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80

 

Chapter 4 Radiation by the Nonuniform Component of Surface Sources

 

 

 

 

or

 

 

 

 

 

 

 

 

 

 

 

 

 

eik√

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

∞

 

∞

 

 

 

 

 

u(0)

 

 

u

eikz cos γ

ik1 sin ϕ0

e−ik1ξ cos ϕ0 dξ

 

 

(x−ξ )2+y2+s2

 

ds,

=

 

 

 

 

 

eiks cos γ

 

 

 

 

 

 

 

 

 

 

 

s

0

 

 

2π

 

 

 

'

 

 

 

 

 

 

 

 

 

 

 

 

0

 

−∞

(x

−

ξ

)

2

+

y2

+

s2

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

(4.41)

 

 

 

 

 

 

 

 

 

 

 

eik√

 

 

 

(0)

 

 

 

 

1 ∂

∞

e−ik1ξ cos ϕ0 dξ

∞

 

 

(x−ξ )2+y2+s2

 

 

uh

 

= −u0eikz cos γ

 

 

 

 

eiks cos γ

 

 

 

 

 

 

 

 

 

 

 

ds.

 

2π ∂y

 

 

 

 

 

 

 

 

 

 

 

 

 

0

'(x −

 

 

2

+

y2

+

s2

 

 

 

 

 

 

 

 

 

 

 

−∞

ξ )

 

 

(4.42)

The integrals over the variable s still contain the wave number k for the normal

incidence and need further investigation. Let us rewrite them as

√

∞

eik

D2

s2

 

 

 

 

ds, with D = ((x − ξ )2 + y2.

 

−∞ eiks cos γ

 

 

 

 

+

(4.43)

'

 

 

 

D2

 

s2

 

 

 

+

 

 

 

 

We then return to the integral (3.8) for the Hankel function and make the following

changes: w = s, z = t, d = −ip, k = −iD, q =

p2 − t2

with D > 0 and Im q > 0.

After these manipulations it follows from (3.8) that

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

√

 

 

 

 

 

 

 

 

 

'

 

 

 

 

 

 

√

 

 

 

 

 

 

 

∞

 

 

 

 

2

+s

2

 

 

 

 

 

 

 

 

 

∞

 

 

 

2

+s

2

 

H(1)

(qD)

1

e−ist

eip

 

D

 

ds

1

 

 

eist

eip

 

D

 

ds. (4.44)

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

0

= iπ

 

 

 

 

 

 

2

+ s

2

 

 

 

 

 

π

 

 

 

 

 

2

+ s

2

 

 

−∞

 

 

'D

 

 

 

 

 

 

= i

 

 

 

−∞

 

 

'D

 

 

 

 

 

 

 

 

 

 

 

 

 

 

2

2

 

 

 

 

 

 

 

 

 

 

By setting here t = k cos γ , p = k, q = 2'p2

− t

= k sin γ = k1, we find

 

 

 

∞

 

 

 

 

 

 

 

√

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

iks cos γ eik

 

D

+s

 

 

 

 

 

(1)

 

 

 

 

 

 

 

 

 

 

 

−∞ e

 

 

 

 

 

 

 

 

ds = iπ H0

 

(k1D).

 

 

 

(4.45)

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

D2 + s2

 

 

 

 

This relationship allows one to rewrite the PO fields (4.41) and (4.42) in the form

 

 

 

 

 

 

 

'

 

 

 

 

 

 

 

 

 

 

 

k1 sin ϕ0

∞

 

 

 

k1

 

 

 

dξ ,

 

(0)

 

 

 

(1)

((x − ξ )2 + y2

 

us

= −u0eikz cos γ

 

 

 

 

 

 

e−ik1ξ cos ϕ0 H0

(4.46)

2

 

0

 

 

i ∂

∞

 

 

 

k1

 

 

 

 

dξ .

 

(0)

= −u0eikz cos γ

e−ik1ξ cos

(1)

((x − ξ )2 + y2

 

us

 

 

 

 

 

ϕ0 H0

(4.47)

2 ∂y

0

Comparison with expressions (3.10) and (3.11) finally confirms that the PO fields really satisfy the above rule for the transition to oblique incidence.

Thus, it has been proved that this rule is applicable both to the exact solution and to the PO approximation. Hence, this rule is also applicable to their difference, which is the field us,h(1) generated by the nonuniform component js,h(1) of the surface sources of the diffracted field. By the application of this rule to Equations (4.12) and (4.13), one can easily find the field us,h(1) generated under the oblique incidence:

(1)

u0 f

(1)

 

ei(k1r+π/4)

 

ikz cos γ

 

 

us

 

(ϕ, ϕ0, α)

√

 

 

e

 

,

(4.48)

 

2π k1r

 

TEAM LinG

4.3 Oblique Incidence of a Plane Wave at a Wedge 81

Figure 4.4 Cone of diffracted rays.

(1)

u0 g

(1)

 

ei(k1r+π/4)

 

ikz cos γ

 

 

uh

 

(ϕ, ϕ0, α)

√

 

 

e

 

,

(4.49)

 

2π k1r

 

where the functions f (1) and g(1) are defined in Section 4.1 and the angle ϕ0 is determined by Equation (4.36).

The waves (4.48) and (4.49) have the form of conic waves and can be interpreted in terms of diffracted rays. Indeed, their eikonal

S = z cos γ + r sin γ

(4.50)

describes the phase fronts in the form of conic surfaces where S = const. The gradient of the eikonal

S = zˆ cos γ + rˆ sin γ

(4.51)

indicates the directions of the edge diffracted rays. They are distributed over a cone surface shown in Figure 4.4. The axis of this cone is directed along the edge. All rays form the same angle γ with the edge as the incident ray.

