Файл: Ufimtsev P. Fundamentals of the physical theory of diffraction (Wiley 2007)(348s) PEo .pdf

Скачать файл
Заказать решение

ВУЗ: Не указан

Категория: Не указан

Дисциплина: Не указана

Добавлен: 28.06.2024

Просмотров: 947

Скачиваний: 0

ВНИМАНИЕ! Если данный файл нарушает Ваши авторские права, то обязательно сообщите нам.

58 Chapter 2 Wedge Diffraction: Exact Solution and Asymptotics

2.15This problem is analogous to Problem 2.14. Analyze the grazing diffraction of the wave

Hzinc = H0z exp(ikx) at a perfectly conducting wedge. Calculate the surface current induced on the face ϕ = 0.

(a) Use the exact solution (2.41), (2.51),

(b) Apply the stationary phase technique and show that

1

 

π eikr+iπ/4

 

 

jx ≈ H0z 1 −

 

cot

 

 

√

 

 

,

with kr 1.

n

n

 

2π kr

TEAM LinG

Chapter 3

Wedge Diffraction:

The Physical Optics Field

The relationships us(0) = Ez(0) and uh(0) = Hz(0) exist between the PO acoustic and

electromagnetic fields. An exception is for an oblique incidence (see Section 4.3).

The exact expressions for the scattered field were derived in the previous chapter. In the present chapter, we calculate the Physical Optics (PO) part of the scattered field, that is, the field generated by the uniform component of the induced surface sources. It will be used in the next chapter to examine the field radiated by the nonuniform sources as the difference between the exact and PO fields.

3.1ORIGINAL PO INTEGRALS

The geometry of the problem is shown in Figure 3.1. A perfectly reflecting wedge located in a homogeneous medium is excited by a plane wave

uinc = u0e−ikr cos(ϕ−ϕ0) = u0e−ik(x cos ϕ0+y sin ϕ0),

(3.1)

where we assume that 0 ≤ ϕ0 ≤ π . In the PO approximation, the surface sources (induced on the face ϕ = 0) are determined according to Equation (1.31):

j(0)

= −

u

2ik sin ϕ e−ikx cos ϕ0

,

j(0)

=

u

2e−ikx cos ϕ0

s

0

0

 

h

0

 

Equation (1.32) determines the field radiated by these sources:

√

u(0)

 

u

 

ik sin ϕ0

∞

∞

 

eik (x−ξ )2+y2+ζ 2

=

0

 

 

e−ikξ cos ϕ0 dξ

 

 

s

 

2π

0

−∞

'

(x − ξ )2 + y2 + ζ 2

Fundamentals of the Physical Theory of Diffraction. By Pyotr Ya. Ufimtsev Copyright © 2007 John Wiley & Sons, Inc.

. (3.2)

dζ (3.3)

59

TEAM LinG

60 Chapter 3 Wedge Diffraction: The Physical Optics Field

Figure 3.1 A wedge and related coordinates.

and

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

eik√

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

(0)

1

 

∞

e−ikξ cos ϕ0 dξ

 

∞ ∂

 

(x−ξ )2+y2+ζ 2

 

uh

= u0

 

 

 

 

 

 

 

 

 

 

 

dζ .

(3.4)

2π

 

 

−∞ ∂n

'

 

 

 

0

 

 

(x − ξ )2 + y2 + ζ 2

Here, we use the denotations

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

∂

f (r)

=

f (r)

n

 

 

f (r)

n

∂

f (r)

(3.5)

 

 

 

∂n

 

 

 

 

 

 

 

 

 

·

ˆ = −

 

 

 

 

· ˆ = −

∂y

 

where and are the gradient operators applied to coordinates of the integration and observation points, respectively. In view of Equation (3.5), the field uh(0) can be written as

 

 

 

 

 

 

 

eik√

 

 

 

 

(0)

= −u0

1 ∂

∞

e−ikξ cos ϕ0 dξ

∞

(x−ξ )2+y2+ζ 2

dζ .

