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190Chapter 7 Elementary Acoustic and Electromagnetic Edge Waves

•EEWs in the directions of the diffraction cone (Fig. 4.4). These directions are determined by ϑ = π − γ0. According to Equations (7.37), (7.74), (7.75), and (7.95),

cos σ1 = − cos ϕ

and

cos σ2 = − cos(α − ϕ).

(7.112)

The quantities σ1,2 belong to the interval 0 ≤ σ1,2 ≤ π . Therefore,

 

 

 

 

 

 

π

ϕ,

for 0

≤

ϕ

≤

π

(7.113)

 

 

 

 

σ1 = ϕ −

π ,

for π

ϕ

2π

 

 

 

 

 

−

 

 

≤

 

≤

 

 

and

=

 

 

−

−

 

≤

−

≤

 

2

π

 

 

 

 

 

 

− (α − ϕ),

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

σ

 

π

 

for 0 α

ϕ

π

(7.114)

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

− ϕ − π ,

 

for π ≤ α − ϕ ≤ 2π .

 

 

 

α

 

 

The first row in Equation (7.114) relates to the region inside the wedge (α ≤ ϕ ≤ 2π ). Utilizing these relationships one can show that the directivity patterns Fs,h(1) transform into

Fs(1)(π − γ0, ϕ) = f (1)(ϕ, ϕ0, α)

and

Fh(1)(π − γ0, ϕ) = g(1)(ϕ, ϕ0, α)

outside the wedge (0 ≤ ϕ ≤ α), and into

 

(7.115)

 

 

Fs(1)(π − γ0, ϕ) = −f (0)(ϕ, ϕ0, α)

and

Fh(1)(π − γ0, ϕ) = −g(0)(ϕ, ϕ0, α)

(7.116)

inside the wedge (α < ϕ < 2π ).

The last equation indicates that the total field of EEWs inside the wedge

equals zero:

 

 

 

Fs,h(1)(π − γ0, ϕ) + Fs,h(0)(π − γ0, ϕ) = 0.

(7.117)

Here, functions

 

 

 

Fs(0)(π − γ0, ϕ) = f (0)(ϕ, ϕ0, α)

and

Fh(0)(π − γ0, ϕ) = g(0)(ϕ, ϕ0, α)

relate to EEWs radiated by the uniform scattering sources js,h(0). This is the result of the screening of the region α < ϕ < 2π by the perfectly reflecting facets of the wedge.

The functions f (1), g(1) are defined in Equations (4.14) and (4.15). According to Equations (3.56) and (3.57) the functions f (0), g(0) are determined by

f (0)(ϕ, ϕ

, α)

 

ε(ϕ0) sin ϕ0

 

ε(α − ϕ0) sin(α − ϕ0)

(7.118)

= cos ϕ + cos ϕ0

+ cos(α − ϕ) + cos(α − ϕ0)

0

 

 

TEAM LinG

 

 

 

 

 

 

 

 

 

 

 

 

7.7 Numerical Calculations of Elementary Edge Waves 191

and

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

g(0)(ϕ, ϕ

 

, α)

 

 

 

 

 

ε(ϕ0) sin ϕ

 

 

 

 

 

ε(α − ϕ0) sin(α − ϕ)

, (7.119)

 

0

= − cos ϕ + cos ϕ0

− cos(α − ϕ) + cos(α − ϕ0)

 

 

 

 

 

 

 

 

with ε(x) defined in Equation (7.48).

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

The functions f , g and f (0), g(0) are singular at the boundaries of the incident

and reflected geometric optics rays, that is, in the directions ϕ = π ± ϕ0,

ϕ

=

2α

−

π

−

ϕ

0

. As shown in Section 4.1, these singularities completely

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

(1)

= f − f

(0)

, g

(1)

 

= g

− g

(0)

are always

cancel each other and functions f

 

 

 

 

 

 

 

 

 

 

 

finite.

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Notice also that the EEWs in the directions ϑ

=

 

π

−

γ

0

generated by the

total scattering sources j

(t)

(0)

 

 

 

 

(1)

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

(t)

s,h

= js,h + js,h

satisfy the reciprocity principle. Their

directivity patterns Fs,h do not change after the permutations ϑ ↔ γ0, ϕ ↔ ϕ0.

The situation with this principle in the general case is discussed in Chapter 8.

In conclusion, we present the acoustic EEWs for the directions belonging to the

diffraction cone (ϑ = π − γ0):

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

dus(0)

 

 

 

 

 

 

 

 

 

 

f (0)(ϕ, ϕ

 

 

, α)

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

du(1)

 

 

uinc(ζ )

dζ

 

f (1)(ϕ, ϕ0

, α)

 

eikR

,

 

 

 

 

(7.120)

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

s

 

 

=

 

 

 

2π

 

 

0

 

 

 

 

R

 

 

 

 

 

 

 

 

 

 

 

 

 

(t)

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

dus

 

 

 

 

 

 

 

 

 

 

f (ϕ, ϕ , α)

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

0

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

and

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

du(0)

 

 

 

 

 

 

 

 

 

 

g(0)(ϕ, ϕ

0

, α)

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

h

 

 

 

 

 

dζ

 

 

 

 

 

 

 

 

 

 

eikR

 

 

 

 

 

 

 

 

 

 

 

 

du(1)

 

 

uinc(ζ )

 

g(1)(ϕ, ϕ

0

, α)

 

 

 

.

