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7.5 General Asymptotics for Elementary Edge Waves

185

The above estimations allow one to obtain the asymptotic expressions for the

integrals Iu,v defined by Equations (7.64) and (7.65):

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

eikR

+

 

4αε(ϕ0) sin ϕ0

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Iu = −2

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

R

p2,st + k2 cos ϕ0

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

2π

 

cot

π(σ1 + ϕ0)

 

 

−

cot

π(σ1 − ϕ0)

 

 

,

 

 

 

(7.72)

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

+ k2

 

 

 

2α

 

 

 

 

2α

 

,

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

eikR

+−

4αε(ϕ0)

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Iv = 2ik sin β1

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

R

 

p2,st + k2 cos ϕ0

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

2π

 

cot

π(σ1 + ϕ0)

+

cot

π(σ1 − ϕ0)

 

,

 

(7.73)

 

 

 

 

 

+ k2 sin σ1

 

 

 

 

 

 

 

 

 

 

 

 

 

 

2α

 

 

 

 

 

2α

 

 

,

 

 

where

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

p2,st

= pst − k cos2 γ0 = k(cos β1 − cos2 γ0)

 

 

 

(7.74)

and

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

cos β1 = sin γ0 sin ϑ cos ϕ − cos γ0 cos ϑ .

 

 

 

 

(7.75)

Parameter σ1 is determined according to Equations (7.38) and (7.42), (7.43) as

σ

 

 

π

 

arccos

 

cos β1 − cos2 γ0

 

 

,

 

 

with 0

 

β

 

 

β

,

 

(7.76)

 

=

−

 

 

 

 

 

 

≤

 

≤

 

 

1

 

 

 

 

 

 

 

 

 

 

sin2 γ0

 

 

 

 

 

 

 

 

1

k

 

 

 

and

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

σ1 = i ln +cos2 γ0 − cos β1 + (

 

,

 

 

 

(cos2 γ0 − cos β1)2 − sin4 γ0

 

 

 

 

 

− 2i ln(sin γ0),

 

 

 

with βk ≤ β1 ≤ π ,

 

 

 

 

 

 

 

 

(7.77)

where

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

2γ0,

 

γ0),

with 0 ≤ γ0 ≤ π/2

 

 

 

 

(7.78)

 

 

 

 

 

βk = 2(π

−

with π/2

≤

γ0

≤

π .

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

By substituting Equations (7.72) and (7.73) into Equations (7.25) and (7.26) we find the asymptotic estimations for the field generated by elementary strip 1:

du1(1)

dv1(1)

=u0e−ikζ

=u0e−ikζ

cos γ0 dζ F(1)(ϑ , ϕ)

2π s,1

cos γ0 dζ F(1)(ϑ , ϕ)

2π h,1

eikR

,

R

eikR

R

(7.79)

(7.80)

TEAM LinG

186 Chapter 7

Elementary Acoustic and Electromagnetic Edge Waves

 

where kR

1 and

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Fs,1(1) = −U(σ1, ϕ0) sin2 γ0,

 

 

 

 

 

 

 

(7.81)

 

 

 

 

 

 

Fh(1,1) = −V (σ1, ϕ0) sin γ0 sin ϑ sin ϕ,

 

 

(7.82)

with

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

U(σ1

, ϕ0) = Ut (σ1, ϕ0) − U0(β1, ϕ0),

 

 

 

 

 

 

 

 

 

 

(7.83)

V (σ1

, ϕ0) = Vt (σ1, ϕ0) − V0(β1, ϕ0),

 

 

 

 

 

 

 

 

 

 

(7.84)

U

(σ

, ϕ

)

 

 

π

 

cot

π(σ1 + ϕ0)

 

cot

π(σ1 − ϕ0)

,

(7.85)

= 2α sin2 γ0

 

 

−

 

 

 

t

1

0

 

 

 

 

2α

 

 

 

 

 

2α

 

 

U0(β1

, ϕ0) = −

 

 

ε(ϕ0) sin ϕ0

 

 

 

 

 

,

 

 

 

 

(7.86)

cos β1 − cos2 γ0 + sin2 γ0 cos ϕ0

 

 

 

 

V

(σ

, ϕ

)

 

 

π

 

 

 

cot

π(σ1 + ϕ0)

+

cot

π(σ1 − ϕ0)

, (7.87)

= 2α sin2 γ0 sin σ1

 

 

 

t

1

0

 

 

 

2α

 

 

2α

 

and

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

V0(β1, ϕ0) =

 

 

 

 

ε(ϕ0)

 

 

 

 

 

 

.

