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5.1 Diffraction at a Strip

85

5.1.1Physical Optics Part of the Scattered Field

The Physical Optics (PO) part of the scattered field generated by the uniform components (1.31) of the surface sources is determined by the integrals (1.32). However, this integration process can be avoided. It turns out that the far field can be immediately calculated as the sum of two edge waves described in general form by Equations (3.53) and (3.54):

(0)

= u0

f

(0)

ei(kr1+π/4)

+ f

(0)

ei(kr2+π/4)

,

 

us

 

(ϕ1, ϕ01)

√

 

 

 

 

 

(ϕ2, ϕ02)

√

 

 

 

 

(5.2)

 

2π kr1

 

2π kr2

(0)

 

 

 

 

ei(kr1+π/4)

 

 

 

ei(kr2+π/4)

 

 

uh

= u0

g(0)(ϕ1, ϕ01)

√

 

 

+ g(0)(ϕ2, ϕ02)

√

 

 

.

(5.3)

2π kr1

2π kr2

For the far zone (r

)

us(0) = u0 f (0)

)

uh(0) = u0 g(0)

where

ka2), these expressions can be simplified:

(1)eika(sin φ0

−sin φ) + f (0)(2)e−ika(sin φ0−sin φ)*

ei(kr+π/4)

 

 

√

 

 

 

 

,

(5.4)

2π kr

 

−sin φ) + g(0)(2)e−ika(sin φ0−sin φ)*

i(kr

π/4)

 

 

(1)eika(sin φ0

e√

+

 

 

 

,

(5.5)

2π kr

f (0)(1) ≡ f (0)(ϕ1

, ϕ01),

f (0)(2) ≡ f (0)(ϕ2, ϕ02),

(5.6)

and

 

 

 

g(0)(1) ≡ g(0)(ϕ1

, ϕ01),

g(0)(2) ≡ g(0)(ϕ2, ϕ02).

(5.7)

In accordance with Equation (3.55), functions f (0) and g(0) are defined in terms of the basic coordinates φ and φ0 as

f (0)(1) = −f (0)(2) =

cos φ0

(5.8)

sin φ0 − sin φ

 

and

 

g(0)(1) = −g(0)(2) =

 

cos φ

 

 

,

(5.9)

sin φ0 − sin φ

with −π/2 < φ < 3π/2 and −π/2 < φ0 < π/2.

The field expressions (5.4) and (5.5) possesses a wonderful property. Although all functions (5.8) and (5.9) are singular for the directions φ = φ0 and φ = π − φ0,

their combinations in Equations (5.4) and (5.5) are always finite due to the relationships f (0)(1) = −f (0)(2) and g(0)(1) = −g(0)(2). This property of expressions

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

(5.4) and (5.5) becomes obvious when they are written in the explicit form

 

(0)

 

 

 

(0)

 

 

 

 

ei(kr+π/4)

 

 

 

 

 

us

 

 

 

= u0 s

(φ, φ0)

√

 

 

,

 

 

(5.10)

 

 

 

 

2π kr

 

(0)

 

 

 

(0)

 

 

 

 

ei(kr+π/4)

 

 

 

 

 

uh

 

 

 

= u0 h

(φ, φ0)

√

 

 

,

 

 

(5.11)

 

 

 

 

2π kr

where

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

(0)

(φ, φ

0

)

=

i2 cos φ

0

sin[ka(sin φ − sin φ0)]

,

(5.12)

 

 

s

 

 

 

 

 

sin φ − sin φ0

 

and

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

(0)(φ, φ

0

)

=

i2 cos φ

sin[ka(sin φ − sin φ0)]

.

(5.13)

 

h

 

 

 

 

 

 

 

sin φ − sin φ0

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Now it is clear that

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

s(0)(φ0, φ0) = h(0)(φ0, φ0) = i2ka cos φ0

(5.14)

for the forward direction φ = φ0, and

 

 

 

 

 

 

 

 

 

 

 

s(0)(π − φ0, φ0) = − h(0)(π − φ0, φ0) = i2ka cos φ0

(5.15)

for the specular direction φ = π − φ0.

In accordance with the 2-D form of the optical theorem (Ufimtsev, 2003), the

total scattering cross-section is defined as

 

2

 

 

σ tot =

 

Im (φ0, φ0),

(5.16)

k

which, following Equation (5.14), equals

 

σs(0)tot = σh(0)tot = 2A,

(5.17)

where A = 2a cos φ0 is the “width” of the incident wave part intercepted by the strip (Fig. 5.3). Equation (5.17) can be also interpreted as the total scattered power per unit length of the strip in the z-axis direction.

It is expedient to introduce the 2-D bistatic scattering cross-section σ by a relationship similar to Equation (1.24):

Pavsc

=

σ · Pavinc

(5.18)

 

2π r

 

 

 

 

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5.1 Diffraction at a Strip

87

Figure 5.3 Cross-section A of the incident wave intercepted by the strip.

where

 

1

 

 

2

 

 

1

 

 

2

 

Pavinc =

2

k2Z u0

 

 

,

Pavsc =

2

k2Z usc

.

(5.19)

Therefore in view of Equations (5.10) and (5.11), the bistatic scattering cross-section is defined by

1

 

 

2

 

σs,h(0) =

 

s,h(0)

(φ, φ0) .

(5.20)

k

 

 

 

 

 

 

This is a general definition of the bistatic scattering cross-section σ for any 2-D scattered fields represented in the form of Equations (5.10) and (5.11).

In the case of backscattering when φ = π + φ0, the directivity patterns are given by the simple expression

s(0) = − h(0) = i cot φ0 sin(2ka sin φ0).

