Acoustic diffraction by a material cone

Wave Motion 42 (2005) 127–141

Acoustic diffraction by a material cone Aladin H. Kamel∗ P.O. Box 433, Heliopolis Center 11757, Cairo, Egypt Received 29 September 2004; received in revised form 20 October 2004; accepted 16 December 2004 Available online 20 January 2005

Abstract The boundary-value problem for the Helmholtz equation connected with an infinite circular cone has been analyzed. The proposed scheme of solution includes application of the Kontorovich–Lebedev (KL) transform. The fields excited by a rotationally invariant ring source have been considered. Uncoupled singular integral equations (SIEs) satisfied by the spectral functions were derived. The singularities of the spectral functions were deduced. Some asymptotic approximation for the field with source or observation point near the tip of the cone was obtained. Alternative representations for the far fields were derived. The conditions of validity of the derived field representations for a given set of problem parameters were studied. © 2004 Elsevier B.V. All rights reserved. Keywords: Kontorovich–Lebedev transform; Singular integral equations; Diffraction; Material cone

1. Introduction Acoustic wave propagation from one medium into a different infinite cone-shaped medium is a very difficult problem to solve in closed-form analytically. To this day, there is no explicit closed-form solution for a cone with an arbitrary refractive index. A Kontorovich–Lebedev (KL)-based formulation has been used before for the equivalent electromagnetic problem (dielectric cone) for the special case of an almost diaphanous cone [1]. The formulation presented here is an attempt to extend the technique to the scattering and diffraction by a large class of material cones. In Section 2, the problem is formulated. Section 3 derives the singular integral equations (SIEs) satisfied by the KL spectra. In Section 4, the field representations in the cone region are given and those for the medium external to the cone are discussed in Section 5. Conclusions are given in Section 6. Appendix A presents a scheme to numerically solve the integral equations. In Appendix B the singularities of the KL spectra of are identified and quantified. ∗

Tel.: +20 122 113 608; fax: +187 726 077 44. E-mail address: [email protected] (Aladin H. Kamel).

0165-2125/$ – see front matter © 2004 Elsevier B.V. All rights reserved. doi:10.1016/j.wavemoti.2004.12.003

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A.H. Kamel / Wave Motion 42 (2005) 127–141

2. Formulation We consider the problem of scattering of harmonic acoustic waves by an infinite material cone of circular crosssection imbedded in an infinite medium. In what follows (r, θ, φ) denotes the usual spherical coordinates such that on the surface of the cone one has θ = β with 0 < β < π. The region C1 = {0 < r < ∞, 0 ≤ θ < β, −π ≤ φ ≤ π} will be referred to as the medium I while the region C2 = {0 < r < ∞, β < θ ≤ π, −π ≤ φ ≤ π} (i.e. the material cone) will be called the medium II. A time factor exp{−iωt} is assumed and omitted throughout. k1 , ρ1 and λ1 are respectively the wave number, density and incompressibility of C1 . k2 , ρ2 and λ2 are corresponding quantities inside C2 and N = ( kk21 ). Since the problem under consideration is one of scattering and diffraction the wave numbers k1,2 are real (complex) for the lossless (lossy) case. However, for a while we shall assume that k1,2 are such that φ1,2 = arg k1,2 =

π . 2

(1)

The condition in Eq. (1) has been shown by Osipov [2] to guarantee the convergence of the subsequent KL integral representations, which in turn is required for the possibility to use the boundary conditions conveniently, leading to the derivation of integral equations on the KL spectra. Once integral equations are derived and a numerical scheme is proposed to solve them, we examine the restrictions under which k1,2 could be extended to real or complex values (to tackle the original scattering and diffraction problem) while maintaining the validity of the numerical scheme. It should be noted that the analysis carried out here is for the case |N| > 1. Modifications for the case |N| < 1 will be carried out in a future publication. The acoustic field is describable by the pressure p and velocity V obeying the Euler field equations [3]. Acoustic field excitation is provided by an impressed rotationally invariant unit force density located at (r0 , θ0 ) in C1 . With ρ1,2 and λ1,2 assumed constant, the acoustic pressure in C1 , p1 (r, θ), satisfies the inhomogeneous Helmholtz equation (∇ 2 + k12 )p1 (r, θ) = −

δ(r − r0 )δ(θ − θ0 ) , r0 sin θ0

(2)


where k1 = cω1 , c1 = λρ11 is the acoustic speed in C1 and ∇ 2 stands for the Laplacian. The acoustic pressure in C2 , p2 (r, θ), satisfies the homogeneous Helmholtz equation (∇ 2 + k22 )p2 (r, θ) = 0,
where k2 = cω2 and c2 = λρ22 is the acoustic speed in C2 . The velocity fields are derived from the acoustic pressures as   ∂ 1 ∂ −i V1,2 (r, θ) = p1,2 (r, θ). r0 + θ 0 ωρ1,2 ∂r r ∂θ

(3)

(4)

Pressure and normal velocities are continuous across the surface of the cone, namely p1 (r, β) = p2 (r, β), 0 ≤ r < ∞,

(5)

V1θ (r, β) = V2θ (r, β), 0 < r < ∞.

