Sarabjeet Singh (1), Colin H. Hansen (2) and Carl Q. Howard (2)

Transcription

1 Acoustics 8 Geeong, Victoria, Austraia 4 to 6 November 8 Acoustics and ustainabiity: How shoud acoustics adapt to meet future demands? A detaied tutoria for evauating in-duct net acoustic power transmission in a circuar duct with an attached cyindrica Hemhotz resonator using method arabjeet ingh (1), Coin H. Hansen () and Car Q. Howard () (1) Vipac Engineers and cientists Ltd., PO Box 419, Kent Town, A 571, Austraia () choo of Mechanica Engineering, The University of Adeaide, A 5, Austraia ABTRACT The method has been extensivey expored by many researchers for anaysing acoustic duct systems. However, there does not exist a comprehensive source that eaborates the use of method for evauating the performance of acoustic fiters. The current approach describes a detaied step-by-step method for evauating induct net acoustic power transmission for a harmonic pane wave traveing in a circuar duct with an attached cyindrica Hemhotz resonator using the method. The net acoustic power transmission is evauated using two different methods: (1) the product of acoustic pressure and acoustic voume veocity at the duct exit, and () estimates of the in-duct sound fied using the two-microphone technique. INTRODUCTION The method (aso known as transmission matrix or four-poe parameter presentation) has been widey accepted as a too for anaysing compex systems due to its computationa efficiency and fexibiity. Many previous researchers (Munja 1987, Pierce 1989, nowdon 1971, and To and Doige 1978) have used the method for anaysing acoustic duct systems. Athough many research papers, in addition to standard acoustic text books (Beranek and Ver 199, Kinser et a. 199, and Pierce 1989), provide a brief overview of the method and its associated uses, they do not incorporate detaied anayses of a particuar system. This paper describes in detai a step-by-step method for approximating the downstream net acoustic power transmission for a harmonic pane wave traveing inside a circuar duct to which a cyindrica Hemhotz resonator (HR) is attached. It begins with a description of the of a uniform circuar duct which incudes a the steps and pertinent cacuations in order to estimate the in-duct net acoustic power transmission. Then, the buiding of the of a couped duct-hr system is described and the net acoustic power transmission inside the duct downstream of the HR is approximated. The net acoustic power transmission inside the duct was estimated by using the estimates of acoustic pressure and acoustic voume veocity at the terminating end of the duct. However, in addition to the above stated method, net acoustic power transmission inside the duct was aso estimated using the two-microphone method deveoped by Chung and Baser (Chung and Baser 198) and extended by Åbom (Åbom 1989). The purpose of estimating the acoustic power transmission using the atter method is to incorporate additiona detaied cacuations using the method. Furthermore, the atter method provides a means to verify the estimations of the former method. UNIFORM CIRCULAR DUCT Figure 1 shows a schematic of a circuar duct of (physica) ength, radius a, and cross-sectiona area. v p s piston-driven (source) end open end radiating into free space x x v p Figure 1. A schematic of a uniform circuar duct modeed as being driven by a constant ampitude piston source mounted at one (eft) end and open at the other (right). Acoustics 8 1

2 4-6 November 8, Geeong, Austraia Proceedings of ACOUTIC 8 Aso shown at the end of the duct is the end-correction factor,, which wi be discussed ater. The eft end of the duct, referred to as a source end, was modeed as being driven by a constant ampitude piston with a unit acoustic voume veocity, and the right end of the duct, referred to as a duct exit, was modeed as an open end radiating into free space. Pane Wave Assumption As the method works on a principe of pane wave propagation inside the duct, it is important to estimate the cut-on frequency, which is defined as the frequency beow which ony pane waves propagate inside the duct. The cut-on frequency is given by: 1.841c f c (1) πa where, c is the speed of sound in air (343 m/s at C), and a is the radius of the duct. For the current paper, the