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

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

Figure 6.13 Generatrix of the body of revolution.

The incident plane wave (6.1) propagates in the positive direction of the z-axis, which represents the symmetry axis of a scattering object. We use two systems of coordinates: cylindrical coordinates ρ, ϕ, z and spherical coordinates r, ϑ , ϕ. The generatrix is given as a function ρ = ρ(z). It is assumed that d2ρ/dz2 =0 for the illuminated side (0 ≤ z ≤ l) of the object. This condition ensures that the Gaussian curvature of this surface is not zero. We also utilize the following denotations related to the edge points (ρ = a): dρ/dz = tan ω for the illuminated side (z = l − 0) and dρ/dz = − tan for the shadowed side (z = l + 0). The shadowed side is an arbitrary smooth surface with 0 ≤ ≤ π − ω. In the limiting case = π − ω, the scattering object is an infinitely thin (but still perfectly reflecting) screen ρ = ρ(z) with 0 ≤ z ≤ l.

The principal radii of curvature of the scattering surface are determined according to the differential geometry (Bronshtein and Semendyaev, 1985)

1 = {

+

|[ρ (z)|

2 =

'

+ [

]

1

ρ (z)]2

}3/2

,

R

and

R

ρ(z) 1

ρ

(z)

2

(6.121)

where ρ = dρ(z)/dz, ρ = d2ρ(z)/dz2. The radius R1 relates to the normal section of the surface by the plane (ρ, z). The radius R2 relates to the orthogonal normal section. As ρ = tan θ with ω ≤ θ ≤ π/2, the principal radii can be represented as

R1 =	1		R2 =	ρ(z)
		and			.	(6.122)
	|ρ (z)| cos3 θ			cos θ

The radius R2 is shown in Figure 6.13. The Gaussian curvature is determined by

kG = R1R2	=	ρ(z)	cos4 θ .	(6.123)
1		ρ (z)

The above expressions for R1, R2, and kG become indefinite at the point z = 0. In order to disclose these indefinitenesses, one can use the alternative expressions

1 + [z (ρ)]2

3/2

R

and

R

ρ

1 + [z (ρ)]2

,

(6.124)

=

5

z (ρ)

6

=

'

z (ρ)

1

2

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6.4 Backscattered Focal Fields 143

where z = z(ρ), z = dz(ρ)/dρ, and z = d2z(ρ)/dρ2. It follows from these equations that at the point ρ = z = 0

1
R1 = R2 = z (ρ) .	(6.125)

The equality of the two principal radii means that the vertex point ρ = z = 0 of the scattering surface is an umbilic point.

6.4.1PO Approximation

This is the field radiated by the uniform components (1.31) of the scattering sources

js(0) = iu02knzeikz

and

jh(0) = u02eikz.

(6.126)

According to the differential geometry (Bronshtein and Semendyaev, 1985),

cos ψ

sin ψ

ρ (z)

nˆ = xˆ

+ yˆ

− zˆ

(6.127)

'

'

'

or

1 + [ρ (z)]2

1 + [ρ (z)]2

1 + [ρ (z)]2

nˆ = xˆ cos θ cos ψ + yˆ cos θ sin ψ − zˆ sin θ .

(6.128)

Here, we use the letter ψ for the polar coordinate of the scattering surface and retain the letter ϕ for the polar coordinate of the field point.

The scattered field in the far zone is determined by Equations (1.16) and (1.17),

where one should set
and ds = ρ(z)'		dzdψ ,	r sin ϑ = ρ(z),	r cos ϑ = z
	1 + (ρ )2				(6.129)
	mˆ = r = xˆ sin ϑ cos ϕ + yˆ sin ϑ sin ϕ + zˆ cos ϑ .				(6.130)

Due to the axial symmetry of the scattered field, it is sufficient to calculate the field only in the meridian plane ϕ = π/2. Taking this into account, one has

m

n

sin ϑ sin ψ − ρ

(z) cos ϑ ,

ˆ · ˆ =

ik '

1 + [ρ (z)]2

2π

eikR

l

u(0)

=

u

eikz(1−cos ϑ )ρ(z)ρ (z)dz

e−ikρ(z) sin ϑ sin ψ dψ

0 2π

s

R

0

0

and

uh(0) = −u0

ik

eikR

l

0 eikz(1−cos ϑ )ρ(z)dz

2π

R

2π

e−ikρ(z) sin ϑ sin ψ [sin ϑ sin ψ − ρ (z) cos ϑ ]dψ

×

0

(6.131)