(6)

The fields are required to decay exponentially to zero as r → ∞. This replaces the Sommerfeld radiation condition for the Im k = 0 case. The field behavior near the tip of the cone, owing to the condition of Meixner [3], is p = O(rν− 2 ), 1

|V| = O(rν− 2 ), 3

r → 0, ν > 0.

(7)

A.H. Kamel / Wave Motion 42 (2005) 127–141

129

We propose to solve the problem by means of a KL transform. Let us introduce the KL transform [3]:  ∞ (1) p(r, θ)h 1 (kr)dr, P(ν, θ) = ν− 2

0

p(r, θ) =

k π



i∞

−i∞

where jν− 1 (kr) and h 2

 jν (z) =

π 2z

νjν− 1 (kr)P(ν, θ)dν, are spherical Bessel functions given by

1 2

(9)

2

(1) (kr) ν− 21

(8)

 Jν+ 1 (z), h(1) ν (z) 2

=

π 2z

1 2

H

(1) (z). ν+ 21

(1)

Jν (z) and Hν (z) are the standard Bessel and Hankel functions respectively and ν is purely imaginary. Since (1) (1) H−ν (z) = eiπν Hν (z) the definition in Eq. (8) implies that P(−ν, θ) = eiπν P(ν, θ).

(10)

There are two additional ways of writing the inverse transform of Eq. (9):  i∞ k (1) νh (kr)P(ν, θ)dν. p(r, θ) = 2π −i∞ ν− 21  −ik i∞ iπν (1) p(r, θ) = νe sin πνh 1 (kr)P(ν, θ)dν. ν− 2 π 0

(11) (12)

The velocity field is derived from Eq. (4). Relevant to the problem under consideration is the velocity component normal to the cone surface, namely Vθ (r, θ)  i∞ −k 1 ∂ (1) νeiπν sin πνh 1 (kr)P(ν, θ)dν. (13) Vθ (r, θ) = ν− 2 πωρ r ∂θ 0 Thus, −k 1 Vθ (r, θ) = πωρ r



i∞

νeiπν sin πνh

0

d (1) (kr) P(ν, θ)dν. ν− 21 dθ

(14)

Applying the KL transform to Eqs. (2) and (3), we get the ordinary differential equations 

(1)     h 1 (k1 r0 )δ(θ − θ0 ) 1 d d 1 ν− 2 2 sin θ , + ν − P1 (ν, θ) = − sin θ dθ sin θ0 dθ 4      d 1 1 d sin θ + ν2 − P2 (ν, θ) = 0, β < θ ≤ π, sin θ dθ dθ 4

0 ≤ θ < β,

(15) (16)

whose general solutions are the Legendre functions [3] Pν− 1 (± cos θ), with Pν− 1 (cos θ) and Pν− 1 (− cos θ) bounded 2 2 2 at θ = 0 and θ = π, respectively, but not at the opposite endpoint. The boundary condition in Eq. (5) now reads  i∞  i∞ (1) (1) iπν k1 νe sin πνh 1 (k1 r)P1 (ν, β)dν = k2 νeiπν sin πνh 1 (k2 r)P2 (ν, β)dν. (17) 0

ν− 2

0

ν− 2

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A.H. Kamel / Wave Motion 42 (2005) 127–141

From Eqs. (4) and (14), the boundary condition in Eq. (6) now reads   k1 i∞ iπν d k2 i∞ iπν d (1) (1) νe sin πνh 1 (k1 r) P1 (ν, β)dν = νe sin πνh 1 (k2 r) P2 (ν, β)dν, ν− ν− ρ1 0 dβ ρ2 0 dβ 2 2

(18)

d d )P1,2 (ν, β) stands for ( dθ )P1,2 (ν, θ)|θ=β , which notation will be used throughout. where ( dβ (1)

(1)

We represent the field in C1 as the sum over an unperturbed field p01 , V10 plus a scattered field p1 , V1 due to the presence of the cone. Hence, (0)

(1)

p1 (r, θ) = p1 (r, θ) + p1 (r, θ), (0)

(19)

(1)

P1 (ν, θ) = P1 (ν, θ) + P1 (ν, θ).

(20)

2.1. The unperturbed field (0)

P1 (ν, θ) satisfies the source conditions of Eq. (15) and is required to be bounded at θ = 0, π. Hence, (1)

(0) P1 (ν, θ)

h 1 (k1 r0 ) π ν− 2 = P 1 (cos θ< )Pν− 1 (− cos θ> ), 2 2 sin θ0 sin(ν − 21 )π ν− 2

(21)

where θ> |θ< is the greater|lesser of θ and θ0 . (0) From Eqs. (14) and (21), V1θ (r, θ) is given by  −k1 1 i∞ iπν d (0) (0) (1) V1θ (r, θ) = νe sin πνh 1 (k1 r) P1 (ν, θ)dν, ν− πωρ1 r 0 dθ 2

(22a)

(1)

h 1 (k1 r0 ) d (0) π d ν− 2 P1 (ν, θ) = P 1 (cos θ0 ) Pν− 1 (− cos θ), 2 dθ 2 sin θ0 sin(ν − 21 )π ν− 2 dθ

θ > θ0 ,

(22b)

θ < θ0 .

