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: 968

Скачиваний: 0

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

Chapter 8

Ray and Caustics Asymptotics for Edge Diffracted Waves

This chapter is based on the papers by Ufimtsev (1989, 1991).

8.1RAY ASYMPTOTICS

The following relationships exist between the acoustic and electromagnetic diffracted rays:

us = Et , if uinc(ζ ) = Etinc(ζ );

uh = Ht ,

if uinc(ζ ) = Htinc(ζ ),

where ˆt is the tangent to the edge at the diffraction point

ζ .

8.1.1 Acoustic Waves

The theory of EEWs is applied here for calculation of scattering at a smoothly curved edge L with a slowly changing angle α(ζ ) between its faces (Fig. 8.1). In a small vicinity of the point ζ on the edge, an arbitrary incident field

uinc(ζ ) = u0(ζ )eikφ i (ζ )

(8.1)

can be locally considered as a plane wave propagating in the direction

ki

=

φi

=

grad

φi.

(8.2)

ˆ

 

 

 

 

Therefore, replacing the quantity uinc(ζ ) in Equations (7.89) and (7.90) and Equations (7.96) and (7.97) by Equation (8.1), we obtain the asymptotic expressions

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

Copyright © 2007 John Wiley & Sons, Inc.

213

TEAM LinG

214 Chapter 8 Ray and Caustics Asymptotics for Edge Diffracted Waves

Figure 8.1 Element of a scattering edge L with a curvilinear coordinate ζ along the edge; ˆt is the unit vector tangential to the edge at the point ζ .

for EEWs generated by an arbitrary incident wave. The resulting diffracted wave arising at the edge L and created by the nonuniform/fringe sources js,h(1) is a linear superposition of EEWs (7.89), (7.90),

us,h(1) =

1

L u0(ζ )Fs,h(1)(ζ , mˆ )

eik (ζ )

dζ ,

(8.3)

2π

R(ζ )

and the edge wave generated by the total scattering linear superposition of EEWs (7.96), (7.97)

sources js,h(t) = js,h(0) + js,h(1) is a

us,h(t) =

1

L u0(ζ )Fs,h(t)(ζ , mˆ )

eik (ζ )

dζ .

(8.4)

 

2π

R(ζ )

Here,

 

 

 

 

 

 

 

 

mˆ = R,

= φi + R,

 

(8.5)

and R is the distance between the edge point ζ and the observation point P(x, y, z). Notice that the differential operator in Equation (8.2) acts on coordinates of the edge point ζ , but the operator in Equation (8.5) acts on coordinates x, y, z of the observation point P.

A high-frequency approximation (with k 1) of the scattered field can be obtained by the stationary-phase technique (Copson, 1965; Murray, 1984), whose details have been already considered in Section 6.1.2. The stationary point ζst is determined by the equation

 

dζ

=

 

 

· ˆ

=

 

 

+

 

· ˆ =

ˆ

− ˆ

· ˆ =

 

 

 

 

d

 

 

 

t

 

 

(φi

 

R)

t

(ki

m)

t

 

0.

(8.6)

 

 

 

 

 

 

 

ks the unit vector m directed from the stationary point ζ

st

to the observation

Denote by ˆ

 

 

ˆ

 

 

 

 

 

 

 

 

 

 

 

 

 

point . Then Equation (8.6) can be rewritten as

 

 

 

 

 

 

 

 

 

 

 

 

 

ˆ

 

t

 

ˆ

· ˆ

= −

 

0

 

 

 

 

 

 

 

 

 

 

 

ks

· ˆ =

ki

cos γ .

 

 

 

 

 

(8.7)

 

 

 

 

 

 

 

t

 

 

 

 

 

 

 

TEAM LinG

 

8.1 Ray Asymptotics 215

ks form a cone with its axis along the tangent t to the

Thus, the scattering directions ˆ

ˆ

edge at the stationary point. Such a cone is shown in Figure 4.4.

The function describes the distance between the points Q and P along the straight lines Qζ and ζ P (Fig. 8.1). Hence, Equation (8.6) indicates that this distance is extremal (minimal or maximal) when the point ζ is stationary. In other words, the location of the stationary point ζst on edge L satisfies the Fermat principle.

