Helmholtz Resonator with Extended Neck

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Helmholtz resonator with extended necka) Ahmet Selametb) and Iljae Lee Department of Mechanical Engineering and The Center for Automotive Research, The Ohio State University, Columbus, Ohio 43210-1107 ͑Received 23 August 2002; revised 3 January 2003; accepted 13 January 2003͒ Acoustic performance of a concentric circular Helmholtz resonator with an extended neck is investigated theoretically, numerically, and experimentally. The effect of length and shape of, and the perforations on the neck extension is examined on the resonance frequency and the transmission loss. A two-dimensional analytical method is developed for an extended neck with constant cross-sectional area, while a three-dimensional boundary element method is applied for the variable area and perforated extension. Lumped and one-dimensional approaches are also included to illustrate the effect of the higher order modes. For a piston-driven model, predicted resonance frequencies using lumped, one-dimensional, and two-dimensional analytical methods are compared with those from multidimensional boundary element method. Analytical and computational transmission loss predictions for pipe-mounted model are compared to the experimental data obtained from an impedance tube setup. It is shown that the resonance frequency may be controlled by the length, shape, and perforation porosity of the extended neck without changing the cavity volume. © 2003 Acoustical Society of America. ͓DOI: 10.1121/1.1558379͔ PACS numbers: 43.50.Gf ͓ANN͔

LIST OF SYMBOLS u p Piston velocity V Cavity volume a1 Neck radius c a2 Cavity radius x1 , x2 , x3 Coordinates An Modal amplitudes in domain I x, y, z Cartesian coordinates Bn Modal amplitudes in domain II Y m Bessel function of the second kind and order m Cn Modal amplitudes in domain III ZH Acoustic impedance of a Helmholtz resonator c Speed of sound ␣ ␣ ϭ 0 n Roots of J1( n) 0 f r Resonance frequency ␤ ͑ ͒ n Roots of Eq. 9 G Green’s function ⌫ Acoustic domain boundary J Bessel function of the ﬁrst kind and order m m ␦ End correction k Wave number n ␳ Density of air k Wave number of mode ͑0,0͒ 0 0 ␴ k Wave number of mode ͑0,n͒ Duct porosity n ␺ ᐉ n Eigenfunctions 1 Base neck length ᐉ ␻ Angular velocity 2 Neck extension length ᐉ n Total neck length Superscript ᐉ c Cavity length ϩ ᐉ ᐉ Ϫᐉ Traveling in the positive direction 3 c 2 Ϫ n Unit normal vector in the outward direction Traveling in the negative direction P Acoustic pressure i Inlet o Outlet q1 , q2 Points in domains r Coordinate r Rigid wall S Cross-sectional area of acoustic domains Subscript Sn Neck cross-sectional area A, I Domain I S p Pipe cross-sectional area T Impedance matrix B, II Domain II Tij Transfer matrix elements C, III Domain III TL Transmission loss c Cavity u Acoustic velocity M Main duct I. INTRODUCTION combination of cavity and neck and their relative orientation. Helmholtz resonator is an effective acoustic attenuation The classical lumped analysis of this attenuator gives the ϭ ␲ ͱ ᐉ ϩ␦ device at low frequencies with its resonance dictated by the resonance frequency as f r (c0/2 ) Sn /͕Vc( n n)͖, where c0 is speed of sound, Sn the neck cross-sectional area, a͒A preliminary version of this manuscript has been presented at the 142nd ᐉ ␦ Vc the resonator volume, n the neck length, and n the end meeting of the Acoustical Society of America at Fort Lauderdale, FL, 3–7 December 2001. correction to account for higher modes excited at the discon- b͒Electronic mail: [email protected] tinuities, which can be determined by the geometry and

