ON SOUND RADIATION FROM AN OPEN-ENDED NON-UNIFORMLY LINED CYLINDRICAL NOZZLE

Transcription

1 24th INTERNATIONAL CONGRESS OF THE AERONAUTICAL SCIENCES ON SOUND RADIATION FROM AN OPEN-ENDED NON-UNIFORMLY LINED CYLINDRICAL NOZZLE L. M. B. C. Campos, J. M. G. S. Oliveira Secção e Mecânica Aeroespacial, Instituto Superior Técnico, Lisboa Coex, Portugal Keywors: uct, impeance, noise attenuation, soun raiation Abstract Acoustic liners are wiely use in jet engine inlet an exhaust ucts, as a passive means of noise reuction. One possible way of liner optimization is the use of a non-uniform acoustic impeance istribution. It has been shown that such a liner can lea to an attenuation in the acoustic amplitue of some of the moes present in the uct. However, the soun attenuation perceive by an observer in the far-fiel is arguably the most important effect to be achieve. These issues may be illustrate by consiering the raiation of soun from a cylinrical uct with impeance varying circumferentially, axially or in both irections an the acoustic pressure at the far-fiel in each situation. The raiation of soun from an open pipe is represente by a pressure istribution on a isk, viz. the exit plane. The raiation of soun in free space, i.e. without obstacles, is specifie by a Kirchhof integral. The source istribution on the uct exit plane, which allows the evaluation of raiation integrals, is specifie by the raial, axial, an circumferential moes, in the cylinrical nozzle. The evaluation of the raiation integrals show that (i) the total acoustic fiel consists of a spherical wave multiplying a sum of raial moes n = 1,..., an azimuthal moes of o orer only m = 1,3,5,...; (ii) each moe consists of a monopole term an a ipole term an epen on the frequency an the raial wavenumbers, etermine by the bounary conition at the uct wall. When the impeance istribution varies circumferentially or axially, the evaluation of the wavenumbers involves the etermination of the roots of an infinite eterminant, while when the impeance aries both axially an circumferentially, the roots of a oubly infinite eterminant have to be calculate. In the case of external noise of an aircraft, the observer is on the groun, at a istance much greater than the uct iameter, an the raiation integrals for an observer in the far-fiel can be simplifie, since the ipole term is weak. The evaluation of the acoustic pressure for an observer in the far-fiel, shows that it epens on the raial wavenumbers in the nozzle, which are specifie by the wall bounary conitions, an thus epen on the acoustic impeance istribution. This allows comparison of har-walle nozzles, with liners with constant impeance an non-uniform liners, the latter with impeance istribution varying circumferentially, axially or in both irections. The far-fiel acoustic pressure is specifie by a irectivity factor that is calculate for several values of the parameters for each of the cases mentione above. 1 Introuction In a cylinrical nozzle noise can be absorbe in its interior by vortical flow [1 1] an at the walls by acoustic liners, which may have uniform [11 16] or non-uniform [17 24] impeances. The soun fiel receive by an observer in the farfiel is etermine by raiation out of the open 1

2 L. M. B. C. CAMPOS, J. M. G. S. OLIVEIRA en of the nozzle [25 32] or inlet of a fan [33,34]. In the case of nozzle there is a refraction effect which can be represente either by a vortex sheet [35] or an irregular shear layer [1,2] issuing from the lip. The aim of the present paper is to relate the acoustic fiel raiate through the nozzle exit to the far-fiel to the acoustic moes in a nozzle with non-uniform wall impeance. The raiation from the isk in the nozzle exit plane, to an observer in the far-fiel, consists, to leaing orer, of monopole an ipole terms. The acoustic pressure istribution in the nozzle exit plane, is expresse in terms of uct moes, allowing