Complex resonance frequencies of a finite, circular radiating duct with an infinite flange
aa r X i v : . [ phy s i c s . c l a ss - ph ] J a n Complex resonance frequencies of a finite, circularradiating duct with an infinite flange
B. Mallaroni a, ∗ , P.-O. Mattei a , J. Kergomard a a Laboratoire de M´ecanique et d’Acoustique, UPR CNRS 7051, 31 chemin Joseph Aiguier,13402 Marseille cedex 20, France
Abstract
Radiation by solid or fluid bodies can be characterized by resonance modes.They are complex, as well as resonance frequencies, because of the energy lossdue to radiation. For ducts, they can be computed from the knowledge of theradiation impedance matrix. For the case of a flanged duct of finite length ra-diating on one side in an infinite medium, the expression of this matrix wasgiven by Zorumski, using a decomposition in duct modes. In order to calculatethe resonance frequencies, the formulation used in Zorumski’s theory must bemodified as it is not valid for complex frequencies. The analytical developmentof the Green’s function in free space used by Zorumski depends on the integralsof Bessel functions which become divergent for complex frequencies. This pa-per proposes first a development of the Green’s function which is valid for allfrequencies. Results are applied to the calculation of the complex resonance fre-quencies of a flanged duct, by using a formulation of the internal pressure basedupon cascade impedance matrices. Several series of resonance modes are found,each series being shown to be related to a dominant duct mode. Influence ofhigher order duct modes and the results for several fluid densities is presentedand discussed.
Keywords:
Acoustics, radiation impedance, cylindrical pipe, resonancefrequencies
PACS:
1. Introduction
For the problem of duct radiation, many works have been done concerningthe calculation of radiation for a given duct mode, but to the author’s knowkedgeno study has been done concerning the resonance modes. We will treat theproblem, in order to get a better insight of the coupling of the duct and the ∗ Corresponding author
Email addresses: [email protected] (B. Mallaroni), [email protected] (P.-O. Mattei), [email protected] (J. Kergomard)
Preprint submitted to Elsevier November 2, 2018 urrounding space by radiation. We choose the case of a duct with infiniteflange, because of its relative simplicity. One difficulty is due to the fact thatresonance frequencies (and modes) are complex, because radiation is a formof dissipation. Notice that in the literature, resonance modes are also calledeigenmodes: they must be distinguished from the duct modes used in the presentpaper for the purpose of the calculation.Green’s functions are widely used in many physical situations and notablyin acoustics, for the calculation of the pressure field radiated by physical sources( e.g. speakers, musical instruments, vibrating structures,...). For instance, thesolution given by Rayleigh [1] to the classical problem of a plane piston radiatinginto an infinite flange involves the Green’s function in free space. Zorumski [2]extended this result to know the radiation of a semi-infinite flanged duct in theform of matrix impedance, giving the coupling between duct modes and used theSonine’s infinite integral ( Ref. [3], p.416, Eq.4) to develop the Green’s functionin free space. In Refs [4, 5], formulations based on the Zorumski’s method ofthis radiation impedance are obtained for a larger class of problems. However,the development fails when the frequency becomes complex: the correspondinginfinite integral, involving a Bessel function whose argument is a product of thefrequency and the dummy argument, becomes divergent for complex frequen-cies. In many studies, the Zorumski’s radiation matrix is used as a boundarycondition at the end of the duct in order to calculate input impedances, lengthcorrections or reflection coefficients (see e.g. Refs. [6] or [7]). It is worth notingthat complex resonance frequencies can occur in various situations ( e.g. dissi-pative fluid, radiation, complex impedance wall boundary conditions such as inRefs. [8] or [9]). In section 2 of this paper, we present a new expression for theGreen’s function in free space for complex frequencies. In section 4, an appli-cation of this result is devoted to the determination of the complex resonancefrequencies of a cylindrical duct, closed at its input, considering the influenceof higher order duct modes. In the same secion, some results are given and dis-cussed. For this purpose, the internal Green’s function is previously calculatedin section 3 with a method of cascade impedance.