In accordance with the stationary-phase technique, the first term of the asymptotic expression for the field (8.3), (8.4) equals

 

(1)

 

1

uinc(ζ

 

 

(1)

 

 

, ks)

eikR ∞

k (ζst )

(ζ −ζst )

2

 

u

s,h

=

 

st

)F

s,h

(ζ

st

 

ei

2

dζ ,

(8.8)

2π

R

 

 

 

 

ˆ

−∞

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

where (ζst ) = d2 (ζst )/dζ 2, and R is the distance between the stationary point ζst and the observation point P. Due to the equality

 

 

 

 

 

∞

2

 

 

 

 

 

i

π

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

e±ix dx = √π e±

 

4 ,

 

 

 

 

 

 

 

 

−∞

 

 

 

 

 

 

 

 

 

 

 

Equation (8.8) can be written as

 

 

 

 

 

 

 

 

 

 

 

(1)

 

inc

(1)

 

ks)

 

 

 

eiπ/4

eikR

 

 

 

 

 

 

 

 

,

us,h

= u

 

 

(ζst )Fs,h

(ζst , ˆ

√

 

 

 

 

 

 

 

R

 

 

2π k (ζst )

where

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

'

 

= '

 

ei π2 ,

 

 

 

if (ζst ) < 0.

(ζst )

| (ζst )|

 

 

 

(8.9)

(8.10)

(8.11)

In terms of the local spherical coordinates R, ϑ , ϕ (introduced in Fig. 7.3), the

 

(1)

 

have the directions ϑ

=

π

−

γ

 

 

, 0

≤

ϕ

≤

2π . For these directions,

 

ks

 

 

0

 

 

unit vectors ˆ

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

functions F

s,h

 

ks) are determined by Equations (7.115) and (7.116). Hence

 

(ζst , ˆ

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

us(1) = uinc

(ζst ) f (1)

(ϕ, ϕ0, α)

 

 

 

 

 

eiπ/4

 

 

 

 

 

eikR

(8.12)

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

'

 

 

 

 

 

R

and

 

 

2π k (ζst )

 

 

 

uh(1) = uinc

(ζst )g(1)(ϕ, ϕ0, α)

 

 

 

 

 

eiπ/4

 

 

 

eikR

 

(8.13)

 

 

 

'

 

 

 

 

 

R

 

 

 

2π k (ζst )

 

 

 

in the directions 0 ≤ ϕ ≤ α, and

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

us(1) = −uinc

(ζst ) f (0)(ϕ, ϕ0, α)

√

 

eiπ/4

 

 

 

 

 

eikR

(8.14)

 

 

 

 

 

R

 

 

 

2π k (ζst )

TEAM LinG

216 Chapter 8

Ray and Caustics Asymptotics for Edge Diffracted Waves

 

 

 

and

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

uh(1) = −uinc(ζst )g(0)(ϕ, ϕ0, α)

√

eiπ/4

eikR

 

 

(8.15)

 

 

 

 

 

 

 

 

 

R

 

 

 

2π k (ζst )

 

 

in the directions α < ϕ < 2π , related to the region inside the tangential wedge.

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

(1)

 

(0)

The total diffracted field us,htot radiated by the total sources j s,htot = j s,h

+ j s,h

is described

by

Equations (8.12)

and

(8.13), where

 

one

should replace

f (1)(ϕ, ϕ0, α),

g(1)(ϕ, ϕ0, α) by functions

f (ϕ, ϕ0, α),

g(ϕ, ϕ0, α). In

the

region

α < ϕ < 2π (inside the tangential wedge),

the total

diffracted

field

u(1)

+

u(0)

asymptotically equals zero, because u

(1)

= −u

(0)

 

 

 

 

 

 

 

 

 

 

 

 

in accordance with Equations (8.14)

and (8.15).

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

The above asymptotics for edge diffracted waves can be presented in another form that reveals their ray structure. To do this, we utilize the following differential

operations:

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

=

 

d

=

t

 

 

(φinc

+

R)

=

t

ki

t

·

R

= −

cos γ

t

R, (8.16)

 

dζ

 

 

 

 

 

 

 

 

 

ˆ ·

 

 

 

 

 

 

ˆ · ˆ

+ ˆ

 

0

+ ˆ ·

 

 

 

 

d

 

 

 

 

 

 

 

dγ0

 

d

 

R

 

 

 

 

 

dt

 

 

 

 

 

 

 

 

 

 

 

 

 

 

sin γ

 

 

 

 

 

 

 

 

 

t

 

R

 

ˆ

,

 

 

(8.17)

= dζ

 

 

=

 

+

 

 

dζ

 

 

 

 

· dζ

 

 

 

 

 

 

 

 

 

[

 

0 dζ

 

 

 

 

 

· ˆ +

 

 

 

 

 

 

 

 

dζ

 

 

 

·

ˆ =

R

 

− ˆ ·

 

 

 

 

]

 

 

 

 

 

 

 

 

 

 

 

 

 

 

d R

 

t

 

1

 

1

 

(t

 

R)2

 

,

 

 

 

 

 

 

 

 

 

 

 

(8.18)

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

dˆt

 

 

 

vˆ

.