 

 

 

 

(7.121)

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

h

 

 

=

 

 

2π

 

 

 

 

 

 

 

R

 

 

 

 

 

 

 

 

 

 

 

 

 

(t)

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

duh

 

 

 

 

 

 

 

 

 

 

g(ϕ, ϕ , α)

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

0

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

7.7 NUMERICAL CALCULATIONS OF ELEMENTARY EDGE WAVES

In this section we present the results of numerical calculations of the elementary edge-diffracted waves radiated by the nonuniform scattering sources js,h(1). The analytical expressions for the directivity patterns of these waves are given in the previous

sections. The quantities 10 log Fs,h(1)

are calculated for the parameters α = 315◦,

γ0 = 45◦, and ϕ0 = 45◦. Figure

7.6

shows the directivity

patterns in the

plane

perpendicular to the edge (ϑ = 90◦, 0

◦ ≤ ϕ ≤ 360◦).

 

 

Figure 7.7 demonstrates the

directivity patterns in the

bisecting plane

con-

taining the edge. In this figure, the polar angle θ is defined through the spherical coordinate ϑ as

ϑ ,

 

ϑ ,

for ϕ

=

α/2

 

180◦.

θ = 2π

−

for ϕ

α/2

+

 

 

 

=

 

 

TEAM LinG

192 Chapter 7 Elementary Acoustic and Electromagnetic Edge Waves

This means that 0◦ ≤ θ ≤ 180◦ when ϕ = α/2 and 180◦ ≤ θ ≤ 360◦ when ϕ = α/2 + 180◦.

Figures 7.6 and 7.7 show the elementary edge waves both outside and inside the wedge. The presence of elementary waves inside the wedge does not contradict the fact that the wedge is nontransparent. The PTD is based on the Helmholtz equivalency principle (Equations (1.10), (1.54), and (1.59)). According to this principle, a real perfectly reflecting object is replaced by the equivalent scattering sources distributed (in free space (!)) over the geometrical surface conformal to the actual scattering surface. These sources generate the field everywhere, including the region inside the object, where this field completely cancels the incident field and ensures the zero total field there. The nonzero elementary edge waves inside the wedge also cancel each other.

Figure 7.8 demonstrates the elementary edge waves propagating in the directions along the diffraction cone.

The following comments are pertinent regarding these calculations:

•As ϕ0 = 45◦, only the wedge face ϕ = 0 is illuminated. Because of this, only the functions Ut (σ1, ϕ0), Vt (σ1, ϕ0), U0(β1, ϕ0), and V0(β1, ϕ0) are singular,

Figure 7.6 Directivity patterns of elementary edge waves (radiated by the nonuniform/fringe sources js,h(1)) in the plane ϑ = 90◦. The interval 315◦ ≤ ϕ ≤ 360◦ relates to the region inside the wedge.

TEAM LinG

7.7 Numerical Calculations of Elementary Edge Waves 193

Figure 7.7 Directivity patterns of elementary edge waves (radiated by the nonuniform/fringe sources) in the bisecting plane. The interval 180◦ ≤ θ ≤ 360◦ relates to the region inside the wedge.

and their linear combinations in the functions U(σ1, ϕ0) and V (σ1, ϕ0) are finite. To treat these singularities, one should apply expressions (7.104) and (7.105), keeping in mind that now γ0 = 45◦ and 2 sin2 γ0 = 1. Together with Equations (7.85) and (7.87), they lead to the following Taylor approximations:

 

U(σ

, ϕ

)

≈

π

cot

 

π(σ1 + ϕ0)

 

−

cot

 

σ1 + ϕ0

+

A

 

−

B

(7.122)

 

 

 

 

 

2

 

 

 

 

 

1

0

 

α

 

 

 

 

2α

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

and

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

V (σ

, ϕ

)

 

 

 

π

 

 

 

cot

π(σ1 + ϕ0)

 

 

 

 

 

 

1

 

 

cot

σ1 + ϕ0

 

 

1

(A

 

B),

≈

α sin σ1

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

− sin σ1

−

1

0

 

 

 

 

 

 

 

 

2α

 

− sin σ1

2

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

(7.123)

where

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

A

=

 

π

 

 

π(σ1 − ϕ0)

 

+

1

 

π 3(σ1 − ϕ0)3

 

 

 

(7.124)

 

 

 

 

 

 

 

 

α

 

45

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

6α

 

 

 

 

 

 

8α3

 

 

 

 

 

 

 

 

and

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

B

=

σ1 − ϕ0

+

 

1

 

 

(σ1 − ϕ0)3

.

 

 

 

 

 

(7.125)

 

 

 

 

 

 

 

 

 

 

 

 

6

45

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

8

 

 

 

 

 

 

 

 

 

 

 

TEAM LinG

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