2. Calculations of Green’s function for the Helmholtz equation in freespace
Many studies on sound radiation by cylindrical ducts can be found in theliterature. For the case of an infinite flange, Norris and Sheng [10] or Nomura[11] used a Green’s function integral to find an appropriate formulation for theexternal field. We can also cite the classical work by Levine and Schwinger[12] for the case of an unflanged pipe. Zorumski [2] extended the results for theplanar mode to obtain a multimodal radiation impedance which is a combinationof the duct modes present in the duct. In this section, these calculations arebriefly recalled, exhibiting the difficulty related to complex frequencies. Thus, anew analytical formula for the Green’s function, valid for a dissipative problem,is presented. 2 .1. Zorumski’s radiation impedance
We consider the radiation of sound into an infinite half space from a circularduct (with radius b and length L ), with an infinite flange at z = 0 (the index 0corresponds to the cross section S at the end of the duct) and we have chosento work with circular coordinates where the vector r is denoted ( z, r, θ ) as shownby Fig. 1. b b -L z z’ z Z e Z ray receiver source l l z =0 S zr θ Figure 1: Schema and coordinates of the duct.
The acoustic pressure in the infinite medium ( z ≥
0) is given by a Helmholtzintegral (the time factor exp( − iωt ) is omitted throughout this paper): p ( r ) = − iωρ π Z π Z b r v ( r , θ ) e ikh h dr dθ , (1)where h = [ r + r − rr cos( θ − θ ) + z ] , ρ the ambient density and thewavenumber k = ω/c (with ω the circular frequency and c the speed of sound).The pressure p and the velocity v inside the duct ( z <
0) are expressed as aseries of duct eigen modes, so in z = 0: p ( r, θ, z = 0) = ρc X m X n ψ mn ( kr ) e imθ P mn , (2) v ( r, θ, z = 0) = c X m X n ψ mn ( kr ) e imθ V mn , (3)where ψ mn ( kr ) e imθ is the transverse function for the mode mn with ψ mn ( kr ) = J ( kr ) /N mn . The λ mn are the eigenvalues, solutions of J ′ ( λ mn kb ) = 0, and thenorm N mn is chosen similarly to that used by Zorumski [2].Substituting Eq. (3) into Eq. (1) gives the pressure for z ≥ V mn : p ( r, θ, z ) = − iωρc π X m X n V mn Z π e imθ Z b r e ikh h ψ mn ( kr ) dr dθ . (4)Zorumski expressed the free space Green’s function in equation (4) in termsof a Sonine’s infinite integral ([3], p. 416, Eq. 4) and wrote for z = 0 in theexpression of h : e ikh h = k Z ∞ τ ( τ − − J ( τ kh ) dτ. (5)3ext, he introduced a concept of ”generalized radiation impedance matrix Z ray ”for a semi-infinite duct with an infinite flange to describe the relation betweenthe modal pressure and velocity amplitudes: P mn = ∞ X l =1 Z mnl V ml , (6)where m , l and n are, respectively, the orders of circumferential, radial incidentand reflected modes. The element Z mnl of the radiation impedance matrixgives the contribution of the velocity mode ml to the pressure mode mn (forreasons of symmetry, the coupling is possible only for a duct mode with the sameazimuthal dependance). The expression of the radiation impedance is obtainedas: Z mnl = − i Z ∞ τ ( τ − − D mn ( τ, k ) D ml ( τ, k ) dτ, (7)with, for a hard wall condition: D mn ( τ, k ) = kb τ ψ mn ( kb ) J ′ m ( τ kb ) λ mn − τ . (8) The following asymptotic form (see Ref. [13], Eq. 9.2.1, p. 364) occurs when ν is fixed and | κ | → ∞ : J ν ( κ ) = r πκ [ cos ( κ − νπ − π ) + e |ℑ ( κ ) | O ( | κ | − )] , (9)with | arg κ | < π (in this paper, the real part and imaginary part are repre-sented, respectively, by the symbols ℜ and ℑ ). As a consequence, for τ → ∞ with k ∈ C we have J ( τ kh ) → ∞ when ℑ ( k ) = 0, thus relation (5) and theradiation impedance (7) given by Zorumski are divergent integrals for all nonreal frequencies.In order to have a Green’s function for the Helmholtz equation in free spacevalid for complex frequencies, we use another form of the Sonine’s infinite in-tegral to develop this Green’s function, expressed by Watson [3] (p. 416, Eq.4): e ikh h = Z ∞ τ ( τ − k ) − J ( τ h ) dτ. (10)This integral remains convergent even for k complex. A difficulty occurs since k is a branch point of the square root. For a time factor exp( − iωt ), the integrationpath on the real axis must remain below k . However the complex resonancefrequencies ω = ck have a negative imaginary part (see explanation in section4), so the previous formula must be adapted because of the branch cut. Theintegration path below k is classically deformed, as shown in Fig. 2:4 c k0 ℜ ( k ) b k b I I Ic Figure 2: Deformation of the integration contour.