J. Acoust. Soc. Am. 113 (4), Pt. 1, April 2003 0001-4966/2003/113(4)/1975/11/$19.00 © 2003 Acoustical Society of America 1975 location of the neck relative to the volume and main duct.1,2 Ingard1 investigated the effect of neck geometry, such as cross-sectional area shape, location, and size, on the resonance frequency of a Helmholtz resonator with circular or rectangular cross-sectional area for volume. He developed end corrections for both single and double holes to account for the higher order modes at the interface between neck and cavity. Chanaud2 examined the effect of both orifice and cavity geometry on the resonance frequency of a Helmholtz resonator using Ingard’s end correction. The effect of the depth and width for variable and fixed volumes was studied. He presented the limitations of simple lumped and transcen- dental models based on the predictions, and concluded that for a fixed volume and orifice size, the orifice position changed the resonance frequency substantially, while the orifice shape was not significant. References 1 and 2 considered only very short neck length compared to wavelength. Tang and Sirignano3 studied a Helmholtz resonator with a neck comparable to wavelength as an application to reducing combustion instability. They developed a general formula- tion and applied it to quarter wave resonators and Helmholtz resonators with various neck lengths. Recently, Selamet and co-workers4–6 employed several approaches to examine the effect of cavity volume and neck locations. They illustrated the effect of length-to-diameter ratio of the volume on the resonance frequency and transmission loss characteristics using lumped and one-dimensional 4 ͑1D͒ approaches. Earlier works have been extended further FIG. 1. Helmholtz resonators without ͑a͒ and with extended necks ͑b͒–͑d͒. by studying a number of circular concentric configurations with lumped, one-dimensional radial and axial, two- cylindrical walls varies the alignment of cut-outs, therefore dimensional ͑2D͒ analytical approaches, and three- effectively changing the neck length. dimensional ͑3D͒ boundary element method ͑BEM͒.5 They The objective of the present study is, in the absence of ͑ ͒ also developed a 3D analytical approach to investigate the mean flow, to 1 investigate theoretically, numerically, and experimentally the acoustic attenuation performance of a effect of neck offset on the behavior of circular asymmetric concentric circular Helmholtz resonator with an extended Helmholtz resonator.6 neck; and ͑2͒ examine the effects of length, shape, and per- While the Helmholtz resonator is known to be an effec- foration of the neck extension on the resonator behavior. A tive acoustic reflector at low frequencies, the use of it may, at two-dimensional analytical method is developed for an ex- times, be limited due to volume constraints. Thus, it is im- tended neck with constant cross-sectional area, while a three- portant to lower the resonance frequency without increasing dimensional boundary element method is applied for the volume or reduce volume without increasing resonance fre- variable-area and perforated extension. Lumped and one- quency. While a wealth of literature exists on the effect of dimensional approaches are also presented to illustrate the neck cross-sectional area and opening location to volume, effect of the higher order modes. For a piston-driven model, the impact of neck extension remains to be investigated. The predicted resonance frequencies using lumped, 1D, and 2D present work therefore concentrates on the effect of neck analytical methods are compared with those from 3D BEM. ͑ ͒ extension geometry Fig. 1 on the Helmholtz resonator be- Analytical and computational transmission loss predictions havior. The neck extension length will shift the resonance for pipe-mounted model are compared to the experimental frequency without changing the volume. Such an extension data obtained from an impedance tube setup. also acts like a quarter wave resonator in the cavity resulting Following this Introduction, Sec. II develops a two- in additional transmission loss peaks at higher frequencies dimensional analytical approach and Sec. III summarizes a compared to a resonator without extension. Furthermore, the 3D BEM. Section IV compares the analytical and computa- neck shape involving either variable cross-sectional area or tional predictions with experiments, and discusses the effect perforations can readily change the acoustic characteristics of geometry of the neck extension on the resonance fre- of the resonator. Such an ability to modify the resonance quency and transmission loss characteristics. Section V con- frequency without changing the volume may be desirable in cludes the study with final remarks. a variety of applications, including adaptive passive-noise control.7–9 Among other designs, Cheng and co-workers,8 for II. TWO-DIMENSIONAL ANALYTICAL APPROACH example, suggested two concentric cylindrical neck exten- A two-dimensional analytical approach is introduced sions with preferably parabolic cut-outs. The rotation of the next to determine the acoustic characteristics of a piston-

1976 J. Acoust. Soc. Am., Vol. 113, No. 4, Pt. 1, April 2003 A. Selamet and I. Lee: Helmholtz resonator with extended neck and ␣ 2 ␣ ͱ 2Ϫͩ n ͪ Ͼ n k0 , k0 a1 a1 k ϭ ͑5͒ A,n ␣ 2 ␣ Ά n n Ϫͱ 2Ϫͩ ͪ Ͻ k0 , k0 a1 a1

the axial wave number of the mode (0,n), and k0 wave number of the mode ͑0,0͒. The negative sign in Eq. ͑5͒ is as- Ϫ jkA,nx1 signed so that e decays exponentially in x1 . The particle velocity may then be written, in terms of the linearized momentum equation, as ϱ 1 ϩ Ϫ ͑ ͒ϭ ͓ jkA,nx1 uA r,x1 ␳ ␻ ͚ kA,n An e 0 nϭ0 Ϫ Ϫ jkA,nx1͔␺ ͑ ͒ ͑ ͒ An e A,n r , 6 ␳ ␻ where 0 is the medium density and the angular velocity.