(7.88)

 

 

 

 

cos β1 − cos2 γ0 + sin2 γ0 cos ϕ0

Here, the quantities Ut , Vt relate to the field generated by the total scattering sources js,htot = js,h(1) + js,h(0), and U0, V0 represent the field radiated by their uniform components js,h(0). The quantity ε(ϕ0) is determined using Equation (7.48).

Asymptotic expressions (7.79) and (7.80) describe the field generated by the scattering sources js,h(1) induced on strip 1 located at the wedge face ϕ = 0. Utilizing Equations (7.79) and (7.80) and the relationships (7.31), one can easily obtain asymptotics for the field du2(1), dv2(1) generated by elementary strip 2 belonging to the wedge face ϕ = α. The total field of the EEW created by both strips equals

 

 

dζ

eikR

 

 

dus(1)

= uinc(ζ )

 

Fs(1)(ϑ , ϕ)

 

,

(7.89)

2π

R

 

 

dζ

eikR

 

 

duh(1)

= uinc(ζ )

 

Fh(1)(ϑ , ϕ)

 

,

(7.90)

2π

R

where kR 1 and

 

 

 

 

 

 

Fs(1)(ϑ , ϕ) = −[U(σ1, ϕ0) + U(σ2, α − ϕ0)] sin2 γ0,

 

(7.91)

Fh(1)(ϑ , ϕ) = −[V (σ1, ϕ0) sin ϕ + V (σ2, α − ϕ0) sin(α − ϕ)] sin γ0 sin ϑ

(7.92)

with

 

 

 

 

 

 

U(σ2, α − ϕ0) = Ut (σ2, α − ϕ0) − U0(β2, α − ϕ0)

(7.93)

TEAM LinG

7.6 Analytic Properties of Elementary Edge Waves

187

and

 

V (σ2, α − ϕ0) = Vt (σ2, α − ϕ0) − V0(β2, α − ϕ0).

(7.94)

Here, the quantity uinc(ζ ) stands for the incident field at the point ζ on the scattering edge. The parameter σ2 is defined by Equations (7.76) and (7.77) after the substitution β1 → β2. The angle β2 is determined by the expression

cos β2 = sin γ0 sin ϑ cos(α − ϕ) − cos γ0 cos ϑ,

(7.95)

which follows from Equation (7.75) after the substitution of ϕ by α − ϕ.

In view of the observation following Equations (7.87) and (7.88), the EEWs

generated by the total scattering sources js,h(t) = js,h(1) + js,h(0) can be obtained by a simple modification of Equations (7.89) and (7.90):

dus(t) = uinc(ζ )

dζ

eikR

 

 

 

Fs(t)(ϑ , ϕ)

 

,

(7.96)

2π

R

 

dζ

eikR

 

 

duh(t) = uinc(ζ )

 

Fh(t)(ϑ , ϕ)

 

,

(7.97)

2π

R

where

 

 

 

 

 

Fs(t)(ϑ , ϕ) = −[Ut (σ1, ϕ0) + Ut (σ2, α − ϕ0)] sin2 γ0

(7.98)

and

 

 

 

 

 

Fh(t)(ϑ , ϕ) = −[Vt (σ1, ϕ0) sin ϕ + Vt (σ2, α − ϕ0) sin(α − ϕ)] sin γ0 sin ϑ .

