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7.9 Improved Theory of Elementary Edge Waves 199

Figure 7.9 Grazing incidence on the wedge with ϕ0 = π and grazing scattering in the direction ϕ = 0. No singularity exists in the reverse situation, when ϕ0 = 0 and ϕ = π .

deﬁnitions (1.31) and (7.7), (7.8) for uniform and nonuniform components are not adequate for the actual surface ﬁeld in this case. In particular, component jh(1) does not vanish away from the edge, but instead it contains a plane wave there. Besides, component jh(0) includes the absent reﬂected wave.

To remove the grazing singularity we introduce, in the following, new deﬁnitions for components j(0) and j(1), which are more appropriate, when the incident wave illuminates both faces of the edge and propagates in the direction close to the face orientation.

7.9.1 Acoustic EEWs

This theory is based on the paper by Uﬁmtsev (2006a).

The appropriate candidate for a new deﬁnition of jh,s(0) is the ﬁeld jh,shp induced by the incident wave on the illuminated side of the tangential half-plane. On strip 1 (Fig. 7.3)

of the half-plane ϕ = 0, these components are determined as

jh(0) ≡ jhhp = 2u0 e−ikζ cos γ0 eikξ1 cos2 γ0

× −e−ikξ1 sin

e−π/4

∞

γ0 cos ϕ0

√

sin γ0 cos

ϕ0

eit dt + e−ikξ1 sin

γ0 cos ϕ0

2kξ1

(7.155)

and

j(0)

≡

jhp

e−ikζ cos γ0 eikξ1 cos2 γ0

× ik sin

γ0 sin ϕ0 e−ik

ξ1 sin

γ0 cos ϕ0

e−iπ/4

∞

√

sin γ0 cos

ϕ0

eit dt

2kξ1

e−iπ/4 eikξ1 sin2 γ0

ϕ0

sin

√

− ik sin γ0 sin ϕ0 e−ikξ1 sin

γ0 cos ϕ0

ξ1

(7.156)

TEAM LinG

200 Chapter 7 Elementary Acoustic and Electromagnetic Edge Waves

The last terms in these equations (together with the factor 2 in front of the brackets) relate to the sum of the incident and reﬂected plane waves. When ϕ0 → π , these equations transform into

jh(0) = u0 e−ikζ cos γ0 eikξ1				(7.157)
and

js(0) = u0 e−ikζ cos γ0	2k		−π/4),
		ei(kξ1		(7.158)
	π ξ1	ei(kξ1		(7.158)

where jh(0) is actually the grazing incident wave, and js(0) represents the edge wave. The quantities jh,s(0) on strip 2 (of the half-plane ϕ = α) can be found from Equations (7.157) and (7.158) with the replacement of ϕ0 by α − ϕ0 and ξ1 by ξ2.

The new nonuniform components jh,s(1) are deﬁned as the difference between the total scattering source jh,s (induced on the wedge faces) and the uniform component jh,s(0) ≡ jh,shp (induced on the tangential half-planes). These nonuniform components are determined by Equations (7.12) and (7.13), where one should replace

αw(η + ψ ) by αw(η + ψ ) − 2π whp(η + ψ )	(7.159)
with
whp(x) = 1 − eix/2.	(7.160)

The ﬁeld generated by the new component jh,s(1) is found with a procedure completely similar to that described in Sections 7.3 to 7.5. The associated ﬁeld integrals are deﬁned by Equations (7.44) and (7.45), where one should apply the replacement (7.159). It is worth noting that now the contribution to these integrals caused by the residues at the poles η = ±ϕ0 is equal to zero. The ﬁeld of EEWs found in this way can be represented again in the form of Equations (7.89) and (7.90), with the new functions Fh,s(1):

	dζ		eikR
duh(1) = uinc(ζ )		Fh(1)(ϑ , ϕ)		,
	2π		R
	dζ		eikR
dus(1) = uinc(ζ )		Fs(1)(ϑ , ϕ)		,
	2π		R

and

Fh(1)(ϑ , ϕ) = {[−Vt (σ1, ϕ0) + Vthp(σ1, ϕ0)] sin ϕ + [−Vt (σ2, α − ϕ0) + ε(α − ϕ0)Vthp(σ2, α − ϕ0)] sin(α − ϕ)} sin γ0 sin ϑ ,

