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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)

= z→0

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	= π R1,22	(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.

TEAM LinG

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

ei2kl

eikR

2ω

= − 2

(0) −

cos

− 1

− cos

(6.140)

and

2 sin

ikR

ei2kl

2ω

z (0) +

cos

− 1

+ cos

− cos

(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

TEAM LinG

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

ω ei2kl e

2 sin

ikR

(6.144)

= −

−

cos

− 1

− cos

and

ei2kl

2 sin

eikR

2ω

(6.145)

cos

− 1

+ cos

− cos

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),	if Hy(0) = uinc.	(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)
us	= −uh	= −u0	2	tan ω(1			)	R	(6.147)
									TEAM LinG

6.4 Backscattered Focal Fields

147

and

tan ω

ei2kl

2 sin

ikR

2ω

= −

−

cos

− 1

− cos

(6.148)

tan ω

ei2kl

2 sin

eikR

2ω

cos

− 1

+ cos

− cos

(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)

and

ika

sin

eikR

cot

(6.151)

+ n

+ cos n

− 1

ika

sin

eikR

= −

cot

(6.152)

n − cos

+ n

− 1

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

TEAM LinG

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

and the ﬁrst-order PTD by

−

2ω

2 sin

σs

π a2

n −

−

tan ω

ei2kl

cos

and

2ω

2 sin

σh

π a2

n −

−

tan ω

ei2kl

cos

with n

(ω

)/π .

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

σ (0) = π a2(ka)2,

+ n

π a2 ika

sin

σs

cot

−

cos

and

+ n

n −

ika

sin

σh

π a2

cot

−

cos

(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.

TEAM LinG

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