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5.2 Diffraction at a Triangular Cylinder

99

as the part of the true edge waves arising due to the secondary diffraction. The true secondary edge waves are described by the TED approximation (5.47), which leads to the following asymptotic expression

TED(φ, φ )

=

g(1)eika(sin φ0−sin φ)

+

g(2)e−ika(sin φ0−sin φ)

 

h

0

+

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

eika(sin φ0+sin φ)

+

 

e−ika(sin φ0+sin φ)

 

 

 

√

 

 

 

√

 

√

 

√

 

 

 

 

1 + sin φ)

1 + sin φ0

1 − sin φ

1 − sin φ0

 

 

 

×

ei(2ka+π/4)

 

 

 

 

 

 

 

 

 

 

 

 

√

 

.

 

 

 

 

 

 

 

(5.70)

 

 

 

 

π ka

 

 

 

 

 

 

 

The function trh (φ, φ0) contains the terms singular in the specular direction φ = π − φ0. However, such singular terms cancel each other and generate the finite quantity for this function:

htr (π − φ0, φ0)

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

√

 

 

 

 

 

 

 

 

 

 

 

 

 

 

√

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

eiπ/4

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

2ka(1

+

sin φ0)

2

 

 

 

 

2ka(1

−

sin φ0)

= i2ka cos φ0 − cos φ0

4ka

√

 

 

 

 

 

 

 

 

 

 

 

 

eit

dt +

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

π

0

 

 

 

 

 

 

 

0

 

 

 

 

 

 

+

 

 

 

 

 

ei2ka(1+sin φ0)

 

 

 

 

ei2ka(1−sin φ0)

−

ei3π/4

 

 

 

 

 

 

 

2ka

ei3π/4

+

 

 

 

 

 

 

 

 

 

 

√

 

 

 

√

 

 

 

√

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

π

 

1 + sin φ0

1 − sin φ0

π

 

 

 

 

 

 

 

 

×

 

 

 

 

 

 

√

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

√

 

 

 

 

 

 

 

 

 

1

 

 

2ka(1

+

sin φ0)

eit

2

dt +

 

 

1

 

 

 

 

 

2ka(1

−

sin φ0)

eit

2

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

dt

1 + sin φ0

 

0

 

 

 

 

 

 

 

 

 

1 − sin φ0

0

 

 

 

 

 

 

 

eit2 dt

.

(5.71)

Expressions (5.68) and (5.71) were used to calculate the normalized scattering crosssection (5.45). The results are plotted in Figure 5.8. It is seen that the truncated PTD is in good agreement with the exact asymptotic theory identified here as the TED (Ufimtsev, 2003). The significant improvement compared to the nontruncated version of PTD (Fig. 5.6) has been achieved in the vicinity of the directions φ = 90◦ and φ = 270◦.

5.2DIFFRACTION AT A TRIANGULAR CYLINDER

The relationships us = Ez and uh = Hz exist between the acoustic and electromagnetic

fields for this problem.

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

Figure 5.8 Scattering at a hard strip predicted by the truncated version of PTD (Hz -polarization of the electromagnetic field).

For simplicity we consider here the diffraction at a cylinder with the cross-section in the form of an equilateral triangle (Fig. 5.9). Two special cases will be investigated: (a) Symmetric case, when the incident plane wave propagates in the direction parallel to the bisector of the triangle; (b) Backscattering, when the scattered field is evaluated for the direction from which the incident wave comes. First, we study these problems utilizing the PO approximation, and after that it will be corrected by taking into account the first-order edge waves generated by the nonuniform scattering sources js,h(1).

Figure 5.9 Cross-section of the scattering cylinder. Numbers 1, 2, and 3 denote the edges; r and φ are polar coordinates of the field point.

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5.2 Diffraction at a Triangular Cylinder

101

5.2.1Symmetric Scattering: PO Approximation

The incident wave,

uinc = u0eikx ,

(5.72)

generates the identical scattering sources js,h(0) at faces 1–2 and 1–3 of the cylinder (Fig. 5.9), which are symmetric with respect to the x-axis. The soft (1.5) or hard (1.6) boundary conditions are imposed on the faces. The scattering cylinder is equilateral; the width of each face is equal to l, and each internal angle between faces equals 60◦. The Cartesian coordinates of the edges 1, 2, and 3 are (0, 0), (h, a), and (h, −a), respectively, where h = l cos(π/6) and a = l/2. Due to the symmetry of the problem, it is sufficient to calculate the scattered field in the directions 0 ≤ φ ≤ π .