(7.99)

As was expected, the elementary edge wave (EEW) (at large distances (kR 1) from its origin on the edge) is a spherical wave with the directivity pattern described by Equations (7.91), (7.92) and (7.98), (7.99). This wave can also be interpreted as a set of elementary edge-diffracted rays. In the next section we study the analytical properties of EEWs.

7.6 ANALYTIC PROPERTIES OF ELEMENTARY EDGE WAVES

•It is seen that, for real parameters σ1,2, the directivity patterns Fs,h(1) of EEWs are real functions. One can verify that they also remain real for imaginary param-

eters σ1,2. Thus, the functions Fs,h(1) are always real, although their arguments can be complex quantities.

•The field dus(1) of EEWs for acoustically soft scatters equals zero for the grazing incidence (ϕ0 = 0 or ϕ0 = α). Consider for example the case ϕ0 = 0. Indeed,

TEAM LinG

188 Chapter 7 Elementary Acoustic and Electromagnetic Edge Waves

according to Equations (7.85) and (7.86), Ut (σ1, 0) = 0 and U0(β1, 0) = 0. The function Ut (σ2, α) is also equal to zero because

cot

π(σ2 + α)

=

cot

π(2α + σ2 − α)

 

2α

2α

 

 

 

 

 

 

=

 

2α

 

 

 

 

cot

π(σ2 − α)

.

(7.100)

 

 

 

 

This property is the result of the fact that the incident plane wave cannot propagate along the wedge face. Due to the boundary condition, it is completely cancelled by the reflected plane wave. Therefore, the field uinc(ζ ) incident on the edge equals zero – no incident field, no diffracted field.

• It is obvious that functions Ut (σ1, ϕ0), Vt (σ1, ϕ0) and Ut (σ2, α − ϕ0), Vt (σ2, α − ϕ0) are singular at the points σ1 = ϕ0 and σ2 = α − ϕ0, respectively. At these points, functions U0(β1, ϕ0), V0(β1, ϕ0) and U0(β2, α − ϕ0), V0(β2, α − ϕ0) are also singular, because in this case β1 = β10 for σ1 = ϕ0 and β2 = β20 for σ2 = α − ϕ0, where

cos β10 = cos2 γ0 − sin2 γ0 cos ϕ0

(7.101)

and

 

cos β20 = cos2 γ0 − sin2 γ0 cos(α − ϕ0).

(7.102)

One can show that the singular terms of functions Ut , Vt are cancelled by the singular terms of functions U0, V0, and as a result the functions U = Ut − U0, V = Vt − V0 remain finite. We demonstrate this property for the functions U(σ1, ϕ0) = Ut (σ1, ϕ0) − U0(β1, ϕ0) and V (σ1, ϕ0) = Vt (σ1, ϕ0) − V0(β1, ϕ0).

According to Equations (7.37) and (7.74),

cos σ

1

=

 

cos2 γ0 − cos β1

and

cos β

1

=

cos2 γ

0

−

sin2 γ

0

cos σ

1

.

sin2 γ0

 

 

 

 

 

 

 

 

(7.103)

Utilizing the last equality, one can represent the functions U0(β1, ϕ0) and V0(β1, ϕ0) in the form

 

U

(β

, ϕ

)

=

 

 

1

 

 

 

cot

σ1 + ϕ0

 

−

cot

σ1 − ϕ0

 

 

(7.104)

 

2 sin2 γ0

 

 

 

 

 

 

 

 

 

and

0

1

 

0

 

 

 

 

 

 

 

 

2

 

 

 

 

 

 

 

 

 

2

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

V

(β

, ϕ

)

=

 

 

 

1

 

 

 

 

 

 

cot

σ1 + ϕ0

 

+

cot

σ1 − ϕ0

.