dimensions of the circuar duct were taken as: diameter.1555 m and ength 3 m. The cut-on frequency for the above dimensioned duct is 19 Hz. Therefore, the pane wave mode is the ony mode which propagates beow 19 Hz. Radiation Impedance For the purpose of modeing the end of the duct as open and radiating into free space, the cacuation of radiation impedance is required. It is approximated by assuming a circuar piston ocated in an infinite baffe, and is the ratio of the force exerted by the piston on the acoustic fied to the veocity of the piston. Radiation impedance is given by (Beranek and Ver 199, Bies and Hansen 3, Pierce 1989, Kinser et a. 198, Munja 1987): r R + jx () The rea part of radiation impedance, R, is termed as radiation reactance and represents the energy radiated away from the open end in the form of sound waves. The imaginary part, X, is termed as radiation admittance and represents the mass oading of the fuid (air) just outside the open end. It is we known that the theoretica expression for the impedance of an unfanged open duct, with pane waves propagating inside it, is given by (Beranek and Ver 199, Bies and Hansen 3, Pierce 1989, Kinser et a. 198, Munja 1987): ( ka) + j(. 6) ka 4 where, is the radiation impedance, ρ is the density of the fuid medium, c is the speed of sound, k is the wave number, and a is the radius of the duct. End-Correction The term end-correction refers to an additiona ength to the actua ength of a tube or an orifice that accounts for the entrained mass of the fuid that vibrates outside the tube s or orifice s opening. The expression for the end-correction of an unfanged open end of a duct radiating into free space can be found in standard acoustica text books (Beranek and Ver 199, Bies and Hansen 3, Kinser et a. 199, and Munja 1987). It is given by: (3) where, a is the radius of the duct. Transfer Matrix of a Circuar Duct The of a circuar duct of uniform crosssectiona area,, and ength,, is given by (Munja 1987): p cos r j qr sin Yr ( k ˆ ) jy ( ˆ ) r sin k pr 1 ( k ˆ ) cos( k ˆ ) q r 1 The subscript r impies that the duct is defined/denoted as an c eement r. The quantity Y r is the characteristic impedance, and k ˆ k( 1+ iη) is the compex wave number. Com- pex wave numbers are introduced in order to convenienty incorporate damping in the system, even though this formuation impies hysteretic damping when in fact the system is better described as viscousy damped. The quantity η is the oss factor, which is equivaent to twice the critica damping ratio for a viscous system. The quantities, p r, p r-1 and q r, q r-1 represent the acoustic pressures and acoustic mass veocities at the extreme ends of the duct, respectivey (input and output sides). These quantities can be r.m.s vaues, ampitudes or instantaneous, provided they are a the same type. Rewriting equation (5) by reating acoustic voume veocity and acoustic pressure instead of acoustic mass veocity and acoustic pressure, respectivey gives: v cos r pr j sin ( ˆ k) j sin( k ˆ ) ( k ˆ ) cos( k ˆ ) vr 1 pr 1 where, v r and v r-1 represent acoustic voume veocities at the input and output sides of the duct (eement r), respectivey. The above shown transition in equation (6) affected the definition of the eements of the compared to those defined by Munja (1987) (in equation (5)). A pivota reason for impementing this transition is to enabe a quick comparison between theoretica predictions and experimenta resuts. Athough no experimenta resuts are discussed in this paper, in most of the reevant experimenta investigations reported in the iterature, a oudspeaker backed by a sma air-tight cavity, as an acoustic source, is attached to one end of the duct. And, generay, sound pressure eve measurements inside the duct are normaised by the acoustic voume veocity of the oudspeaker. Therefore, from a practica point of view, a reation between acoustic voume veocity and acoustic pressure is much preferred over a reation between acoustic mass veocity and acoustic pressure. Athough acoustic mass veocity is directy proportiona to acoustic partice veocity which, in turn, is directy proportiona to acoustic voume veocity, the equations shown in this paper eiminate the need to derive a reationship between acoustic voume veocity and pressure. As indicated in figure 1, v and p are acoustic voume veocity and pressure, respectivey, at the input side of the duct. imiary, v and p have the same definition at the output side. Reating these state variabes gives the resutant transfer matrix of the duct, which is described beow: (5) (6). 6a (4) Acoustics 8

