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Multimodal analysis of acoustic propagation in three-dimensional bends
Wave Motion 36 (2002) 157–168
Multimodal analysis of acoustic propagation in three-dimensional bends S. Félix∗ , V. Pagneux Laboratoire d’Acoustique de l’Université du Maine, UMR-CNRS 6613, Université du Maine, Avenue Olivier Messiaen, 72085 Le Mans Cedex 09, France Received 9 October 2001; received in revised form 10 December 2001; accepted 16 January 2002
Abstract An exact multimodal formalism is proposed for acoustic propagation in three-dimensional rigid bends of circular crosssection. Two infinite systems of first-order differential equations are constructed for the components of the pressure and axial velocity in the bend, projected on the local transverse modes. These equations are numerically unstable, due to the presence of evanescent modes, and cannot be integrated directly. An impedance matrix is defined, which obeys a Riccati equation, numerically workable. With this nonlinear first-order differential equation, the impedance can be calculated everywhere in the bend, allowing a direct characterization of its acoustical properties or allowing the acoustic field to be integrated. An exact algebraic formulation of the reflection and transmission matrices is carried out to allow bends and more complex duct systems to be characterized. This result is applied to calculate the reflection and transmission of a typical bend, and also to obtain the resonance frequencies of closed tube systems. © 2002 Elsevier Science B.V. All rights reserved. Keywords: Bend; Multimodal acoustic propagation; Riccati equation; Impedance matrix; Modal scattering properties
1. Introduction The theoretical study of sound propagation in bends of circular cross-section is difficult since the toroidal coordinate system required to suit the boundary conditions at the walls is not one of the coordinate systems in which the Helmholtz equation is separable [1] and therefore amenable to an analytical solution. Thus, relatively few works have been published on this subject and mostly approximate methods or purely numerical methods have been adopted. One can refer to [2] for a review. Cummings [3], who was one of the first to mention in acoustics the difficulty of solving the wave equation in non-separable toroidal coordinates, suggested, on the basis of experimental measurements, to visualize the pipe bend as being split up into a superposition of “slices” of rectangular section ducts, all in the plane of the bend. More simply, he also suggested to consider an equivalent rectangular section duct whose width equals the diameter of the circular cross-section and whose height is such that both ducts have the same cross-sectional area. This “slice” method has been adopted later by Keefe and Benade [4] to study the wave propagation in strongly curved ducts in the long wavelength limit. Wave propagation in slender curved three-dimensional tubes of arbitrary cross-section was investigated by Ting and Miksis [5], who bypassed the difficulty mentioned by Cummings by using a perturbation ∗
Corresponding author. Tel.: +33-24383-3553; fax: +32-24383-3520. E-mail addresses: [email protected] (S. F´elix), [email protected] (V. Pagneux). 0165-2125/02/$ – see front matter © 2002 Elsevier Science B.V. All rights reserved. PII: S 0 1 6 5 - 2 1 2 5 ( 0 2 ) 0 0 0 0 9 - 4
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Fig. 1. Geometry of the duct system with bend and system of coordinates.
method, assuming that the tube is weakly curved. Their analysis was carried out for four different scalings of the wavenumber. A numerical analysis of acoustic propagation in toroidal bends has been published by El-Raheb and Wagner [6], using a boundary elements method. Surface elements for a toroidal boundary were defined, allowing the acoustic field to be calculated under certain end condition for a water-filled bend joining two finite straight sections. Another method was used by Firth and Fahy [7], who gave an interesting study, using a spectral method to evaluate acoustic torus modes. They assumed solutions of the form A(r, φ) exp(jks s), where (r, φ, s) are the toroidal coordinates (see Fig. 1). The solutions A(r, φ), i.e. the torus modes, which are not separable in r and φ, were sought in terms of a series expansion on the transverse modes of a straight tube. They evaluated the mode shapes and propagation constants of these torus modes, and the reflection and transmission coefficients for a 90◦ bend with a plane wave incident and a non-reflecting termination for both end conditions, upstream and downstream from the bend. Later, in their study of modal wave propagation in circularly curved waveguides, Furnell and Bies [8] obtained the transverse field eigenfunctions by an approximate method based on the calculus of variations. They used these solutions to calculate the matrices characterizing the modal reflection and transmission properties of the bend. These approaches to determine the non-separable torus modes are not easy to implement since they require the calculation of the modes at each frequency and because of the difficulty to match the solutions obtained in the bend with the solutions in the waveguides upstream and downstream from the bend. This paper is concerned with the formulation of an exact approach, based on multimodal decomposition, to analyze acoustic propagation in three-dimensional bends of circular cross-section, or pipe bends. This approach is of particular interest, in both theoretical and practical points of view, since it is not limited to any domain of geometrical parameters or frequency and leads to an exact algebraic formulation of the modal scattering properties of bends and as a consequence of almost every practical duct systems. The multimodal method has been already validated for varying cross-section waveguides of straight longitudinal axis [9,10] and more recently for two-dimensional curved ducts of constant or varying cross-section [11], giving an efficient alternative to “classical” methods (e.g. separation of variables). This method consists in constructing two infinite first-order differential equations for the components of the pressure and axial velocity projected on the basis of the transverse modes in a straight cylindrical duct with the same cross-section. These two equations are numerically unstable, due to the presence of evanescent modes, and cannot be integrated directly. A impedance matrix is defined, which obeys a Riccati equation, numerically workable. With this first-order differential equation, the impedance can be calculated everywhere in the bend, allowing a direct characterization of its acoustical properties or the acoustic field to be integrated. Here the formulation of the multimodal method is developed for three-dimensional bends of circular cross-section. Such extension may indeed offer a larger field of applications, since practical duct systems are in general constituted
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with circular ducts. Furthermore, besides the fact that no exact analytical solution can be deduced from the wave equation in toroidal coordinates, alternatives to the multimodal method are much more constraining: the “torus” modes, the determination of which is not easy as discussed before, or purely numerical methods that involves 3D mesh construction and heavy implementations. The first part of the paper is devoted to the definition of the problem to be analyzed and the formulation of the multimodal method. The appropriate quantities are introduced and the main stages of the method are presented, with mention to previous works [9–11], where detailed information can be found. In a second part, an example of calculation of the pressure field is presented and used to evaluate the convergence of the method. The use of the multimodal method to calculate the modal scattering properties of bends and notably the exact algebraic formulation of the reflection and transmission matrices is detailed in the third part, with a generalization to complex duct systems in the last part.
2. Formulation Consider the propagation of sound in a circularly curved section of a duct system of circular cross-section as shown in Fig. 1. The parameters of the toroidal section are its mean radius R0 , the radius of the cross-section r0 and its length sf , measured along its circular axis. The duct walls are assumed to be rigid. Assuming adiabatic lossless linear media, acoustic velocity vˆ and pressure pˆ satisfy the linearized Euler equations jωvˆ = −
1 ∇p, ˆ ρ0
(1)
and ∇ · vˆ = −
jω p, ˆ ρ0 c02
(2)
where ρ0 is the density of air and c0 the speed of sound. The accent (ˆ) denotes dimensional quantities and time dependence exp(jωt) is omitted. The reference pressure ρ0 c02 and velocity c0 are used to reduce Eqs. (1) and (2) to dimensionless forms and we introduce the frequency k = ω/c0 so that these equations become −jkv = ∇p,
(3)
∇ · v = −jkp.