Now, the integral in Eq. (10) is written as: e ikh h = Z |ℜ ( k ) | J ( τ h ) τ ( τ − k ) − dτ + I ( k, h )+ I c + I ( k, h )+ Z ∞|ℜ ( k ) | J ( τ h ) τ ( τ − k ) − dτ, (11)with I ( k, h ) = I ( k, h ) = − Z k |ℜ ( k ) | J ( τ h ) τ ( τ − k ) − dτ and I c is zero according to Jordan’s lemma.After calculations (similar to those developed by Morse and Feshbach [14],p.410), the following five cases can be distinguished:i) ℜ ( k ) ≤ ℑ ( k ) > e ikh h = Z ∞ J ( τ h ) τ ( τ − k ) − dτ. (12)ii) ℜ ( k ) ≤ ℑ ( k ) ≤ e ikh h = − i Z |ℜ ( k ) | J ( τ h ) τ √ k − τ dτ + Z ∞|ℜ ( k ) | J ( τ h ) τ √ τ − k dτ − Z k |ℜ ( k ) | J ( τ h ) τ √ τ − k dτ. (13)iii) ℜ ( k ) > ℑ ( k ) > e ikh h = + i Z |ℜ ( k ) | J ( τ h ) τ √ k − τ dτ + Z ∞|ℜ ( k ) | J ( τ h ) τ √ τ − k dτ. (14)iv) ℜ ( k ) > ℑ ( k ) = 0 e ikh h = J ( kh ) k √ ǫ (1 − i ) + i Z | k | (1 − ǫ )0 J ( τ h ) τ √ k − τ dτ + Z ∞| k | (1+ ǫ ) J ( τ h ) τ √ τ − k dτ, (15)5ith ǫ << ℜ ( k ) > ℑ ( k ) < e ikh h = + i Z |ℜ ( k ) | J ( τ h ) τ √ k − τ dτ + Z ∞|ℜ ( k ) | J ( τ h ) τ √ τ − k dτ − Z k |ℜ ( k ) | J ( τ h ) τ √ τ − k dτ. (16)Contrary to the original Zorumski’s formulation, these results involve con-vergent integrals when the frequency is complex. Initially, the previous result will be checked for the real case, in order tocompare the original Zorumski’s result, noted in Eq. (7), and that obtainedusing the expansion of exp( ikh ) /h for the case (iv) where ℑ ( k ) = 0. Eq. (15)leads to the following results: Z mnl = − ik [ ˜ D mn ( k ) ˜ D ml ( k ) k √ ǫ (1 − i ) + i Z | k | (1 − ǫ )0 τ √ k − τ ˜ D mn ( τ ) ˜ D ml ( τ ) dτ + Z ∞| k | (1+ ǫ ) τ √ τ − k ˜ D mn ( τ ) ˜ D ml ( τ ) dτ ] , (17)where ˜ D mn ( τ ) = b τ ψ mn ( b ) J ′ m ( τ b ) λ mn − τ . (18)The radiation impedance for the planar mode ( m = n = 0 with l = 0)with Zorumski’s formulation (7) and formulation (17) (with ǫ = 10 − ) are verysimilarly and thus, confirms the validity of formula (17) with the identical com-putational cost. This formula is used in Ref. [15] to calculate an approximationof the reflection coefficient and of the length correction, taking into account theeffect of the higher order duct modes below the first cut-off frequency.The comparison between complex Zorumski’s formulation for the planar mode(m=n=0 and l=0) and Rayleigh’s radiation impedance of a flanged plane pistonconfirm the validity of Z for all the frequencies (see subsection 4.1), becauseRayleigh’s radiation impedance is by definition the same quantity as Z .In what follows, we show the interest of the radiation impedance valid for com-plex frequencies when calculating the complex resonance frequencies of a flangedfinite length duct terminating in a Zorumski’s radiation condition. In a first in-stance, the internal Green’s function must be determined, with a method ofcascade impedances. 