3 Proceedings of ACOUTIC 8 v cos j sin ( k ˆ eff ) j sin( k ˆ eff ) ( k ˆ ) cos( k ˆ ) eff eff v p A very important point which needs to be considered is the end-correction factor of the unfanged open end of the duct. Whie estimating the eements of the, the endcorrection factor,, must be added to the physica ength of the duct. Therefore, in equation (7), eff respresents the effective ength of the duct, which corresponds to the sum of physica duct ength,, and the end-correction factor,. In order to make the evauation of equation (7) easier for reevant future cacuations, the eements of the above described were assigned aphabetica characters as shown beow: A C ( ) B v D p where, A D cos kˆ ( ), B ˆ j sin k( + ) + ( ) c ρ. and C ˆ j sin k( + ) (7) (8) ( ), ρ c As described earier, the in-duct net acoustic power transmission is cacuated using two different methods, (1) the product of acoustic pressure and veocity at the duct exit, and () the two-micrphone method. These methods are described in the foowing text. METHOD 1: PRODUCT OF ACOUTIC PREURE AND VOLUME VELOCITY AT THE DUCT EXIT The equations shown beow describe the procedure to estimate the acoustic pressure and voume veocity at the duct exit. Estimation of these two variabes wi in turn faciitate the estimation of in-duct net acoustic power transmission. From equation (8), we get v A v + B p (9) p By using, equation (9) can be written as: v p v A + B p (1) oving equation (1) for p with respect to v gives p v A + B (11) Equation (11) gives the estimate of the ratio of acoustic pressure p at the duct exit to the input voume veocity v. is the radiation impedance of an unfanged open end of the duct and can be cacuated using equation (3). 4-6 November 8, Geeong, Austraia B v A B + v A (1) Equation (1) gives the measure of the ratio of acoustic voume veocity v at the duct exit to the input voume veocity v. If v is set to unity (1 m 3 /s), then p and v woud be p (13) A + B B A + B v A (14) Equations (13) and (14) represent the estimates of acoustic pressure and acoustic voume veocity, respectivey, at the duct exit; the duct being driven by a constant ampitude piston at the source end with a unit voume veocity. In-Duct Net Acoustic Power Transmission The net acoustic power fux associated with a harmonic pane wave, which takes into account the pertinent incident and refected panes waves, traveing inside a duct is given by (Fahy, 1995): 1 W * net Re p v (15) where, p and v represent the compex ampitudes of pressure and voume veocity, respectivey, and the * indicates the compex conjugate. This expression can be written as: 1 W net PV cos( φ) (16) where, P and V are the scaar ampitudes that are the modui of the compex ampitudes of the pressure and voume veocity, respectivey shown in equation (15), and φ is the phase ange between them. Equation (16) represents the net acoustic power transmission inside the duct as the product of acoustic pressure and voume veocity ampitudes. ubstituting the vaues of equations (13) and (14) in equation (16) provides the estimate of net acoustic power transmission in the duct. Assuming that the anaysis in the preceding sections used ampitude quantities, equation (16) cacuates the estimate of the net acoustic power transmission in the duct when the fuid present inside the duct is assumed to be driven by a constant ampitude harmonic piston source with a unit voume veocity. Figure shows a pot of the net acoustic power transmission in the uniform circuar duct of diameter equas.1555 m and ength equas 3 m. imiary, soving equation (9) for v with respect to v, which incorporates the use of equation (11), gives: Acoustics 8 3