(4)
The coordinates adapted to suit the boundary conditions of a circular cross-section curved duct are the toroidal coordinates (r, φ, s) (see Fig. 1). Eqs. (3) and (4) in toroidal coordinates become (see Appendix A) −jkvr = ∂vr + ∂r
∂p , ∂r
−jkvφ =
1 κ cos φ − r 1 − κr cos φ
1 ∂p , r ∂φ
vr +
−jkvs =
1 ∂p , 1 − κr cos φ ∂s
1 ∂vφ κ sin φ 1 ∂vs + vφ + = −jkp, r ∂φ 1 − κr cos φ 1 − κr cos φ ∂s
(5)
(6)
where v has components (vr , vφ , vs ) and κ = 1/R0 is the curvature of the bend axis. By eliminating vr and vφ , these equations give ∂p = −jk(1 − κr cos φ)vs , ∂s
(7)
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∂vs 1 ∂ 2p 1 ∂p = (1 − κr cos φ)k 2 p + (1 − κr cos φ) 2 + (1 − 2κr cos φ) ∂s jk r ∂r ∂r 2 ∂p 1 ∂ p 1 . +(1 − κr cos φ) 2 2 + κ sin φ r ∂φ r ∂φ
(8)
Pressure p and axial velocity vs are now projected on the complete orthonormal basis (ψα ) of the classical transverse modes of a straight tube of radius r0 (see Appendix B): p(r, φ, s) = Pα (s)ψα (r, φ), (9) α
vs (r, φ, s) =
Uα (s)ψα (r, φ),
(10)
α
where Pα and Uα are scalar coefficients. The parameter α being a triplet of integers, p and vs are expressed in the form a triple series that may be simplified to a simple one. Indeed, since the duct has a plane of symmetry parallel to the plane of the torus, a source distribution which is symmetric with respect to this plane will produce an acoustic field which is also symmetric; similarly an antisymmetric source distribution will produce an antisymmetric field [7]. In other words, there is no coupling between the modes characterized by a symmetry index σ = 1 (symmetric) and those characterized by σ = 0 (antisymmetric). We can thus eliminate the summation on this index and consider two similar and independent problems. Without this symmetry, a coupling between the symmetric and antisymmetric modes would occur, without compromising the suitability of the multimodal method. Furthermore, the double series on the circumferential and radial indexes can be simplified to a simple one by sorting the modes ψα in increasing order of their cutoff frequency. Therefore, from this point, σ is a fixed value depending on the source distribution and α is a positive integer (associated to the indexes m and n), with ψ0 corresponding to the plane wave mode. The modal decompositions (9) and (10) are written as p = t ψP,
(11)
vs = t ψU,
(12)
where P = (Pα )α≥0 , U = (Uα )α≥0 and ψ = (ψα )α≥0 are column vectors. Then, by projecting (7) and (8) on the functions ψα , the following equations are obtained: P = −jkBU, U =
1 (C + KB)P, jk
in which K is a matrix given by 2 γmn 2 Kαβ = k − 2 δαβ ∀(α, β) ∈ N2 , r0
(13) (14)
(15)
where γmn is the (n + 1)th zero of Jm , the derivative of the mth-order Bessel function of the first kind (see Appendix B). An important property of Eqs. (13) and (14) is that the matrices B, C, given in Appendix B, depend only on the geometrical parameters r0 and κ. Thus, they do not depend on k and should not be calculated for each frequency. As the coupled differential equations (13) and (14) are written, their form is identical to that in a two-dimensional circular bend [11]. Thus, we will give only the main stages in the formulation of the multimodal method. The reader can refer to previous works [9–11] for more details.
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Because of the presence of evanescent modes (corresponding to Kαβ < 0), Eqs. (13) and (14) are numerically unstable and cannot be integrated directly. Therefore, an impedance matrix Z is defined, fulfilling P = ZU. This matrix Z obeys the Riccati equation Z = −jkB −
1 Z(C + KB)Z, jk
(16)
which is numerically workable. With this nonlinear first-order differential equation and a radiation condition (given as an impedance Zr ), it is then possible to calculate the impedance everywhere in the bend and in particular at the inlet, to get the input impedance and reflection coefficient (see [9]). Moreover, once the impedance is known, the pressure and velocity fields can be calculated using Eqs. (13) and (14), reformulated as P = −jkBYP, U =
(17)
1 (C + KB)ZU, jk
(18)
where Y is the admittance matrix (Y = Z −1 ). Owing to the previous calculation of Z, Eqs. (17) and (18) are not numerically divergent. Such formalism allows us to extend the integration of Z, P and U in any type of waveguide of circular cross-section upstream and downstream from the bend and therefore to analyze complex waveguide systems.
3. Calculation of the acoustic field: study of the convergence In this section and in the following, we assume that the duct downstream from the bend (region (III) in Fig. 1) is a semi-infinite straight duct of constant cross-section. The radiation condition Zr is then the characteristic impedance Zc , which is diagonal and given by Zcα =
k kα
∀α ∈ N,
(19)
where
γmn 2 2− k r0 kα = γmn 2 − k2 −j r0
for propagative modes, (20) for evanescent modes.
In order to evaluate the rate of convergence of the multimodal method, two different quantities are proposed for the error to be measured: 1/2 2 φ=0,π p − pref (1 − κr) dr ds
, (21) %1 = 2 φ=0,π pref (1 − κr) dr ds %2 =
s=sf
p − pref 2 r dr dφ
s=sf
pref 2 r dr dφ
1/2 ,
(22)
where the reference solution pref is the one obtained for 30 modes introduced in the multimodal calculation, supposing that the convergence is reached. %1 is the error in the plane represented in Fig. 2 (on the left), while %2 the error on the Section A–A (on the right). A plane piston source for the pressure is imposed at the inlet (s = 0), and the parameters are R0 /r0 = 2.5, θf = sf /R0 = π/2 and kr0 = 2.4.
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Fig. 2. Contour of real part of pressure; R0 /r0 = 2.5, θf = sf /R0 = π/2, kr0 = 2.4.