6 . Calculation of the finite flanged duct internal Green’s functionwith a method of cascade impedances We search for the internal Green’s function G ( M, M ′ , ω ) at a point M ( r, θ, z )with a source in M ′ ( r ′ , θ ′ , z ′ ), satisfying:(∆ M + k ) G ( M, M ′ , ω ) = − πr δ ( r − r ′ ) δ ( θ − θ ′ ) δ ( z − z ′ ) , (19)with ∂ n G ( M, M ′ , ω ) = 0 on the walls. It is classically (see e.g. Morse andFeshbach [14]) expanded in a series of duct modes (the boundary conditions forthe variable z will be given later on): G ( M, M ′ , ω ) = ∞ X m =0 ∞ X n =0 φ mn ( r, θ ) φ mn ( r ′ , θ ′ ) g mn ( z, z ′ , ω ) , (20)where φ mn ( r, θ ) = ψ mn ( r ) e imθ is the transverse function for the mode mn and g mn ( z, z ′ , ω ) is the longitudinal function for the mode mn . In what follows, thedependance in ω of g mn ( z, z ′ , ω ) is omitted for simplicity. It is worth notingthat now we have the transverse function with respect to r and not kr , as perZorumski [2]. Therefore, ψ mn ( r ) = J m ( λ mn r ) N mn where the λ mn are solutions of J ′ m ( λ mn b ) = 0, with λ mn = γ mn /b . The γ mn are the ( n + 1) th zeros of the firstderivative of the Bessel’s function J m . The following norm N mn is chosen (Ref.[2] with respect to b and not kb ): N mn = b √ π s − m λ mn b J m ( λ mn b ) . (21)The transverse modes φ mn ( r, θ ) satisfy:(∆ ⊥ + λ mn ) φ mn ( r, θ ) = 0 , with ∆ ⊥ = r ∂∂r ( r ∂∂r ) + r ∂ ∂θ , thus: ∆ ⊥ φ mn ( r, θ ) = − λ mn φ mn ( r, θ ). Studiessuch as Refs. [6] or [7] show that a small finite number of terms is necessary for anumerical estimate of the summation and it can be truncated: M m is defined asthe number of circumferential modes m and N n as the number of radial modes n . Introducing the previous expression (20) in (19) gives: M m X m =0 N n X n =0 ( ∂ ∂z + k mn ) φ mn ( r, θ ) φ mn ( r ′ , θ ′ ) g mn ( z, z ′ ) = − πr δ ( r − r ′ ) δ ( θ − θ ′ ) δ ( z − z ′ ) , (22)where k mn = k − λ mn . Then, multiplying the left and right sides of (22) by φ ∗ m ′ l ( r, θ ) and integrating the resulting equality on the surface S, the orthogo-nality condition ( R S φ mn φ ∗ m ′ l dS = δ mm ′ δ nl ), leads to:( ∂ ∂z + k mn ) g mn ( z, z ′ ) = − δ ( z − z ′ ) . (23)7ith a conservative Neumann or Dirichlet boundary condition at the ex-tremities of the duct, the resonance wavenumbers can be easily computed. Butwith a dissipative boundary condition like those occurring for the sound radi-ation of a flanged cylinder, there is no simple analytical solution. Therefore,in the next section, the elements g mn ( z, z ′ ) are calculated with a method ofcascade impedances presented in Refs. [6],[16] or [17]. The calculation of the duct impedance at an abscissa z with respect toanother abscissa z is based on the following transfer matrix relationship (seee.g. Ref. [17]): (cid:18) P mn ( z ) V mn ( z ) (cid:19) = ( M T ) (cid:18) P mn ( z ) V mn ( z ) (cid:19) , (24) M T = cosh( ik mn ( z − z )) Z c,mn sinh( ik mn ( z − z ))sinh( ik mn ( z − z )) Z c,mn cosh( ik mn ( z − z )) , where Z c,mn is an element of the diagonal matrix of characteristic impedance Z c : Z c,mn = kρck mn . Moreover, matrices formulation is chosen: Φ ( r, θ ) is a column vector constitutedby the M m N n elements φ mn , verifying R S Φ ( r, θ ) Φ T ( r, θ ) dS = I , P ( z ) is acolumn vector constituted by the M m N n elements P mn and V ( z ) is a columnvector constituted by the M m N n elements V mn . Thus, relation (24) is nowwritten: (cid:18) P ( z ) V ( z ) (cid:19) = (cid:18) C Z c SZ − S C (cid:19) (cid:18) P ( z ) V ( z ) (cid:19) , (25)where C and S are diagonal matrices constituted by the elements cosh( ik mn ( z − z )) and sinh( ik mn ( z − z )), respectively.The transfer matrix formulation (25) is now transformed into an impedance ma-trix formulation. The