B. Solution for annular volume „domain II… For a concentric annular duct with inner and outer duct ͑ ͒ diameters of a1 and a2 , the solution to Eq. 1 can be written in domain II as ϱ ϩ Ϫ Ϫ ͑ ͒ϭ ͑ jkB,nx2ϩ jkB,nx2͒␺ ͑ ͒ ͑ ͒ FIG. 2. Wave propagation in a piston-driven Helmholtz resonator with a PB r,x2 ͚ Bn e Bn e B,n r 7 ϭ neck extended into the cavity volume. n 0 with driven circular Helmholtz resonator with an extended neck, r J ͑␤ ͒ r which consists of circular and annular ducts ͑Fig. 2͒. For a ␺ ͑ ͒ϭ ͩ ␤ ͪ Ϫ 1 n ͩ ␤ ͪ ͑ ͒ B,n r J0 n ͑␤ ͒ Y 0 n , 8 two-dimensional axisymmetric wave propagation in a circu- a2 Y 1 n a2

lar or annular duct, the Helmholtz equation is given, in cy- where Y m is the Bessel function of the second kind and order ͑ ͒ ␤ lindrical coordinates r,x ,by m, n the root satisfying the boundary condition of 2 2 P͑r,x͒ϩk P͑r,x͒ϭ0, ͑1͒ a J ͑␤ ͒ a ٌ ͩ ␤ 1 ͪ Ϫ 1 n ͩ ␤ 1 ͪ ϭ ͑ ͒ J1 n ␤ ͒ Y 1 n 0 9 where P and k are the acoustic pressure and the wave num- a2 Y 1͑ n a2 ͑ ͒ ber, respectively. The solution to Eq. 1 is obtained next in and three domains: I—neck, II—annular, and III—circular in the volume. ␤ 2 ␤ ͱ 2Ϫͩ n ͪ Ͼ n k0 , k0 a2 a2 k ϭ ͑10͒ A. Solution for circular neck „domain I… B,n ␤ 2 ␤ Ά n n Ϫͱ 2Ϫͩ ͪ Ͻ The solution to Eq. ͑1͒ in domain I or circular neck ͑Fig. k0 , k0 . a2 a2 2͒ can be written as10 Again the negative sign in Eq. ͑10͒ is assigned so that ϱ Ϫ jkB,nx2 ϩ Ϫ Ϫ e decays exponentially in x2 . The particle velocity is ͑ ͒ϭ ͑ jkA,nx1ϩ jkA,nx1͒␺ ͑ ͒ ͑ ͒ PA r,x1 ͚ An e An e A,n r , 2 nϭ0 then ϩ Ϫ ϱ where PA is the acoustic pressure, An and An the modal 1 ϩ Ϫ ͑ ͒ϭ ͓ jkB,nx2 uB r,x2 ␳ ␻ ͚ kB,n Bn e amplitudes corresponding to components traveling in the 0 nϭ0 positive and negative x directions in domain I, respectively, 1 Ϫ ␺ Ϫ jkB,nx2͔␺ ͑ ͒ ͑ ͒ kA,n the wave number, and A,n(r) the eigenfunctions. For Bn e B,n r . 11 this circular duct, the eigenfunctions are given by C. Solution for circular volume domain III r „ … ␺ ͑ ͒ϭ ͩ ␣ ͪ ͑ ͒ A,n r J0 n , 3 The solution to Eq. ͑1͒ in domain III or circular volume a1 ͑Fig. 2͒ can be written as where Jm is the Bessel function of the ﬁrst kind and order m, ␣ ϱ a1 the duct diameter, and n the roots satisfying the rigid ϩ Ϫ Ϫ ͑ ͒ϭ ͑ jkC,nx3ϩ jkC,nx3͒␺ ͑ ͒ ͑ ͒ PC r,x3 ͚ Cn e Cn e C,n r 12 wall boundary condition of nϭ0 Ј͑␣ ͒ϭ ͑␣ ͒ϭ ͑ ͒ J0 n J1 n 0, 4 with