the evaluation of raiation integrals. The latter involve the acoustic eigenfunction, corresponing to the eigenvalues for propagating or cut-on moes, in the case of rigi walls or walls with uniform impeance, an also walls with impeance varying circumferentially, axially or in both irections. A numerical application is mae, incluing the calculation of raial eigenvalues an plotting o acoustic eigenfunctions for uniform liner uct in comparison with weakly an strongly circumferentially non-uniform liners; the benefits of the non-uniform lining are assesse, not incluing the effects of transmission across the irregular an turbulent shear layer issuing from the jet nozzle lip. 2 Soun raiation from an open pipe The raiation of soun from an open pipe (Fig. 1) is represente by a pressure istribution on a isk (2.1), viz. the exit plane; the raiation integrals are simplifie (2.3) for an observer in the far-fiel (2.2). 2.1 Source istribution on a circular isk The raiation of soun in free space, i.e. without obstacles, is specifie by the Kirchhof integral: p(x) = 1 4π D 1 x y e iω(t x y /c) q(y) 3 y, (1) for a spatial source istribution of strength q at position y in the omain D, with frequency ω, y r α z Fig. 1 Soun raiation from the isk on the exit plane of a cylinrical pipe to an observer in the far-fiel an raiation to the observer at x, in a homogeneous meium at rest, for which the soun spee is c. In the case of a isk of raius a on the XOY - plane with centre at the origin, the position of the source is written in polar coorinates (R,α): α 2π, R a; D θ ρ y = R(e x cosα + e y sinα), (2) an the position of the observer in spherical coorinates (r, θ, ϕ): x = r(e x cosθ + e z sinθ), (3) where ϕ = because the XOZ-plane can be taken through the observer. The raiation integral (1) becomes in this case p(r,θ,t) = e iωt 4π 2π R α R D(R,α) ei(ω/c)d(r,α) q(r,α) (4) where q(r,α) is the source istribution, an D(R,α) = x y = R 2 +r 2 2Rr sinθcosα 1/2, (5) is the istance between observer an source. 2.2 Reception by an observer in the far-fiel In the case of external noise of an aircraft, the observer is on the groun, at a istance much P x 2

3 On soun raiation from an open-ene non-uniformly line cylinrical nozzle greater than the nozzle iameter, R 2 a 2 << r 2 : D(R,α) = r Rsinθcosα + O(R 2 /r 2 ), (6a) an the istance may be simplifie by (6a), an the inverse istance by: 1/D(R,α) = 1/r + R/r 2 sinθcosα + O(R 2 /r 2 ). (6b) Substitution of (6a, 6b) in the raiation integral (4) specifies p(r,θ,t) = e iωt 4π R r 2π α R q(r,α) [1 + Rr ] sinθcosα e i(ω/c)(r Rsinθcosα) (7) the acoustic fiel receive by the observer in the far-fiel. 2.3 Decomposition into monopole an ipole terms The acoustic pressure receive in the far-fiel (7) may be written p(r,θ,t) = eiω(r/c t) (I 1 + I 2 ), (8) 4πr as a spherical wave raiate from the origin (or isk or nozzle centre) to the observer, multiplie by monopole (9a) an ipole (9b) terms I 1 = I 2 = 2π 2π α Rq(R,α)Re i(ωr/c)sinθcosα) (9a) α R q(r,α) Re i(ωr/c)sinθcosα) (R/r)sinθcosα where the latter is relate to the former by I 2 = 1 r (9b) (iω/c) I 1, (1) since ifferentiation with regar to iω/c is equivalent to multiplication by Rsinθcosα. Substitution of (9a) an (1) in (8) yiels: { } p(r,θ,t) = eiω(r/c t) 1 + i 4πr (ωr/c) 2π α Rq(R,α)Re i(ωr/c)sinθcosα, (11) as the acoustic pressure receive by an observer in the far-fiel, from a source istribution q(r, α) on a isk. The latter is specifie next in terms of the acoustic moes of a nozzle. 3 Moal structure in a cylinrical nozzle The source istribution on the nozzle exit plane (3.2), which allows the evaluation of raiation integrals (3.3), is specifie by the raial, axial, an circumferential moes, in the cylinrical nozzle. 3.1 Raial, axial an circumferential moes The moes in a cylinrical nozzle are specifie by the solution of the classical wave equation in cylinrical coorinates: { 1 R R R R R 2 α z } c 2 t 2 Q =, (12) in the absence of mean flow. Since the coefficients of the wave equation epen only on raius R, it is convenient to use a Fourier ecomposition in (z,α,t), viz.: Q(R,α,z,t) = e iωt + + e imα k e ikz P m (R,k), m= (13) a wave of frequency ω, with longituinal wavenumber k an circumferential wavenumber m. Substitution of (13) in (12) shows that: [R 2 2 R 2 + R R + κ2 R 2 m 2 ] P m (R,k) =, (14) the raial epenence is specifie, for a cylinrical nozzle, by a Bessel function of orer m: P m (R,k) = J m (κr), (15a) 3