The traditional integration technique for calculation of the PO scattered field can be avoided here. Indeed, in this approximation, the scattering cylinder can be considered as the combination of the strips 1–2 and 1–3. As was shown in Section 5.1.1, the far field scattered by a strip consists of two edge waves determined by Equations (3.53) and (3.54). We omit all routine calculations related to the transition from the local polar coordinates (r1,2,3, ϕ1,2,3) (used for description of individual edge waves) to the basic coordinates (r, φ), and provide the final equations for the scattered field in the region r kl2, 0 ≤ φ ≤ π :

(0)

(0)

 

ei(kr+π/4)

 

 

us,h = u0

s,h

(φ, 0)

√

 

 

,

(5.73)

2π kr

where

 

 

 

 

 

 

 

s(0) = f (0)(1) + f (0)(2)eiψ2 + f (0)(3)eiψ3

(5.74)

and

 

 

 

 

 

 

 

h(0) = g(0)(1) + g(0)(2)eiψ2 + g(0)(3)eiψ3 ,

(5.75)

with

 

 

 

 

 

 

 

ψ2 = kh(1 − cos φ) − ka sin φ

and

ψ3 = kh(1 − cos φ) + ka sin φ.

(5.76)

 

It is understood here that the uniform sources js,h(0) induced on each face (1–2 and 1–3),

radiate the field in the entire surrounding space (0 ≤ φ ≤ 2π ).

 

 

 

 

 

 

 

 

 

Functions f (0) and g(0) determine the directivity patterns of the edge waves identified by the corresponding numbers in the argument of these functions. They are given as

 

 

 

 

sin

π

 

 

 

 

 

π

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

sin

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

f (0)(1)

 

 

 

6

 

 

 

 

 

 

6

 

 

 

,

(5.77)

= cos %

π

− φ& − cos

π

+ cos %

π

+ φ& − cos

π

 

 

 

 

6

6

 

6

6

 

 

 

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Here we assume that the incident wave
uinc = u0eik(x cos φ0+y sin φ0)

102 Chapter 5

First-Order Diffraction at Strips and Polygonal Cylinders

 

 

 

 

 

sin

%

π

 

 

 

 

 

 

 

 

sin

%

π

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

g(0)(1)

 

 

6

 

− φ&

 

 

 

6 + φ&

 

 

 

,

 

 

 

 

 

(5.78)

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

= cos %

π

− φ& − cos

π

+ cos %

π

+ φ& − cos

π

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

6

 

6

 

6

6

 

 

 

 

 

&

 

 

 

 

 

 

 

 

 

 

sin

 

π

 

 

 

 

 

 

 

 

 

 

 

 

sin

π

−

φ

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

6

 

 

 

 

 

 

 

 

 

 

 

 

 

6

 

 

 

f (0)(2)

 

 

 

 

 

 

 

 

 

 

 

,

g(0)(2)

 

 

 

 

 

%

 

 

,

(5.79)

= cos

π

 

− cos %

π

− φ&

= cos

π

− cos %

π

− φ&

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

6

 

6

 

 

 

 

 

6

 

6

 

 

 

 

 

 

 

 

 

sin

 

π

 

 

 

 

 

 

 

 

 

 

 

 

sin

π

+

φ

&

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

6

 

 

 

 

 

 

 

 

 

 

 

 

 

6

 

 

 

f (0)(3)

 

 

 

 

 

 

 

 

 

 

 

,

g(0)(3)

 

 

 

 

 

%

 

 

.

(5.80)

= cos

π

 

− cos %

π

+ φ&

= cos

π

− cos %

π

+ φ&

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

6

 

6

 

 

 

 

 

6

 

6

 

 

All these functions have singularities in the

forward direction φ

=

0, which cancel

(0)

 

each other, being substituted into the functions s,h . This results in the expression

s(0)(0, 0) = h(0)(0, 0) = ikl.

