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6.2 Scattering at a Disk 129

Equations we obtain in this way reveal a ray structure of the ﬁeld, which consists of two diffracted rays coming from the stationary points 1 and 2:

us(0) = √ u0a

2π ka sin ϑ

and

uh(0) = √ u0a

2π ka sin ϑ

where the functions

and

)f (0)(1)e−ika sin ϑ +i π4 + f (0)(2)eika sin ϑ −i π4 * eikR

)g(0)(1)e−ika sin ϑ +i π4 + g(0)(2)eika sin ϑ −i π4 * eikR ,

f (0)(1) = −f (0)(2) = − 1

sin ϑ

g(0)(1) = −g(0)(2) = − cos ϑ sin ϑ

(6.60)

(6.61)

(6.62)

(6.63)

determine the directivity patterns of diffracted rays in the PO approximation.

Electromagnetic Waves

To show the relationship between the acoustic and electromagnetic diffracted ﬁelds, we present here the PO approximation for the ﬁeld scattered at a perfectly conducting disk (Uﬁmtsev, 1962):


Eϑ(0) = Z0Hϕ(0) = −iaZ0H0x sin ϕ	cos ϑ				eikR
			J1(ka sin ϑ )
	sin ϑ				R
and
1						eikR
Eϕ(0) = −Z0Hϑ(0) = −iaZ0H0x cos ϕ				J1(ka sin ϑ )			.	(6.64)
		sin ϑ				R
This ﬁeld is due to the diffraction of the incident wave
Eyinc = −Z0Hxinc = −Z0H0x eikz = E0yeikz,								(6.65)

where Z0 = √μ0/ε0 = 120π ohms is the impedance of free space. In view of the equations

E0ϕ = E0y cos ϕ = −Z0H0x cos ϕ,				(6.66)
the expressions (6.64) can be rewritten as
Eϕ(0) = E0ϕ	ia		eikR
		J1(ka sin ϑ )
	sin ϑ		R

TEAM LinG

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

and

Hϕ(0) = H0ϕ	ia		eikR
		cos ϑ J1(ka sin ϑ )		.	(6.67)
	sin ϑ		R

Their comparison with Equations (6.56) and (6.57) reveals the following equivalence existing between the acoustic and electromagnetic diffracted ﬁelds:

us(0) = Eϕ(0),	if u0 = E0ϕ ,	(6.68)
uh(0) = Hϕ(0),	if u0 = H0ϕ .

These equations are in an agreement with relationships (1.100) and (1.101).

6.2.2 Field Generated by Nonuniform Scattering Sources

The ﬁeld generated by the nonuniform scattering sources js,h(1) was investigated in Section 6.1.4. Here we reproduce the related approximations


us(1) = u0	a eikR			0 f (1)(1)[J0(ka sin ϑ ) − i J1(ka sin ϑ )]
	2		R
+ f (1)(2)[J0(ka sin ϑ ) + i J1(ka sin ϑ )]1						(6.69)
and
uh(1) = u0		a eikR		0g(1)(1)[J0(ka sin ϑ ) − i J1(ka sin ϑ )]
		2	R
+ g(1)(2)[J0(ka sin ϑ ) + i J1(ka sin ϑ )]1					,	(6.70)

where 0 ≤ ϑ ≤ π . The ray-type asymptotics of (6.69) and (6.70) are shown in Equations (6.21) and (6.22). The focal asymptotics of (6.69) and (6.7) are determined by

Equations (6.41) and (6.42).

General expressions for functions f (1)(1), f (1)(2) and g(1)(1), g(1)(2) are given by Equations (6.25) to (6.31). For the scattering disk, they are written below. Functions f (1)(1) and g(1)(1) are described in the entire region 0 ≤ ϑ ≤ π by

	1	1			1			1
f (1)(1) = −					+			+		(6.71)
	2		sin	ϑ		cos	ϑ		sin ϑ

TEAM LinG

6.2 Scattering at a Disk 131

and

cos ϑ

g(1)(1) =

−

(6.72)

sin

cos

sin ϑ

Functions f (1)(2) and g(1)(2) are described by

f (1)(2) =

−

(6.73)

sin

cos

sin ϑ

and

cos ϑ

g(1)(2) =

−

(6.74)

sin

cos

sin ϑ

in the region 0 ≤ ϑ ≤ π/2, and by

	1				1			1
		1
f (1)(2) =		−			+			−		(6.75)
	2		sin	ϑ		cos	ϑ		sin ϑ

and

cos ϑ

g(1)(2) = −

−

(6.76)

sin

cos

sin ϑ

	2	ϑ	≤	π .
in the region π/	(1≤)		≤		(1)	(2) are symmetric with respect to the disk plane,
Functions f	(1) and f					(2) are symmetric with respect to the disk plane,
f (1)(1, π − ϑ ) = f (1)(1, ϑ ),							f (1)(2, π − ϑ ) = f (1)(2, ϑ ),	(6.77)
but functions g(1)(1) and g(1)(2) are antisymmetric,
g(1)(1, π − ϑ ) = −g(1)(1, ϑ ),							g(1)(2, π − ϑ ) = −g(1)(2, ϑ ).	(6.78)

Here, the fun ctions with the argument π − ϑ correspond to the left half-space (z < 0). It follows from Equations (6.71) to (6.76) that

f (1)(1) = f (1)(2) = −	1	, for ϑ = 0 and ϑ = π ,	(6.79)
	2

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

and

	1		0
g(1)(1) = g(1)(2) = ±		,	for ϑ = π	.	(6.80)
	2

After substitution of these values into Equations (6.41) and (6.42), it follows that

us(1)	= −	u0a eikR		,	for ϑ = 0 and ϑ = π ,		(6.81)
		2	R
and
(1)		u0a eikR			0
uh	= ±			,	for ϑ = π	.	(6.82)
		2	R

It is worth noting that the amplitude of the focal ﬁeld generated by the nonuniform sources does not depend on frequency.

6.2.3 Total Scattered Field

The sum

us,h = us,h(0) + us,h(1)

(6.83)

provides the ﬁrst-order PTD approximation for the scattered ﬁeld:

•Quantities us,h(0) and us,h(1) are deﬁned by Equations (6.56) and (6.57) and Equations (6.69) and (6.70).

•Their rays-type asymptotics are determined in Equations (6.60), (6.61) and (6.21), (6.22).

When they are included in Equation (6.83), the functions f (0), g(0) contained both in (6.60) and (6.61) and in (6.21) and (6.22) cancel each other. As a result, the ray asymptotics for the total ﬁeld contain only the Sommerfeld functions f and g: