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160 Chapter 6

 

Axially Symmetric Scattering of Acoustic Waves at Bodies of Revolution

and

 

 

 

 

 

 

 

 

 

 

 

uh(0) = u0

 

'

 

eik 1(zst )

 

 

 

 

 

R1(zst )R2(zst )

 

 

 

 

2

 

 

 

 

 

a

1

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

+

 

 

{g(0)(1)[J0(ka sin ϑ ) − iJ1(ka sin ϑ )]

 

 

 

 

2

eR

,

(6.193)

+ g(0)(2)[J0(ka sin ϑ ) + iJ1(ka sin ϑ )]} eikl(1−cos ϑ )

 

 

 

 

 

 

 

 

 

ikR

 

 

where J0 and J1 are the Bessel functions. These asymptotics are valid away from the focal line (ϑ = π ) under the condition ka sin ϑ 1. The focal field is described by the asymptotics (6.138), which can be rewritten as

 

−

1

 

 

 

a

 

eikR

 

 

us(0) = −uh(0) = u0

'R1(0)R2(0) +

 

 

 

 

tan ω ei2kl

 

,

(6.194)

2

2

R

where R1(0) = R2(0) = 1/z (0).

When ϑ → π the above asymptotics (6.192) and (6.193) exactly transform into the focal asymptotics (6.194). Therefore, the expressions (6.192) and (6.193)

can be considered as the appropriate approximations valid in the entire region

π − ω ≤ ϑ ≤ π .

6.5.3 Bessel Interpolations for the PTD Field in the Region π − ω ≤ ϑ ≤ π

The PTD field consists of the sum of the PO field and the field us,h(1) generated

by the nonuniform components js,h(1) of the scattering sources. The components js,h(1) caused by the smooth bending of the scattering surface generate the far field of

order k−1 (Schensted, 1955), and those caused by the sharp edge create the field of order k−1/2 (as shown in Equations (6.21) and (6.22)). Therefore, in the first-order PTD approximation, one can retain only the dominant contributions generated by the edge-type sources js,h(1). The uniform asymptotics for these contributions in the region π − ω ≤ ϑ ≤ π are given by the expressions (6.47) and (6.48), where one should include the additional factor exp[ikl(1 − cos ϑ )] due to the shift of the coordinates’ origin. By the summation of the modified Equations (6.47) and (6.48) with the PO asymptotics (6.192), (6.193), one obtains

 

1

 

 

 

 

 

 

 

 

 

 

 

usPTD = u0

−

 

 

R1(zst )R2(zst ) eik 1(zst )

 

2

 

 

a

'

 

 

 

+

 

{ f (1)[J0(ka sin ϑ ) − iJ1(ka sin ϑ )]

 

2

 

+ f (2)[J0(ka sin ϑ ) + iJ1(ka sin ϑ )]} eikl(1−cos ϑ )

ikR

 

e

(6.195)

R

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6.5 Axially Symmetric Bistatic Scattering 161

and

 

 

 

 

 

 

 

 

 

 

uhPTD = u0

 

 

'

 

eik 1(zst )

 

 

 

 

 

R1(zst )R2(zst )

 

 

 

 

2

 

 

 

 

 

a

 

1

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

+

 

{g(1)[J0(ka sin ϑ ) − iJ1(ka sin ϑ )]

 

 

 

 

2

eR .

(6.196)

+ g(2)[J0(ka sin ϑ ) + iJ1(ka sin ϑ )]} eikl(1−cos ϑ )

 

 

 

 

 

 

 

 

ikR

 

 

The functions f (1, 2) and g(1, 2) are defined by Equations (6.25), (6.26), and (6.30), and (6.31), where one should omit the last terms, which are exactly cancelled by the terms f (0)(1, 2) and g(0)(1, 2) during the summation of modified Equations (6.47) and (6.48) with Equations (6.193) and (6.194).

6.5.4 Asymptotics for the PTD Field in the Region 2ω < ϑ ≤ π − ω away from the GO Boundary ϑ = 2ω

In this observation region, the stationary edge point 2 (Fig. 6.21) is not visible (when< 2ω), and its first-order contribution to the scattered field equals zero. We also assume that the observation directions are far from the last GO ray (reflected at the point 1 and shown in Fig. 6.21), where functions f (1) and g(1) are singular. Under these conditions, one can use the obvious modifications of Equations (6.195) and (6.196) for the scattered field:

usPTD = u0

−

1

'R1(zst )R2(zst ) eik 1(zst )

2

+a f (1)[J0(ka sin ϑ ) − iJ1(ka sin ϑ )] eikl(1−cos ϑ )

2

and

uhPTD = u0

 

2

'R1(zst )R2(zst ) eik 1(zst )

 

 

1

 

 

 

+a g(1)[J0(ka sin ϑ ) − iJ1(ka sin ϑ )] eikl(1−cos ϑ )

2

eikR

(6.197)

R

eikR

.(6.198)

R

6.5.5 Uniform Approximations for the PO Field in the Ray Region 2ω ≤ ϑ ≤ π − ω Including the GO Boundary ϑ = 2ω

The above ray asymptotics (6.197), (6.198) are not applicable in the vicinity of the geometrical optics boundary ϑ = 2ω, where the wave field does not have a ray structure. In this so-called transition region, the process of the transverse diffusion of the

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

wave field happens to actually give birth to diffracted rays (Ufimtsev, 2003). The singularities of the functions f (1), g(1) and the singularity of the factor 1/ (l) in Equation (6.185) for the direction ϑ → 2ω (in this case, ξst → l and (l, P) → 0) are the mathematical evidence of the existence of this process. For the calculation of the field integral (6.184) in this region, one should apply a more accurate method of the stationary phase that allows the approach of the stationary phase to the end point (Felsen and Marcuvitz, 1972). In the following we present the basics of this technique.