(22c)

(1)

h 1 (k1 r0 ) d (0) π d ν− 2 P1 (ν, θ) = Pν− 1 (− cos θ0 ) Pν− 1 (cos θ), 1 2 2 dθ 2 sin θ0 sin(ν − 2 )π dθ 2.2. The scattered field (1)

Since P1 (ν, θ) must be bounded at θ = 0 we represent P1 (ν, θ) by (1)

P1 (ν, θ) = A1 (ν)Pν− 1 (cos θ). 2

(23)

In C2 , since P2 (ν, θ) must be bounded at θ = π we represent it by P2 (ν, θ) = A2 (ν)Pν− 1 (− cos θ). 2

(24)

A1 (ν) and A2 (ν) are KL spectra to be determined from the boundary conditions in Eqs. (17) and (18). Eq. (10) together with [3] Pν− 1 (x) = P−ν− 1 (x), 2

2

(25)

enforce A1,2 (−ν) = eiπν A1,2 (ν).

(26)

A.H. Kamel / Wave Motion 42 (2005) 127–141

131

Additionally, the convergence of the KL integrals [2] at θ = β implies that, when Im ν → +∞, the spectral functions must vanish as   A1 (ν) = O exp(−β Im ν) , (27a)   (27b) A2 (ν) = O exp(−(π − β) Im ν) . From the above we obtain  −ik1 i∞ iπν (1) (1) νe sin πνh 1 (k1 r)A1 (ν)Pν− 1 (cos θ)dν, p1 (r, θ) = ν− 2 2 π 0  i∞ −ik2 (1) νeiπν sin πνh 1 (k2 r)A2 (ν)Pν− 1 (− cos θ)dν, p2 (r, θ) = ν− 2 2 π 0  i∞ −k1 1 d (1) (1) νeiπν sin πνh 1 (k1 r)A1 (ν) Pν− 1 (cos θ)dν, V1θ (r, θ) = ν− 2 2 πωρ1 r 0 dθ  i∞ −k2 1 d (1) V2θ (r, θ) = νeiπν sin πνh 1 (k2 r)A2 (ν) Pν− 1 (− cos θ)dν. ν− 2 2 πωρ2 r −i∞ dθ

(28a) (28b) (28c) (28d)

Substituting from Eqs. (20), (21), (23) and (24) into Eq. (17)  i∞ (1) νeiπν sin πνh 1 (k1 r)[F1 (ν) + A1 (ν)Pν− 1 (cos β)]dν 0

 =N

ν− 2

i∞

2

νeiπν sin πνh

(1) (k r)A2 (ν)Pν− 1 (− cos β)dν, ν− 21 2 2

0

(29a)

(1)

h 1 (k1 r0 ) π ν− 2 F1 (ν) = P 1 (cos θ0 )Pν− 1 (− cos β). 2 2 sin θ0 sin(ν − 21 )π ν− 2 Substituting from Eqs. (20), (22b), (23) and (24) into Eq. (18)  i∞ d (1) νeiπν sin πνh 1 (k1 r)[F2 (ν) + A1 (ν) Pν− 1 (cos β)]dν ν− 2 dβ 2 0  i∞ Nρ1 d (1) = νeiπν sin πνh 1 (k2 r)A2 (ν) Pν− 1 (− cos β)dν, ν− 2 ρ2 0 dβ 2

(29b)

(30a)

(1)

h 1 (k1 r0 ) π d ν− 2 F2 (ν) = P 1 (cos θ0 ) Pν− 1 (− cos β). 2 2 sin θ0 sin(ν − 21 )π ν− 2 dβ

(30b)

3. Derivation of the singular integral equation (1) (k r) and integrate with respect to r from 0 to ∞. To that end we make µ− 21 1

We multiply Eqs. (29a) and (30a) by h use of [1]  ∞ 0

h

(1) (1) (kr)h 1 (kNr)dr µ− 21 ν− 2

= D(ν, µ) + I(ν, µ),

(31a)

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A.H. Kamel / Wave Motion 42 (2005) 127–141

√ cosh(µ ln N) −iµπ D(ν, µ) = πk N [e δ( Im ν − Im µ) + δ( Im ν + Im µ)], µ sin µπ √   F (1 + 21 (µ + ν), 1 + 21 (µ − ν); 2; 1 − (1/N 2 )) 1 −πk N −i(µ+ν)π/2 e 1 − 2 N −µ , I(ν, µ) = 2 cos πν − cos πµ N

(31b)

(31c)

where |N| > 1 and ν, µ imaginary. F (a, b; c; z) is the Gauss Hypergeometric function. For the particular case which is considered here ( Im µ > 0, Im ν > 0), the δ( Im ν + Im µ) in Eq. (31b) will be removed. We obtain F1 (µ) + A1 (µ)Pµ− 1 (cos β) 2   = N N1∗ (µ)A2 (µ)Pµ− 1 (− cos β) + N2∗ (µ)v.p. 2