4 4-6 November 8, Geeong, Austraia Proceedings of ACOUTIC 8 Acoustic Impedance at Arbitrary Location Inside a Duct From standard acoustic text books (Beranek and Ver 199, Bies and Hansen 3, Kinser et a. 199, and Pierce 1989), the specific acoustic impedance at any ocation inside a duct can be expressed as x jkx + jkx c a+ exp + a exp jkx + jkx a+ exp a exp ρ (17) where, a + and a - are the moda ampitudes of incident and refected acoustic waves, respectivey, the - and + signs represent the propagation of acoustic wave in +ve and ve x directions, respectivey, and x is the ocation aong the duct. Figure. Net acoustic power transmission as a function of harmonic frequency in a uniform circuar duct driven by a constant ampitude harmonic piston source (of unit acoustic voume veocity) mounted at one (eft) end and open at the other (right). Athough the above discussed method showed the equations and procedure required to estimate the in-duct net acoustic power transmission, the use of the method is better iustrated in the second anaysis method to foow. METHOD : TWO-MICROPHONE TECHNIQUE The two-microphone technique was deveoped by Chung and Baser (Chung and Baser 198) and requires an estimate of acoustic pressure at two different ocations inside a duct. The foowing paragraphs detai the procedure for estimating the acoustic pressure at different ocations inside a duct. Acoustic Pressure and Veocity at Arbitrary Locations Inside a Duct Figure 3 shows a schematic of a uniform duct of ength depicting the end-correction,, at the open end of the duct. v piston-driven (source) end v x open end radiating into free space v Using equation (17), the vaues of acoustic impedance at ocations x and x are given by: a+ + a (18) a+ a jk + jk c a+ exp + a exp jk + jk a+ exp a exp ρ (19) Eiminating constants a + and a - faciitates a reationship between and, which is given by: + j tan( k) () 1+ j tan( k) If the impedance at a duct termination is known, the vaue of the acoustic impedance at any point inside the duct can be cacuated using equation (). In order to distinguish between the source and radiation impedance, equation () was rewritten with a sight modification in terms of nomencature. Denoting the acoustic impedance at x by s (source impedance) and at x by r (radiation impedance), gives: r + j tan( k) s (1) 1+ j r tan( k) p p x x p Finay, the vaue of acoustic impedance at ocation x ooking into the duct from the source end (x ) can be given by (utiising equation (1)): x -x x x Figure 3. A schematic of a uniform duct depicting ocation x aong the duct where acoustic pressure and acoustic voume veocity need to be estimated. uppose that acoustic pressure and acoustic voume veocity at ocation x need to be estimated, and are denoted by p x and v x, respectivey. As evident from equations (11) and (1), the estimation of acoustic pressure and voume veocity at a particuar ocation requires the corresponding vaue of acoustic impedance at that particuar ocation. Therefore, estimation of acoustic pressure and voume veocity at ocation x requires estimating the acoustic impedance at ocation x, which is detaied beow. s j tan( kx) x () j s tan( kx) The aternate expression for cacuating the vaue of x, by using the vaue of r instead of s, is given by: r + j tan x 1+ j s tan ( k( x) ) ( k( x) ) (3) Equation (3) shows the acoustic impedance at ocation x ooking into the duct from the open end (x ). 4 Acoustics 8

5 Proceedings of ACOUTIC 8 End-Corrections Before using equations () and (3) for cacuating the vaue of x, the end-correction of an unfanged open end of a duct must be accounted. After incorporating the vaue of the end-correction in equations () and (3), x is expressed as: s j tan x j s tan r + j tan x 1+ j r tan ( k( x )) ( k( x )) ( k( x + )) ( k( x + )) (4) (5) Once the vaue of acoustic impedance at a particuar ocation in the duct is known, the acoustic pressure and veocity can be cacuated using equations (11) and (1), respectivey. The acoustic pressure and voume veocity at ocation x with respect to unit voume veocity, v, are given by: p x x (6) Ax + Bx x Bx x Ax + Bx v x x Ax where, ( kx ˆ cos ) and j sin( kx ˆ ) A x B x. (7) Knowedge of the acoustic pressure at two appropriatey spaced different ocations inside a duct estimates the in-duct net acoustic power transmission by using the decomposition of sound fied which is discussed beow. Decomposition of the ound Fied Decomposition is a technique for determining the ampitudes of the acoustic waves propagating each way inside a duct. It faciitates the separation of modes into incident and refected parts. Once the acoustic power