J. Acoust. Soc. Am., Vol. 113, No. 4, Pt. 1, April 2003 A. Selamet and I. Lee: Helmholtz resonator with extended neck 1977 r mounted Helmholtz resonators, respectively. Assuming de- ␺ ͑r͒ϭJ ͩ ␣ ͪ , ͑13͒ ϭᐉ C,n 0 na cay of the higher order modes through neck from x1 n to 2 ϭ ͑ 5͒ x1 0 Selamet et al. , the piston oscillating with velocity ␣ ϭ where n are the roots satisfying the rigid wall boundary magnitude u p at x1 0 gives condition of Eq. ͑4͒, and 1 ͑ ϩϪ Ϫ͒ϭ ͑ ͒ ␣ 2 ␣ ␳ A0 A0 u p . 21 ͱ 2Ϫͩ n ͪ Ͼ n 0c0 k0 , k0 a2 a2 ͑ ͒ ͑ ͒ ϭ k ϭ ͑14͒ Once Eqs. 16 – 21 are solved using s n and letting C,n ␣ 2 ␣ ␳ ϭ Ά n n 0c0u p 1, the acoustic impedance of a Helmholtz resonator Ϫͱ 2Ϫͩ ͪ Ͻ k0 , k0 . ϭ 6 a2 a2 at x1 0 is calculated by p The particle velocity is then ϭ A ϭ ϩϩ Ϫ ͑ ͒ ZH ␳ A0 A0 , 22 ϱ 0c0u p 1 ϩ Ϫ ͑ ͒ϭ ͓ jkC,nx3 uC r,x3 ͚ kC,n Cn e which then allows the evaluation of resonance frequency. ␳ ␻ ϭ 0 n 0 For a pipe-mounted Helmholtz resonator, transmission Ϫ Ϫ jkC,nx3͔␺ ͑ ͒ ͑ ͒ loss can be determined, with the assumptions of anechoic Cn e C,n r . 15 termination at the exit of the main duct and plane wave D. Boundary conditions propagation through the main duct, by6 The coefﬁcients in Eqs. ͑2͒, ͑7͒, and ͑12͒ are determined S 1 ϭ ϭᐉ ϭ ͯ ϩ n ͩ ͪͯ ͑ ͒ next by the boundary conditions. At x2 0 and x3 3 , the TL 10 log10 1 , 23 ϭ ϭ 2S p ZH rigid boundary conditions, uB 0 and uC 0, give, in view of Eqs. ͑11͒ and ͑15͒, where Sn and S p are the area of the neck and main pipe, respectively. ϩϭ Ϫ ͑ ͒ Bn Bn 16 and III. THREE-DIMENSIONAL BOUNDARY ELEMENT Ϫ ϩ Ϫ ᐉ ϭ 2 jkC,n 3 ͑ ͒ Cn Cn e . 17 METHOD ϭᐉ ϭ At x1 n or x3 0, the pressure continuity condition be- The wave propagation is governed by the Helmholtz ϭ ͑ 11 tween domains I and III, PA PC , results in Miles and equation in Cartesian coordinates, Selamet and Ji12͒ 2P͑x,y,z͒ϩk2P͑x,y,z͒ϭ0, ͑24ٌ͒ ϩ Ϫ ᐉ Ϫ ᐉ ͑ jkA,s nϩ jkA,s n͒͗␺ ␺ ͘ As e As e A,s , A,s A and the Green’s theorem results in boundary integral equa- ϱ tion as follows: ϭ ͑ ϩϩ Ϫ͒͗␺ ␺ ͘ ͑ ͒ ͒ P͑qץ ͚ Cn Cn C,n , A,s A , 18 nϭ0 ͑ ͒ ͑ ͒ϭ ͵ ͫ ͑ ͒ 2 C q1 P q1 G q1 ,q2 nץ ⌫ where ͗͘s indicates the integration over area S, and s ͒ ץ ϭ ϱ 0,1,2,..., , which is deferred to Appendix A. Similarly, at G͑q1 ,q2 ϭᐉ ϭ ϪP͑q ͒ ͬd⌫͑q ͒, ͑25͒ n 2ץ x2 2 or x3 0, the pressure continuity between domains II 2 and III, P ϭP , leads to B C ⌫ ϩ Ϫ ᐉ Ϫ ᐉ where q1 and q2 are points on the boundary surface , C(q1) ͑B e jkB,s 2ϩB e jkB,s 2͒͗␺ ,␺ ͘ s s B,s B,s B coefﬁcient, and G(q1 ,q2) the Green’s function or fundamen- ϱ tal solution given, for a three-dimensional acoustic domain, ϭ ͑ ϩϩ Ϫ͒͗␺ ␺ ͘ ͑ ͒ by ͚ Cn Cn C,n , B,s B . 19 nϭ0 Ϫ ͉ Ϫ ͉ e jk q1 q2 ϭᐉ ϭᐉ G͑q ,q ͒ϭ . ͑26͒ At x1 n or x2 2 , the volume velocity continuity, uASA 1 2 4␲͉q Ϫq ͉ ϩ ϭ 1 2 uBSB uCSC , gives ϱ A. Piston-driven model ϩ Ϫ ᐉ Ϫ ᐉ ͑ jkA,n nϪ jkA,n n͒͗␺ ␺ ͘ ͚ kA,n An e An e A,n , C,s A Discretizing the boundary surfaces into a number of el- nϭ0 ements and applying numerical integration to Eq. ͑25͒ ϱ yields13 for domain I ϩ Ϫ ᐉ Ϫ ᐉ ϩ ͑ jkB,n 2Ϫ jkB,n 2͒͗␺ ␺ ͘ ͚ kB,n Bn e Bn e B,n , C,s B ͑ ϭ ͒ ͑ ϭ ͒ nϭ0 PI x1 0 uI x1 0 ͑ ϭᐉ ͒ ϭ ͑ ϭᐉ ͒ ͑ ͒ ϩ Ϫ ͭ PI x1 n ͮ ͓T1͔ͭ uI x1 n ͮ , 27 ϭ ͑ Ϫ ͒͗␺ ␺ ͘ ͑ ͒ kC,s Cs Cs C,s , C,s C . 20 r r PI uI ϩ E. Resonance frequency and transmission loss and for the domain II III, P ϩ ͑x ϭ0͒ u ϩ ͑x ϭ0͒ The expressions for the resonance frequency and trans- ͭ II III 3 ͮ ϭ͓ ͔ͭ II III 3 ͮ ͑ ͒ r T2 r , 28 mission loss are given next for the piston-driven and pipe- PIIϩIII uIIϩIII