Fig. 3 shows how the errors %1 and %2 decrease as more modes are taken into account, following a 1/N 2.5±0.3 law, where N is the number of modes. Contributions of each mode introduced in the calculation to the convergence of the solution are naturally different. Thus, apart from the first modes, the 7th, 13th or 22th modes in our example have a significant contribution, while the contribution of other modes (e.g. the 8th, 11th, 14th, 16th, etc.) seems to be negligible. The 7th, 13th and 22th modes are those for which the circumferential index m is equal to 1, i.e. those having symmetry properties similar to those of the transverse solution (see the Section A–A in Fig. 2). On the contrary, the modes whose contribution seems to be negligible are mostly circumferential, with relatively great values of m. In view of the smooth variations of the solution with the coordinate φ, it is clear that these modes give information of “secondary” importance. Here, the modes have been sorted in increasing order of their cutoff
Fig. 3. Convergence of the multimodal method.
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frequency; it appears with the remarks above that another way of sorting the modes would increase the convergence rate, subject to a prior knowledge of the symmetry of the solution.
4. Modal scattering properties The sequel of this paper concerns the use of the multimodal method to calculate the modal scattering properties of pipe bends, in the present section, and of more complex duct systems in the following section. Let P(i) , P(r) and P(t) be the wave incident, reflected and transmitted by the bend, respectively. The reflection matrix R and transmission matrix T are defined by P(r) = RP(i) and P(t) = T P(i) . Once the impedance is known at the inlet of the bend, the reflection matrix R can be easily calculated by R = (Zc + Z)−1 (Zc − Z) (see [9] for details), without prior calculation of the acoustic field or hypothesis on the source. Here, Z is the input impedance and Zc the characteristic impedance. Moreover, since the formulation of the multimodal method for pipe bends is similar to that for two-dimensional bends, an exact algebraic formulation of the reflection and transmission matrices can be carried out. This formulation, developed in [11], is very attractive, since it allows us to get the scattering matrix by simple inversion of matrices of order Nmodes × Nmodes . Fig. 4 shows the modulus of the amplitude reflection coefficients R00 , R10 , R20 and transmission coefficients T00 , T10 , T20 for the same bend as considered in Section 3, i.e. with geometrical parameters R0 /r0 = 2.5 and θf = π/2. The frequency is varying from 0 to twice the first cutoff frequency: kr0 ∈ [0, 2γ10 ]. For α in N, Rα0 (respectively Tα0 ) gives the contribution of the incident plane wave mode to the reflected (respectively transmitted) (α + 1)th
Fig. 4. Amplitude reflection and transmission coefficients.
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mode. The parameters are chosen to enable the comparison with results published by Firth and Fahy: it appears that the agreement is very good.
5. Resonances in complex cavities Besides the fact that the algebraic formulation, the reflection and transmission matrices, as given in [11], is a simple and direct method to obtain the modal scattering properties of a pipe bend, it allows more complex duct systems to be characterized, and notably their resonance frequencies to be calculated, in case of closed annular ducts. 5.1. Formulation Consider a simple section of duct. We define its scattering matrix S as follows: T R S= , R T
(23)
where R and T are the reflection and transmission matrices when considering a right-going incident wave and R and T are the reflection and transmission matrices when considering a left-going incident wave. For symmetrical waveguides, R = R and T = T . In the case of a bent section, which is a symmetrical waveguide, the reflection and transmission matrices R = R and T = T are calculated algebraically as discussed before. In the case of a straight section, the R and T matrices are recalled in Appendix B. Consider now a system composed of a succession of different sections of duct. If the scattering matrix of each elementary part is known, we can calculate analytically the global scattering matrix Ssys. of the system (see Appendix B), and thus characterize its reflection and transmission properties. Moreover, if this duct system is closed such that it becomes an annular cavity, the closing condition imposes a relation dispersion det(Ssys. − I ) = 0
(24)
with I the identity matrix. The frequency solutions of Eq. (24) are the resonance frequencies of the cavity. Two results of calculation of resonance frequencies are presented in the following for two different shapes of annular duct: first a simple torus and then a more complex annulus, composed of a succession of straight and bent sections. 5.2. Resonances in a torus The torus is a bend of overall angle 2π whose scattering matrix Storus can be calculated directly using the algebraic formulation of R and T . The solutions of (24) with Ssys. = Storus , denoting that the torus is closed, give its resonance frequencies. Table 1 shows in the first column the resonance frequencies in a torus whose dimensions are such that R0 /r0 = 2.5. Only the frequencies less than the first cutoff frequency are given. The second column shows the same results, but without taking into account the curvature, i.e. when considering a straight tube with the same axis length 2π R0 and the same closing condition (24). As a validation, these results are compared with values deduced from the dispersion curve for the first torus mode published by Firth and Fahy [7], showing that, although the discrepancy between the exact approach and the straight tube approximation is not important, the exact approach give results closer to those deduced from [7]. 5.3. Resonances in an annulus Consider now a more complex annular duct, as shown in Fig. 5. This annulus is composed of straight sections of length l1 and l2 and curved sections of angle π/2. The scattering matrix of each of this sections being known, the
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Table 1 Resonance frequencies, adimensioned by γ10 /r0 , in a torus with R0 /r0 = 2.5a det(Storus − I ) = 0b
det(Sapprox. − I ) = 0c
Firth and Fahy [7]d
0.2210 0.4367 0.6434 0.8397
0.2173 0.4345 0.6518 0.8690
0.22 0.44 0.64 0.84
Only the frequencies less than the first cutoff frequency γ10 /r0 are given. Storus is the scattering matrix of a bend of overall angle 2π , calculated with 15 modes. cS approx. is the scattering matrix of a straight tube of length 2π R0 , calculated with 15 modes. d Results deduced from the dispersion curve for the first torus mode published by Firth and Fahy [7]. a
b
Fig. 5. Modulus of det(Sann. − I ) (plain) and modulus of det(Sapprox. − I ) (dashed-dotted) vs frequency, where Sapprox. is the scattering matrix of a straight tube of length 2l1 + 2l2 + 2π R0 . On the left, the shape of the annulus. R0 /r0 = 1.25, l1 /r0 = 1.5, l2 /r0 = 0.75.
overall scattering matrix Sann. of the annulus is calculated iteratively using the result given in Appendix B. Zeros of det(Sann. − I ) can thus be found, giving the resonances frequencies of the annulus. Fig. 5 shows the modulus of det(Sann. − I ) for an annulus with parameters R0 /r0 = 1.25, l1 /r0 = 1.5 and l2 /r0 = 0.75. Again we compare these results with those obtained when the curvature is not taken into account, i.e. when considering a straight duct with the same axis length 2l1 + 2l2 + 2πR0 and the same closing condition (24). Since the curvature is relatively large, the discrepancy between the two results becomes rapidly noticeable when increasing the frequency. One can also notice in Fig. 5 that each zero of the plain curve, i.e. each resonance frequency of the annulus splits into two values. When calculating the resonance frequencies of a torus, we do not observe such splitting. The symmetry properties of each geometry explain these two different results: the degeneracy of the eigenvalues k is double for the torus, whose geometry is invariant to any rotation around its axis normal to the plane of the torus. Because of the straight parts, the annulus shown in Fig. 5 has not this symmetry property and as a consequence the degeneracy is lifted. If we consider now the same duct with l1 = l2 , the annulus will have a supplementary symmetry plane and therefore every second eigenvalue will be degenerated again.