(3.6)

uh

 

 

 

 

 

 

 

2π

∂y

 

−∞

'

 

 

0

(x − ξ )2 + y2 + ζ 2

In Equations (3.3) and (3.6), the integral over the variable ζ can be expressed through the Hankel function. We utilize two integral forms for this function. The first form

 

 

∞ eik√

 

 

 

 

 

(1)

1

d2+ζ 2

 

H0

(kd) =

 

 

 

 

 

dζ

(3.7)

iπ

−∞

'

 

 

d2 + ζ 2

follows from the table formula 8.421.11 of Gradshteyn and Ryzhik (1994). The second form

(1)

 

 

 

1

∞ ei(vd−wz)

 

 

%k'd2 + z2

& =

dw

(3.8)

H0

 

 

 

π

−∞ v

TEAM LinG

3.2 Conversion of the PO Integrals to the Canonical Form 61

√

(where v = k2 − w2, Im v > 0, and d > 0) can be verified by its conversion to the Sommerfeld formula (Sommerfeld, 1935)

(1)

(ρ) =

1

δ−i∞

e

iρ cos β

dβ, 0

≤ δ ≤ π ,

H0

 

 

 

π

−δ+i∞

 

 

 

 

 

 

 

 

√

by setting w = k sin t, v = k cos t, and k d2 + z2 = ρ. Application of Equation (3.7) leads to

 

 

k sin ϕ0

∞

 

 

 

 

 

 

 

 

 

(0)

 

e−

ikξ

 

(1)

k((x − ξ )2 + y2

dξ

us

= −u0

 

 

 

 

 

 

 

 

cos ϕ0 H0

2

 

 

0

 

and

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

i ∂

∞

 

 

 

 

 

 

 

 

 

 

(0)

 

e−ikξ cos

(1)

k((x − ξ )2 + y2

dξ .

uh

= −u0

 

 

 

 

 

 

ϕ0 H0

2 ∂y

0

 

Then we use Equation (3.8) and find

u(0)

u

k sin ϕ0

∞ eiv|y|

 

 

 

 

 

 

e−iwx dw

s

= −

0

2π

 

−∞

v

and

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

(0)

 

1

 

∞

ei(v|y|−wx) dw

uh

= sgn( y)u0

 

 

2π

−∞

 

 

 

 

 

 

 

 

∞

ei(w−k cos ϕ0)ξ dξ

0

∞

ei(w−k cos ϕ0)ξ dξ .

0

(3.9)

(3.10)

(3.11)

(3.12)

(3.13)

To ensure the convergence of the internal integrals, we impose the condition Im(w − k cos ϕ0) > 0 and obtain

 

 

 

i sin ϕ0

 

 

1

 

 

 

 

us(0) = u0

 

 

Is,

uh(0) = sgn( y)u0

 

Ih

 

 

2π

i2π

where

 

 

 

 

 

 

 

 

 

 

 

∞

ei(v|y|−wx)

 

 

∞ ei(v|y|−wx)

Is = k

−∞

 

dw,

Ih = −∞

 

dw

v · (k cos ϕ0 − w)

k cos ϕ0 − w

and the integration contour skirts above the pole w = k cos ϕ0.

3.2 CONVERSION OF THE PO INTEGRALS TO THE CANONICAL FORM

(3.14)

(3.15)

In integrals Is and Ih, we introduce the polar coordinates by the relationships

x = r cos ϕ, |y| =

r

sin ϕ,

with ϕ < π

(3.16)

 

r sin ϕ,

with ϕ > π

 

−

 

 

 

TEAM LinG

62 Chapter 3 Wedge Diffraction: The Physical Optics Field

and change the integration variable w by ξ , setting w = −k cos ξ , v = k sin ξ ,

Then,

kr cos(ξ − ϕ),

v|y| − wx =

kr cos(ξ + ϕ),

The equations

Im v > 0.

(3.17)

with ϕ < π

(3.18)

with ϕ > π .

w = −k cos ξ cosh ξ + ik sin ξ sinh ξ ,

(3.19)

v = k sin ξ cosh ξ + ik cos ξ sinh ξ

 

and the condition

 

 

 

 

 

 

 

Im v = k cos ξ sinh ξ

> 0

(3.20)

determine the integration path F in the complex plane ξ = ξ + iξ

as shown in

Figure 3.2.

 

 

 

 

 

After these manipulations we obtain the following expressions:

 

Is =

 

eikr cos(ξ −ϕ)

dξ ,

with ϕ < π ,

(3.21)

F cos ξ + cos ϕ0

Is =

 

eikr cos(ξ +ϕ)

dξ ,

with ϕ > π ,

(3.22)

F cos ξ + cos ϕ0

Figure 3.2 Integration contour F in the complex plane ξ = ξ + iξ .

TEAM LinG

Смотрите также файлы