1978 J. Acoust. Soc. Am., Vol. 113, No. 4, Pt. 1, April 2003 A. Selamet and I. Lee: Helmholtz resonator with extended neck FIG. 3. Wave propagation inside a pipe-mounted Helmholtz resonator with ᐉ ϭ a neck extended into the cavity volume. FIG. 4. Sample boundary element mesh generated by IDEAS ( 1 8.5 cm, ᐉ ϭ ͒ 2 10 cm, and 10° expansion . where superscript r indicates rigid wall boundary. Using rϭ r ϭ which defines the transfer matrix elements, Tij . Assuming a rigid wall boundary condition, uI uIIϩIII 0, and then applying acoustic pressure and particle velocity continuities at main duct with constant cross-sectional area, transmission ϭᐉ ϭ ͑ ͒ ͑ ͒ loss can then be calculated from the transfer matrix by x1 n or x3 0, Eqs. 27 and 28 can be combined into an impedance matrix as ϭ ͑ 1͉ ϩ ϩ ϩ ͉͒ ͑ ͒ TL 20 log10 2 T11 T12 T21 T22 . 32 ͕P ͑x ϭ0͖͒ϭ͓T ͔͕u ͑x ϭ0͖͒. ͑29͒ I 1 Z I 1 The mesh for BEM calculations, with a typical size of 2.5 The average of acoustic pressure and particle velocity at cm, are generated using IDEAS.14 Figure 4 shows a sample ᐉ ϭ nodes yields the acoustic impedance of the Helmholtz reso- mesh for the Helmholtz resonator with 2 10 cm and 10° nator as a function of frequency. The resonance frequency at expanded conical extension. which the acoustic impedance is minimum can be calculated. B. Pipe-mounted model IV. RESULTS AND DISCUSSION For a pipe-mounted model as shown in Fig. 3, the main duct ͑M͒ and neck ͑I͒ form an acoustic domain (MϩI) and A cylindrical Helmholtz resonator has been fabricated ϭ ᐉ ϭ the relationship between acoustic pressure and velocity at the with fixed radius a2 7.62 cm, length c 20.32 cm, and a ᐉ ϭ boundaries is given by base neck of length 1 and fixed radius a1 2.00 cm. For the i i extended neck into the chamber, four configurations with PMϩI uMϩI various extension lengths and shapes are considered as sum- Po uo MϩI ϭ͓ ͔ MϩI ͑ ͒ marized in Table I: configuration 1—base neck with fixed ͭ r ͮ T3 ͭ r ͮ , 30 ᐉ ϭ PMϩI uMϩI length of 1 8.5 cm and straight extensions with different ϭᐉ ͒ ϭᐉ ͒ ᐉ ϭ PMϩI͑x1 n uMϩI͑x1 n lengths, 2 0 – 18 cm; configuration 2—fixed total (base ϩ ᐉ ϭᐉ ϩᐉ ϭ extended) neck length of n 1 2 18.5 cm and vary- where superscripts i and o denote inlet and outlet of the main ᐉ ϭ ing extension length from 2 0 to 18 cm, with correspond- duct, respectively, and r the rigid walls. Using rigid boundary ing base neck length from ᐉ ϭ18.5 to 0.5 cm; configuration r 1 condition at the walls of the main duct and neck, u ϩ ϭ0, ᐉ ϭ M I 3—base neck with fixed length of 1 8.5 cm and neck ex- and then applying acoustic pressure and velocity continuity tension with ᐉ ϭ10 cm and conical shapes ͑5° contraction or ϭᐉ ͑ ͒ ͑ ͒ 2 at x1 n , Eqs. 28 and 30 yield 10° expansion͒; and configuration 4—base neck with fixed ᐉ ϭ ᐉ ϭ i o length of 1 8.5 cm, and neck extension with 2 10 cm P T11 T12 P ͫ ͬϭͫ ͬͫ ͬ ͑ ͒ and perforations. The main duct is built of square cross- ␳ i ␳ o , 31 0c0u T21 T22 0c0u section (4.3 cmϫ4.3 cm) for clear identification of the neck