4 L. M. B. C. CAMPOS, J. M. G. S. OLIVEIRA with raial wavenumber κ 2 = (ω/c) 2 k 2, (15b) specifie by a bounary conition at the nozzle wall R = a, which specifies the raial moes κ nm with n = 1,...,, which may be istinct for each circumferential orer m. 3.2 Amplitues of soun generation in a nozzle Thus the acoustic fiel in the nozzle consists of a superposition (13, 15a) of: Q(R,α,z,t) = e iωt + m= e imα J m (κ mn R)e ikmnz, (16) n=1 where m is the azimuthal an n the raial orer, the raial wavenumbers κ mn are specifie by the bounary conition at the nozzle wall R = a, the axial wavenumbers are relate by (15b), viz.: k mn = (ω/c) 2 (κ mn ) 2, (17) an the amplitues of each moe are specifie by the source istribution in the nozzle. The pressure istribution q(r,α)e iωt in the nozzle exit plane z = is thus specifie by: q(r,α) = Q(R,α,,) = + e imα J m (κ mn R), (18) m= n=1 which may be substitute in the raiation integral (11) to specify the soun fiel receive by the observer in the far-fiel. Substitution of (17) into (11) shows that the soun fiel receive by the observer in the far-fiel p(r,θ,t) = eiω(r/c t) 4πr + m= n=1 p mn (r,θ), (19a) is specifie by a superposition of raiation integrals for each moe: { } p mn (r,θ) = 1 i (ωr/c) 2π α RRe i(ωr/c)sinθcosα e imα J m (κ mn R), which are evaluate next. (19b) 3.3 Evaluation of raiation integrals for each moe The acoustic pressure in the far-fiel for the n-th raial an m-th azimuthal moe is given by an integration in R: { } p mn (r,θ) = 1 + i (ωr/c) RRJ m (κ mn R)I m ((ωr/c)sinθ), where the integration in α appears in ζ = (ωr/c)sinθ : I m (ζ) = 2π The integral may be re-written: e imα+iζcosα α. (2a) (2b) π ( I m (ζ) = e imα e iζcosα + ( 1) m e iζcosα) α π [ = cos(mα) e iζcosα + ( 1) m e iζcosα] α, which is expressible in terms of Bessel functions [36]: by: J m (ζ) = i m π π e izcosα cos(mα)α, (21) I m (ζ) = πi m [J m (ζ) + ( 1) m J m ( ζ)]. (22) The series expansion for the Bessel functions J m (ζ) = (ζ/2) m { ( ζ 2 /4) s /(s!(s + m)!) }, s= (23) 4

5 On soun raiation from an open-ene non-uniformly line cylinrical nozzle implies that so that J m ( ζ) = ( ) m J m (ζ), (24) I m (ζ) = 2πi m J m (ζ). (25) 4 Acoustic effects of non-uniform impeance The evaluation of the acoustic pressure for an observer in the far-fiel (4.1), shows that it epens on the raial wavenumbers in the nozzle, which are specifie by the wall bounary conitions (4.2). This allows comparison of har-walle nozzles, with liners with constant impeance an non-uniform liners, the latter with impeance istribution varying circumferentially, axially or in both irections. 4.1 Selection of cut-off an cut-on moes When substituting equations (25) in (19b): { } p mn (r,θ) = 1 + i (ωr/c) J m (κ mn R)J m ((ωr/c)sinθ)rr, (26) the property [37] of Bessel functions: implies that J m (ζ) = ( ) m J m (ζ), (27) p mn (r,θ) = p m,n (r,θ), (28) so that the total pressure fiel (19a) simplifies to: p(r,θ,t) = eiω(r/c t) 2πr n=1 + m= The secon term of (26) involves: (ωr/c) J m((ωr/c)sinθ) = R r p mn (r,θ). (29) (ωr/c) J m((ωr/c)sinθ) = R r sinθj m((ωr/c)sinθ), (3) where prime enotes erivative of the Bessel function with regar to its argument: J m(ζ) = J m 1 (ζ) m ζ J m(ζ). (31) Substitution of (31) in (3) yiels (ωr/c) J m((ωr/c)sinθ) = ( ) R ωr r sinθj m 1 c sinθ mc ωr J m ( ) ωr c sinθ, (32) which specifies the ipole term in