calculation, presented in Ref. [17], is recalled in AppendixA. This gives the following matrix equation: (cid:18) P ( z ) P ( z ) (cid:19) = (cid:18) Z − Z Z − Z (cid:19) (cid:18) V ( z ) V ( z ) (cid:19) , (26)where Z = Z c S − C , Z = Z c S − , Z = Z c S − and Z = Z c S − C . P (z’) First, the pressure vector at the source position z ′ is calculated. For thispurpose, we calculate a right-side matrix impedance Z + ( z ′ ) at z ′ with respectto Z ray , a left-side matrix impedance Z − ( z ′ ) at z ′ with respect to Z e and fi-nally the connection between these two matrices at z ′ , the abscissa of the source.8 tep 1: Right-side matrix impedance Z + ( z ′ )Let us denote C s and S s the diagonal matrix constituted by the elementscosh( ik mn l ) and sinh( ik mn l ), respectively, where l (see Fig. 1) is the distancebetween the abscissa z ′ and the extremity ( z = 0). The radiation impedance Z ray , constituted by the elements Z mnl calculated in the previous section, ver-ifies P ( z ′ + l ) = Z ray V ( z ′ + l ). Thus Eq. (25) leads to: V ( z ′ + l ) = ( Z ray + Z ) − Z V ( z ′ ) , (27)and, with Eqs. (26) and (27), to: P ( z ′ ) = Z V ( z ′ ) − Z ( Z ray + Z ) − Z V ( z ′ ) . With Z + verifying P ( z ′ ) = Z + ( z ′ ) V ( z ′ ), we have finally: Z + ( z ′ ) = Z − Z ( Z ray + Z ) − Z , (28)or: Z + ( z ′ ) = Z c S − C s − Z c S − [ Z − Z ray + S − C s ] − S − . (29) Step 2: Left-side matrix impedance Z − ( z ′ )We choose a Neumann condition for z = − L (thus V ( z ′ − l ) = 0). Here, C e and S e are the diagonal matrices constituted by the elements cosh( ik mn l ) andsinh( ik mn l ), respectively. l (see Fig. 1) is the distance between the point z = − L and the point z ′ . The impedance Z e is calculated at z = − L .Relation (26) is written for the present case as: P ( z ′ ) = Z c S − V ( z ′ − l ) − Z c S − C e V ( z ′ ) . (30)Thus, with the Neumann condition in z = − L and with Eq. (30), using P ( z ′ ) = Z − ( z ′ ) V ( z ′ ), the following result is obtained: Z − ( z ′ ) = − Z c S − C e . (31) Step 3: Connection between the impedance matrices Z + and Z − at z ′ Let us denote P ± mn ( z ′ ) = [ g mn ( z, z ′ )] z = z ′ ± ǫ . Using the continuity of the Green’sfunction at z = z ′ , leads to when ǫ → P + mn ( z ′ ) = P − mn ( z ′ ) = P mn ( z ′ ) , (32)Integrating relation (23) on an interval of width 2 ε between z ′ + ε and z ′ − ε gives: Z z ′ + εz ′ − ε ( ∂ ∂ z + k mn ) g mn ( z, z ′ ) dz = − ε → ∂ z P + mn ( z ′ ) − ∂ z P − mn ( z ′ ) = − , (34)9ith ∂ z P ± mn ( z ′ ) = [ ∂ z g mn ( z, z ′ )] z = z ′ ± ǫ .Euler’s dimensionless equation ρc V mn ( z ′ ) = − iωρ ∂ z P mn ( z ′ ) implies ∂ z P mn ( z ′ ) = ikV mn ( z ′ ), thus, using Eq. (34): V + mn ( z ′ ) − V − mn ( z ′ ) = − ik . (35)We introduce a column vector W of M m N n lines whose elements equal 1. Equa-tion (35) may be expressed as:( Z + ( z ′ )) − P + ( z ′ ) − ( Z − ( z ′ )) − P − ( z ′ ) = − ik W , and using Eq. (32), the pressure at the source is written as follows: P ( z ′ ) = − ik (cid:2) ( Z + ( z ′ )) − − ( Z − ( z ′ )) − (cid:3) − W . (36) g (z,z’) We introduce a column vector g ( z, z ′ ) constituted by the M m N n elements g mn ( z, z ′ ). They are two possible configurations with respect to the relativepositions of receiver at z and source at z ′ : First configuration: z > z ′ Let l r = z − z ′ be the distance between the receiver and the source (see Fig. 1), C l r the diagonal matrix constituted by the elements cosh( ik mn l r ), and S l r thediagonal matrix constituted by the elements sinh( ik mn l r ). Relation (24) givesfor z > z ′ : g ( z, z ′ ) = C l r P ( z ′ ) − Z c S l r V ( z ′ ) , then, with P ( z ′ ) = Z + ( z ′ ) V ( z ′ ): g ( z, z ′ ) = S l r (cid:2) S − r C l r − Z c ( Z + ( z ′ )) − (cid:3) P ( z ′ ) . (37) Second configuration: z ′ > z Let l l = z ′ − z be the distance between the receiver and the source(see Fig. 1), C l l the diagonal matrix constituted by the elements cosh( ik mn l l ), and S l l thediagonal matrix constituted by the elements sinh( ik mn l l ). Relation (24) givesfor z ′ > z : g ( z, z ′ ) = C l l P ( z ′ ) + Z c S l l V ( z ′ ) , then, with P ( z ′ ) = Z − ( z ′ ) V ( z ′ ): g ( z, z ′ ) = S l l (cid:2) S − l C l l + Z c ( Z − ( z ′ )) − (cid:3) P ( z ′ ) . (38)Finally, the Green’s function of a finite duct with an infinite flange is given as: G ( M, M ′ ) = Φ ( r, θ ) Φ ( r ′ , θ ′ ) g ( z, z ′ ) . (39)10 . Application to complex resonance frequencies of a flanged, finitelength duct Resonances of a flanged, finite length duct are interesting as they containimportant information about the coupling between internal and external fluids.Their calculation is based on the fact that the internal pressure becomes infiniteat each resonance. Newton’s method is used to compute the zeros of the inverseof the pressure. Since the resonances of a dissipative problem are complex, acomplex formulation of the impedance radiation is needed. As a time depen-dence exp( − iωt ) has been chosen, the imaginary part needs to be negative forresonance frequencies in order to ensure that the amplitude remains boundedfor all times t . Using the integrals (13) and (16), for ℜ ( k ) > ℑ ( k ) < Z mnl becomes: Z mnl = − ik [ i Z |ℜ ( k ) | τ √ k − τ ˜ D mn ( τ ) ˜ D ml ( τ ) dτ + Z + ∞|ℜ ( k ) | τ √ τ − k ˜ D mn ( τ ) ˜ D ml ( τ ) dτ − Z k |ℜ ( k ) | τ √ τ − k ˜ D mn ( τ ) ˜ D ml ( τ ) dτ ] . (40)This expression is used as a radiation condition Z ray at the end of the finitelength duct. In a first instance, we consider the resonance wavenumbers of the planarduct mode (m=n=0) without the influence of higher order duct modes (l=0).With radiation, the j th resonance wavenumbers of duct mode mn are denoted k jmn,r and denoted k jmn without radiation.In order to validate the complex formulation of the radiation impedance, wecompare the resonances obtained with radiation impedance given by relation(40) for m = n = 0 and l = 0, with the resonances obtained with the radiationimpedance of a flanged plane piston given by Rayleigh’s formulation as (see Ref.[14] p.1458): Z ≃ ρc [1 − kb J (2 kb ) − ikb S (2 kb )] (41)The radiation impedance for m=n=l=0 calculated with relation (40) andthat of a flanged plane piston (41) are identical. Thus, resonance wavenumberscalculated with these two formulations are so identical, as observed in Fig. 3.It is worth noting in Table 1 that without radiation, the resonance frequen-cies are those obtained by the usual longitudinal resonances of a cylindrical ductwith one side ”closed” and the other side ”open” (Neumann/Dirichlet problem)for: k j = (2 j + 1) π L , j = 0 , , , ...