J. Acoust. Soc. Am., Vol. 113, No. 4, Pt. 1, April 2003 A. Selamet and I. Lee: Helmholtz resonator with extended neck 1979 TABLE I. Helmholtz resonators with various neck lengths ͓cm͔ and shapes.

Extended Base neck Neck extension Total neck length ᐉ ᐉ ᐉ ϭᐉ ϩᐉ Conﬁguration neck shape length ( 1) length ( 2) ( n 1 2) 1 Straight 8.5 0–18 8.5–26.5 solid 2 Straight 18.5–0.5 0–18 18.5 solid 3 Conical 8.5 10 18.5 straight solid 4 Straight 8.5 10 18.5 perforated

FIG. 5. Resonance frequency predictions for Helmholtz resonators with various neck extension lengths ᐉ ϭ ( 1 8.5 cm).

FIG. 6. Measured transmission loss for Helmholtz resonators with neck ᐉ ϭ extension ( 1 8.5 cm).

1980 J. Acoust. Soc. Am., Vol. 113, No. 4, Pt. 1, April 2003 A. Selamet and I. Lee: Helmholtz resonator with extended neck FIG. 7. Comparison of predictions with experiment for a Helmholtz resonator without neck extension ᐉ ϭ ᐉ ϭ ( 1 8.5 cm and 2 0cm).

length. The square main duct is then connected to the circu- than the other models. In the lumped model, the volume of a lar impedance tube with smooth transitions that retain a con- neck extension is subtracted from the total resonator volume. stant cross-sectional area development. The relatively large difference for short extensions among For a piston-driven Helmholtz resonator of configuration the analytical methods may be attributed to increasing rela- 1, Fig. 5 compares a number of predictions for resonance tive significance of higher order modes ͑due to decreasing frequencies. In general, the resonance frequency decreases as total neck length͒ generated at the interface between neck the extension length ᐉ increases from 0 to 18 cm. 2D ana- 2 and volume. lytical and BEM predictions exhibit a good agreement, as An impedance tube test setup5 is used to obtain trans- expected. The difference between the resonance frequencies using 2D analytical approach with five and ten higher order mission loss of pipe-mounted resonators applying the two- modes is within 0.3%, thus only the predictions with five microphone technique. Figure 6 shows the measured trans- modes are presented. Setting sϭ0 in Eqs. ͑16͒–͑20͒ gives mission loss for different neck extension lengths. As the 1D axial model predictions, leading to overestimation of extension length increases, the resonance frequency de- resonance frequencies. Lumped model predictions without creases and the attenuation band becomes narrower. Note end correction demonstrate higher resonance frequencies that a 15 cm change in the extension length results in a 33 Hz