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

(8.19)

 

 

 

 

 

dζ

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

= a

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Here vˆ is the unit vector of the principal normal to the edge L, and a is the radius of

curvature of the edge.

 

R

= −ˆ

 

 

 

ˆ ·

R

= −ˆ · ˆ

=

 

 

At the stationary point,

 

 

 

cos γ0

. Therefore,

 

 

 

ks and t

 

 

t ks

 

 

 

 

d R

 

t

 

sin2

γ0

,

 

 

 

(8.20)

 

 

 

 

 

 

 

R

 

 

 

 

 

 

 

dζ

· ˆ =

 

 

 

 

 

 

 

 

 

 

 

 

dt

 

 

ks

v

 

 

 

 

 

 

 

 

 

 

 

 

 

ˆ

 

 

 

 

 

 

 

 

 

R

 

 

 

ˆ

 

 

 

 

· ˆ

.

 

 

 

(8.21)

 

 

 

· dζ

= −

 

 

 

 

 

 

 

a

 

 

 

 

 

In view of relationships (8.16)–(8.19) and (8.20), (8.21),

 

 

 

 

1

 

 

 

 

 

 

 

 

 

 

 

 

(ζst ) =

R

1

+

 

ρ

sin2 γ0,

(8.22)

where

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

1

1

 

dγ

0

 

 

 

ks

v

 

 

 

 

 

 

 

ˆ

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

· ˆ

.

(8.23)

 

ρ = sin γ0

dζ

 

 

 

 

 

 

 

 

− a sin γ0

 

TEAM LinG

8.1 Ray Asymptotics 217

The quantity ρ is a caustic parameter; it determines the distance (R = −ρ) along the ray from the edge to the caustic.

Now the edge diffracted field can be written in the ray form

us,h(1) = uinc(ζst ) · (DF) · (DC) · eikR,

(8.24)

where

 

 

 

 

 

DF =

√

 

1

 

 

(8.25)

 

 

 

R|1 + R/ρ|

 

is the rays’ divergence factor, and

 

 

 

 

 

 

 

 

e±i π4

f (1)(ϕ, ϕ0, α)

 

DC =

sin γ0

√

 

g(1)(ϕ, ϕ0, α)

(8.26)

2π k

can be interpreted as the diffraction coefficient. The quantities R and R + ρ are

the two principal radii of curvature of the diffracted phase front. The upper sign in e±iπ/4 is taken if (ζst ) > 0 and the lower one if (ζst ) < 0. The last multiplier in Equation (8.24), eikR, is the phase factor.

The divergence factor shows how the edge waves, being cylindrical-like waves in the vicinity of the edge (R |ρ|),

 

 

 

(DF)eikR ≈

eikR

 

 

 

 

 

 

 

√

 

 

 

,

 

 

 

(8.27)

 

 

R

 

 

transform into spherical waves,

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

(DF)eikR ≈ '

 

 

 

 

 

 

ikR

 

 

 

 

 

 

 

 

e

,

 

 

 

 

|ρ|

 

 

(8.28)

 

 

 

 

R

 

 

at a large distance from the edge (R

ρ).

 

 

 

 

 

 

 

 

 

 

 

The total edge diffracted fields can be also represented in ray form (with kR 1)

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

(t)

1

ei π4

 

 

 

 

 

 

f (ϕ, ϕ

0

, α)

 

us,h

= uinc(ζst )

√

 

sin γ0√

 

+g(ϕ, ϕ0, α), eikR.

(8.29)

R(1 + R/ρ)

2π k

We note that all variable parameters and coordinates in Equation (8.29) relate to the stationary point ζst .

Notice that the PTD ray asymptotics (8.14), (8.15) with the second derivative(ζ ) is much easier for the calculation than the GTD form (8.29), which involves complicated calculations of the caustic parameter ρ(ζ ).

TEAM LinG

Смотрите также файлы