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

112 Chapter 5 First-Order Diffraction at Strips and Polygonal Cylinders

Figure 5.14 Backscattering at an acoustically hard cylinder (Hz -polarization of the electromagnetic wave).

PROBLEMS

5.1Derive the PO approximation for the field (in the far zone) scattered at a soft strip as shown in Figure 5.1. The incident wave is given by Equation (5.1). Start with the general expressions (1.32), use Equation (3.7) for the Hankel function, apply its asymptotic form (2.29). Represent the scattered field in terms of edge waves and compare the result with Equation (5.4). Verify that they are identical.

5.2Derive the PO approximation for the field (in the far zone) scattered at a hard strip as shown in Figure 5.1. The incident wave is given by Equation (5.1). Start with the general expressions (1.32), use Equation (3.7) for the Hankel function, apply its asymptotic form (2.29). Represent the scattered field in terms of edge waves and compare the result with Equation (5.5). Verify that they are identical.

5.3Derive the PO approximation for the field (in the far zone) scattered at a perfectly conducting strip as shown in Figure 5.1. The incident wave is given as

Ezinc = E0zeik(x cos ϕ0+y sin ϕ0)

Start with the general expressions (1.87) and (1.97). Use Equation (3.7) for the Hankel function, apply its asymptotic form (2.29). Represent the scattered field in terms of edge waves and compare the result with Equation (5.4). Formulate the relationship between acoustic and electromagnetic diffracted waves.

5.4Derive the PO approximation for the field (in the far zone) scattered at a perfectly conducting strip as shown in Figure 5.1. The incident wave is given as

Hzinc = H0zeik(x cos ϕ0+y sin ϕ0).

Start with the general expressions (1.88) and (1.97). Use Equation (3.7) for the Hankel function, apply its asymptotic form (2.29). Represent the scattered field in terms of edge

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Problems 113

waves and compare the result with Equation (5.5). Formulate the relationship between acoustic and electromagnetic diffracted waves.

5.5Use Equation (5.35). Prove Equation (5.39). Apply Equation (5.16) and calculate the total cross-section of a soft strip.

5.6Use Equation (5.37). Prove Equation (5.40). Apply Equation (5.16) and calculate the total cross-section of a hard strip.

5.7Investigate the diffraction of a cylindrical wave at a soft strip (Fig. 5.4). Related coordinates are shown in Figure 5.1. The incident wave is given by

eikr0		((x − x0)2 + y2 and k|x0| 1.
uinc = u0 √kr0	, with r0 =

Derive the PTD approximation for the scattered field as a sum of two separate components: the PO field us(0) and the fringe field us(1). Start the calculation of the PO field with the general expressions (1.32), use Equation (3.7) for the Hankel function, apply its asymptotic form (2.29). Represent the far field us(0) in integral form. Use Equation (4.12) and write the asymptotic expressions for the edge waves generated by the nonuniform/fringe sources js(1). Verify that these waves are finite at the shadow boundaries. Retrace the transition (in the field expressions) from the cylindrical wave excitation to the incident plane wave by setting

		eikr0
x0 lim	u0	√		= eikx .
→−∞			kr0

5.8Investigate the diffraction of a cylindrical wave at a hard strip (Fig. 5.4). Related coordinates are shown in Figure 5.1. The incident wave is given by

eikr0		((x − x0)2 + y2 and k|x0| 1.
uinc = u0 √kr0	, with r0 =

Derive the PTD approximation for the scattered field as a sum of two separate components: the PO field uh(0) and the fringe field uh(1). Start the calculation of the PO field with the general expressions (1.32), use Equation (3.7) for the Hankel function, apply its asymptotic form (2.29). Represent the far field uh(0) in integral form. Use Equation (4.13) and write the asymptotic expressions for the edge waves generated by the nonuniform/fringe sources jh(1). Verify that these waves are finite at the shadow boundaries. Retrace the transition (in the field expressions) from the cylindrical wave excitation to the incident plane wave by setting

		eikr0
x0 lim	u0	√		= eikx .
→−∞			kr0

5.9Investigate the diffraction of a cylindrical wave at a perfectly conducting strip (Fig. 5.4). Related coordinates are shown in Figure 5.1. The incident wave is given by

eikr0		((x − x0)2 + y2 and k|x0| 1.
Ezinc = E0z √kr0	, with r0 =

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114 Chapter 5 First-Order Diffraction at Strips and Polygonal Cylinders