FIG. 8. Comparison of predictions with experiment for a Helmholtz resonator with neck extension ᐉ ϭ ᐉ ϭ ( 1 8.5 cm and 2 15 cm).

J. Acoust. Soc. Am., Vol. 113, No. 4, Pt. 1, April 2003 A. Selamet and I. Lee: Helmholtz resonator with extended neck 1981 FIG. 9. Predicted resonance frequencies for Helmholtz resonators ᐉ ϭᐉ ϩᐉ with ﬁxed total length, n 1 2 ϭ18.5 cm.

shift in the resonance frequency, which is a significant varia- neck and main duct interface. As the neck extension length ᐉ tion at low frequencies considered here. 2 increases, the differences tend to diminish since the total In addition to the resonance frequency, transmission loss neck length increases. Both 2D analytical and BEM have of the pipe-mounted Helmholtz resonator can be predicted peaks higher than the experiment since damping or dissipa- using Eqs. ͑23͒ and ͑32͒. For a Helmholtz resonator without tion effects are neglected in the predictions. ᐉ ϭ ͓ ͑ ͒ ͑ ͒ ͑ ͒ ᐉ and with ( 2 15 cm) neck extension Figs. 1 a and b , For a fixed total neck length configuration 2 n configuration 1͔, Figs. 7 and 8 compare such transmission ϭ18.5 cm, the predicted resonance frequency shifts as a ᐉ loss predictions with experiments. Since the transmission function of the extension length 2 as shown in Fig. 9, which ᐉ ϭ loss behavior for 2 5 and 10 cm is similar to Figs. 7 and 8 include lumped, 1D, and 2D analytical models. For a given except for the resonance frequencies, the comparisons for geometry, the variation in resonance frequency remains ᐉ ϭ 2 5 and 10 cm are excluded. While the 2D analytical pre- within 3 Hz for all analytical approaches. dictions shift to higher frequencies by 0.5–2 Hz, the 3D Figure 10 considers a 10-cm-long conical neck exten- BEM shows a close agreement with the experiments. The sion ͓Fig. 1͑c͒, configuration 3͔ and compares the BEM pre- difference between the 2D analytical method and 3D BEM is dictions with experiments. Due to the complexity of the ge- due to the neglect of higher order modes in the former at the ometry, only the BEM predictions are presented here.

FIG. 10. Comparison of BEM predictions with experiment for a Helmholtz resonator with conical neck extension ᐉ ϭ ᐉ ϭ ( 1 8.5 cm and 2 10 cm).

1982 J. Acoust. Soc. Am., Vol. 113, No. 4, Pt. 1, April 2003 A. Selamet and I. Lee: Helmholtz resonator with extended neck FIG. 11. Comparison of BEM predictions with experiment for a Helmholtz resonator with perforated neck exten- ᐉ ϭ ᐉ ϭ sion ( 1 8.5 cm and 2 10 cm).