associated with the incident and refected waves is determined, the net acoustic power transmission is simpy their difference. Åbom (Åbom 1989) extended the two-microphone technique deveoped by Chung and Baser (Chung and Baser 198) and documented the scheme for in-duct moda decomposition. However, in this paper we are not concerned with higher order modes and we are ony decomposing the sound fied into right-traveing and eft-traveing pane waves. Generay, for a uniform straight duct with rigid was and an arbitrary cross-sectiona shape carrying an axia mean fow, the sound fied in the duct can be expressed as: n ik x ik x [ aˆ n + + a n + exp + ˆ exp ] Ψ n pˆ (8) where, pˆ is the acoustic pressure, aˆ +, aˆ are the moda ampitudes of acoustic pressure associated with the incident (+ve x direction) and refected waves (-ve x direction), respectivey, k ˆ + n, k ˆ n are the axia wave numbers in the positive and 4-6 November 8, Geeong, Austraia negative directions, respectivey, n is the moda number, and Ψ is the eigenfunction for mode n. n For the case of a uniform circuar duct with rigid-was and carrying no axia fow, with ony pane waves propagating inside it, equation (8) can be written as: ikx ikx pˆ + aˆ + exp + aˆ exp (9) The terms aˆ + and aˆ can be cacuated by either estimating or experimentay measuring the acoustic pressure at two ocations in a duct and using the two-microphone technique. Expressing the resuts of such a measurement in a matrix formuation yieds (Åbom 1989): where, pˆ is a [ 1] pˆ Maˆ (3) coumn vector containing the estimates or measures of the acoustic pressures at two different oca- moda matrix containing the propagation tions, M is a [ ] terms, and â is a [ 1] coumn vector containing the unknown moda ampitudes. Writing vector pˆ in terms of transfer function between the two pressure measuring or estimating ocations yieds: pˆ pˆ r H (31) where, pˆ r is the acoustic pressure at one of the two pressure measuring or estimating ocations, and H is the [ 1] coumn vector containing the transfer function between the two ocations. Using equation (31), equation (3) can be written as: oving equation (3) for â gives: aˆ p ˆ rh Maˆ (3) 1 p ˆ M H (33) r aˆ pˆ r + ˆ iks iks (34) a exp exp H1 For this anaysis, pˆ r was taken as the acoustic pressure estimate at the ocation coser to the source end, H 1 represents the pressure transfer function between two pressure estimates, and s is the axia distance between the two pressure estimating ocations. oving equation (34), for the estimates of â+ and â, yieds: + iks ( exp ) pˆ r H aˆ iks iks exp exp iks ( exp ) pˆ r H aˆ 1 + iks iks exp exp (35) (36) The acoustic power associated with the incident and refected waves is expressed as: aˆ + W+ (37) Acoustics 8 5

6 4-6 November 8, Geeong, Austraia Proceedings of ACOUTIC 8 aˆ W (38) The in-duct net acoustic power transmission in x direction is estimated as the difference between W + and W -, and is expressed as: W net W+ W aˆ + aˆ In-Duct Net Acoustic Power Transmission (39) Assuming that the anaysis in the preceding sections used ampitude quantities, equation (39) cacuates the estimate of the net acoustic power transmission in the duct when the fuid present in the duct is assumed to be driven by a constant ampitude harmonic piston source with a unit voume veocity. The net acoustic power estimates from the two mehods described above exacty match with one another. COUPLED DUCT HR YTEM Mounting a resonator on a duct reduces the noise transmission at a particuar frequency which is a characteristic of the resonator s geometry. Comparison of the acoustic power transmission inside the duct without and with the resonator can be used to evauate its performance. In order to deveop the compete of the duct- HR system, it was discretised into three eements: (1) eement - section of the duct upstream of the HR, () eement 1 - section of the duct downstream of the HR, and (3) eement HR - the Hemhotz resonator. Figure 4 shows a schematic of a HR mounted on to a duct of ength at ocation x. The figure aso iustrates the three eements described above, aong with the neck-cavity and neck-duct interfaces. eement HR s p s, v s v HR p s1, v s1 v, p v 1, p 1 neck-cavity interface neck-duct interface eement eement 1 v p r A key issue reated to the theoretica