 

(5.81)

A similar situation with compensation of the singularities occurs for the specular direction φ = π/3, when

s(0)%

π

, 0&

 

l

 

π

h(0)%

π

, 0&

 

l

1

 

 

 

 

= ik

 

+ tan

 

eiψ3 ,

 

= −ik

 

+

 

 

 

eiψ3 . (5.82)

3

2

6

3

2

cos

π

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

6

 

 

Due to the symmetry of the problem, the same field is scattered in the direction of the specular reflection from the face 1–3:

s(0) %−

π

, 0&

= s(0)%

π

, 0& ,

h(0)%−

π

, 0&

= h(0) %

π

, 0& . (5.83)

 

 

 

 

3

3

3

3

It is worth noting that the sum of the dominant terms in the reflected fields (5.82) and (5.83) equals ±ikl and conforms to the field (5.81) scattered in the forward direction. This result is in agreement with the fundamental law that the total power of the reflected field (1.79) is asymptotically (with kl 1) equal to the total power of the shadow radiation (1.83).

5.2.2Backscattering: PO Approximation

(5.84)

propagates in the direction determined by the angle φ0 given in the interval −π ≤ φ0 ≤ 0, and the scattered field is evaluated in the opposite direction φ = π + φ0.

The basic feature of the PO approximation for the backscattering follows from the properties of functions f (0) and g(0) defined in Equations (3.55) to (3.57) through

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5.2 Diffraction at a Triangular Cylinder

103

the local polar angles ϕ and ϕ0. These angles determine the directions to the field point and to the source of the incident wave, respectively. For the backscattering direction ϕ = ϕ0, it turns out that f (0)(ϕ0, ϕ0) = −g(0)(ϕ0, ϕ0). This means that the fields scattered back to the source by the acoustically soft and hard cylinders differ only in sign, and therefore their directivity patterns differ in this way as well:

s(0) = − h(0).

(5.85)

It is clear from the previous section that the scattered field consists of the sum of the edge waves. We again omit simple routine calculations of these waves and present the final expressions for the directivity patterns of the total scattered field. We have different expressions for different intervals of observation because of the different number of contributions to the scattered field.

In the interval 0 ≤ φ ≤ π/6, only two diffracted waves exist incoming from edges 2 and 3 (Fig. 5.9). In this interval,

(s0)(φ) = f (0)(2)eiψ2 + f (0)(3)eiψ3

where

 

 

 

 

 

 

 

1

 

 

f (0)(2) = −f (0)(3) = −

 

cot φ,

2

ψ2 = −2k(h cos φ + a sin φ),

ψ3 = −2k(h cos φ − a sin φ)

with h = l cos(π/6) and a = l/2.

 

 

 

 

 

 

 

Equation (5.86) can be rewritten in the form

 

 

 

 

(0)(φ)

=

i cos φ

sin(2ka sin φ)

e−i2kh cos φ

 

s

 

 

 

sin φ

 

 

 

 

which predicts the value

 

 

 

 

 

 

 

 

 

 

(0)(0)

=

i2kae−i2kh

=

ikle−i2kh.

s

 

 

 

 

 

 

 

 

(5.86)

(5.87)

(5.88)

(5.89)

(5.90)

In the interval π/6 < φ < π/2, the scattered field consists of the three edge

waves, and respectively

 

 

 

 

 

 

 

 

 

 

 

 

 

s(0)(φ) = f (0)(1) + f (0)(2)eiψ2 + f (0)(3)eiψ3 ,

(5.91)

where

 

 

 

 

tan %

 

 

 

 

− φ& ,

 

 

f (0)(1) = −

1

π

 

(5.92)

 

 

 

 

 

 

 

2

 

6

 

 

 

 

 

1

 

 

 

π

 

 

f (0)(2) =

 

 

)tan %

 

 

 

− φ& − cot φ

* ,

(5.93)

2

6

and

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

1

 

 

 

 

 

f (0)(3) =

 

cot φ.

 

(5.94)

 

2

 

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