First, we modify the canonical integral (6.184). For the sake of simplicity, the symbol P (related to coordinates of the observation point) is omitted. We notice that the edge point 2 (Fig. 6.21) is not visible and therefore its first-order contribution to the scattered field equals zero. In the integral (6.184), this point corresponds to the end point ξ = −l. To exclude its contribution, we set ξ = −∞ for the lower limit of integration. Then we introduce a new variable t with the equation

(ξ ) = (ξst ) + t2.

(6.199)

Notice that, according to Equation (6.176), the

second derivative (ξst ) =

−ρ (ξst ) sin ϑ is positive and therefore the quantity (ξst ) is the minimum of function (ξ ). Taking this into account, we define the variable t as the continuous and differentiable function of the old variable ξ ,

√

 

 

,

 

for ξ ≥ ξst

 

(ξ ) − (ξst )

 

t(ξ ) = −√

 

,

for ξ ≤ ξst ,

(6.200)

(ξ ) − (ξst )

where the radical is understood in the arithmetic sense. In the vicinity of the stationary point, where (ξst ) = 0, one can use the Taylor approximations

 

1

(ξst )(ξ − ξst )2

 

1

(ξst )(ξ − ξst )3

 

(ξ ) = (ξst ) +

 

 

+

 

 

 

 

+ · · ·

2

!

3

!

and

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

t(ξ ) = (ξ − ξst )

 

 

 

 

 

 

1 (ξst )

 

1

 

 

 

 

 

 

 

(ξst )

1

+

 

 

 

(ξ − ξst ) .

 

2

6

(ξst )

 

(6.201)

(6.202)

Now the canonical integral (6.184) can be represented in this form

 

 

I = eik (ξst )

t(l)

eikt2 G(t)dt,

 

 

 

 

 

−∞

 

 

(6.203)

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

where

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

dξ

 

 

 

2t(ξ )

 

 

 

 

G(t) = F(ξ )

 

= F(ξ )

 

 

 

 

 

 

(6.204)

 

dt

(ξ )

 

 

and

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

G(0)

=

lim

2t(ξ )

F(ξ )

=

2

 

F(ξ

st

).

(6.205)

 

 

 

 

 

 

 

 

 

 

ξ →ξst (ξ )

 

 

(ξst )

 

 

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6.5 Axially Symmetric Bistatic Scattering 163

The next idea is to extract (in the explicit form!) the Fresnel integral from Equation

(6.203). It is accomplished with a simple procedure:

 

 

 

 

 

 

 

I = eik (ξst ) G(0)

t(l)

eikt2 dt +

t(l)

eikt2 [G(t) − G(0)]dt

(6.206)

 

 

 

 

 

 

 

 

 

 

−∞

 

 

 

 

−∞

 

 

 

 

 

 

 

or

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

√

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

1

 

kt(l)

 

 

G

t(l)] − G(0)

 

 

 

1

 

I

 

eik (ξst )G(0)

eix2 dx

 

eik (l)

 

O

. (6.207)

 

 

 

 

 

 

 

 

 

 

 

= √

k

 

√

 

−∞

 

 

+

[

i2kt(l)

+

 

k2

 

Under the condition

 

 

1, when the observation point is far from the geomet-

kt(l)

 

rical optics boundary ϑ = 2ω, this expression is reduced asymptotically to the first two terms in Equation (6.185).

When the observation point approaches the boundary (ϑ = 2ω + 0 and t = +0), one should utilize Equations (6.201) and (6.202) and the additional approximations

1

 

1

 

 

1

 

(ξst )

 

 

 

 

 

 

 

=

 

 

 

 

 

 

+1 −

 

 

 

 

 

 

(ξ − ξst ) + O)(ξ − ξst )2*, ,

(6.208)

 

(ξ )

 

(ξ − ξst ) (ξst )

2

(ξst )

 

2t(ξ )

 

2

 

 

 

1 (ξst )

 

 

 

 

 

 

 

=

 

+1 −

 

 

 

 

 

 

(ξ − ξst ) + O[(ξ − ξst )2], ,

 

(6.209)

 

(ξ )

(ξst )

3

(ξst )

 

 

F(ξ ) = F(ξst ) + F (ξst )(ξ − ξst ) + · · · ,

 

 

 

 

 

(6.210)

and

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

1

 

(ξst )

 

 

 

 

 

 

 

 

 

 

 

2

 

 

 

 

 

 

 

 

 

G(t) − G(0) = (ξ − ξst )

 

F (ξst ) −

 

 

F(ξst )

 

.

(6.211)

 

(ξst )

3

(ξst )

These relationships lead to the following value of the canonical integral at the boundary ϑ = 2ω + 0:

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

I =

π

 

F(l)eik (l)+iπ/4

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

2 k (l)

 

 

 

 

 

.

 

 

1

 

 

1

 

 

 

 

(l)

 

 

 

1

 

+

 

F

(l) −

 

 

 

F(l)

 

eik (l) + O

 

(6.212)

ik (l)

3

(l)

k2

This technique is applied further to the calculation of the PO field:

 

 

 

 

 

 

 

 

 

 

 

ik

eikR

 

 

 

 

 

 

 

us(0) = u0

√

 

 

eiπ/4Is

 

 

 

 

 

 

(6.213)

 

 

 

 

R

 

 

 

 

 

 

2π k sin ϑ

 

 

 

and

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

uh(0)

 

 

 

 

 

 

 

ik

eikR

 

 

 

 

 

 

= −u0

√

 

eiπ/4Ih

 

,

 

 

(6.214)

 

 

R

 

 

 

 

2π k sin ϑ

 

 

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