F2 (µ) + A1 (µ) =

i∞ 0

 I˜ ∗ (ν, µ)A2 (ν)Pν− 1 (− cos β)dν , 2

(32a)

d P 1 (cos β) dβ µ− 2

   i∞ d d Nρ1 I˜ ∗ (ν, µ)A2 (ν) Pν− 1 (cos β)dν , N1∗ (µ)A2 (µ) Pµ− 1 (cos β) + N2∗ (µ)v.p. 2 2 ρ2 dβ dβ 0

√ N cosh(µ ln N),   1 −1 √ −iµπ/2 ∗ 1 − 2 N −µ , Ne N2 (µ) = 2 N N1∗ (µ) =

I˜ ∗ (ν, µ) = νeiπν sin πνI ∗ (ν, µ), I ∗ (ν, µ) = e−iνπ/2

(32b)

(32c) (32d) (32e)

F (1 + 21 (µ + ν), (1 + 21 )(µ − ν); 2; 1 − (1/N 2 )) cos πν − cos πµ

(32f)

and v.p. in front of the integral sign denotes that the Cauchy principal value is to be taken. d )Pµ− 1 (cos β) and Eq. (32b) by Pµ− 1 (cos β) then subtracting the two equations, we Multiplying Eq. (32a) by ( dβ 2 2 eliminate A1 (µ) and reach the SIE on A2 (µ)  i∞ A2 (µ) = Q(µ) − C(µ)v.p. νeiπν sin πνK(ν, µ)A2 (ν)dν, Im µ
0, Reµ = 0, (33) 0

Q(µ) = −

(1) h 1 (k1 r0 ) µ− 2

sin θ0 sin β

Pµ− 1 (cos θ0 )/NN1∗ (µ)M(µ), 2

C(µ) = N2∗ (µ)/N1∗ (µ)M(µ), K(ν, µ) = I ∗ (ν, µ)[Pν− 1 (− cos β) 2

M(µ) = Pµ− 1 (− cos β) 2

(34a) (34b)

d ρ1 d P 1 (cos β) − Pµ− 1 (cos β) Pν− 1 (− cos β)], 2 2 dβ µ− 2 ρ2 dβ

d ρ1 d P 1 (cos β) − Pµ− 1 (cos β) Pµ− 1 (− cos β). 2 2 dβ µ− 2 ρ2 dβ

(34c) (34d)

A.H. Kamel / Wave Motion 42 (2005) 127–141

133

4. Field representations in the cone region 4.1. Near Field in the cone region For r < r0 , making use of the KL representation in Eq. (9) we obtain  k2 i∞ νj 1 (k2 r)A2 (ν)Pν− 1 (− cos θ)dν. p2 (r, θ) = 2 π −i∞ ν− 2

(35)

Closing contours of Eq. (35) in the right-hand side of the complex ν-plane and collecting residue contributions, we express νpl jνpl − 1 (k2 r) Res [A2 (νpl )]Pνpl − 1 (− cos θ), (36) p2 (r, θ) = −2ik2 2

νpl

2

where the residues Res[A2 (νpl )] are given in Eqs. (B.9) and (B10) in Appendix B. Eq. (36) is valid for first order poles only. Modifications are required for higher order poles if they exist. For instance, the second order poles that may be present indicate (see [4]) that the acoustic pressure (velocity) field near the tip of the cone may contain the 1 3 logarithm of the distance in addition to its power, namely r µ− 2 log r (rµ− 2 log r). Since it is possible for the poles νpl to accumulate at infinity, forming a dense set on the real line, in addition to the possibility of higher order poles, either of these possibilities could avoid the convergence of the residue sum. The truncated residue sum should, therefore, be understood as giving some asymptotic approximation to the field in terms of first identified poles. Next we analyze the behavior of the summand as νpl → ∞. From the identity [5, formula 8.733(1)]    d 1 1 (37a) Pν− 1 (± cos θ) = ∓ ν − ν+ P −11 (± cos θ), ν− 2 2 dθ 2 2 where P −11 (± cos θ) are Associated Legendre functions, from [5, formula 8.721 (4)] ν− 2



   2 1 π Pν (cos φ) ∼ cos ν + φ− , ν → ∞, νπ sin φ 2 4

   1 π 2 −1 cos ν + φ− , ν → ∞, νPν (cos φ) ∼ νπ sin φ 2 2 from [1]   µ ν µ ν 1 F 1 + + , 1 + − , 2, 1 − 2 ∼ N µ I1 (µ ln n2 )/µ ∼ N µ eµn2 /µ3/2 , 2 2 2 2 N where I1 (z) is the modified Bessel function of the first kind with order one and  n2 = N + N 2 − 1,

(37b)

(37c)

µ → ∞,

(38a)

(38b)

and from [6, formula 15.7.2] 

1 F µ + 1, 1, 2, 1 − 2 N

 ∼

e

µ(1−

µ

1 ) N2

,

µ → ∞,

substituting Eqs. (37)–(39) into Eqs. (B.9a), (B10) and Eq. (36), it is revealed that:

(39)

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A.H. Kamel / Wave Motion 42 (2005) 127–141

For l = 0, νp0 → ∞ (p → ∞) the series summand is dominated by 1 Jν (k2 r)Hν(1) (k r ), ∗ p0 1 0 N1 (νp0 ) p0

νp0 → ∞.