20 40 60 80 100 120 140−2.5−2−1.5−1−0.50 ℜ (k) ℑ ( k ) Figure 3: Evolution of resonance wavenumbers k j ,r depending on wavenumber k (with b =0 . m and L = 1 m ) with the radiation impedance defined by Eq. (40) for m = n = 0 and l = 0 (+) and with the radiation impedance of a flanged plane piston (Eq. 41) ( (cid:3) ). We can observe in Table 1 of Appendix B that the real part of resonance fre-quency decreases when radiation is taken into account: this is normal behaviorbecause the reactive effect of radiation can roughly be described as an increasein the duct length. The semi-infinite duct length correction is estimated by thefollowing formula (see Ref. [10] with the modification given in private commu-nication): △ Lb = 0 . kb . kb . + ( kb . ) . (42)The difference between the real part of the j first resonance wavenumbers k j ,r ofa finite radiating cylindrical duct (below the first cut off wavenumber k cut =3 . /b ) and those estimated using the length correction defined by Eq. (42)( k j , △ L = (2 j + 1) π/ [2( L + △ L )]), tends to zero when the ratio L/b increases.The values agree well ( ≤
1% of difference) for
L/b ≥ L . The principal interest of the complex Zorumski’s formulation is that wecan observe the influence of higher order duct modes (also denoted by H.Mafterward). In this paper, we taken into account only the axisymmetric modes( m = 0) (but the non-axisymmetric are very easy to calculate with the samemethod). Figure 4 shows resonance wavenumbers k j ,r , k j ,r , k j ,r . We observethree series of resonances. Each series starts at the cutoff frequency of a ductmode. The first one corresponds to a domination of the planar duct mode, thesecond one to a domination of duct mode 01 and the third one to a dominationof duct mode 02 (see Appendix B, the Green’s function profile is plotted around k and k ). Fig. 5 shows the influence of the two first higher order duct modes( m = 0 , l = 1 and m = 0 , l = 2) on the resonance wavenumbers of the first series.It is worth noting that below the first cut off wavenumber k cut = 3 . /b , only12ne higher order duct mode is sufficient to accurately describe the resonances(see some values in Table 1); between the first and second cut off wavenumber k cut = 7 . /b , only two higher order duct modes are enough and similarly tothe higher order duct modes: between the n th and ( n + 1) th cut off, only ( n + 1)higher order duct modes are enough. ℜ (k) ℑ ( k ) k k Figure 4: Resonance wavenumbers k j ,r (+), k j ,r ( △ ), k j ,r ( ∗ ) for L = 1 m and b = 0 . m .Figure B.8 in Appendix C shows the mode profiles around the two wavenumbers indicated byan arrow. ℜ (k) ℑ ( k ) Figure 5: Resonance wavenumbers k j ,r of the first series ( m = 0, n = 0) with an influence of0 ( l = 0: ∗ ), 1 ( l = 1: △ ) and 2 ( l = 2: +) H.M for L = 1 m and b = 0 . m k j k j ,r with 0 H.M k j ,r with 1 H.M0 1.5708 1.449 - 0.0095i 1.451 - 0.0096i1 4.712 4.369-0.077i 4.375-0.0776i2 7.854 7.333-0.182i 7.345-0.183i3 10.996 10.336-0.296i 10.345-0.299i4 14.137 13.365-0.412i 13.4-0.415i5 17.279 16.409-0.526i 16.45-0.53i6 20.42 19.463-0.645i 19.523-0.644i7 23.562 22.523-0.773i 22.608-0.76i8 26.7 25.59 -0.924i 25.707-0.881i9 29.85 28.674-1.12i 28.824-1.01i10 32.99 31.822-1.408i 31.965-1.152i11 36.13 35.377-1.711i 35.149-1.317i12 39.27 38.747-1.409i 38.395-1.579i Table 1: Values of the j first resonance wavenumbers without radiation ( k j ) and with radi-ation ( k j ,r ) for 0 and 1 H.M, with b = 0 . m and L = 1 m . j first resonance wavenumbers with respect to radiation In the present section, we show the evolution of the j first resonance wavenum-bers when only the planar duct mode propagates with respect to radiation andas shown in the previous section, we take into account the effect of one higherorder duct mode. For this purpose, we introduce a multiplicative coefficient onthe radiation impedance. Physically, this coefficient η ρ can be regarded as theratio between the external fluid density ρ ext and the internal fluid density ρ int ,such as: η ρ = ρ ext ρ int , (43)the density of the external fluid ρ ext varying from a vacuum to water density,the sound celerity, 340 m.s − , remaining constant.Figures 6 and 7 show that when the parameter η ρ increases, a Neumann/Neumannproblem is obtained: the real part tends to jπ/L and the imaginary part tendsto zero. This behavior corresponds to a system without losses, the external fluidbecoming a perfectly reflective surface.Fig. 7 shows that the absolute value of the imaginary part of the resonancefrequency goes through a maximum, corresponding to the maximum of the ra-diation. Similarly results have been observed for non planar modes. So, we canconclude that energy radiation losses evolves with the densities of internal andexternal fluid and goes through a maximum for a specific densities ratio.