anaysis of a duct-hr system is the incorporation of two end-correction factors for the neck of the HR; (1) the neck-cavity interface endcorrection factor, and () the neck-duct interface end correction factor, both of which must be added to the physica neck ength to get the effective neck ength. The expression for the neck-cavity interface end-correction factor cosey approximates the geometry of a cyindrica piston radiating in a tube. It is given by (eamat and Ji ): r δ e.8r (4) R where, r denotes the radius of the neck of a HR, and R denotes the radius of the cavity of a HR. Another equation which has been widey used in the iterature for the neck-cavity interface end-correction factor was presented by Ingard (Ingard 1953). His equation sighty differs from equation (4) and was for an orifice in an anechoicay terminated duct. However, equation (4) is considered to be better than the equation presented by Ingard and has been used in this current paper. On the other hand, the neck-duct interface end-correction factor has not been modeed anayticay due to the difficuty in interpreting the non-panar sound fied in the region of the duct adjacent to the neck opening. However, Ji (5) derived two-curve fitting expressions for the circuar neck-duct interface end-correction factor based on the ratio of neck and duct diameter using Boundary Eement Anaysis. The corresponding expressions are given by: r r δ i r a a ; r.4 a r r r ; >.4 a a (41) where, r denotes the radius of the neck of a HR, and a denotes the radius of a duct to which a HR is attached. Transfer Matrix of a HR Figure 4 shows a uniform duct with a HR attached as a side branch to one its was at ocation x, aong with the reevant notations required to deveop the compete of the duct-hr system. The compex ampitude of the voume veocity entering the duct, before the HR's ocation, is denoted by v. imiary, it is v 1 in the section after the resonator's ocation and v HR fowing into the HR. From the continuity condition, the voume veocity and pressure are ocay conserved at junction x. Therefore, ( x) v ( x) v ( x) v HR + 1 (4) ( x) p ( x) p ( x) p HR 1 (43) x x x Figure 4. A schematic of a couped duct-hr system depicting its division into three eements aong with neck-duct and neck-cavity interfaces, and reevant notations. End-Corrections of a HR s Neck Putting equations (4) and (43) in a matrix form yieds: vhr + v p 1 vhr v p1 p 1 (44) (45) 6 Acoustics 8

7 Proceedings of ACOUTIC 8 From equation (43), p ( x) p ( x) HR 1 HR and aso, HR vhr p. Here, HR is the acoustic impedance just outside the opening of the resonator, which can be referred to as the input point impedance. Therefore, equation (45) can be re-written as: 1 1 v HR p 1 (46) 1 Finay, the matrix 1 represents the HR 1 of the Hemhotz resonator. Input Point Impedance of a HR The impedance at the opening of the HR, i.e. HR, can be estimated by appying the equation to the neck and cavity of the HR. The neck section of the resonator is denoted by s 1 having cross-sectiona area A n and ength 1, and the cavity section is denoted by s having cross-sectiona area A c and ength. The ength 1 represents the effective ength of the neck, which incudes two end-corrections at each side of the neck, which were described in the previous section beow figure 4 (equations (4) and (41)). Reating the acoustic voume veocity and pressure at the opening of the neck v s1, p s1 (see figure 4) and at the end of the cavity v s, p s (see figure 4), the equation for the Hemhotz resonator is expressed as: or vs1 vs ps1 ps of the neck of the cavity vs1 As1 ps1 Cs1 Bs1 As Ds1 Cs Bsvs Ds ps where, ( ˆ A D cos k ), j sin( kˆ ) C s 1 B s s 1 s1 1 (47) (48) B s 1 1, and j sin( kˆ 1 ). imiary, A ( ˆ s Ds cos k ) sin( ˆ j k ), and j sin( kˆ ). C s, Let us denote the resutant eements of equation (48) as: vs1 As ps1 Cs Bsvs Ds ps (49) As the end of the resonator (cavity) is bocked (rigidy terminated), there wi be no fow at the end of the resonator, hence, v. ubstituting v in equation (49), the s s acoustic pressure and voume veocity at the opening of the neck becomes v B p (5) s1 s s 4-6 November 8, Geeong, Austraia p D p (51) s1 s s Dividing equation (51) by (5), gives the estimate of the input point impedance of the HR, as: ps1 Ds HR vs1 Bs Compete