(40)

From [3]  2ν −ν Jν (z) ∼ √ , ν → ∞, 2πν ez  ν 1 2ν Hν(1) (z) ∼ √ , ν → ∞, 2πν ez 

1

Eq. (40) reduces to  νp0 1 r , νp0 r0

(41a) (41b)

νp0 → ∞.

(42)

For νpl → ∞ (l → ∞, p → ∞) the dominant behavior of the summand in Eq. (36) is √  νp0  2l νp0 k2 r 1 (N + N 2 − 1)νp0 +2l ek2 r , l → ∞, p → ∞. 1+l [νp0 + 2l]k1 r0 2[νp0 + 2l] N 2(νp0 +l)

(43)

The field representation in Eq. (36) has the advantage of being valid for all observation angles in the cone region and the disadvantage of cumbersome expressions for the residues of the poles. However, if the main interest is to compute the singular behavior of the field near the tip of the cone, then one could use νp0 < 21 (s) p2 (r, θ)

= −2ik2



νp0 jνp0 − 1 (k2 r) Res [A2 (νp0 )]Pνp0 − 1 (− cos θ), 2

2

(44)

(s)

where p2 (r, θ) is the singular field near the tip of the cone, together with the corresponding sums from Eq. (4) for the velocity field with the sums running to νp0 < 23 . One then needs to compute only the residues of A2 (ν) for νp0 < 23 . It should be noted that the pole at ν = 21 (from M(1/2) = 0) is excluded from the sum in Eq. (44) since its corresponding field is non-singular. Another field representation is available by substituting for A2 (ν), from Eq. (33), into Eq. (35), leading to p2 (r, θ) = p21 (r, θ) + p22 (r, θ),  k2 i∞ νj 1 (k2 r)Q(ν)Pν− 1 (− cos θ)dν, p21 (r, θ) = 2 π −i∞ ν− 2  −k2 i∞ νj 1 (k2 r)C(ν)I(ν)Pν− 1 (− cos θ)dν, p22 (r, θ) = 2 π −i∞ ν− 2  i∞  ν eiπν sin πν K(ν , ν)A2 (ν )dν . I(ν) = v.p.

(45) (46) (47a) (47b)

0

Closing contours of Eq. (46) in the right-hand side of the complex ν-plane and collecting residue contributions, we express νp0 jνp0 − 1 (k2 r) Res [Q(νp0 )]Pνp0 − 1 (− cos θ), (48a) p21 (r, θ) = −2ik2 νp0

2

2

A.H. Kamel / Wave Motion 42 (2005) 127–141

Res [Q(νp0 )] = −

135

  d Pνp0 − 1 (cos θ0 )/ NN1∗ (νp0 ) M(νp0 ) . 2 sin θ0 sin β dνp0 (1) (k r ) νp0 − 21 1 0

h

(48b)

The series summand, as νp0 → ∞, behaves as 1 νp0



r r0

νp0

,

νp0 → ∞.

(49)

Regarding the integral in Eq. (41a), from [1]   ν ν ν ν 1 N ν I1 (ν ln n2 ) F 1 + + , 1 + − , 2, 1 − 2 ∼ , 2 2 ν 2 2 N

ν → ±i∞,

(50)

together with I1 (z) = e∓πi/2 J1 (ze±πi/2 ),   1 3π J1 (z) ∼ cos z − /|z| 2 , 4

(51a) (51b)

leading to     ν ν Nν 3π ν ν F 1 + + , 1 + − ; 2; 1 − N −2 ∼ 3/2 cos |ν| ln n2 − , 2 2 2 2 4 |ν|

ν → ±i∞,

(52)

from Eq. (37a) and [5, formula 8.721(1)] Pν− 1 (cos θ) ∼ 2

e|ν|θ 1

|ν| 2

Pν− 1 (− cos θ) ∼

ν− 2

1

|ν| 2

e|ν|θ 3

|ν| 2

P −11 (− cos θ) ∼ ν− 2

ν → ±i∞,

e|ν|(π−θ)

2

P −11 (cos θ) ∼

,

,

,

(53a)

ν → ±i∞,

ν → ±i∞,

e|ν|(π−θ) 3

|ν| 2

,

(53b)

(53c)

ν → ±i∞

(53d)

and from[3] Jν (z) ∼

e|ν|(π/2)+iν arg z 1

|ν| 2

,

ν → ±i∞,

(54)

we obtain I(ν) ∼

e−|ν|(π−| arg n2 |−β)+iν(φ2 −φ1 ) , |ν|

ν → ±i∞,

(55a)

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A.H. Kamel / Wave Motion 42 (2005) 127–141

φ1,2 = arg k1,2 ,

(55b) πν

C(ν) ∼ e−|ν|(π+|φ2 −φ1 |)−iν(φ2 −φ1 )−i( 2 ) ,

ν → ±i∞.

(56)

We thus infer that the integrand in Eq. (47a) decays exponentially as π

νπ

e−[|ν|(( 2 )+θ−β+|φ2 −φ1 |−| arg n2 |)]+iνφ2 −i( 2 ) , ν → ±i∞. |ν| Since for the problem under consideration 0 ≤ φ1,2 ≤ exists when

π 2

(57)

and since 0 ≤ (| arg n2 | − |φ2 − φ1 |) < π2 , the integral

θ > β + | arg n2 | − |φ2 − φ1 | − φ2 .