Derive the PTD approximation for the scattered field as a sum of two separate components: the PO field Ez(0) and the fringe field Ez(1) . Start the calculation of the PO field with the general expression (1.87), use Equation (3.7) for the Hankel function, apply its asymptotic form (2.29). Represent the far field Ez(0) in integral form. Use Equation (4.12) (adapted for electromagnetic waves) and write the asymptotic expressions for the field

Ez(1). Verify that these waves are finite at the shadow boundaries. Retrace the transition (in the field expressions) from the cylindrical wave excitation to the incident plane wave by setting

		eikr0
x0 lim	E0z	√		= eikx .
→−∞			kr0

5.10Investigate the diffraction of a cylindrical wave at a perfectly conducting strip (Fig. 5.4). Related coordinates are shown in Figure 5.1. The incident wave is given by

eikr0		((x − x0)2 + y2 and k |x0| 1.
Hzinc = H0z √kr0	, with r0 =

Derive the PTD approximation for the scattered field as a sum of two separate components: the PO field Hz(0) and the fringe field Hz(1). Start the calculation of the PO field with the general expression (1.88), use Equation (3.7) for the Hankel function, apply its asymptotic form (2.29). Represent the far field Hz(0) in integral form. Use Equation (4.13) (adapted for electromagnetic waves) and write the asymptotic expressions for the field Hz(1). Verify that these waves are finite at the shadow boundaries. Retrace the transition (in the field expressions) from the cylindrical wave excitation to the incident plane wave by setting

eikr0

lim H0z √ = eikx . x0→−∞ kr0

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Chapter 6

Axially Symmetric Scattering

of Acoustic Waves at Bodies

of Revolution

A similar problem for electromagnetic waves is considered in Chapter 2 of Ufimtsev

(2003).

This chapter develops the first-order PTD for acoustic waves scattered at bodies of revolution with sharp edges. Axially symmetric scattering is studied. This situation occurs when an incident plane wave propagates in the direction along the symmetry axis of a body of revolution. An edge of a body of revolution is a circle. When its diameter is much greater than a wavelength, then the nonuniform scattering sources js,h(1) induced near the edge are asymptotically identical to those near the edge of the tangential conic surface consisting of two parts (Fig. 6.1). Diffraction at this surface is an appropriate canonical problem, which is studied in Section 6.1. Its solution is used in the next sections to determine the field scattered at certain bodies of revolution.

6.1 DIFFRACTION AT A CANONICAL CONIC SURFACE

The geometry of the problem is illustrated in Figures 6.1 and 6.2. The solid lines in Figure 6.1 show a general view of a body of revolution with a circular edge. The dashed tangent lines belong to the tangential conic surface. The cross-section of this surface by the meridian plane and some related denotations are presented in Figure 6.2. Here, ξ is the distance from the edge along the generatrix; r , ϑ , ψ the spherical coordinates and ρ, ψ the polar coordinates of the point on the conic surface; R, ϑ , ψ the coordinates of the observation point; ϕ0 the angle of incidence measured

Fundamentals of the Physical Theory of Diffraction. By Pyotr Ya. Ufimtsev

Copyright © 2007 John Wiley & Sons, Inc.

115

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

Figure 6.1 A body of revolution (solid lines) with a circular edge and a conic surface (dashed lines) tangential to a body at the edge points.

from the illuminated side of the conic surface; and α − ϕ0 the angle of incidence measured from the shadowed side; the meaning of the angles ω and is clear; the edge points 1 and 2 are symmetrical.

Figure 6.2 Cross-section of the canonical conic surface.

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6.1 Diffraction at a Canonical Conic Surface 117

6.1.1Integrals for the Scattered Field

It is supposed that the incident plane wave

uinc = u0eikz

(6.1)

propagates along the symmetry axis of the conic surface. The incident wave undergoes diffraction at this surface and creates there the nonuniform scattering sources

js(1) = u0Js(1),

jh(1) = u0Jh(1).