the acoustic pressure receive in the far-fiel: i a r p mn (r,θ) = a 2 1 { J m (κ mn as) J m (sωsinθ) [ m ] } Ω J m(sωsinθ) sinθj m 1 (sωsinθ) ss, (33) where a imensionless raial istance (34a) an a imensionless frequency (34b) were introuce: s = R/a Ω = ωa/c. (34a) (34b) Thus: (i) the total acoustic fiel (29) consists of a spherical wave multiplying a sum of raial moes n = 1,..., an azimuthal moes of orer m =,1,..., ; (ii) each moe (33) consists of a monopole term (first in the curly brackets) an a ipole term (in square brackets); (iii) the parameters are the imensionless frequency (34b) calculate from the nozzle raius a, the ratio of the latter to the istance of the observer a/r in the ipole term (which is weak because a 2 << r 2 for observers in the far-fiel), an κ mn a which are the raial wavenumbers, etermine by the bounary conition at the uct wall. 4.2 Rigi, impeance an non-uniform walls The simplest case (I) is a nozzle with rigi walls, for which the normal velocity at the wall is zero, 5

6 L. M. B. C. CAMPOS, J. M. G. S. OLIVEIRA implying from the momentum equation that the normal erivative of the pressure is zero: 1 p = iωv n (R = a) = ρ r, (35) r=a where ρ is the mass ensity. Thus: = J m(κ mn a), (36) so that the raial wavenumbers κ mn are etermine by the zeros j mn of the erivative of the Bessel function J m : J m( j mn ) = : κ mn = j mn /a. (37) Since these zeros are real, the raial wavenumbers κ mn are real, an the corresponing (17) axial wavenumbers: k mn a = Ω 2 ( j mn ) 2, (38) are: (i) either real, for propagating or cut-on moes, if j mn Ω; (ii) or imaginary, for evanescent or cut-off moes, if j mn > Ω. Since the zeros of the erivative of the Bessel function J m form an unboune sequence j m1, j m2,..., there is a finite number of cut-on moes, larger for larger imensionless frequency. The cut-off moes make a negligible contribution to raiation to the far-fiel, so the sum in the total acoustic pressure (29) is restricte to cut-on moes. The amplitue of the latter cut-on moes has been foun to be weakly epenent on moe orer for turbomachinery noise, an thus may be taken as a constant factor, an omitte together with the spherical wave term e iω(r/c t) /2πr an a 2, which are common factors, regarless of the wall conition, an thus o not affect the comparison between rigi line walls. For a rigi wall, the farfiel acoustic pressure is thus specifie by a irectivity factor: P(θ) = m= j mn <Ω 1 J m ( j mn s)j m (sωsinθ)s s, n=1 (39) where only the monopole term was consiere, since it ominates the ipole term. In the case (II) of a wall with uniform impeance Z, the raial wavenumbers are specifie by the roots of: iz J m(κ mn a) = J m (κ mn a), (4) where Z is the specific impeance, i. e. impeance ivie by that of a plane wave: the Z = Z /ρc. (41) In this case the raial wavenumbers κ mn are generally complex, an the axial wavenumbers (17) also: k mn a = Ω 2 (κ mn a) 2. (42) Although the istinction between cut-on an cutoff moes is not so clear in the case of an impeance wall, the sum for the total acoustic fiel is taken for the cut-on moes of a rigi wall, as in (39), but for the complex raial wavenumbers, so that (39) remains vali, an can be evaluate (cfr. [36]). The acoustic pressure receive in the far-fiel is: p mn (θ) = 2ik mn a [ (κ mn a) 2 (Ωsinθ) 2] 1 [ Ω sinθjmn (κ mn a)j m(ωsinθ) κ mn aj m (Ωsinθ)J (κ mn a) ] (43) We procee to consier the calculation of raial eigenvalues k mn in the case of liners of circumferentially or axially non-uniform impeance or both. 4.3 Axially an /or circumferentially nonuniform liners This expression (43) can also be use in the case of non-uniform wall impeance Z(θ, z), except that raial wavenumbers are no longer etermine as the roots of (4). In the case of a circumferentially non-uniform istribution, which is represente by the first two terms of a Fourier series: Z A (θ) = Z (1 + 2εcosθ), (44) the raial wavenumbers are specifie exactly by the roots of an infinite eterminant, [ et i Ω ] κa J m(κa)δ mm Z m mj m(κa) =, (45) 6