Compared to the one with constant cross-sectional area ex- and narrows the transmission loss band. With the total neck tension, the resonator with 10° expansion has increased the lengths retained, the shape of the neck extension, e.g., coni- resonance frequency and broadened the attenuation band, cal contraction and expansion, also changes the resonance whereas the 5° contraction has lowered the resonance fre- frequency and transmission loss. Thus, modification of the quency and narrowed the attenuation band. The directional neck extension length or shape may be an effective method behavior may be qualitatively interpreted in terms of increas- to control the resonance frequency of a Helmholtz resonator ing and decreasing effective cross-sectional areas, respec- without changing the cavity volume. Also adding perforation tively. The BEM predictions capture these trends with satis- to the neck extension can change the resonance frequency factory accuracy. and transmission loss behavior. As demonstrated in Figs. 6–10, the resonance frequency For a piston-driven Helmholtz resonator, the 2D analyti- and transmission loss behavior can be modified by changing cal method shows a good agreement with BEM, while sim- the neck length or shape. The resonance frequency can be plified lumped and 1D models overpredict the resonance fre- further controlled by the perforation of the neck extension. quencies. For a pipe-mounted resonator, transmission loss Figure 11 shows transmission loss predicted by BEM for predictions by 2D analytical model show reasonable agree- Helmholtz resonators with perforated neck extension ͓Fig. ment with experiments. The deviation of 2D analytical trans- 1͑d͒, configuration 4͔. The top of the extension is fully open mission loss prediction from experiment is mainly due to the ᐉ to the cavity volume and the cylindrical extension with 2 neglect of higher order modes in the vicinity of neck and ϭ10 cm is now perforated with porosity ␴.As␴ increases, main duct interface. As the total neck length increases, the the resonance frequency increases, and the magnitude of the difference between 2D analytical method and experiments transmission first decreases followed by an increase. Sulli- decreases. van’s empirical expression15 is used for perforation imped- The extension of neck into the cavity can shift the reso- ance in these predictions. As ␴ increases, the extension nance frequency down without increasing the volume. ᐉ ϭ ᐉ Х length effectively varies from 2 10 cm to 2 0cmat Changing the length and shape of or adding perforations to high porosities, therefore the corresponding resonance fre- the neck extension can move the resonance frequency and quency varies from the limit of solid extension to the no- modify the transmission loss behavior. Such trends can be extension case with accompanying increase in frequency. appealing in noise control applications where designers may Also, as ␴ varies from 1 to 99%, the resistance or damping be limited by the space constraints. of the perforation impedance decreases, leading to an increase in transmission loss. APPENDIX A: INTEGRATION IN EQS. „18…–„20… V. CONCLUDING REMARKS The following integral relations for Bessel functions are ͑ 16͒ Acoustic characteristics of a Helmholtz resonator with considered Abramowitz and Stegun : extended neck into cavity have been studied analytically, nu- ͑␣ ͒ ͑␤ ͒ merically, and experimentally. Predictions by lumped, 1D ͵ xJ0 x J0 x dx axial, 2D analytical, and 3D BEM models are compared to ͕␣ ͑␣ ͒ ͑␤ ͒Ϫ␤ ͑␣ ͒ ͑␤ ͖͒ each other and the experimental results. Extension of neck x J1 x J0 x J0 x J1 x ϭ , ͑A1͒ into the cavity lowers the resonance frequency substantially ␣2Ϫ␤2

J. Acoust. Soc. Am., Vol. 113, No. 4, Pt. 1, April 2003 A. Selamet and I. Lee: Helmholtz resonator with extended neck 1983 2 2 x x where Jn and Y n are the ﬁrst and second kind Bessel func- ͵ 2͑␣ ͒ ϭ ͕ Ј͑␣ ͖͒2ϩ ͕ ͑␣ ͖͒2 xJ x dx J x J0 x ͗͘ 0 2 0 2 tions, respectively. The integration denoted by S in Eqs. ͑ ͒ ͑ ͒ ͑ ͒ ͑ ͒ x2 18 – 20 can be calculated from Eqs. A1 – A3 and the ϭ ͕J2͑␣x͒ϩJ2͑␣x͖͒, ͑A2͒ orthogonality of Bessel functions as follows: 2 1 0

͵ ͑␣ ͒ ͑␣ ͒ xJ0 x Y 0 x dr

a 2 1 1 r a1 ϭ 2 ␣ ͒ ␣ ͒ϩ ␣ ͒ ␣ ͒ ͑ ͒ ͗␺ ␺ ͘ ϭ ͵ 2ͩ ␣ ͪ ϭ 2͑␣ ͒ ͑ ͒ x ͕J0͑ x Y 0͑ x J1͑ x Y 1͑ x ͖, A3 A,s , A,s A rJ0 s dr J0 s , A4 2 0 a1 2

␣ a n ͑␣ ͒ ͩ ␣ 1 ͪ a1 J0 s J1 n a1 r r a a ␣ ␣ ͵ ͩ ␣ ͪ ͩ ␣ ͪ ϭ 2 2 ͩ n Þ s ͪ rJ0 n J0 s dr 2 2 0 a a ␣ ␣ a a ␺ ␺ ϭ 2 1 ͩ s ͪ Ