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272 Chapter 13 Backscattering at a Finite-Length Cylinder

Figure 13.2 Backscattering at the ﬁnite cylinder according to the PO approximations PO-1 (Equation (13.12)) and PO-2 (Equation (13.13)).

negligible in the case when d = 3λ, as is clearly seen in Figure 13.3. Because of that we use in the following calculations the simplest approximation (13.12). There is another reason in favour of using Equation (13.12). It matches an analogous type of approximation for the ﬁeld (13.22), (13.23) generated by the nonuniform/fringe source j(1). This ﬁeld is investigated in the following section.

Figure 13.3 Backscattering at the ﬁnite cylinder according to the PO approximations PO-1 (Equation (13.12)) and PO-2 (Equation (13.13)).

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13.1 Acoustic Waves 273

13.1.2 Backscattering Produced by the Nonuniform Component j (1)

There are three types of nonuniform components j(1) of the scattering sources on a ﬁnite cylinder. The edge/fringe component jfr(1) concentrates near the edges. The component jcr(1) associated with creeping waves concentrates near the shadow boundary on the cylindrical part of the object and exponentially attenuates away from this boundary. The third component jdif(1) is caused by the transverse diffusion of the wave ﬁeld between the adjacent rays reﬂected from the cylindrical surface. This component jdif(1) exists on the illuminated part of the cylindrical surface away from the shadow boundary. Among these components of j(1), a main contribution to the backscattering is provided by the fringe component jfr(1) and this contribution is investigated here. Notice also that the ﬁeld generated by jdif(1) is comparable with that produced by jfr(1), but this situation occurs only in the direction of the specular rays reﬂected from the cylindrical surface. This topic is considered in the next chapter.

First we analyze the ﬁeld generated by the fringe component located near the left

edge (z = −l, x2 + y2 = a). According to Equation (8.3) it is determined as

(1)left

= u0

i2kl cos ϑ eikR

2π

(1)

(ϕ )e

i2ka sin ϑ sin ϕ

us,h

Fs,h

dϕ

(13.16)

2π

In the direction ϑ = π , functions Fs,h(1) transform into functions f (1) and g(1) as shown in Equations (7.120) and (7.121). Recall that these functions are deﬁned in Section 4.1. Hence for this direction,

us(1)left	f (1)		eikR
uh(1)left	= u0a g(1)	e−i2kl		,	(ϑ = π ).	(13.17)
			R
For other directions ϑ , which satisfy the condition 2ka sin ϑ						1, the integral

in Equation (13.16) is evaluated asymptotically by the stationary-phase technique. There are two stationary points (ϕst,1 = π/2 and ϕst,2 = 3π/2) in the integrand of Equation (13.16). At these points, functions Fs,h(1) also transform into functions f (1) and g(1). The resulting asymptotic approximations for the ﬁeld (13.16) are given as

(1)left

i2kl cos ϑ eikR

= u0

√

π ka sin ϑ

× [ f (1)(1)ei2ka sin ϑ −iπ/4 + f (1)(2)e−i2ka sin ϑ +iπ/4]

(13.18)

and

(1)left

= u0

i2kl cos ϑ eikR

√

π ka sin ϑ

[g(1)(1)ei2ka sin ϑ −iπ/4 + g(1)(2)e−i2ka sin ϑ +iπ/4].

(13.19)

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274 Chapter 13 Backscattering at a Finite-Length Cylinder

Here, the functions f (1)(1), g(1)(1) relate to the stationary point 1 (ϕst,1 = π/2) and

the functions f (1)(2), g(1)(2) relate to the stationary point 2 (ϕst,2 = 3π/2). These points are shown in Figure 13.1.

In order to construct the ﬁeld approximations in the entire region π/2 ≤ ϑ ≤ π , we apply the idea suggested in Sections 6.1.4 and 13.1.1. In otherwords, we substitute the asymptotics


	ei2ka sin ϑ −iπ/4			≈ J0	(2ka sin ϑ ) + iJ1
	√					(2ka sin ϑ ),	(13.20)
			π ka sin ϑ
e−i2ka sin ϑ +iπ/4				≈ J0	(2ka sin ϑ ) − iJ1
	√					(2ka sin ϑ )	(13.21)
		π ka sin ϑ

into Equations (13.18) and (13.19) and then extend the obtained ﬁeld expressions to the entire region π/2 ≤ ϑ ≤ π . It turns out that the resulting expressions

	a			eikR	0f (1)(1)[J0(2ka sin ϑ ) + i J1(2ka sin ϑ )]
us(1)left = u0			ei2kl cos ϑ
	2			R
+ f (1)(2)[J0(2ka sin ϑ ) − i J1(2ka sin ϑ )]1						(13.22)
and
	a			eikR	0g(1)(1)[J0(2ka sin ϑ ) + i J1(2ka sin ϑ )]
uh(1)left = u0			ei2kl cos ϑ
		2		R
+ g(1)(2)[J0(2ka sin ϑ ) − i J1(2ka sin ϑ )]1						(13.23)

exactly transform into Equation (13.17) when ϑ → π . Therefore, these expressions can be considered as appropriate approximations for the scattered ﬁeld in all directions

π/2 ≤ ϑ ≤ π .

The contribution of the right edge (z = +l,

x2 + y2

= a) to the ﬁeld in the

region π/2 < ϑ < π is described by the

expression

(1)right

i2kl cos ϑ eikR

2π

(1)

i2ka sin ϑ sin ϕ

us,h

= u0

e−

Fs,h (ϕ )e

dϕ

(13.24)

2π

analogous to Equation (13.16). Its asymptotic approximation found by the stationaryphase technique is determined as

uh	=	2	g		(3)
us(1)right			f (1)		(3)
		u0 a		(1)
(1)right		u0 a		(1)

e−i2ka sin ϑ +iπ/4		e−i2kl cos ϑ	eikR	,	(13.25)
√
	π ka sin ϑ		R

where 2ka sin ϑ 1 and functions f (1)(3), g(1)(3) relate (ϕst,3 = 3π/2) shown in Figure 13.1. With the application

to the stationary point 3 of Equation (13.21), this

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13.1 Acoustic Waves 275

expression is extended to all directions π/2 < ϑ < π and provides the following approximations to the ﬁeld (13.24):

us(1)right = u0

f (1)(3)

[J0(2ka sin ϑ ) − iJ1(2ka sin ϑ )]e−i2kl cos ϑ

eikR

(13.26)

and

eikR

uh(1)right = u0

g(1)(3) [J0(2ka sin ϑ )

− iJ1(2ka sin ϑ )]e−i2kl cos ϑ

(13.27)

Thus, in the ﬁrst approximation, the total ﬁeld produced by the component

jfr(1) equals

a eikR

us(1)

= u0

0f (1)(1) [J0(2ka sin ϑ ) + iJ1(2ka sin ϑ )]ei2kl cos ϑ

+ [ f (1)(2)ei2kl cos ϑ + f (1)(3)e−i2kl cos ϑ ][J0(2ka sin ϑ ) − iJ1(2ka sin ϑ )] ,

(13.28)

and

a eikR

uh(1)

= u0

0g(1)(1) [J0(2ka sin ϑ ) + iJ1(2ka sin ϑ )]ei2kl cos ϑ

+ [g(1)(2)ei2kl cos ϑ + g(1)(3)e−i2kl cos ϑ ][J0(2ka sin ϑ ) − iJ1(2ka sin ϑ )] .

(13.29)

The functions f (1) and g(1) are determined according to Section 4.1 as

sin

cos ϑ

f (1)(1)

(13.30)

cos

− cos

cos

π − 2ϑ

2 sin ϑ

n −

−

sin

cos ϑ

g(1)(1)

(13.31)

cos

+ cos

cos

π − 2ϑ

−

2 sin ϑ

n −

−

sin

1 cos ϑ

1 sin ϑ

f (1)(2)

2ϑ

cos

− 1

− cos

−

2 sin ϑ −

2 cos ϑ

(13.32)

sin

cos ϑ

1 sin ϑ

g(1)(2)

2ϑ

cos

− 1

+ cos

− cos

2 sin ϑ +

2 cos ϑ

(13.33)

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276 Chapter 13

Backscattering at a Finite-Length Cylinder

sin

sin ϑ

f (1)(3)

(13.34)

−

n −

− cos

and

cos

π + 2ϑ

2 cos ϑ

sin

sin ϑ

g(1)(3)

(13.35)

−

n −

+ cos

−

cos

π + 2ϑ

2 cos ϑ

with n = 3/2.

Although certain terms in these functions are singular in the directions ϑ = π/2 and ϑ = π , these singularities always cancel each other, and the functions f (1), g(1) remain ﬁnite. In the direction ϑ = π/2 they have the values

sin

f (1)(1) = 0,

f (1)(2) = f (1)(3) =

cot

(13.36)

cos

− 1

and

sin

g(1)(1) =

g(1)(2) = g(1)(3) =

−

cot

, (13.37)

cos

− 1

cos

− 1

and in the direction ϑ = π they are determined as

sin

f (1)(1) = f (1)(2) =

cot

f (1)(3) = 0

(13.38)

cos

− 1

and

sin

g(1)(1) = g(1)(2) =

−

g(1)(3) =

cot

(13.39)

cos

− 1

cos

− 1

We also obtain the expressions for the functions f (1)(4) and g(1)(4) related to the

stationary point 4 (Fig. 13.1), which

becomes visible in the directions ϑ

π/2 and

(1)

ϑ = π . For both directions, functions f

(4) and g

(4) have the same values,

sin

f (1)(4) = 0

and

g(1)(4) =

(13.40)

cos

− 1

As the contribution from point 4 equals zero for the soft cylinder, the approximation (13.28) can be used in the entire region π/2 ≤ ϑ ≤ π . In the case of

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