(6.2)

It is obvious that, due to the symmetry of the problem, these sources do not depend on the polar angle ψ . The scattered field is described by general Equations (1.16) and (1.17). In the particular case of the conic surface, the quantities involved in these equations are determined by the following expressions. The quantities

ds− = (a − ξ sin ω)dξ dψ
and
ds+ = (a − ξ sin )dξ dψ	(6.3)

are the differential elements of the conic surface on its illuminated (z < 0) and shadowed (z > 0) sides, respectively;

r sin ϑ		a − ξ sin ω			for the points with z < 0,	(6.4)
r cos ϑ	= −		ξ cos ω
	=
r sin ϑ	=	a − ξ sin			for the points with z > 0;	(6.5)
r cos ϑ
	= ξ cos

ˆ	· ˆ	=	sin ϑ cos ω cos(ψ	−	ψ )	−	sin ω cos ϑ ,	(6.6)
(m	n)−							for the points with z < 0,
ˆ	· ˆ	=	sin ϑ cos cos(ψ	−	ψ )	+	sin cos ϑ ,	(6.7)
(m	n)+							for the points with z > 0.

In this section, we use the symbol for the angle shown in Figure 6.2. In Equations (1.16) and (1.17), the same symbol was used for another angle (Fig. 1.2). We note this to avoid possible confusion.

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

In view of the above comments, the scattered field (1.16–1.17) can be represented in this form:

1		eikR			ξeff
us(1) = −u0						Js(1)(kξ , ϕ0)eikξ cos ω cos ϑ (a − ξ sin ω)dξ
	4π		R		0

2π
× 0	e−ik −(ψ	,ψ ) dψ	+

2π

× e−ik +(ψ ,ψ ) dψ ,

ξeff

Js(1)(kξ , α − ϕ0)e−ikξ cos cos ϑ (a − ξ sin )dξ

0

(6.8)

0

uh(1) = −u0		ik eikR
		4π			R
×		ξeff					2π
	0		Jh(1)(kξ , ϕ0)eikξ cos ω cos ϑ (a − ξ sin ω)dξ				0	e−ik −(ψ	,ψ )(mˆ	· nˆ )− dψ
+	ξeff		Jh(1)(kξ , α − ϕ0)e−ikξ cos cos ϑ (a − ξ sin )dξ
	0
	2π		e−ik +(ψ ,ψ )(mˆ · nˆ )+ dψ ,
×	0									(6.9)
where
					−(ψ , ψ ) = (a − ξ sin ω) sin ϑ cos(ψ − ψ )
and
					+(ψ , ψ ) = (a − ξ sin ) sin ϑ cos(ψ − ψ ).					(6.10)

The first integrals in the brackets of Equations (6.8) and (6.9) relate to the illuminated surface (z < 0), and the second integrals relate to the shadowed surface (z > 0). These integrals are calculated over the interval 0 ≤ ξ ≤ ξeff , as the nonuniform sources Js,h(1)(kξ ) decrease with increasing ξ and can be neglected at a certain distance ξ > ξeff from the edge. In the next sections we present the asymptotic estimates for the scattered field.

6.1.2Ray Asymptotics

First we investigate the field in the observation points, which are visible from all edge points (0 ≤ ψ ≤ 2π ). This happens for the two intervals of the observation directions: 0 ≤ ϑ ≤ and π − ω ≤ ϑ ≤ π . We assume that

k(a − ξeff sin ω) sin ϑ	1
and
k(a − ξeff sin ) sin ϑ	1	(6.11)

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6.1 Diffraction at a Canonical Conic Surface 119

and apply the stationary phase technique (Copson, 1965; Murray, 1984) to the integrals over the variable ψ . The stationary points ψ1,2 are found from the condition

d ±(ψ , ψ )

= 0

(6.12)

dψ

which leads to the equation

d cos(ψ − ψ )

= −

sin(ψ

−

ψ )

=

0,

at ψ

=

ψ

.

(6.13)

dψ

st

In the interval 0 ≤ ψ ≤ 2π , two stationary points exist:

ψ1 = ψ

and

ψ2 = π + ψ .

(6.14)

In accordance with this asymptotic method, the function cos(ψ − ψ ) contained in± is approximated by

cos(ψ − ψ ) ≈ 1 −		1	(ψ			− ψ1)2	in the vicinity of the point ψ = ψ1 (6.15)
	2
and
cos(ψ − ψ ) ≈ −1 +				1	(ψ − ψ2)2		in the vicinity of the point ψ = ψ2.
				2
The slowly varying factor (m							(6.16)
						n)± is approximated by its value at the stationary points.
				ˆ		· ˆ

The initial integral over the variable ψ asymptotically equals the sum of two integrals calculated in the vicinity of each stationary point. The intervals of integration in these integrals are extended from −∞ to +∞. These standard manipulations lead to the asymptotic expression