Fs(1) = {[−Ut (σ1, ϕ0) + Uthp(σ1, ϕ0)]

(7.161)

(7.162)

(7.163)

+ [−Ut (σ2, α − ϕ0) + ε(α − ϕ0)Uthp(σ2, α − ϕ0)]} sin2 γ0. (7.164)

TEAM LinG

7.9 Improved Theory of Elementary Edge Waves

201

designation (7.48). Functions V

, U

Here, we assume that 0 ≤ ϕ0 ≤ π and utilize the hp

are deﬁned in Equations (7.87) and (7.85), but Vt

and Ut

are the similar functions

associated with the tangential half-plane:

V hp

(σ , ψ )

cot

σ + ψ

cot

σ − ψ

(7.165)

4 sin2

γ0 sin σ

Uhp

(σ , ψ )

cot

σ + ψ

−

cot

σ − ψ

(7.166)

4 sin2

γ0

For the directions of the diffraction cone (ϑ = π − γ0), functions (7.163) and (7.164) can be represented in the form

Fh(1)(π − γ0, ϕ) = g(ϕ, ϕ0, α) − ghp(ϕ, ϕ0)

(7.167)

and

Fs(1)(π − γ0, ϕ) = f (ϕ, ϕ0, α) − f hp(ϕ, ϕ0),

(7.168)

where f and g are the Sommerfeld functions (2.62) and (2.64), and

cot

−

ghp(ϕ, ϕ0) = −

+ cot

π −

−

2α

ε(α

)

cot

+ ϕ −

ϕ0

cot

ϕ + ϕ0

, (7.169)

cot

−

f hp(ϕ, ϕ0) =

−

− cot

−

+ 4 −

−

2α

ε(α

)

cot

ϕ0

cot

ϕ + ϕ0

(7.170)

are the functions associated with the ﬁeld scattered by the tangential half-planes. Notice also that for analytic analysis and numeric calculation, the cotangent forms

of functions f and g are more convenient than the Sommerfeld expressions (2.62) and

(2.64). These forms are determined as

cot

π −

−

g(ϕ, ϕ0, α) = −

−

+ cot

π +

+ cot

−

(7.171)

and

cot

−

f (ϕ, ϕ0, α) =

−

− cot

π + ϕ − ϕ0

+ cot

− cot

(7.172)

with n = α/π .

TEAM LinG

202 Chapter 7	Elementary Acoustic and Electromagnetic Edge Waves
Functions f ,	g and f hp, ghp are singular in the directions of the incident

and reﬂected plane waves (ϕ = π + ϕ0, ϕ = π − ϕ0, ϕ = 2α − π − ϕ0), but their difference is ﬁnite. For instance,

(1)

(π − γ0, π

− ϕ0) = −

cot

ϕ0

cot

ϕ0

cot

π − ϕ0

ε(α

)

cot

π − ϕ0

−

cot

α − π

ε(α

)

cot

α − π

(7.173)

(1)

(π

−

, π

−

)

= −

cot

ϕ0

cot

ϕ0

−

cot

π − ϕ0

ε(α

)

cot

π − ϕ0

cot

α − π

−

ε(α

)

cot

α − π

(7.174)

(1)

(π

−

, 2α

−

)

= −

cot

ϕ0 − (α − π )

cot

ϕ0 − (α − π )

cot

α − π

−

cot

α − π

−

cot

α − ϕ0

cot

α − ϕ0

(7.175)

and

(1)(π

−

, 2α

−

)

= −

cot

ϕ0 − (α − π )

cot

ϕ0 − (α − π )

−

cot

α − π

cot

α − π

−

cot

α − ϕ0

cot

α − ϕ0

(7.176)

F(1)

These equations clearly show that functions h,s are free from the grazing singularity. Indeed, for the grazing directions of the incident wave (ϕ0 = π , ϕ0 = α − π ), these functions have the ﬁnite values

Fh(1)

(π − γ0, 0) = Fh(1)

(π − γ0, α) = −

cot

tan

(7.177)

and

Fs(1)(π − γ0, 0) = Fs(1)(π − γ0, α) = −

tan

(7.178)

It also follows from these equations that functions Fh,s(1) are equal to zero when α = 2π and the wedge transforms into the half-plane. This result is the obvious consequence

TEAM LinG

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