In the case of electromagnetic waves

Ezinc = E0zeikz cos γ e−ik1r cos(ϕ−ϕ0)

(4.52)

and

 

Hzinc = H0zeikz cos γ e−ik1r cos(ϕ−ϕ0)

(4.53)

TEAM LinG

82 Chapter 4 Radiation by the Nonuniform Component of Surface Sources

incident at a perfectly conducting wedge, the following aymptotics diffracted far field (Ufimtsev, 2003):

(0)

= [E0z f

(0)

(ϕ, ϕ0) − H0z cos γ ]

eik1r+iπ/4

ikz cos γ

 

Ez

 

 

√

 

e

 

,

 

 

2π k1r

 

(0)

= H0zg

(0)

 

 

eik1r+iπ/4

ikz cos γ

 

 

 

 

Hz

 

 

(ϕ, ϕ0)

√

 

e

 

 

,

 

 

 

 

 

2π k1r

 

 

 

 

 

describe the

(4.54)

(4.55)

(1)

= [E0z f

(1)

(ϕ, ϕ0, α) + H0z cos γ ]

eik1r+iπ/4

ikz cos γ

 

 

Ez

 

 

√

 

e

 

,

(4.56)

 

 

2π k1r

 

(1)

= H0zg

(1)

 

 

eik1r+iπ/4

ikz cos γ

 

 

 

 

 

Hz

 

 

(ϕ, ϕ0, α)

√

 

e

 

 

.

 

 

 

(4.57)

 

 

2π k1r

 

 

 

 

 

The second term inside the brackets in Equation (4.54) shows the polarization coupling in the PO field under the oblique incidence. It also reveals itself in the second term of Equation (4.56). However, the field generated by the total surface current

(t) = (0) + (1)

j j j is free from this polarization coupling,

(t)

 

eik1r+iπ/4

ikz cos γ

 

 

Ez

= E0z f (ϕ, ϕ0, α)

√

 

 

e

 

,

(4.58)

2π k1r

 

(t)

= H0zg(ϕ, ϕ0, α)

eik1r+iπ/4

ikz cos γ

 

 

Hz

√

 

 

e

 

.

(4.59)

2π k1r

 

One should mention that the diffraction cone was first discovered theoretically for arbitrary curved edges by Rubinowicz (1924). In the case of curved edges, the axis of the diffraction cone is directed along the tangent to the edge at the diffraction point. He made this discovery by the asymptotic evaluation of the Kirchhoff diffraction integral (Rubinowicz, 1924). He also established that the edge diffracted rays satisfy Fermat’s principle. Later on, this concept of edge diffracted rays was included into the Geometrical Theory of Diffraction (Keller, 1962). The ray interpretation of the edge diffracted waves (4.48) and (4.49) was also suggested in the PTD (Ufimtsev, 1962). Senior and Uslenghi (1972) proved the existence of these rays in experiments with the diffraction of laser radiation.

PROBLEMS

Functions f (ϕ, ϕ0, α), g(ϕ, ϕ0, α) as well as functions f (0)(ϕ, ϕ0), g(0)(ϕ, ϕ0) are singular at the geometrical optics boundaries of the incident and reflected rays. Verify that their differences

f(1)(ϕ, ϕ0, α) and g(1)(ϕ, ϕ0, α) are finite there.

4.1Prove Equation (4.16).

4.2Prove Equation (4.17).

4.3Prove Equation (4.18).

4.4Prove Equation (4.19).

TEAM LinG

Chapter 5

First-Order Diffraction at

Strips and Polygonal

Cylinders

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

fields for these problems.

In Chapters 3 and 4, we have built a foundation for the solution of 2-D diffraction problems. General asymptotic expressions have been derived for the first-order edge diffracted waves generated both by the uniform and nonuniform components of the surface sources. In the present chapter, this general theory is applied to highfrequency diffraction at strips and cylinders with triangular cross-sections. These specific diffraction problems have been comprehensively studied in the existing literature. In particular, the uniform asymptotic expressions (with arbitrary high asymptotic precision) for the directivity pattern and for the surface field at the strips have been derived in Ufimtsev (1969, 1970, 2003). In these publications one can also find many other references related to the strip diffraction problem. Among them one should note the first and classical solution by Schwarzschild (1902). High-frequency diffraction at polygonal cylinders was investigated in Morse (1964) and Borovikov (1966). We now consider these problems again, to demonstrate the first applications of PTD.

5.1DIFFRACTION AT A STRIP

The geometry of the problem is shown in Figure 5.1. The soft (1.5) or hard (1.6) boundary conditions are imposed at the strip. The incident wave is given by

uinc = u0eik(x cos φ0+y sin φ0),

with − π/2 < φ0 < π/2.

(5.1)

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

Copyright © 2007 John Wiley & Sons, Inc.

83

TEAM LinG

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

Figure 5.1 Cross-section of the strip by the plane z = const.

The diffracted field is investigated around the strip in the directions −π/2 <

φ < 3π/2.

For description of edge waves we utilize the local coordinates r1

, ϕ1, ϕ01

and r2, ϕ2, ϕ02, which are measured from the illuminated side of

the strip

(Fig. 5.2).

 

 

 

 

 

 

 

 

 

 

 

Figure 5.2 Local coordinates for edge waves.

TEAM LinG

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