6. Conclusion An exact multimodal method for the analysis of wave propagation in three-dimensional rigid bends of circular cross-section has been formulated. This method, thanks to the calculation of the impedance matrix along the duct, is convenient both for the formulation of radiation conditions and when considering junctions of the bend with any other waveguides.
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Whatever the frequency of the source or the dimensions of the bend, it can be characterized in a very simple manner once the impedance is known, by its input impedance or modal scattering properties, without prior calculation of the acoustic field or hypothesis on the source. An algebraic formulation of the reflection and transmission matrices has also been carried out, allowing bends and more complex duct systems to be characterized accurately.
Appendix A The toroidal coordinates used are r ∈ [0, r0 ], φ ∈ [0, 2π [ and s ∈ [0, sf ] (see Fig. 1). They are orthogonal and we have dr = hr dr uˆ r + hφ dφ uˆ φ + hs ds uˆ s ,
(A.1)
where the scale factors are given by hr = 1,
hφ = r,
hs = 1 − κr cos φ.
(A.2)
The gradient and divergence operators in this coordinate system are thus 1 ∂f ∂f 1 ∂f uˆ r + uˆ φ + uˆ s , ∂r r ∂φ 1 − κr cos φ ∂s 1 1 ∂Fφ 1 κ sin φ κ cos φ ∂Fs ∂Fr + + − Fr + Fφ + . ∇·F= ∂r r 1 − κr cos φ r ∂φ 1 − κr cos φ 1 − κr cos φ ∂s
∇f =
(A.3) (A.4)
Appendix B The transverse modes in a rigid straight tube of radius r0 are the solutions ψ(r, φ) = R(r)Φ(φ) of the separable transverse problem 1 dR 1 d2 Φ 1 d2 R 2 + + k⊥ = 0, + R dr 2 rR dr r 2 Φ dφ 2 satisfying the homogeneous Neumann condition on the walls ∂ψ = 0. ∂r r=r0 These modes are written as γmn r σπ ψα (r, φ) = Aα Jm sin mφ + , r0 2
(B.1)
(B.2)
(B.3)
where Jm is the mth-order Bessel function of the first kind and γmn the (n + 1)th zero of Jm . Their cutoff frequency is γmn /r0 . The parameter α denotes a triplet of integers (m, n, σ ), where m is the circumferential index (m ≥ 0), n the radial index (n ≥ 0) and σ the symmetry index (σ = 0, 1). The basis of functions (ψα ) is orthogonal and can be normed by taking 1 if m = 0, πr02 J02 (γ0n ) Aα = (B.4) 1 if m > 0. (π r 2 /2)(1 − m2 /γ 2 )J 2 (γ ) mn mn m 0
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Appendix C The parameter α is associated to circumferential and radial indexes m and n, respectively, and β is associated to µ and ν: r0 γµν r γmn r 2 Bαβ = Aα Aβ Jmµ Jµ r Jm dr, (C.1) r0 r0 0 Cαβ = Aα Aβ
γmn κJmµ r0
r0
γµν r γmn r Jµ dr r0 r0 r0 γµν r γmn r Jm Jµ dr, r0 r0 0
rJm+1 0 −Aα Aβ mκ(Jmµ − Hmµ )
in which
Imµ = Jmµ = and
2π 0 2π
0
Hmµ = =
2π 0
σπ σπ sin mφ + sin µφ + dφ = π δmµ (1 + (δσ 1 − δσ 0 )δm0 ), 2 2 σπ σπ π cos φ sin mφ + sin µφ + dφ= δ|m−µ|,1 (1 + (δσ 1 − δσ 0 )(δm0 + δµ0 )), 2 2 2
(C.2)
(C.3) (C.4)
σπ σπ sin φ cos mφ + sin µφ + dφ 2 2
π (δµ−m,1 (1 − (δσ 1 − δσ 0 )δm0 ) − δm−µ,1 (1 + (δσ 1 − δσ 0 )δµ0 )). 2
(C.5)
Appendix D Consider a finite straight tube with length l; its reflection matrices R and R are null and its transmission matrices T and T are diagonal and given by Tα = Tα = e−jkα l
∀α ∈ N.
(D.1)
We give now the calculation of the global scattering matrix of a duct system composed of two sections (Fig. 6), when the scattering matrix of each section is known. The scattering matrices of the sections (1–2) and (2–3) are given, respectively, by T12 R12 T23 R23 , S23 = , (D.2) S12 = R12 T12 R23 T23
Fig. 6. Duct system composed of two sections.
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such that + A2 A1−
= S12
A1+ A2−
,
A3+ A2−
= S23
A2+ A3−
.
(D.3)
A1− and A3+ are then given as functions of A1+ and A3− , leading to the following expression for the global scattering matrix S13 : R )−1 T + T (1 − R R )−1 R T T23 (1 − R12 R23 23 23 12 12 23 12 23 S13 = . (D.4) (1 − R R )−1 R T (1 − R R )−1 T R12 + T12 T12 23 12 23 12 23 12 23 The scattering matrix of a duct composed of n parts will be obtained by iterating n − 1 times the calculation above. References [1] [2] [3] [4] [5] [6] [7] [8]
P.M. Morse, H. Feshbach, Methods of Theoretical Physics, McGraw-Hill, New York, 1953. W. Rostafinski, Monograph on Propagation of Sound Waves in Curved Ducts, NASA Reference Publication 1248, 1991. A. Cummings, Sound transmission in curved duct bends, J. Sound Vib. 35 (4) (1974) 451–477. D.H. Keefe, A.H. Benade, Wave propagation in strongly curved ducts, J. Acoust. Soc. Am. 74 (1) (1983) 320–332. L. Ting, M.J. Miksis, Wave propagation through a slender curved tube, J. Acoust. Soc. Am. 74 (2) (1983) 631–639. M. El-Raheb, P. Wagner, Acoustic propagation in a rigid torus, J. Acoust. Soc. Am. 71 (6) (1982) 1335–1346. D. Firth, F.J. Fahy, Acoustic characteristics of circular bends in pipes, J. Sound Vib. 97 (2) (1984) 287–303. G.D. Furnell, D.A. Bies, Characteristics of modal wave propagation within longitudinally curved acoustic waveguides, J. Sound Vib. 130 (3) (1989) 405–423. [9] V. Pagneux, N. Amir, J. Kergomard, A study of wave propagation in varying cross-section waveguides by modal decomposition. I. Theory and validation, J. Acoust. Soc. Am. 100 (4) (1996) 2034–2048. [10] N. Amir, V. Pagneux, J. Kergomard, A study of wave propagation in varying cross-section waveguides by modal decomposition. II. Results, J. Acoust. Soc. Am. 101 (5) (1997) 2504–2517. [11] S. Felix, V. Pagneux, Sound propagation in rigid bends: a multimodal approach, J. Acoust. Soc. Am. 110 (3) (2001) 1329–1337.