5. Conclusion
A development of the Green’s function for the Helmholtz equation in a freespace valid for complex frequencies is possible and leads to a new formula in Zo-14 −4 −2 η ρ ℜ ( k , r j ) j = 0j = 1j = 2j = 3j = 4j = 5 Figure 6: Evolution of the real part of the j first longitudinal resonance wavenumbers k j ,r with respect to η ρ , for L = 1 m and b = 0 . m . −4 −2 −0.7−0.6−0.5−0.4−0.3−0.2−0.10 η ρ ℑ ( k , r j ) j = 1j = 2j = 3j = 4j = 5 j = 0 Figure 7: Evolution of the imaginary part of the j first longitudinal resonance wavenumbers k j ,r with respect to η ρ , for L = 1 m and b = 0 . m . rumski’s radiation impedance. The interest has been shown for an applicationexample, dedicated to the calculations of the complex resonance frequencies ofa radiating flanged cylindrical duct. It has been shown that length correctioncalculated for a semi-infinite duct is a good estimate of a finite duct radiationwhen the ratio L/b ≥ n th and ( n + 1) th cut off, only ( n + 1) higher order duct modes areenough. In the last part, it has been observed a maximum of radiation for aspecific densities ratio. In the future, it will be interesting to use a BEM methodto study more complicated geometries and to observe the resonances with anexperimental method. This work is a first step to study the relation between15adiation and several parameters in order to optimize geometry for minimizing(e.g. for noise pollution) or maximizing the sound radiation (e.g. for wind in-struments). Acknowledgements
The authors wish to thank P. Herzog and F. Silva for their useful discussionpoints.
Appendix A. Matricial calculation
The main steps required to obtain the results of section 3.2 are presentedhere (see Ref. [17]).For a cylinder, the general solutions at point z can be described with respectto the values of P and V at point z such as described by relation (24). For allthe modes mn , the following matrix problem is obtained: P ( z ) = CP ( z ) + Z c SV ( z ) , (A.1) V ( z ) = Z − SP ( z ) + CV ( z ) , (A.2) C being a diagonal matrix constituted by the elements cosh( ik mn ( z − z )) et S a diagonal matrix constituted by the elements sinh( ik mn ( z − z )). Eq. (A.2)implies: P ( z ) = Z c S − V ( z ) − Z c S − CV ( z ) . (A.3)Introducing Eq. (A.3) in Eq. (A.1) and using the commutativity of thediagonal matrices, we obtain: P ( z ) = Z c S − CV ( z ) − [ Z c S − CC − Z c S ] V ( z ) , with Z c S − CC − Z c S = Z c S − ( I + SS ) − Z c S = Z c S − + Z c S − Z c S = Z c S − , thus: P ( z ) = Z c S − CV ( z ) − Z c S − V ( z ) . (A.4)Therefore, with Eqs. (A.3) and (A.4), we obtain the relation (26).16 ppendix B. Green’s function profile in the duct around resonancefrequencies Figure B.8 shows that around the resonance frequency k , the profile ofGreen’s function corresponds to the profile of the planar duct mode even if ductmode 01 is propagating and similarly around the resonance frequency k , theprofile of Green’s function corresponds to the profile of the first non planarduct mode even if planar duct mode is propagating. The same comportment isobserved around other resonance frequencies. Therefore, it is worth noting thateach series observed in Fig. 4 corresponds to a predominant duct mode evenif other duct modes are propagating. Notice that the evanescent duct modesexist mainly near to the source (at z = − . −0.1 0 0.1−1−0.8−0.6−0.4−0.20−202 x 10 bm=0 and n=3,l=3 with k=44.63−2.246i z ℜ ( G ( M , M ’ , z )) −0.1 0 0.1−1−0.8−0.6−0.4−0.20−505 x 10 bzm=0 and n=3,l=3 with k=44.65−0.235i ℜ ( G ( M , M ’ , z )) Figure B.8: Profile in the duct of the real part of Green’s function around k with 3 H.M(upper figure) and around k with 3 H.M (lower figure)with 3 H.M (lower figure)