(6.132)

(6.133)

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

It follows from these equations that in the forward focal direction (ϑ = 0),

ika2 eikR
us(0) = uh(0) = u0 2 R .	(6.134)

For the backscattering direction (ϑ = π ), the expressions (6.132) and (6.133) are reduced to

		u(0)			eikR	l	(z)dz.
u(0)	= −		=	u ik		ei2kzρ(z)ρ		(6.135)
s		h		0	R	0

By integrating by parts, one obtains the following asymptotic estimations:

us(0) = −uh(0) = −u0 2	ρ(0)ρ (0) − ρ(l)ρ (l)ei2kl + O	k	R	. (6.136)
1		1	eikR

The indefiniteness for the first term in the square brackets is disclosed by the following manipulations:

ρ(0)ρ

(0)

lim ρ(z)

dρ(z)

dz

= z→0

1

1

=

ρ

=

lim

lim

.

(6.137)

ρ→0 dz(ρ)/dρ

ρ→0 d2z(ρ)/dρ2

= z (0)

In view of Equation (6.125), scattering surface at the point

this quantity determines the radius of curvature of the z = 0. Now Equation (6.136) can be written as

1		1			eikR
us(0) = −uh(0) = −u0				− a tan ωei2kl		.	(6.138)
	2		z (0)		R

Here the first term represents the ordinary ray reflected from the vertex of the body, and the second term is the first-order focal field generated by the uniform components js,h(0) of the scattering sources induced near the circular edge (z = l). Indeed, due to Equations (6.125) and (6.138), the backscattering cross-section (1.26) related to the reflection from the body vertex equals

1	= π R1,22
σ = π [z (0)]2		(6.139)

and totally agrees with Equation (1.27). We note that, according to Equations (1.100) and (1.101), Equations (6.138) and (6.139) are also valid for electromagnetic waves scattered from perfectly conducting objects.

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6.4 Backscattered Focal Fields 145

6.4.2 Total Backscattered Focal Field: First-Order

PTD Asymptotics

In this approximation, the total scattered field in the direction ϑ = π is found by summation of its components (6.138) and (6.41), (6.42) generated by the uniform and nonuniform scattering sources js,h(0) and js,h(1), respectively. Note that functions f (1) and g(1) in Equations (6.41) and (6.42) are determined for the case ϑ = π by Equations (6.102) and (6.103). The summation results in the following asymptotic expressions:

2 sin

π

ei2kl

eikR

u

1

1

1

0

n

us

a

π

π

2ω

= − 2

z

(0) −

n

cos

− 1

− cos

− cos

R

n

n

n

(6.140)

and

2 sin

π

ikR

u0

1

1

1

e

n

uh

a

ei2kl

,

π

π

2ω

=

2

z (0) +

n

cos

− 1

+ cos

− cos

R

n

n

n

(6.141)

with n = 1 + (ω + )/π where 0 ≤ ω ≤ π/2 and 0 ≤ ≤ π − ω. In the next two sections we consider the backscattering from two specific bodies of revolution.

6.4.3Backscattering from Paraboloids

The PO approximations for acoustic and electromagnetic fields in this problem are

identical. Focal asymptotics for the total acoustic and electromagnetic fields are different.

Asymptotics for the Scattered Field

The illuminated surface of the scattering object is a paraboloid with the generatrix given by the equation

ρ2(z) = 2pz,

(6.142)

where 0 ≤ z ≤ l. The focus of a paraboloid is located at the point z = p/2. According to Equation (6.125), the focal parameter p equals the radius of the paraboloid curvature

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

at the vertex point z = 0. The shape of the shadowed side of the scattering object and its other geometric characteristics are shown in Figure 6.13.