Multimodal analysis of acoustic propagation in three-dimensional bends S. Félix∗ , V. Pagneux Laboratoire d’Acoustique de l’Université du Maine, UMR-CNRS 6613, Université du Maine, Avenue Olivier Messiaen, 72085 Le Mans Cedex 09, France Received 9 October 2001; received in revised form 10 December 2001; accepted 16 January 2002
Abstract An exact multimodal formalism is proposed for acoustic propagation in three-dimensional rigid bends of circular crosssection. Two infinite systems of first-order differential equations are constructed for the components of the pressure and axial velocity in the bend, projected on the local transverse modes. These equations are numerically unstable, due to the presence of evanescent modes, and cannot be integrated directly. An impedance matrix is defined, which obeys a Riccati equation, numerically workable. With this nonlinear first-order differential equation, the impedance can be calculated everywhere in the bend, allowing a direct characterization of its acoustical properties or allowing the acoustic field to be integrated. An exact algebraic formulation of the reflection and transmission matrices is carried out to allow bends and more complex duct systems to be characterized. This result is applied to calculate the reflection and transmission of a typical bend, and also to obtain the resonance frequencies of closed tube systems. © 2002 Elsevier Science B.V. All rights reserved. Keywords: Bend; Multimodal acoustic propagation; Riccati equation; Impedance matrix; Modal scattering properties
1. Introduction The theoretical study of sound propagation in bends of circular cross-section is difficult since the toroidal coordinate system required to suit the boundary conditions at the walls is not one of the coordinate systems in which the Helmholtz equation is separable [1] and therefore amenable to an analytical solution. Thus, relatively few works have been published on this subject and mostly approximate methods or purely numerical methods have been adopted. One can refer to [2] for a review. Cummings [3], who was one of the first to mention in acoustics the difficulty of solving the wave equation in non-separable toroidal coordinates, suggested, on the basis of experimental measurements, to visualize the pipe bend as being split up into a superposition of “slices” of rectangular section ducts, all in the plane of the bend. More simply, he also suggested to consider an equivalent rectangular section duct whose width equals the diameter of the circular cross-section and whose height is such that both ducts have the same cross-sectional area. This “slice” method has been adopted later by Keefe and Benade [4] to study the wave propagation in strongly curved ducts in the long wavelength limit. Wave propagation in slender curved three-dimensional tubes of arbitrary cross-section was investigated by Ting and Miksis [5], who bypassed the difficulty mentioned by Cummings by using a perturbation ∗
Corresponding author. Tel.: +33-24383-3553; fax: +32-24383-3520. E-mail addresses: [email protected] (S. F´elix), [email protected] (V. Pagneux). 0165-2125/02/$ – see front matter © 2002 Elsevier Science B.V. All rights reserved. PII: S 0 1 6 5 - 2 1 2 5 ( 0 2 ) 0 0 0 0 9 - 4
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Fig. 1. Geometry of the duct system with bend and system of coordinates.
method, assuming that the tube is weakly curved. Their analysis was carried out for four different scalings of the wavenumber. A numerical analysis of acoustic propagation in toroidal bends has been published by El-Raheb and Wagner [6], using a boundary elements method. Surface elements for a toroidal boundary were defined, allowing the acoustic field to be calculated under certain end condition for a water-filled bend joining two finite straight sections. Another method was used by Firth and Fahy [7], who gave an interesting study, using a spectral method to evaluate acoustic torus modes. They assumed solutions of the form A(r, φ) exp(jks s), where (r, φ, s) are the toroidal coordinates (see Fig. 1). The solutions A(r, φ), i.e. the torus modes, which are not separable in r and φ, were sought in terms of a series expansion on the transverse modes of a straight tube. They evaluated the mode shapes and propagation constants of these torus modes, and the reflection and transmission coefficients for a 90◦ bend with a plane wave incident and a non-reflecting termination for both end conditions, upstream and downstream from the bend. Later, in their study of modal wave propagation in circularly curved waveguides, Furnell and Bies [8] obtained the transverse field eigenfunctions by an approximate method based on the calculus of variations. They used these solutions to calculate the matrices characterizing the modal reflection and transmission properties of the bend. These approaches to determine the non-separable torus modes are not easy to implement since they require the calculation of the modes at each frequency and because of the difficulty to match the solutions obtained in the bend with the solutions in the waveguides upstream and downstream from the bend. This paper is concerned with the formulation of an exact approach, based on multimodal decomposition, to analyze acoustic propagation in three-dimensional bends of circular cross-section, or pipe bends. This approach is of particular interest, in both theoretical and practical points of view, since it is not limited to any domain of geometrical parameters or frequency and leads to an exact algebraic formulation of the modal scattering properties of bends and as a consequence of almost every practical duct systems. The multimodal method has been already validated for varying cross-section waveguides of straight longitudinal axis [9,10] and more recently for two-dimensional curved ducts of constant or varying cross-section [11], giving an efficient alternative to “classical” methods (e.g. separation of variables). This method consists in constructing two infinite first-order differential equations for the components of the pressure and axial velocity projected on the basis of the transverse modes in a straight cylindrical duct with the same cross-section. These two equations are numerically unstable, due to the presence of evanescent modes, and cannot be integrated directly. A impedance matrix is defined, which obeys a Riccati equation, numerically workable. With this first-order differential equation, the impedance can be calculated everywhere in the bend, allowing a direct characterization of its acoustical properties or the acoustic field to be integrated. Here the formulation of the multimodal method is developed for three-dimensional bends of circular cross-section. Such extension may indeed offer a larger field of applications, since practical duct systems are in general constituted
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with circular ducts. Furthermore, besides the fact that no exact analytical solution can be deduced from the wave equation in toroidal coordinates, alternatives to the multimodal method are much more constraining: the “torus” modes, the determination of which is not easy as discussed before, or purely numerical methods that involves 3D mesh construction and heavy implementations. The first part of the paper is devoted to the definition of the problem to be analyzed and the formulation of the multimodal method. The appropriate quantities are introduced and the main stages of the method are presented, with mention to previous works [9–11], where detailed information can be found. In a second part, an example of calculation of the pressure field is presented and used to evaluate the convergence of the method. The use of the multimodal method to calculate the modal scattering properties of bends and notably the exact algebraic formulation of the reflection and transmission matrices is detailed in the third part, with a generalization to complex duct systems in the last part.
2. Formulation Consider the propagation of sound in a circularly curved section of a duct system of circular cross-section as shown in Fig. 1. The parameters of the toroidal section are its mean radius R0 , the radius of the cross-section r0 and its length sf , measured along its circular axis. The duct walls are assumed to be rigid. Assuming adiabatic lossless linear media, acoustic velocity vˆ and pressure pˆ satisfy the linearized Euler equations jωvˆ = −
1 ∇p, ˆ ρ0
(1)
and ∇ · vˆ = −
jω p, ˆ ρ0 c02
(2)
where ρ0 is the density of air and c0 the speed of sound. The accent (ˆ) denotes dimensional quantities and time dependence exp(jωt) is omitted. The reference pressure ρ0 c02 and velocity c0 are used to reduce Eqs. (1) and (2) to dimensionless forms and we introduce the frequency k = ω/c0 so that these equations become −jkv = ∇p,
(3)
∇ · v = −jkp.