7 On soun raiation from an open-ene non-uniformly line cylinrical nozzle where all the impeance Fourier coefficients Z m = except Z an Z ±1 = εz. In the case of an axially non-uniform wall impeance over a length L of uct, represente by the first two terms of a Fourier series: Z B (z) = Z [1 + 2δcos(2πz/L)], (46) the raial wavenumbers are specifie exactly by the roots of an infinite eterminant, [ ] et i 1 + 2πl κa J m(κa)δ ll Z l l J m(κa) =, (47) where all the impeance Fourier coefficients Z l = except Z an Z ±1 = δz. In the case of impeance istribution varying both axially an raially, represente by the first two terms of the ouble Fourier series: Z C (θ,z) = Z (1 + εcosθ)[1 + δcos(2πz/l)], (48) the raial wavenumbers are specifie exactly by the roots of a ouble infinite eterminant, i.e. an infinite eterminant whose terms are infinite eterminants in a combination of (45) an (47). The irectivity of the acoustic pressure in the far-fiel is specifie by (43) also in the case of non-uniform impeance, except that the raial wavenumbers are given by the roots of (45) for circumferentially non-uniform impeance (44), the roots of (47) for axially nonuniform impeance (46), an roots of a combination of (45) an (47) for impeance varying both axially an circumferentially (48). 5 Noise reuction for far-fiel observer The eigenvalues an eigenfunctions are calculate for circumferentially non-uniform liners, to assess the noise reuction benefit relative to uniform liners. 5.1 Eigenvalues for the imensionless raial wavenumbers Arguably the best measure of the effectiveness of an acoustic liner is the effect on the reuction of far-fiel noise. As a numerical example consier a nozzle of raius a = 1m, an a wave frequency f = 1kHz or ω = 2π f = s 1, corresponing, for a soun spee c = 34ms 1, to a imensionless frequency or Helmholtz number Ω = ωr/c = The conition j mn < Ω for the roots j mn of the erivatives of the Bessel functions J ( j mn ) =, specifies the propagating or cut-on moes. The raial wavenumbers can be etermine using (4) for a uniform specific impeance impeance: Z = 2.5 i.4, (49) an (45), for circumferentially non-uniform impeances (44), with relative amplitue of the harmonic ε =.1 +.1i, ε =.3i. (5a) (5b) The former is esignate comparatively the weakly (5a) an the latter the strongly (5b) non uniform liner, although in both cases the impeance is the same (49) on the mean, an the variation from the mean is small in absolute terms in both cases. 5.2 Soun attenuation ue to non-uniform liner In orer to assess the soun attenuation ue to the non-uniform liner, the eigenfunctions for the acoustic pressure (39) with non-uniform p mn (θ;z,ε) an uniform p mn (θ;z,) liner are compare in a logarithmic scale: p (θ,z,ε) mn (θ;z,ε) log 1 p mn (θ,z,), (51) in ecibels; the effect on energy woul be the ouble of that given by (51), an is plotte as a function of azimuthal angle θ in Figs. 2 an 3. In Fig. 2 the axisymmetric moe m = is consiere for increasing raial orers n = 1,...,6. The funamental raial moe n = 1 (top left) is not much affecte by the non-uniform impeance, but the effect becomes visible for the secon 7

8 L. M. B. C. CAMPOS, J. M. G. S. OLIVEIRA a.4 b c e.4 f Fig. 2 Comparison on a ecimal logarithmic scale (51) of the ratio of amplitues of the acoustic pressure (43), for a non-uniform (44) an a uniform (49) liner with the same mean impeance, in the cases of weak (- -) an strong ( ) non-uniformity, as a function of azimuthal angle θ, for axisymmetric moes m = an (a) the funamental raial moe n = 1 an the harmonics (b) n = 2, (c) n = 3, () n = 4, (e) n = 5, an (f) n = 6. n = 2 (top right) an thir n = 3 (mile left) raial moes. Significant amplitue variation occur for the fourth n = 4 (mile right), fifth n = 5 (bottom left), an sixth n = 6 (bottom right) raial moes; these amplitue variations are mostly reuctions, although some increases occur. Broaly similar conclusions can be rawn from Fig. 3, which consiers the funamental raial moe n = 1 for several azimuthal orers m =,2,...,14. The axisymmetric moe m = 8