(5.21)

Finally one should notice another peculiarity of the directivity patterns (5.12) and (5.13) for the field generated by the uniform components of the surface sources. They have exact zeros in the directions where

ka(sin φ0 − sin φ) = ±nπ ,

with n = 1, 2, 3, . . . .

(5.22)

This property is the consequence of the

relationships f (0)(1)

= −

f (0)(2) and

g(0)(1) = −g(0)(2).

 

 

5.1.2 Total Scattered Field

The nonuniform components of scattering sources concentrate near the edges of the strip and generate the two edge waves described by Equations (4.12) and (4.13). Their

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88 Chapter 5

First-Order Diffraction at Strips and Polygonal Cylinders

 

 

sum equals

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

(1)

= u0

f

(1)

 

 

ei(kr1+π/4)

+ f

(1)

 

 

ei(kr2+π/4)

 

 

us

 

(ϕ1, ϕ01, 2π )

√

 

 

 

 

 

(ϕ2, ϕ02, 2π )

√

 

 

 

(5.23)

 

2π kr1

 

2π kr2

and

 

g(1)(ϕ1, ϕ01, 2π )

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

.

 

(1)

= u0

ei(kr1+π/4)

+ g(1)(ϕ2, ϕ02, 2π )

ei(kr2+π/4)

 

uh

 

√

 

 

 

 

√

 

 

 

(5.24)

 

2π kr1

 

2π kr2

As a result of relationships (4.12) and (4.13) and (4.14) and (4.15) these expressions can be written in the form

 

 

us(1) = us − us(0),

 

 

uh(1) = uh − uh(0).

(5.25)

Therefore, the total first-order scattered field is given by

 

 

 

 

 

 

(0)

(1)

= u0 f (ϕ1, ϕ01, 2π )

ei(kr1+π/4)

 

ei(kr2+π/4)

 

us = us

+ us

√

 

 

 

 

+ f (ϕ2, ϕ02, 2π )

√

 

 

 

 

2π kr1

2π kr2

 

 

 

 

 

 

 

 

 

(5.26)

and

 

= u0 g(ϕ1, ϕ01, 2π )

 

 

 

 

 

 

 

 

 

 

 

 

(0)

(1)

ei(kr1+π/4)

 

ei(kr2+π/4)

uh = uh

+ uh

√

 

 

+ g(ϕ2, ϕ02, 2π )

√

 

 

2π kr1

2π kr2

(5.27)

with functions f and g defined in Equations (2.62) and (2.64). For the far zone these field expressions are simplified to

us = u0 )f (1)eika(sin φ0−sin φ) + f (2)e−ika(sin φ0

−sin φ)*

ei(kr+π/4)

 

 

√

 

 

 

 

(5.28)

2π kr

 

and

 

 

 

 

 

 

 

 

uh = u0 )g(1)eika(sin φ0−sin φ) + g(2)e−ika(sin φ0

−sin φ)*

ei(kr+π/4)

 

 

√

 

 

,

(5.29)

2π kr

where

 

f (1) ≡ f (ϕ1, ϕ01, 2π ),

f (2) ≡ f (ϕ2, ϕ02, 2π ),

and

(5.30)

g(1) ≡ g(ϕ1, ϕ01, 2π ),

g(2) ≡ g(ϕ2, ϕ02, 2π ).

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5.1 Diffraction at a Strip

89

In terms of basic coordinates φ and φ0, these functions are determined by

f (1)

 

 

1

 

 

1

 

 

 

 

g(1) =

 

 

 

2

 

 

 

 

 

 

 

 

 

 

 

2

− sin

φ − φ0

 

 

 

 

 

 

 

 

and

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

f (2)

=

1

 

 

 

1

 

 

 

 

 

 

 

 

2

 

 

g(2)

2

 

sin

φ − φ0

 

 

 

 

 

 

 

 

 

 

 

 

 

 

1 , with − π/2 ≤ φ ≤ 3π/2

cos

φ + φ0

2

(5.31)

2

 

 

 

 

1

 

, with π/2

φ

π/2, (5.32)

 

 

cos

φ + φ0

 

−

≤

≤

 

 

 

 

 

but

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

f (2)

 

1

 

 

1

 

 

1

 

 

,

with π/2 φ

3π/2.

 

 

 

2

 

2

 

 

 

 

 

 

 

 

≤

≤

 

 

 

 

 

 

φ − φ0

 

 

 

φ + φ0

 

 

 

g(2) =

2

 

− sin

 

± cos

 

 

 

 

 

 

 

 

 

 

 

 

(5.33)

As can be seen, functions g(2) and ∂f (2)/∂φ are discontinuous in the direction

φ= π/2. This is a consequence of the fact that they relate to the field generated by the scattering sources distributed over the whole half-plane −a < y < ∞. Functions g(1) and ∂f (1)/∂φ are also discontinuous. They have different values in the directions

φ= −π/2 and φ = 3π/2 related to the different sides of the half-plane −∞ < y < a containing the sources of the field.

This discontinuity in the field of the first-order edge waves (5.28) and (5.29) can be eliminated in two ways. First, in the calculation of the field generated by

the nonuniform component js,h(1), one should restrict the integration region by the actual surface of the strip. In other words, one should truncate (outside the strip) the component js,h(1) related to the semi-infinite half-plane. This approach is presented in Section 5.1.4 (see also Michaeli (1987) and Johansen (1996)). Another remedy is the calculation of multiple edge diffraction (Ufimtsev, 2003).

In view of Equations (5.31) to (5.33), the field expressions (5.28) and (5.29) can be written in the form

us,h = u0 s,h(φ, φ0)

ei(kr+π/4)

 

 

√

 

 

,

(5.34)

2π kr

 

 

 

 

 

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