(4)
The coordinates adapted to suit the boundary conditions of a circular cross-section curved duct are the toroidal coordinates (r, φ, s) (see Fig. 1). Eqs. (3) and (4) in toroidal coordinates become (see Appendix A) −jkvr = ∂vr + ∂r
∂p , ∂r
−jkvφ =
1 κ cos φ − r 1 − κr cos φ
1 ∂p , r ∂φ
vr +
−jkvs =
1 ∂p , 1 − κr cos φ ∂s
1 ∂vφ κ sin φ 1 ∂vs + vφ + = −jkp, r ∂φ 1 − κr cos φ 1 − κr cos φ ∂s
(5)
(6)
where v has components (vr , vφ , vs ) and κ = 1/R0 is the curvature of the bend axis. By eliminating vr and vφ , these equations give ∂p = −jk(1 − κr cos φ)vs , ∂s
(7)
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∂vs 1 ∂ 2p 1 ∂p = (1 − κr cos φ)k 2 p + (1 − κr cos φ) 2 + (1 − 2κr cos φ) ∂s jk r ∂r ∂r 2 ∂p 1 ∂ p 1 . +(1 − κr cos φ) 2 2 + κ sin φ r ∂φ r ∂φ
(8)
Pressure p and axial velocity vs are now projected on the complete orthonormal basis (ψα ) of the classical transverse modes of a straight tube of radius r0 (see Appendix B): p(r, φ, s) = Pα (s)ψα (r, φ), (9) α
vs (r, φ, s) =
Uα (s)ψα (r, φ),
(10)
α
where Pα and Uα are scalar coefficients. The parameter α being a triplet of integers, p and vs are expressed in the form a triple series that may be simplified to a simple one. Indeed, since the duct has a plane of symmetry parallel to the plane of the torus, a source distribution which is symmetric with respect to this plane will produce an acoustic field which is also symmetric; similarly an antisymmetric source distribution will produce an antisymmetric field [7]. In other words, there is no coupling between the modes characterized by a symmetry index σ = 1 (symmetric) and those characterized by σ = 0 (antisymmetric). We can thus eliminate the summation on this index and consider two similar and independent problems. Without this symmetry, a coupling between the symmetric and antisymmetric modes would occur, without compromising the suitability of the multimodal method. Furthermore, the double series on the circumferential and radial indexes can be simplified to a simple one by sorting the modes ψα in increasing order of their cutoff frequency. Therefore, from this point, σ is a fixed value depending on the source distribution and α is a positive integer (associated to the indexes m and n), with ψ0 corresponding to the plane wave mode. The modal decompositions (9) and (10) are written as p = t ψP,
(11)
vs = t ψU,
(12)
where P = (Pα )α≥0 , U = (Uα )α≥0 and ψ = (ψα )α≥0 are column vectors. Then, by projecting (7) and (8) on the functions ψα , the following equations are obtained: P = −jkBU, U =
1 (C + KB)P, jk
in which K is a matrix given by 2 γmn 2 Kαβ = k − 2 δαβ ∀(α, β) ∈ N2 , r0
(13) (14)
(15)
where γmn is the (n + 1)th zero of Jm , the derivative of the mth-order Bessel function of the first kind (see Appendix B). An important property of Eqs. (13) and (14) is that the matrices B, C, given in Appendix B, depend only on the geometrical parameters r0 and κ. Thus, they do not depend on k and should not be calculated for each frequency. As the coupled differential equations (13) and (14) are written, their form is identical to that in a two-dimensional circular bend [11]. Thus, we will give only the main stages in the formulation of the multimodal method. The reader can refer to previous works [9–11] for more details.
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Because of the presence of evanescent modes (corresponding to Kαβ < 0), Eqs. (13) and (14) are numerically unstable and cannot be integrated directly. Therefore, an impedance matrix Z is defined, fulfilling P = ZU. This matrix Z obeys the Riccati equation Z = −jkB −
1 Z(C + KB)Z, jk
(16)
which is numerically workable. With this nonlinear first-order differential equation and a radiation condition (given as an impedance Zr ), it is then possible to calculate the impedance everywhere in the bend and in particular at the inlet, to get the input impedance and reflection coefficient (see [9]). Moreover, once the impedance is known, the pressure and velocity fields can be calculated using Eqs. (13) and (14), reformulated as P = −jkBYP, U =
(17)
1 (C + KB)ZU, jk
(18)
where Y is the admittance matrix (Y = Z −1 ). Owing to the previous calculation of Z, Eqs. (17) and (18) are not numerically divergent. Such formalism allows us to extend the integration of Z, P and U in any type of waveguide of circular cross-section upstream and downstream from the bend and therefore to analyze complex waveguide systems.
3. Calculation of the acoustic field: study of the convergence In this section and in the following, we assume that the duct downstream from the bend (region (III) in Fig. 1) is a semi-infinite straight duct of constant cross-section. The radiation condition Zr is then the characteristic impedance Zc , which is diagonal and given by Zcα =
k kα
∀α ∈ N,
(19)
where
γmn 2 2− k r0 kα = γmn 2 − k2 −j r0
for propagative modes, (20) for evanescent modes.
In order to evaluate the rate of convergence of the multimodal method, two different quantities are proposed for the error to be measured: 1/2 2 φ=0,π p − pref (1 − κr) dr ds
, (21) %1 = 2 φ=0,π pref (1 − κr) dr ds %2 =
s=sf
p − pref 2 r dr dφ
s=sf
pref 2 r dr dφ
1/2 ,
(22)
where the reference solution pref is the one obtained for 30 modes introduced in the multimodal calculation, supposing that the convergence is reached. %1 is the error in the plane represented in Fig. 2 (on the left), while %2 the error on the Section A–A (on the right). A plane piston source for the pressure is imposed at the inlet (s = 0), and the parameters are R0 /r0 = 2.5, θf = sf /R0 = π/2 and kr0 = 2.4.
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Fig. 2. Contour of real part of pressure; R0 /r0 = 2.5, θf = sf /R0 = π/2, kr0 = 2.4.
Fig. 3 shows how the errors %1 and %2 decrease as more modes are taken into account, following a 1/N 2.5±0.3 law, where N is the number of modes. Contributions of each mode introduced in the calculation to the convergence of the solution are naturally different. Thus, apart from the first modes, the 7th, 13th or 22th modes in our example have a significant contribution, while the contribution of other modes (e.g. the 8th, 11th, 14th, 16th, etc.) seems to be negligible. The 7th, 13th and 22th modes are those for which the circumferential index m is equal to 1, i.e. those having symmetry properties similar to those of the transverse solution (see the Section A–A in Fig. 2). On the contrary, the modes whose contribution seems to be negligible are mostly circumferential, with relatively great values of m. In view of the smooth variations of the solution with the coordinate φ, it is clear that these modes give information of “secondary” importance. Here, the modes have been sorted in increasing order of their cutoff
Fig. 3. Convergence of the multimodal method.