(7.105)

2 sin2 γ0 sin σ1

 

 

 

 

 

0

1

0

 

 

 

 

 

 

 

 

2

 

 

 

 

 

 

 

2

 

 

 

Now it is easy to prove that

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

U(ϕ

 

 

 

 

 

lim

U

(σ

, ϕ

)

−

U

 

(β

, ϕ )

]

 

 

 

 

 

 

 

0, ϕ0) =

σ1

→

ϕ0[

 

t

 

 

 

1

0

 

 

0

 

 

1

 

0

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

=

 

 

 

π

 

 

 

 

 

cot

π ϕ0

−

 

 

 

 

1

 

 

 

 

cot ϕ0

 

(7.106)

 

 

 

 

 

 

 

2α sin2 γ0

 

 

 

α

 

 

 

2 sin2 γ0

 

 

TEAM LinG

 

 

 

 

 

 

 

 

 

7.6 Analytic Properties of Elementary Edge Waves

189

and

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

V (ϕ0

 

lim

 

V

(σ

, ϕ

)

 

−

V

(β

, ϕ

)

]

 

 

 

 

 

 

 

 

 

 

 

, ϕ0) = σ1

→

ϕ0[

 

t

 

 

1

0

 

 

 

 

 

0

 

1

 

0

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

= U(ϕ0, ϕ0)/ sin ϕ0.

 

 

 

 

 

 

 

 

 

 

 

 

 

(7.107)

These equations also determine the limiting values

 

 

 

 

 

 

 

 

 

 

 

 

 

 

U(α

−

ϕ

, α

 

ϕ

)

 

π

 

 

 

 

cot

π(α − ϕ0)

 

 

 

 

 

1

 

 

 

 

cot(α

 

ϕ

)

−

= 2α sin2 γ0

 

− 2 sin2 γ0

−

 

0

 

 

0

 

 

 

 

 

 

 

 

α

 

 

 

 

 

 

 

0

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

(7.108)

and

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

V (α − ϕ0, α − ϕ0) = U(α − ϕ0, α − ϕ0)/ sin(α − ϕ0).

 

 

(7.109)

It follows from these equations that under the grazing incidence (ϕ0 = 0

or ϕ0 = α),

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

lim

 

 

 

 

 

 

lim

 

U(α

−

ϕ

, α

−

ϕ

)

=

0

 

 

(7.110)

 

 

 

 

ϕ0→0 U(ϕ0, ϕ0) =

ϕ0

→α

 

 

 

 

 

0

 

 

 

 

0

 

 

 

 

 

 

 

 

and

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

lim

V (ϕ

, ϕ

)

=

lim V (α

−

ϕ

, α

−

ϕ

 

)

=

 

 

 

1

 

 

 

1

 

π 2

.

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

6 sin2 γ0

 

 

 

 

 

ϕ0→0

 

0

0

 

ϕ0→α

 

 

0

 

 

 

0

 

 

 

 

 

− % α &

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

(7.111)

However, expressions (7.106), (7.107) and (7.108), (7.109) remain singular in the grazing directions ϕ0 = π and ϕ0 = α − π . The reason for this singularity was explained earlier in conjunction with Equations (4.20) and (4.21) and it is discussed later in Section 7.9, where a new version of PTD is developed that is free from the grazing singularity. This singularity can be also eliminated by the truncation of the elementary strips introduced in Section 7.1 (Johansen, 1996). An example of a similar truncation of scattering sources was considered in Section 5.1.4.

•Notice that the elementary edge-diffracted rays in the directions β1 = β10 and β2 = β20 defined by Equations (7.101) and (7.102) form two conic surfaces

with axes along the elementary strips 1 and 2 (axes x1 and x2 in Fig. 7.3). According to Equation (7.75), the conic surface β1 = β10 contains the diffracted rays in the directions ϑ = π − γ0, ϕ = π + ϕ0 and ϑ = π − γ0, ϕ = π − ϕ0, which are on the shadow boundary of the incident and reflected geometrical

optics rays, respectively (Fig. 2.7). In view of Equations (7.95) and (7.101),

(7.102), the conic surface β2 = β20 contains the diffracted ray on the boundary of the geometrical optics rays (ϑ = π − γ0, ϕ = 2α − π − ϕ0) reflected from the wedge face ϕ = α (Fig. 2.8).

TEAM LinG

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