(58)

From Eq. (58) we infer that for the lossless case, φ1,2 = 0 (also for φ1 = φ2 ), the representation given in Eq. (47a) exists for all the observation angles in the cone region, whereas for the lossy case (except when φ1 = φ2 ) it has a gap, as given by Eq. (58), in the angular domain where it cannot be used. The field representation in Eqs. (47a) and (48a) has the advantage of numerical efficiency and the disadvantages of the gap mentioned above in connection with Eq. (47a) as well as hiding part of the strength of the field singularities near the tip of the cone associated with the poles νp0 < 23 . However, it is possible to recover the full strength of the singularities and close the gap for their field representation by isolating the singular behavior as was given in Eq. (44). 4.2. Far Field in the cone region We propose here two alternatives to calculate the far field. The first alternative is to invoke the reciprocity principle. Thus to calculate the fields when r > r0 we employ νjν− 1 (k1 r0 ) Res [A1 (ν)]Pν− 1 (cos θ0 ), (59) p2 (r, r0 , θ, θ0 ) = −2ik1 2

2

where the sum is on the residues of A1 (ν) with the source located at (r, θ) in C2 and the observer located at (r0 , θ0 ) in C1 and the truncated sum is understood as an asymptotic approximation related to first identified poles. It is worth mentioning that the above residue sum cannot be used to derive far field results for plane wave illumination, since the convergence is an asymptotic result for source near the tip. The second alternative is to substitute in  −ik2 i∞ (1) p2 (r, θ) = ν sin νπeiνπ h 1 (k2 r)A2 (ν)Pν− 1 (− cos θ)dν (60) ν− 2 2 π 0 for A2 (ν), from Eq. (33), to obtain p2 (r, θ) = p21 (r, θ) + p22 (r, θ), p21 (r, θ) = p22 (r, θ) =

ik2 π



−ik2 π

i∞

ν sin νπeiνπ h

0



i∞ 0

(61) (1) (k r)C(ν)I(ν)Pν− 1 (− cos θ)dν, ν− 21 2 2

ν sin νπeiνπ h

(1) (k r)Q(ν)Pν− 1 (− cos θ)dν. ν− 21 2 2

(62)

(63)

A.H. Kamel / Wave Motion 42 (2005) 127–141

137

The contribution from p21 (r, θ) is found by direct numerical integration. From Eqs. (50)–(53) and from [3] (1) (k r) ν− 21 2

h

π

∼ e|ν|(( 2 )−φ2 ) /|ν| 2 , 1

ν → i∞,

(64)

we infer that the integrand in Eq. (62) decays exponentially and that the integral in Eq. (62) exists when the inequality given in Eq. (58) is satisfied. Eq. (63) is rephrased as  k2 i∞ (1) ˆ νh (k2 r)Q(ν)P (65) p22 (r, θ) = ν− 21 (− cos θ) dν, π −i∞ ν− 21 ˆ Q(ν) =−

jν− 1 (k1 r0 ) 2

sin θ0 sin β

Pν− 1 2

(cos θ0 ) . NN1∗ (ν)M(ν)

(66)

By closing contours in the right-hand side of the complex ν-plane and collecting residue contributions, we reach ˜ p0 )h(1) 1 (k2 r)P p22 (r, θ) = Q(ν (67a) νp0 − 1 (− cos θ), νp0

νp0 − 2

2

j 1 (k1 r0 ) Pν− 1 (cos θ0 ) 2 ˜ p0 ) = 2ik2 ν ν− 2 Q(ν |νp0 , d NN1∗ (ν) sin θ0 sin β ( dν )M(ν)

(67b)

where the summand behaves as  νp0 1 r0 , νp0 → ∞. νp0 |N|2 r

(68)

The field representation given in Eq. (59) has the disadvantage of cumbersome computations and the advantage of being valid for all angles. The field representation given in Eqs. (62) and (67a) has the advantage of numerical efficiency and the disadvantage of the angular domain gap mentioned in connection with Eq. (58), where Eq. (62) cannot be used. Far field (k2 r  1) expressions are obtained by utilizing the large argument asymptotic approximation for the Hankel function [3] 2 i(z−νπ/2−π/4) (1) Hν (z) ∼ . (69) e πz Substituting from Eq. (69) in Eqs. (62) and (67a), we get  i i(k2 r−π/4) i∞ p21 (r, θ) = ν sin νπeiνπ/2 C(ν)I(ν)Pν− 1 (− cos θ) dν. e 2 πr 0

(70)

The convergence properties of the integrand in Eq. (70) remain as discussed in connection with Eq. (62). p22 (r, θ) =

1 i(k2 r−π/4) ˜ e Q(νp0 )e−iνp0 π/2 Pνp0 − 1 (− cos θ), 2 k2 r ν

(71)

p0

where the summand behaves as 1

1 Jνp0 (k1 r0 ) ∼ 1 ∗ |ν p0 | |νp0 | 2 N (νp0 ) 1



e |k1 |r0 2|N |νp0

νp0

,

νp0 → ∞.