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frequency; it appears with the remarks above that another way of sorting the modes would increase the convergence rate, subject to a prior knowledge of the symmetry of the solution.
4. Modal scattering properties The sequel of this paper concerns the use of the multimodal method to calculate the modal scattering properties of pipe bends, in the present section, and of more complex duct systems in the following section. Let P(i) , P(r) and P(t) be the wave incident, reflected and transmitted by the bend, respectively. The reflection matrix R and transmission matrix T are defined by P(r) = RP(i) and P(t) = T P(i) . Once the impedance is known at the inlet of the bend, the reflection matrix R can be easily calculated by R = (Zc + Z)−1 (Zc − Z) (see [9] for details), without prior calculation of the acoustic field or hypothesis on the source. Here, Z is the input impedance and Zc the characteristic impedance. Moreover, since the formulation of the multimodal method for pipe bends is similar to that for two-dimensional bends, an exact algebraic formulation of the reflection and transmission matrices can be carried out. This formulation, developed in [11], is very attractive, since it allows us to get the scattering matrix by simple inversion of matrices of order Nmodes × Nmodes . Fig. 4 shows the modulus of the amplitude reflection coefficients R00 , R10 , R20 and transmission coefficients T00 , T10 , T20 for the same bend as considered in Section 3, i.e. with geometrical parameters R0 /r0 = 2.5 and θf = π/2. The frequency is varying from 0 to twice the first cutoff frequency: kr0 ∈ [0, 2γ10 ]. For α in N, Rα0 (respectively Tα0 ) gives the contribution of the incident plane wave mode to the reflected (respectively transmitted) (α + 1)th
Fig. 4. Amplitude reflection and transmission coefficients.
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mode. The parameters are chosen to enable the comparison with results published by Firth and Fahy: it appears that the agreement is very good.
5. Resonances in complex cavities Besides the fact that the algebraic formulation, the reflection and transmission matrices, as given in [11], is a simple and direct method to obtain the modal scattering properties of a pipe bend, it allows more complex duct systems to be characterized, and notably their resonance frequencies to be calculated, in case of closed annular ducts. 5.1. Formulation Consider a simple section of duct. We define its scattering matrix S as follows: T R S= , R T
(23)
where R and T are the reflection and transmission matrices when considering a right-going incident wave and R and T are the reflection and transmission matrices when considering a left-going incident wave. For symmetrical waveguides, R = R and T = T . In the case of a bent section, which is a symmetrical waveguide, the reflection and transmission matrices R = R and T = T are calculated algebraically as discussed before. In the case of a straight section, the R and T matrices are recalled in Appendix B. Consider now a system composed of a succession of different sections of duct. If the scattering matrix of each elementary part is known, we can calculate analytically the global scattering matrix Ssys. of the system (see Appendix B), and thus characterize its reflection and transmission properties. Moreover, if this duct system is closed such that it becomes an annular cavity, the closing condition imposes a relation dispersion det(Ssys. − I ) = 0
(24)
with I the identity matrix. The frequency solutions of Eq. (24) are the resonance frequencies of the cavity. Two results of calculation of resonance frequencies are presented in the following for two different shapes of annular duct: first a simple torus and then a more complex annulus, composed of a succession of straight and bent sections. 5.2. Resonances in a torus The torus is a bend of overall angle 2π whose scattering matrix Storus can be calculated directly using the algebraic formulation of R and T . The solutions of (24) with Ssys. = Storus , denoting that the torus is closed, give its resonance frequencies. Table 1 shows in the first column the resonance frequencies in a torus whose dimensions are such that R0 /r0 = 2.5. Only the frequencies less than the first cutoff frequency are given. The second column shows the same results, but without taking into account the curvature, i.e. when considering a straight tube with the same axis length 2π R0 and the same closing condition (24). As a validation, these results are compared with values deduced from the dispersion curve for the first torus mode published by Firth and Fahy [7], showing that, although the discrepancy between the exact approach and the straight tube approximation is not important, the exact approach give results closer to those deduced from [7]. 5.3. Resonances in an annulus Consider now a more complex annular duct, as shown in Fig. 5. This annulus is composed of straight sections of length l1 and l2 and curved sections of angle π/2. The scattering matrix of each of this sections being known, the
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Table 1 Resonance frequencies, adimensioned by γ10 /r0 , in a torus with R0 /r0 = 2.5a det(Storus − I ) = 0b
det(Sapprox. − I ) = 0c
Firth and Fahy [7]d
0.2210 0.4367 0.6434 0.8397
0.2173 0.4345 0.6518 0.8690
0.22 0.44 0.64 0.84
Only the frequencies less than the first cutoff frequency γ10 /r0 are given. Storus is the scattering matrix of a bend of overall angle 2π , calculated with 15 modes. cS approx. is the scattering matrix of a straight tube of length 2π R0 , calculated with 15 modes. d Results deduced from the dispersion curve for the first torus mode published by Firth and Fahy [7]. a
b
Fig. 5. Modulus of det(Sann. − I ) (plain) and modulus of det(Sapprox. − I ) (dashed-dotted) vs frequency, where Sapprox. is the scattering matrix of a straight tube of length 2l1 + 2l2 + 2π R0 . On the left, the shape of the annulus. R0 /r0 = 1.25, l1 /r0 = 1.5, l2 /r0 = 0.75.
overall scattering matrix Sann. of the annulus is calculated iteratively using the result given in Appendix B. Zeros of det(Sann. − I ) can thus be found, giving the resonances frequencies of the annulus. Fig. 5 shows the modulus of det(Sann. − I ) for an annulus with parameters R0 /r0 = 1.25, l1 /r0 = 1.5 and l2 /r0 = 0.75. Again we compare these results with those obtained when the curvature is not taken into account, i.e. when considering a straight duct with the same axis length 2l1 + 2l2 + 2πR0 and the same closing condition (24). Since the curvature is relatively large, the discrepancy between the two results becomes rapidly noticeable when increasing the frequency. One can also notice in Fig. 5 that each zero of the plain curve, i.e. each resonance frequency of the annulus splits into two values. When calculating the resonance frequencies of a torus, we do not observe such splitting. The symmetry properties of each geometry explain these two different results: the degeneracy of the eigenvalues k is double for the torus, whose geometry is invariant to any rotation around its axis normal to the plane of the torus. Because of the straight parts, the annulus shown in Fig. 5 has not this symmetry property and as a consequence the degeneracy is lifted. If we consider now the same duct with l1 = l2 , the annulus will have a supplementary symmetry plane and therefore every second eigenvalue will be degenerated again.