Due to Equations (6.138) and (6.140), (6.141), the focal backscattered field is described by the following asymptotic expressions, where the first relates to the PO part of the field:

us(0) = −uh(0) = −	u0	%p − a tan ω ei2kl &	eikR
				.	(6.143)
	2		R

The next two expressions represent the first-order PTD approximations:

u0 p

π

ω ei2kl e

2 sin

1

1

ikR

us

a

n

π

(6.144)

π

= −

2

−

n

cos

− 1

− cos

− cos

2

R

n

n

n

and

p

π

ei2kl

u0

2 sin

1

1

eikR

uh

n

π

2ω

,

(6.145)

a

π

=

2

+

n

cos

− 1

+ cos

− cos

R

n

n

n

with n = 1 + (ω + )/π . Comparison with the electromagnetic PO field scattered by a perfectly conducting paraboloid (Equation (2.5.3) in Ufimtsev (2003)) reveals the following relationships:

Ex(0) = us(0),	if Ex(0) = uinc
and	if Hy(0) = uinc.
Hy(0) = uh(0),		(6.146)

This result is in complete agreement with the general relationships (1.100) and (1.101). Taking into account that ρ (l) = p/a = tan ω and p = a tan ω, the above expres-

sions can be rewritten as

(0)	(0)		a		− e	i2kl		eikR
us	= −uh	= −u0		tan ω(1			)		(6.147)
			2					R
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6.4 Backscattered Focal Fields

147

and

tan ω

π

ei2kl

a

2 sin

1

1

e

ikR

n

us

u0

,

π

π

2ω

= −

2

−

n

cos

− 1

− cos

− cos

R

n

n

n

(6.148)

tan ω

π

ei2kl

a

2 sin

1

1

eikR

n

uh

u0

.

π

π

2ω

=

2

+

n

cos

− 1

+ cos

− cos

R

n

n

n

(6.149)

These equations are useful for investigation of the continuous transformation of the paraboloid into the flat disk when ω → π/2. Utilizing the relationship l = a2/2p = (a cot ω)/2, one can show that in the limiting case ω = π/2

us(0)

= −uh(0) = u0

ika2 eikR

(6.150)

2

R

and

ika

2

sin

π

eikR

a

1

π

us

=

u0

cot

n

n

,

(6.151)

2

+ n

n

+ cos n

− 1

R

ika

2

sin

π

eikR

a

1

π

n

uh

= −

u0

cot

n

(6.152)

n − cos

π

2

+ n

n

− 1

R

with n = 3/2 + /π .

The distinguishing feature of the PO field (6.147) is its oscillations with the pure zeros corresponding to parameters kl = mπ where m = 1, 2, 3, . . . . Other properties of the scattered field are illustrated in the next section.

Numerical Analysis of Backscattering

Here we calculate the normalized scattering cross-section (6.112), which is determined in terms of the scattered field by Equations (6.110) and (6.111). According to section (Asymptotics for the Scattered field), one can derive the following expressions for the scattering cross-section.

The PO approximation is given by

σs(0) = σh(0) = π a2	tan ω · (1 − ei2kl )		2
			(6.153)

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

and the first-order PTD by

π

2

=

−

n

−

π

2ω

2 sin

1

σs

π a2

n

n −

n

−

n

tan ω

π

1

ei2kl

cos

1

cos

cos

and

π

2

=

+

n

+

π

2ω

2 sin

1

σh

π a2

n

n −

n

−

n

,

tan ω

π

1

ei2kl

cos

1

cos

cos

with n

=

1

+

(ω

+

)/π .

When l → 0 and ω → π/2, the above expressions transform into

σ (0) = π a2(ka)2,

2

=

+ n

n

+

2

π

π a2 ika

1

π

sin

σs

cot

n

n

,

π

−

n

cos

1

and

=

+ n

n −

2

π

2

ika

1

π

sin

n

σh

π a2

cot

n

,

π

−

n

cos

1

(6.154)

(6.155)

(6.156)

(6.157)

(6.158)

with n = 3/2 + /π .

Here we present three types of calculation similar to those in Section 6.3.2 for the cone. The first type consists of calculations for conformal paraboloids, which differ by their length; the second type relates to the transformation of paraboloids into the disk; and the third type reveals the influence of the shadowed base of paraboloids on backscattering. The calculated quantity is the normalized scattering cross-section (6.112).

•In the study of conformal paraboloids, the focal parameter is set constant (kp = 3π tan 14◦) and the length of paraboloids changes in the interval 6π ≤ kl ≤ 36. In this case, the radius of the paraboloid base and its length are in the intervals

1.5λ ≤ a ≤ 2λ	and	3λ ≤ l ≤ 5.8λ.
For a given frequency (k = const.), the		focal parameter p is constant for

all paraboloids with different l. This condition actually means that all these paraboloids are just the different sections of the same semi-infinite paraboloid. This is why they are called conformal.