9 On soun raiation from an open-ene non-uniformly line cylinrical nozzle a.1 b c e.1 f g.1 h Fig. 3 As Fig. 2, for the funamental axial moe n = 1 an (a) axisymmetric m =, an nonaxisymmetric moes of increasing even azimuthal orer (b) m = 2, (c) m = 4, () m = 6, (e) m = 8, (f) m = 1, (g) m = 12, an (h) m = 14. 9

10 L. M. B. C. CAMPOS, J. M. G. S. OLIVEIRA (top left) is not much affecte by the non-uniform liner. The azimuthal moes m = 2 (top right) an m = 4 (secon row, left) show visible reuctions. The higher orer azimuthal moes m = 6 (thir row, left), m = 1 (thir row, right), m = 12 (bottom left), an m = 14 (bottom right) show a clear amplitue reuction ue to the strongly nonuniform liner over most irections, although an increase is seen for some irections. It is clear that a non-uniform liner, with relative impeance variations in the range 2 3%, can provie an overall soun reuction. 6 Discussion The effect of non-uniform impeance on soun raiation to the far-fiel has been assesse on the basis of the acoustic pressure on the isk corresponing to the nozzle exit plane, without accounting for ege iffraction effects. The latter can be quite important [35], in particular in the presence of a mean flow, when a turbulent an irregular shear layer [1,2] is issue from the nozzle lip. The latter causes spectral an irectional broaening of soun, which further reuces the noise levels. Although the latter effects have not been moelle here, it is clear that an attenuation of the input soun fiel incient on the shear layer from the interior of the jet will result in a further attenuate soun fiel transmitte to an observer in the far-fiel outsie the shear layer. The attenuations shown for the basic in-uct soun fiel are clear even though: (i) the non-uniform liner (44) has the same average impeance as the uniform liner; (ii) only one liner impeance harmonic was use, involving a single parameter ε, which was not optimize. By consiering multi-harmonic impeance istribution ] Z(θ) = Z [1 + L l=1 ε l cos(lθ), (52) an optimizing the parameter ε l, greater noise reuction coul be obtaine. In the present example a relative impeance variation ε of 2 3% was shown to be sufficient to affect significantly the soun fiels. References [1] L. M. B. C. Campos. The spectral broaening of soun in turbulent shear layers. Part 1: The transmission of soun through turbulent shear layers. Journal of Flui Mechanics, 89: , [2] L. M. B. C. Campos. The spectral broaening of soun in turbulent shear layers. Part 2: The spectral broaening of soun an aircraft noise. Journal of Flui Mechanics, 89: , [3] H. E. Plumblee an P. E. Doak. Duct noise raiation through a jet flow. Journal of Soun an Vibration, 65(4): , [4] P. A. Nelson an C. L. Morfey. Aeroynamic soun prouction in low spee flow ucts. Journal of Soun an Vibration, 79(2): , [5] M. S. Howe. The amping of soun by turbulent wall shear layers. Journal of the Acoustical Society of America, 98: , [6] C. K. W. Tam an L. Auriault. The wave moes in ucte swirling flows. Journal of Flui Mechanics, 371:1 2, [7] L. M. B. C. Campos an P. G. T. A. Serrão. On the acoustics of an exponential bounary layer. Philosophical Transactions of the Royal Society of Lonon, Series A, 356: , [8] L. M. B. C. Campos, J. M. G. S. Oliveira, an M. H. Kobayashi. On soun propagation in a linear flow. Journal of Soun an Vibration, 219(5):739 77, [9] L. M. B. C. Campos an M. H. Kobayashi. On the reflection an transmmission of soun in a thick shear layer. Journal of Flui Mechanics, 42:1 24, 2. [1] M. Willatzen. Soun propagation in a moving flui confine by cilinrical walls exact series solutions for raially epenent flow profiles. Acustica, 87: , 21. [11] A. D. Rawlins. Raiation of soun from an unflange rigi cylinrical uct with an acoustically absorbing internal surface. Proceeings of the Royal Society of Lonon, Series A, 361:65 91, [12] K. Ogimoto an G. W. Johnston. Moal raiation impeances for semi-infinite unflange circular ucts incluing flow effects. Journal of 1