Transfer Matrix of a Duct-HR ystem (5) Referring to figure 4, for the convenience of presenting the compete of the duct-hr system, et us denote: (1) eement, the first section of the duct upstream of the HR, by subscript, and () eement 1, the second section of the duct downstream of the HR, by subscript 1. The compete method of the duct-hr system shown in figure 4 can be expressed as: v p of eement of eement HR of eement 1 A C B 1 D 1 A1 HR C 1 1 B1 v D1 p (53) (54) Let the resutant of matrix mutipication of equation (54) be denoted by: A C B v D p (55) In-Duct Net Acoustic Power Transmission Downstream of the HR The dimensions of the duct were kept same as were considered in the anaysis of the uniform circuar duct (diameter of.1555 m and ength of 3 m). The dimensions of the cyindrica Hemhotz resonator were: cavity diameter.131 m, cavity ength.7 m, neck diameter.55 m, and physica neck ength.93 m. The HR was mounted at a distance of.5 m from the source end of the duct. Method 1: Product of the acoustic pressure and voume veocity at the duct exit. The transfer matrices of three eements of the duct-hr system shown in figure 4, eement, eement 1, and eement HR, were written using equation (54) as per the dimensions of the duct-hr system described in the immediate preceding paragraph. The acoustic pressure and acoustic voume veocity at the duct exit (open end radiating into free space), x, were estimated in the simiar way as described in equations (11) and (1), and are given by: p v A + B B v A + B v A (56) (57) Acoustics 8 7

8 4-6 November 8, Geeong, Austraia Proceedings of ACOUTIC 8 where, p and v represent the acoustic pressure and voume veocity at the duct exit, v denotes the input voume veocity, is the radiation impedance at the unfanged open end of the duct (aso denoted by r ), and A, B, C and D are the eements of the resutant compete (equation (55)). Mutipying the estimates of acoustic pressure (equation (56)) and acoustic voume veocity (equation (57)) at the duct exit as per equation (16) resuts in the net acoustic power transmission inside the duct downstream of the HR. Method : Two-Microphone Decompostion of ound Fied The net acoustic power transmission inside the duct downstream of the HR was aso estimated by using the twomicrophone in-duct decomposition method and exacty the same resuts as for Method 1 were obtained. Figure 5 shows a pot of the net acoustic power transmission in the duct downstream of the HR. For comparison purposes, this figure aso incudes the pot of the net acoustic power transmission in the duct without the resonator. factors of the neck of the HR in addition to the actua dimensions of the duct-hr system. The neck-cavity interface endcorrection factor is we documented. However, there exists some uncertainty concerning the end-correction factor at the neck-duct interface, due to the difficuty in anayticay modeing the compex sound fied in the regions of the duct adjacent to the neck opening. However, empirica equations have been formuated by Ji (5). Even though Ji found that his empirica expressions agreed with his experimenta resuts, in another study (ingh et a. 6 and ingh 6), it was observed that the experimenta performance of a duct mounted HR did not match the corresponding theoretica estimations based on the empirica end-correction factor reported by Ji. ingh (6) deveoped and soved a three-dimensiona finite eement mode of the above described duct-hr system using the software package ANY. Unike the theoretica anaysis of a duct-hr system in which the end-correction factors have to be cacuated, ANY automaticay determines and incorporates the end-correction factors during its soution phase, and therefore, ANY resuts are considered more reiabe than the theoretica cacuations which differ sighty from the ANY resuts. In order to further identify the vaidity of the ANY resuts, experiments were conducted, and the experimentay measured performance of the HR mounted on to a duct was found to agree with the ANY resuts. The difficuty in accuratey estimating the neck duct interface end correction factor compromises the abiity of the transfer matrix method to provide an accurate performance estimation of the duct mounted HR. Figure 5. Net acoustic power transmission in a uniform circuar duct with a HR