(72)

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A.H. Kamel / Wave Motion 42 (2005) 127–141

5. Fields outside the cone region By eliminating A2 (µ) between Eqs. (32a) and (32b), we reach  i∞ A1 (µ) = Q1 (µ) − C1 (µ)v.p. νeiπν sin πνK1 (ν, µ)A2 (ν)dν,

(73)

0

where π Q1 (µ) = 2



(1)  h 1 (k1 r0 ) ρ1 (− cos β) d µ− 2 −1 , Pµ− 1 (cos θ0 )Pµ− 1 (− cos β) Pµ− 1 1 2 2 2 ρ2 dβ M(µ) sin θ0 sin(µ − 2 )π

ρ1 NN2∗ (µ) , ρ2 M(µ)   d d ∗ K1 (ν, µ) = I (ν, µ) Pν− 1 (− cos β) Pµ− 1 (− cos β) − Pµ− 1 (− cos β) Pν− 1 (− cos β) . 2 2 2 2 dβ dβ C1 (µ) =

(74a)

(74b) (74c)

Eq. (73) reveals that A1 (µ) is meromorphic with pole singularities at µpl and µs = 21 + s (with sin(µs − 21 )π = 0, s = 0, 1, 2, . . .). It is noted that an apparent second order pole at µ = 21 (from sin(µ − 21 )π = 0 and M(µ) = 0 in Q1 (µ)) is, in fact, a simple pole. Alternative near and far field representations, advantages/disadvantages and conditions of validity of each proceed on lines similar to those detailed in Section 4 and are omitted from the discussion. 6. Conclusion Singular integral equations satisfied by the KL spectra of the acoustic fields in the presence of a material cone, with |N| > 1, have been derived in Section 3, Eq. (33) and Section 5, Eq. (73). An approximate numerical solution for the KL spectral function in Eq. (33) was constructed in Appendix A. Limitations of that solution were given in Eqs. (A.5a) and (A.5b) in the form of a source-boundary separation requirement for the validity of the numerical scheme. In Appendix B, the meromorphic nature of the KL spectral function A2 (ν) was established together with the identification of its pole singularities. Those for A1 (ν) were given in Section 5. Alternative representations for the near field, together with the advantages and disadvantages of each, were given in Section 4.1, Eqs. (36) and (47a), (48a). Situations under which the representation in (36) turns into an asymptotic approximation were discussed in Section 4.1. Similar analysis for the far fields were given in Section 4.2, Eqs. (59), (62) and (67a) and (70) and (71) . Conditions for the validity of each of these representations for a given set of problem parameters have been established in Section 4.1, Eq. (58). The approach of this paper is applicable for 2- and 3-D problems of thermal conductivity, electromagnetics and elastodynamics in a wedge and a cone with boundary conditions on the radial direction. The case of an impedance cone will be dealt with in a follow-up paper. Acknowledgement The author is grateful for the careful reading and the constructive criticism by two anonymous reviewers. Appendix A. Numerical scheme to solve the SIE in Eq. (33) The scheme used here is inspired by the one used by Antipov [7]. Let {νm−1 }, νm−1 = i δ(m − 1)2 , δ > 0, 2 > 0, m = 1, 2, . . . , M + 1 and {µm }, µm = (νm−1 + νm )/2, m = 1, 2, . . . , M be two sets of points. We approximate

A.H. Kamel / Wave Motion 42 (2005) 127–141

139

the SIE in Eq. (33) with the linear system of algebraic equations A 2 = Q∗ − C ∗ A 2 ,

(A.1)

with A2 and Q∗ as vectors and C∗ as the matrix ∗ Cnm = C(µn )K1 (ˆνm , µn )K2 (ˆνm , µn )Inm ,

(A.2)

where

   1 1 1 K1 (ˆνm , µn ) = F 1 + (µn + νˆ m ), 1 + (µn − νˆ m ); 2; 1 − , 2 2 N2

(A.3a)

  d ρ1 d K2 (ˆνm , µn ) = 2iˆνm eiˆνm π/2 Pνˆ m − 1 (− cos β) Pµn − 1 (cos β) − Pµn − 1 (cos β) Pνˆ m − 1 (− cos β) , 2 2 2 2 dβ ρ2 dβ (A.3b) (νm−1 + νm ) , 2   cos(πνm ) − cos(πµn ) −1 ln = / (νm−1 , νm ), , µn ∈ π cos(πνm−1 ) − cos(πµn )   cos(πνm ) − cos(πµn ) −1 ln = , µn ∈ (νm−1 , νm ). π cos(πµn ) − cos(πνm−1 )

νˆ m =

(A.3c)

Inm

(A.3d)

Inm

(A.3e)

Assuming that the inequalities π + |φ2 − φ1 | + φ1 2   π 1 β > + Re cos−1 2 N

θ0 <

(A.4a) (A.4b)

∗ } (n, m = 1, 2, . . . , M), then the approximate are satisfied and that an inverse exists for the matrix {δnm + Cnm (M) ∗ solution A2 converges to the exact one A2 and the rate of convergence is exponential (see [7]). The two inequalities in Eqs. (A.4a) and (A.4b) could be changed by a slight modification to the numerical ˜ 2 (µ)eiµ(( π2 )−θ0 ) . The SIE on A ˜ 2 (µ) is solved using the proposed scheme through a normalization process: A2 (µ) = A numerical scheme which is now valid when the inequalities   −1 1 β − θ0 > Re cos (A.5a) N