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6.4 Backscattered Focal Fields 149
According to the relationship l = a2/2p = (a cot ω)/2, the angle ω
(Fig. 6.13) is determined by the equation cot ω =	√		. For the given values
		2kl/ka

of kl, this angle is in the interval 10.24◦ ≤ ω ≤ 14◦. For the angle (Fig. 6.13), we take its value as 90◦. The results of calculations are plotted in Figure 6.14. The difference between the soft and hard data approaches 16–19 dB. Figure 6.14 demonstrates the rough PO data, which do not depend at all on the boundary conditions and are totally incorrect in the vicinity of minima.

•The next topic is the transformation of the paraboloid into the disk. In this process, each intermediate shape between the initial parabolid and the final disk

is a paraboloid whose focal parameter p depends on its length l. It follows from the equation l = a2/2p that p = a2/2l. To find the angle ω (Fig. 6.13), we use the additional equation p = a tan ω and obtain tan ω = a/2l, or tan ω = ka/2kl. For the parameter ka we take the constant value ka = 3π , which does not depend

on the paraboloid length. In this case, the diameters of all the intermediate paraboloids and the final disk equal 2a = 3λ. For the initial paraboloid (which is transformed into the disk), we take kl = 6π , that is, l = 3λ. The results are plotted in Figure 6.15.

•Now we consider the influence of the paraboloid shadowed base on backscat-

tering. The illuminated part of the object under investigation is the paraboloid with parameters ka = 3π and kl = 6π , when the base diameter and the length of the paraboloid are equal to each other: 2a = l = 3λ. The critical param-

eter of the base, which affects the edge wave, is the angle . It changes from zero to = π − ω, where ω = tan−1(ka/2kl) 14◦. In the limiting

Figure 6.14 Backscattering at conformal paraboloids of different lengths (with constant focal parameter p). According to Equation (6.146), the PO curve also represents scattering of electromagnetic waves from perfectly conducting paraboloids.

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

Figure 6.15 Transformation of the paraboloid into a disk with continuous maintaining of the paraboloidial shape. According to Equation (6.146), the PO curve also represents scattering of electromagnetic waves from perfectly conducting paraboloids.

case = π − ω, the scattering object transforms into the perfectly reflecting infinitely thin screen. The results of this investigaton are shown in Figure 6.16.

The PO approximation does not depend on the shape of the shadowed part of the paraboloid. For the chosen parameter kl = 6π , it predicts the zero value for the scattered field. In the decibel scale it equals minus infinity and it is outside the figure area. As is seen in this figure, the backscattering from a soft paraboloid depends on the angle only a little (the change of scattering

Figure 6.16 Influence of the base shape on backscattering.

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6.4 Backscattered Focal Fields 151

cross-section is about 3 dB), although a strong dependence is observed for the hard paraboloid (about 20 dB).

6.4.4Backscattering from Spherical Segments

The PO approximations for acoustic and electromagnetic fields in this problem are

identical. Focal asymptotics for the total acoustic and electromagnetic fields are different.

Asymptotics for the Scattered Field

The illuminated surface of the scattering object is a spherical segment whose generatrix is shown in Figure 6.17 and given by the equation

z(ρ) = b −

b2

− ρ2,

with 0 ≤ z ≤ l,

(6.159)

where b is the radius of the spherical'

surface. It follows from Equation (6.159) that

z

(ρ)

=

dz(ρ)

ρ

=

cot θ

(6.160)

dρ

= '

and

b2 − ρ2

d2z(ρ)

b2

1

z (ρ) =

=

,

z (0) =

.

(6.161)

dρ2

(b2

−

ρ2)3/2

b

The angle θ (z) in Equation (6.160) is displayed in Figure 6.13. At the point z = l, ρ = a, this angle equals θ (l) = ω. For the given quantities b and a, the angle ω and the segment length l are defined by equations

√

b

2

− a

2

tan ω

=

and l

=

b

b2

−

a2

,

(6.162)

a

− '

Figure 6.17 Generatrix of a spherical segment with an arbitrary shadowed base.

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