11 On soun raiation from an open-ene non-uniformly line cylinrical nozzle Soun an Vibration, 62(4):598 65, [13] W. Koch an W. Möhring. Eigensolutions for liners in uniform mean flow ucts. AIAA Journal, 21(2):2 213, [14] W. Koch. Raiation of soun from a twoimensional acoustically line uct. Journal of Soun an Vibration, 55(2): , [15] S. W. Rienstra. Contributions to the theory of soun propagation in ucts with bulk-reacting lining. Journal of the Acoustical Society of America, 77(5): , May [16] D. A. Bies, C. H. Hansen, an G E Briges. Soun propagation in rectangular an circular cross-section ucts with flow an bulkreacting liner. Journal of Soun an Vibration, 146(1):47 8, [17] M. S. Howe. The attenuation of soun in a ranomly line uct. Journal of Soun an Vibration, 87(1):83 13, [18] W. R. Watson. An acoustic evaluation of circumferentially segmente uct liners. AIAA Journal, 22(9): , September [19] C. R. Fuller. Propagation an raiation of soun from flange circular ucts with circumferentially varying wall amittances, I: semiinfinite ucts. Journal of Soun an Vibration, 93(3):321 34, [2] C. R. Fuller. Propagation an raiation of soun from flange circular ucts with circumferentially varying wall amittances, II: finite ucts with sources. Journal of Soun an Vibration, 93(3): , [21] P. G. Vaiya. The propagation of soun in ucts line with circumferentially non-uniform amittance of the form η + η q exp(iqθ). Journal of Soun an Vibration, 1(4): , [22] B. Regan an J. Eaton. Moelling the influence of acoustic liner non-uniformities on uct moes. Journal of Soun an Vibration, 219(5): , [23] L. M. B. C. Campos an J. M. G. S. Oliveira. On the acoustic moes in a cylinrical nozzle with an arbitrary impeance istribution. In Proceeings of the 23 r ICAS Congress, 22. [24] L. M. B. C. Campos an J. M. G. S. Oliveira. On the optimization of non-uniform acoustic liners on annular nozzles. To be publishe, 23. [25] S. D. Savkar. Raiation of cylinrical uct acoustic moes with flow mismatch. Journal of Soun an Vibration, 42: , [26] W. Koch. Attenuation of soun in multi-element acoustically line rectangular ucts in the absence of mean flow. Journal of Soun an Vibration, 52(4): , [27] S. W. Rienstra. Acoustic raiation from a semiinfinite uct in a uniform subsonic mean flow. Journal of Soun an Vibration, 94(2): , [28] R. Martinez. Diffracting open-ene pipe treate as a lifting surface. AIAA Journal, 26(4):396 44, [29] A. Snakovska, H. Iczak, an B. Bogusz. Moal analysis of the acoustic fiel raiate from an unflange cylinrical uct theory an measurement. Acustica, 82:21 26, [3] A. Snakowska an H. Iczak. Preiction of multitone soun raiation from a circular uct. Acustica, 83: , [31] P. Joseph an C. L. Morfey. Multimoe raiation from an unflange, semi-infinite circular uct. Journal of the Acoustical Society of America, 15(5):259 26, May [32] S. T. Hocter. Exact an approximate irectivity patterns of the soun raiate from cylinrical uct. Journal of Soun an Vibration, 227(2):397 47, [33] S. J. Horowitz, R. K. Sigman, an B. T. Zinn. An iterative metho for preicting turbofan inlet acoustics. AIAA Journal, 2(12): , December [34] H. D. Meyer. Effect of inlet reflections on fan noise raiation. AIAA Journal, 34(9): , September [35] R. M. Munt. Acoustic raiation from a semiinfinite circular uct in an uniform subsonic mean flow. Journal of Flui Mechanics, 83:69 64, [36] M. Abramowitz an I. A. Stegun. Hanbook of Mathematical Functions. Dover, [37] E. T. Whittaker an G. N. Watson. A Course of Moern Analysis. Cambrige University Press,