6. Conclusion An exact multimodal method for the analysis of wave propagation in three-dimensional rigid bends of circular cross-section has been formulated. This method, thanks to the calculation of the impedance matrix along the duct, is convenient both for the formulation of radiation conditions and when considering junctions of the bend with any other waveguides.
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Whatever the frequency of the source or the dimensions of the bend, it can be characterized in a very simple manner once the impedance is known, by its input impedance or modal scattering properties, without prior calculation of the acoustic field or hypothesis on the source. An algebraic formulation of the reflection and transmission matrices has also been carried out, allowing bends and more complex duct systems to be characterized accurately.
Appendix A The toroidal coordinates used are r ∈ [0, r0 ], φ ∈ [0, 2π [ and s ∈ [0, sf ] (see Fig. 1). They are orthogonal and we have dr = hr dr uˆ r + hφ dφ uˆ φ + hs ds uˆ s ,
(A.1)
where the scale factors are given by hr = 1,
hφ = r,
hs = 1 − κr cos φ.
(A.2)
The gradient and divergence operators in this coordinate system are thus 1 ∂f ∂f 1 ∂f uˆ r + uˆ φ + uˆ s , ∂r r ∂φ 1 − κr cos φ ∂s 1 1 ∂Fφ 1 κ sin φ κ cos φ ∂Fs ∂Fr + + − Fr + Fφ + . ∇·F= ∂r r 1 − κr cos φ r ∂φ 1 − κr cos φ 1 − κr cos φ ∂s
∇f =
(A.3) (A.4)
Appendix B The transverse modes in a rigid straight tube of radius r0 are the solutions ψ(r, φ) = R(r)Φ(φ) of the separable transverse problem 1 dR 1 d2 Φ 1 d2 R 2 + + k⊥ = 0, + R dr 2 rR dr r 2 Φ dφ 2 satisfying the homogeneous Neumann condition on the walls ∂ψ = 0. ∂r r=r0 These modes are written as γmn r σπ ψα (r, φ) = Aα Jm sin mφ + , r0 2
(B.1)
(B.2)
(B.3)
where Jm is the mth-order Bessel function of the first kind and γmn the (n + 1)th zero of Jm . Their cutoff frequency is γmn /r0 . The parameter α denotes a triplet of integers (m, n, σ ), where m is the circumferential index (m ≥ 0), n the radial index (n ≥ 0) and σ the symmetry index (σ = 0, 1). The basis of functions (ψα ) is orthogonal and can be normed by taking 1 if m = 0, πr02 J02 (γ0n ) Aα = (B.4) 1 if m > 0. (π r 2 /2)(1 − m2 /γ 2 )J 2 (γ ) mn mn m 0
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Appendix C The parameter α is associated to circumferential and radial indexes m and n, respectively, and β is associated to µ and ν: r0 γµν r γmn r 2 Bαβ = Aα Aβ Jmµ Jµ r Jm dr, (C.1) r0 r0 0 Cαβ = Aα Aβ
γmn κJmµ r0
r0
γµν r γmn r Jµ dr r0 r0 r0 γµν r γmn r Jm Jµ dr, r0 r0 0
rJm+1 0 −Aα Aβ mκ(Jmµ − Hmµ )
in which
Imµ = Jmµ = and
2π 0 2π
0
Hmµ = =
2π 0
σπ σπ sin mφ + sin µφ + dφ = π δmµ (1 + (δσ 1 − δσ 0 )δm0 ), 2 2 σπ σπ π cos φ sin mφ + sin µφ + dφ= δ|m−µ|,1 (1 + (δσ 1 − δσ 0 )(δm0 + δµ0 )), 2 2 2
(C.2)
(C.3) (C.4)
σπ σπ sin φ cos mφ + sin µφ + dφ 2 2
π (δµ−m,1 (1 − (δσ 1 − δσ 0 )δm0 ) − δm−µ,1 (1 + (δσ 1 − δσ 0 )δµ0 )). 2
(C.5)
Appendix D Consider a finite straight tube with length l; its reflection matrices R and R are null and its transmission matrices T and T are diagonal and given by Tα = Tα = e−jkα l
∀α ∈ N.
(D.1)
We give now the calculation of the global scattering matrix of a duct system composed of two sections (Fig. 6), when the scattering matrix of each section is known. The scattering matrices of the sections (1–2) and (2–3) are given, respectively, by T12 R12 T23 R23 , S23 = , (D.2) S12 = R12 T12 R23 T23
Fig. 6. Duct system composed of two sections.
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such that + A2 A1−
= S12
A1+ A2−
,
A3+ A2−
= S23
A2+ A3−
.
(D.3)
A1− and A3+ are then given as functions of A1+ and A3− , leading to the following expression for the global scattering matrix S13 : R )−1 T + T (1 − R R )−1 R T T23 (1 − R12 R23 23 23 12 12 23 12 23 S13 = . (D.4) (1 − R R )−1 R T (1 − R R )−1 T R12 + T12 T12 23 12 23 12 23 12 23 The scattering matrix of a duct composed of n parts will be obtained by iterating n − 1 times the calculation above. References [1] [2] [3] [4] [5] [6] [7] [8]
P.M. Morse, H. Feshbach, Methods of Theoretical Physics, McGraw-Hill, New York, 1953. W. Rostafinski, Monograph on Propagation of Sound Waves in Curved Ducts, NASA Reference Publication 1248, 1991. A. Cummings, Sound transmission in curved duct bends, J. Sound Vib. 35 (4) (1974) 451–477. D.H. Keefe, A.H. Benade, Wave propagation in strongly curved ducts, J. Acoust. Soc. Am. 74 (1) (1983) 320–332. L. Ting, M.J. Miksis, Wave propagation through a slender curved tube, J. Acoust. Soc. Am. 74 (2) (1983) 631–639. M. El-Raheb, P. Wagner, Acoustic propagation in a rigid torus, J. Acoust. Soc. Am. 71 (6) (1982) 1335–1346. D. Firth, F.J. Fahy, Acoustic characteristics of circular bends in pipes, J. Sound Vib. 97 (2) (1984) 287–303. G.D. Furnell, D.A. Bies, Characteristics of modal wave propagation within longitudinally curved acoustic waveguides, J. Sound Vib. 130 (3) (1989) 405–423. [9] V. Pagneux, N. Amir, J. Kergomard, A study of wave propagation in varying cross-section waveguides by modal decomposition. I. Theory and validation, J. Acoust. Soc. Am. 100 (4) (1996) 2034–2048. [10] N. Amir, V. Pagneux, J. Kergomard, A study of wave propagation in varying cross-section waveguides by modal decomposition. II. Results, J. Acoust. Soc. Am. 101 (5) (1997) 2504–2517. [11] S. Felix, V. Pagneux, Sound propagation in rigid bends: a multimodal approach, J. Acoust. Soc. Am. 110 (3) (2001) 1329–1337.