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s(φ0, φ0) = i2ka cos φ0 −

(φ, ϕ0)) do not have

90 Chapter 5

First-Order Diffraction at Strips and Polygonal Cylinders

where

(φ, φ

)

= −

cos[ka(sin φ0 − sin φ)]

sin[ka(sin φ0 − sin φ)]

φ0 + φ

φ0 − φ

cos

sin

with

− π/2 ≤ φ ≤ π/2,

(5.35)

(φ, φ

)

cos[ka(sin φ0 − sin φ)]

−

sin[ka(sin φ0 − sin φ)]

φ0 − φ

φ0 + φ

sin

cos

with π/2 ≤ φ ≤ 3π/2,

(5.36)

and

(φ, φ

)

cos[ka(sin φ0 − sin φ)]

sin[ka(sin φ0 − sin φ)]

φ0 + φ

φ0 − φ

cos

sin

with

− π/2 ≤ φ ≤ π/2,

(5.37)

(φ, φ

)

cos[ka(sin φ0 − sin φ)]

sin[ka(sin φ0 − sin φ)]

φ0 − φ

φ0 + φ

sin

cos

with π/2 ≤ φ ≤ 3π/2.

(5.38)

As can be seen, these functions (in contrast to functions (s,h0) exact zeros in the directions determined by Equation (5.22).

In the following we provide the explicit expressions of functions s,h for certain speciﬁc directions of observation. For the forward direction φ = φ0,

(5.39)

cos φ0

and
1
h(φ0, φ0) = i2ka cos φ0 +				,		(5.40)
	cos φ0
and for the specular direction φ = π − φ0,
1
s(π − φ0, φ0) = i2ka cos φ0 −						(5.41)
		cos φ0
and
1
h(π − φ0, φ0) = −i2ka cos φ0 −					.	(5.42)
			cos φ0
In addition, for the backscattering direction φ = π + φ0,
		sin(2ka sin φ0)
s(π + φ0, φ0) = − cos(2ka sin φ0) + i						(5.43)
				sin φ0

TEAM LinG

5.1 Diffraction at a Strip

and

h(π + φ0, φ0) = − cos(2ka sin φ0) − i	sin(2ka sin φ0)	.	(5.44)
	sin φ0

In conformity with Equations (4.20) and (4.21), functions (5.39) to (5.42) are singular under the grazing incidence (φ0 = ±π/2).

A comparison with Equation (5.14) shows that the ﬁrst term in Equations (5.39) and (5.40) is caused by the uniform component, and the second term by the nonuniform component of the surface sources. In view of Equation (5.16), this observation gives the impression that the ﬁeld generated by the nonuniform component does not contribute to the total scattered power. An interesting question arises. How does it happen that the nonuniform component j(1) generates the ﬁeld (5.23) and (5.24), but does not contribute to the total scattered power? The answer is the following. The ﬁeld (5.23) and (5.24) does contribute to the total scattered ﬁeld, but through the high-order edge waves generated due to multiple diffraction of the primary edge waves (5.34).

An additional comment is necessary on the ﬁeld equations (5.28) and (5.29) and functions f and g involved in these equations. Functions f and g are singular in the forward (φ = φ0) and specular (φ = π − φ0) directions related to the geometrical optics boundaries of the incident and reﬂected rays. However, these singularities cancel each other in the ﬁeld Equations (5.28) and (5.29) and provide ﬁnite values in these special directions, as is seen in Equations (5.39) to (5.42). This cancellation is due to the fact that the incident wave is a plane wave. Because of this, the geometrical optics boundaries related to the different edges of the strip are parallel to each other (Fig. 5.3) and merge in the far zone from the strip. This case represents an exception as compared to a more general situation when the source of the incident wave can be at a ﬁnite distance from the scattering object. For example, in the case of the cylindrical incident wave (Fig. 5.4), the shadow boundaries caused by the strip edges are not parallel and therefore the related singularities of functions f and g are separated in space and cannot cancel each other. In such a case, the traditional PTD procedure consists of the calculation of the ﬁelds generated separately by the uniform and nonuniform

Figure 5.4 Shadow boundaries caused by the strip edges in the ﬁeld of the incident cylindrical wave. At these boundaries, functions f and g are singular: f = ∞ and g = ∞.