mounted on to it at a distance of.5 m from the soure end; the duct being driven by a constant ampitude harmonic piston source mounted at one (eft) end and open at the other (right). DICUION The effect of mounting the HR on to the duct, as evident from Figure 5, minimised the net acoustic power transmission by db at 4 Hz. Generay, a frequency at which the maximum reduction of in-duct net acoustic power transmission downstream of a HR occurs is considered to be the resonance frequency of the HR, which is 4 Hz in the current anaysis. However, when a HR is mounted on to a duct, a couped system is created whose resonance frequency is different to that of the HR as a stand-aone device. Therefore, the frequency at which the maximum reduction of in-duct net acoustic power transmission downstream of a HR occurs does not correspond to the resonance frequency of the HR (ingh et a. 6 and ingh 6). The acoustic performance obtained using the method for the circuar duct-cyindrica HR system shoud ony be considered as approximate. This is because of the duct-hr system reated imitations of the method, which are discussed beow. DUCT-HR YTEM RELATED LIMITATION OF THE TRANFER MATRIX METHOD The key issue in the deveopment of the duct-hr system is the incorporation of the two end-correction Aso, as of the magnitude of the damping that occurs in a practica duct-hr system cannot be accuratey modeed, the method with damping excuded can resut in unreaisticay high estimates of acoustic power reduction. However, in this paper, damping has been accounted for in the method by using the vaue of oss factor, η, equa to.4. CONCLUION A detaied method for evauating the acoustic performance of a cyindrica Hemhotz resonator mounted on to a uniform circuar duct using the method has been presented. The in-duct net acoustic power transmission was estimated using two different methods: (1) product of the acoustic pressure and acoustic voume veocity at the openended duct exit, and () in-duct decomposition of sound fied. The net acoustic power resuts from both methods exacty matched each other. The method for the described duct-hr system appication ony provides a rough estimate of the acoustic performance of the duct mounted HR in terms of the frequency corresponding to the maximum net acoustic power reduction and the amount of net acoustic power reduction. The imitations corresponding to the duct-hr system are aso described. REFERENCE Åbom, M. 1989, Moda decomposition in ducts based on transfer function measurements between microphone pairs, Journa of ound and Vibration, 135(1), Beranek, L.L. and Ver, I.L. 199, Noise and Vibration Contro Engineering, John Wiey and ons. Bies, D.A. and Hansen, C.H. 3, Engineering Noise Contro, pon Press. 8 Acoustics 8

9 Proceedings of ACOUTIC November 8, Geeong, Austraia Chung, J.Y. and Baser, D.A. 198, Transfer function method for measuring in-duct acoustic properties. I theory, II experiments, Journa of Acoustica ociety of America, 68(3), Fahy, F.J. 1995, ound Intensity, E&F pon, econd Edition. Ingard, U. 1953, On the theory and design of acoustic resonators, Journa of Acoustica ociety of America, 5(6), Ji,.L. 5, Acoustic ength correction of a cosed cyindrica side-branched tube, Journa of ound and Vibration, 83, Kinser, L.E., Frey, A.R., Coppens, A.B. and anders, J.V. 198, Fundamentas of Acoustics, John Wiey and ons. Munja, M.L. 1987, Acoustics of Ducts and Muffers, John Wiey and ons. Pierce, A.D. 1989, Acoustics, The Acoustica ociety of America. eamat, A. and Ji,.L., Circuar asymmetric Hemhotz resonators, Journa of the Acoustica ociety of America, 17 (5), ingh,., Hansen, C.H. and Howard, C.Q. 6, The eusive cost function for tuning adaptive Hemhotz resonators, Proceedings of the Austraian Acoustica ociety Conference 6, Christchurch, New eaand ingh,. 6, Tona noise attenuation in ducts by optimising adaptive Hemhotz resonators, Masters Thesis, The University of Adeaide, Austraia. nowdon, J.C. 1971, Mechanica four poe parameters and thier appications, Journa of ound and Vibration, 15(3), To, C.W.. and Doige, A.G. 1978, A transient testing technique for the determination of matrix parameters of acoustic systems. I theory and principes, II experimenta procedures and resuts, Journa of ound and Vibration, 6(), 7-3. Acoustics 8 9