β − θ0 < π − (| arg n2 | − |φ2 − φ1 |)

(A.5b)

are satisfied. These inequalities reveal source-boundary separation requirements imposed by the Kontorovich– Lebedev formulation. Appendix B. Poles and residues of A2 (µ) We continue A2 (µ) analytically into the complex µ-plane in order to analyze its analytic properties. Since [1/(cos νπ − cos µπ)] has poles when µ = ±ν ± 2q, q = 0, 1, 2, . . . we do the analytic continuation in a strip-

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A.H. Kamel / Wave Motion 42 (2005) 127–141

by-strip fashion. The continuation to the qth strip Sq {Reµ ∈ (2q, 2q + 2), Im µ ∈ (−∞, ∞)}q = 0, 1, . . . is given by    i∞ j=q   1 A2 (µ) = + (B.1) s (µ) − R(ν, µ)L(ν, µ)A2 (ν)dν + K+j (µ)A2 (µ − 2j) , µ ∈ Sq ,  M (µ)  −i∞ j=1

where L(ν, µ) =

1 , cos(νπ) − cos(µπ)

(B.2)

M + (µ) = M(µ)7(µ),

(B.3)

(1 − N −2 ) −µ N µ cot µπF (1 + µ, 1; 2; 1 − N −2 ), cosh(µ ln N)  ∗    N2 (µ) µ+ν µ−ν ∗ −iπν/2 −2 R(ν, µ) = ν (µ, ν)e M F 1 + , 1 + , ; 2; 1 − N N1∗ (µ) 2 2

7(µ) = 1 −

K+j (µ) =

(B.4) (B.5)

(−1)j 2(1 − N −2 ) ∗ M (µ, µ − 2j)N −µ (µ − 2j) cot πµF (1 + µ − j, 1 + j; 2; 1 − N −2 ). cosh(µ ln N)

(B.6)

with s (µ) = Q(µ)M(µ), M ∗ (µ, ν) = K(ν, µ)/I ∗ (ν, µ). Hence, for µ ∈ Sq , A2 (µ) has q sets of poles leading to the pole structure: µp with µp ∈ Sq , µp + 2 with µp ∈ Sq−1 ,. . ., µp + 2q with µp ∈ S0 and with µp satisfying M(µp ) = 0.

(B.7)

Eq. (B.7) compares with Idemen’s results [4] for the equivalent electromagnetic problem. Throughout, we will adopt the terminology µpl = µp0 + 2l, l = 0, 1, 2, . . . , ∞; p = 1, 2, 3, . . . , ∞,

(B.8)

with µp0 satisfying Eq. (B.7) to describe the poles structure of A2 (µ). A µp0 pole is either simple or 2nd order. A µpl , l > 0, pole is simple or 2nd or 3rd or 4th order. We note here that second order zeros of M(µp0 ) = 0 have been reported by Idemen [4] who also devised computer algorithms for calculating the zeros. B.1. Residues computation (a) The residue of a pole of the type µp0 in the Sq strip is given by 7(µp0 ) Res [A2 (µp0 )]    i∞ j=q 1 s (µ) − =  R(ν, µ)L(ν, µ)A2 (ν)dν|µp0 + K+j (µ)A2 (µ − 2j) M (µp0 ) −i∞ j=1

M  (µp0 ) =

dM(µ) |µp0 . dµ

(b) The residue of a pole of the type µpl , l > 0, is given by   1 +l K (µ) 7(µpl ) Res [A2 (µpl )] = Res [A2 (µp0 )], M(µ) µpl

,

(B.9a)

µp0

(B.9b)

l = 1, 2, 3, . . . , ∞.

(B.10)

A.H. Kamel / Wave Motion 42 (2005) 127–141

141

The residues of higher order poles are not detailed here but are straightforward and require the utilization of the higher order residue formula instead of the first order formula used in this analysis.

References [1] D.S. Jones, Rawlins’ method and the diaphanous cone, Q. J. Mech. Appl. Math. 53 (1) (2000) 91–109. [2] A.V. Osipov, On the method of Kontorovich–Lebedev integrals for the problems of diffraction in sectorial media, in: Problems of Diffraction and Propagation of Waves, vol. 25, St. Petersburg University Publihers, 1993, 173–219. [3] L.B. Felsen, N. Marcuvitz, Radiation and Scattering of Waves, Prentice-Hall Inc., New Jersey, USA, 1973. [4] M. Idemen, Confluent tip singularity of the electromagnetic field at the apex of a material cone, Wave Motion 38 3 (2003) 251–277. [5] I.S. Gradshteyn, I.M. Ryzhik, Tables of Integrals, Series, and Products, Academic Press, New York, USA, 1980. [6] M. Abramowitz, I. Stegun, Handbook of Mathematical Functions, Dover Publications, New York, USA, 1970. [7] Y.A. Antipov, Diffraction of a plane wave by a circular cone with an impedance boundary condition, SIAM Appl. Math. 62 (4) (2002) 1122–1152.