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

components of the scattering sources with the subsequent summation of these ﬁelds. Each of these ﬁelds is ﬁnite and hence their sum is ﬁnite as well.

5.1.3Numerical Analysis of the Scattered Field

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

ﬁelds for this problem.

In this section we present the numerical analysis of the scattered ﬁeld by utilizing different approaches. Note that the geometry of the problem is shown in Figure 5.1. The soft (1.5) or hard (1.6) boundary conditions are imposed on the strip. The incident wave is the plane wave (5.1). The relationships us = Ez and uh = Hz show the equivalence between the acoustic and electromagnetic ﬁelds studied in this section. Equation (5.20) determines the bistatic scattering cross-section. The calculated quantity is the normalized scattering cross-section

σs,h		(φ, φ )		2
kl2	=	s,h kl	0	,	where l = 2a,	(5.45)

which is plotted in the decibel scale. For parameters ka and φ0 we take the values ka = 3π and φ0 = 45◦, that is, ϕ0 = π/4 rad. In this case, the strip width l = 2a equals 3λ and the application of the asymptotic theory is justiﬁed. The scattering cross-section is investigated in the directions π/2 ≤ φ ≤ 3π/2 around the strip. One should note that the function s is symmetrical with respect to the strip plane [ s(π − φ, φ0) =s(φ, φ0)], and the function h is anti-symmetrical [ h(π − φ, φ0) = − h(φ, φ0)]. That is why it is sufﬁcient to calculate these functions only in the interval π/2 ≤ φ ≤ 3π/2.

Calculations are performed using the following approaches:

•The PO approximation with the functions (s,h0) given in Section 5.1.1;

•The ﬁrst-order PTD approximation with the functions s,h presented in Section 5.1.2;

•The ﬁrst-order TED approximation (Uﬁmtsev, 2003) with the functions

(φ,

φ0) = ˜ 1

)

+ ˜ 1

(

−

α,

− 0

(5.46)

(α, α

α )

and

φ0) = ˜ 1

+ ˜ 1

−

, − 0

(φ,

)

(5.47)

(α, α

(

TEAM LinG

5.1 Diffraction at a Strip 93

where

√

eiχ (α+α0)

√

(α, α

)

1 − α

1 − α0

1 − α

1 + α0

˜ 1

α + α0

−

α + α0

−

− α

ψ (q, α)

1 +

α0

ψ (q,

)

eiq

iχ (α

α0)

−

2Ds

√1

−

ei2q

ψ (q,

)eiχ α0

− '

ψ (q, −α0)ψ (q, 1)eiq−iχ α0

* ψ (q, α)eiχ α ,

1 + α0

(5.48)

√

eiχ (α+α0)

(α, α

)

1 + α

1 + α0

= −

+ α + α0

˜ 1

α + α0

−

)

1 + α

ϕ(q,

)

α)

− α0

ϕ(q, α)

× eiq+iχ (α−α0)

−

ei2q -ϕ(q, α0)eiχ α0

− ϕ(q, −α0)ϕ(q, 1)eiq−iχ α0 . ϕ(q, α)eiχ α .

(5.49)

In these expressions,

α = sin φ,

α0 = − sin φ0,

χ = ka,

q = 2χ = 2ka,

(5.50)

ϕ(q, α)

e−iπ/4e−iq(1+α)

∞

eit2 dt,

(5.51)

= √

√

∂ϕ(q, α)

q(1+α)

ψ (q, α) =

√

(5.52)

∂q

2(1 + α)

and

Ds = 1 − ψ 2(q, 1)ei2q,

Dh = 1 − ϕ2(q, 1)ei2q.

(5.53)

The absolute error of the TED approximation (5.46) and (5.47) is equal to

where

Qs,h(α, α0) = Q˜ s,h(α, α0) + Q˜ s,h(−α, −α0)

(5.54)

Q˜ s(α, α0) =

O(q−1/2)

(5.55)

and

[1 + q(1 + α)][1 + q(1 − α0)]

√

O(√

)

α0

Q˜ h

(α, α

)

(5.56)

= [1 + q(1 + α)][1 + q(1 + α0)]

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