2π

e−ik

0

where

(mˆ · nˆ )+

(mˆ · nˆ )+

2

±(ψ ,ψ )(mˆ · nˆ )±dψ

2π

3

−

ω

3

4

a

ξ sin

sin ϑ

3k

−ik a−ξ sin ω sin ϑ

i π

×

ˆ · ˆ

= 1 ·

·

(m

n)±

e

e 4

ψ ψ

eik a−ξ sin ω sin ϑ

(m n)± ψ

ψ

2

·

·

e−i π4

+ ˆ · ˆ

=

(6.17)

ψ =ψ1

= sin(ϑ + ),

(mˆ · nˆ)− ψ =ψ1

= sin(ϑ − ω),

(6.18)

=

= −

−

ˆ · ˆ

=

= −

+

ψ2

sin(ϑ

),

(m n)−

ψ

ψ2 sin(ϑ

ω).

ψ

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

Equation (6.17) allows one to reduce Equations (6.8) and (6.9) to single integrals over

the variable ξ . These integrals will contain the factors √a − ξ sin ω and √a − ξ sin ,

√

which can be approximated to a under the condition a ξeff .

Before we present the resulting expressions for Equations (6.8) and (6.9), we introduce the local polar coordinates ϕ1 and ϕ2 at the stationary points ψ1 and ψ2. In Figure 6.2 these points are denoted as 1 and 2. The local coordinates are shown in Figure 6.3 for point 1 and in Figures 6.4 and 6.5 for point 2.

Figure 6.3 Local polar coordinates at the stationary point 1 (ψst = ψ ).

Figure 6.4 Local coordinates at the stationary point 2 (ψst = π + ψ ) for the observation directions in the interval π − ω ≤ ϑ ≤ π .

Figure 6.5 Local coordinates at the stationary point 2 (ψst = π + ψ ) for the observation directions in the interval 0 ≤ ϑ ≤ .

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6.1 Diffraction at a Canonical Conic Surface 121

Considering the above relationships between coordinates ϕ1, ϕ2, and ϑ , and utilizing Equation (6.17), one can obtain the following approximations for Equations (6.8) and (6.9):

				a			eikR	π	ξeff
us(1) = −u0		2√						+e−ika sin ϑ +i 4		Js(1)(kξ , ϕ0)e−ikξ
							R
				2π ka sin ϑ					0
+	ξeff		Js(1)(kξ , α − ϕ0)e−ikξ cos(α−ϕ1) dξ
	0
					ξeff
+ eika sin ϑ −i π4					0		Js(1)(kξ , ϕ0)e−ikξ cos ϕ2 dξ
+	0		Js(1)(kξ , α − ϕ0)e−ikξ cos(α−ϕ2) dξ ,
	ξeff

cos ϕ1 dξ

(6.19)

uh(1) = −u0

2√

ika

eikR

R

2π ka sin ϑ

× +e−ika sin ϑ +i π4

sin ϕ1

ξeff

Jh(1)(kξ , ϕ0)e−ikξ cos ϕ1 dξ

0

+ sin(α − ϕ1)

ξeff

Jh(1)(kξ , α − ϕ0)e−ikξ cos(α−ϕ1) dξ

0

+ eika sin ϑ −i π4

sin ϕ2

ξeff

Jh(1)(kξ , ϕ0)e−ikξ cos ϕ2 dξ

0

+ sin(α − ϕ2)

0

Jh(1)(kξ , α − ϕ0)e−ikξ cos(α−ϕ2) dξ ,

(6.20)

ξeff

Under the condition ka 1, the scattering sources Js,h(1) near the edge of the conic surface are asymptotically equivalent to those near the edge of the tangential wedge. Hence, the expressions inside the brackets in Equations (6.19) and (6.20) are also asymptotically equivalent to the similar expressions in Equations (4.29) and (4.30), which relate to the wedge diffraction problem. Utilizing this observation, one can rewrite Equations (6.19) and (6.20) as

(1)

=

√

u0a

)f (1)(1)e−ika sin ϑ +i

π

+ f (1)(2)eika sin ϑ −i

π

*

eikR

us

4

4

,

(6.21)

R

2π ka sin ϑ

(1)

=

√

u0a

)g(1)(1)e−ika sin ϑ +i

π

+ g(1)(2)eika sin ϑ −i

π

*

eikR

uh

4

4

.

(6.22)

R

2π ka sin ϑ

These expressions can be interpreted as the ray asymptotics for the field us,h(1).

They show that, under the condition ka sin ϑ

1, this field consists of two diffracted

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