scattering sources induced by the incident wave on

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14.1 Acoustic Waves 301

the object. Up to now we have calculated the field radiated only by the basic part of the component j(1) that is caused by sharp bending (edges). This is the so-called fringe component j(1)fr . The other component j(1)sm is caused by the smooth bending of the scattering surface and it is asymptotically small compared to j(0) and j(1)fr . In particular on a circular cylinder, the ratio j(1)sm/j(0) is of the order 1/ka. That is why the component j(1)sm is usually neglected for thick cylinders. However, in our papers (Ufimtsev, 1979, 1981, 1989), it was shown that this small component distributed

over the entire generatrix (−l ≤ z ≤ l, ψ = 3π/2, kl						1) creates in the specular
direction	=	2π		γ the co-phased radiation of the same order [(ka)−1/2] as the
			−(1)fr		. For this reason one should include this additional radiation
field generated by j

in the beam field. This is the first fine feature of the theory.

It was also shown (Ufimtsev, 1979, 1981, 1989) that the second term of the asymptotic expansion for the PO field (in the specular direction) is also a quantity of the order (ka)−1/2 and it also should be included in the beam field. Usually, the high-order terms in the PO field are considered incorrect. However, in the framework of PTD, the PO field is the constituent part of the scattered field. Therefore one should incorporate into the field expression the high-order asymptotic terms of the PO field, which are of the same order of magnitude as those taken from the asymptotic expansion of the field generated by the nonuniform sources js,h(1). This is the second fine and important feature of PTD.

Here, these observations are demonstrated in the analytic form for the directivity pattern ( , γ ) introduced by Equation (14.20). The scattered field is evaluated in the vicinity of the specular direction = 2π − γ . The PO field generated by j(0) is described by Equations (14.4) and (14.5). The first term there relates to the field scattered by the left base (disk) of the cylinder. Its contribution to the field in the region 3π/2 < < 2π is created by the vicinity of point 2 (Fig. 14.1). By asymptotic evaluation of this contribution, the expressions (14.4) and (14.5) can be written as

aeiπ/4

cos γ

ei(q−p)

ikal

sin q

2π

(0)

sin γ

eip sin ψ sin ψ dψ ,

= − √

− π

s

2π p

sin γ

−

sin

q

π

(14.43)

and

h(0)

aeiπ/4

cos

ikal

sin q

2π

= −

√

ei(q−p)

−

sin

eip sin ψ sin ψ dψ ,

sin γ

−

sin

π

q

2π p

π

(14.44)

where p = ka(sin γ − sin ) and q = kl(cos − cos γ ). Here, in accordance with the above discussion, we retain the two first terms in the asymptotic expansion for the integrals and obtain

(0)

= −

a

cos γ

ei(q−p)+iπ/4

√2π p

s

sin γ

−

sin

−

ikal

sin γ

sin q

2π

−1 + i

3

e−ip+iπ/4

(14.45)

π

q

p

8p

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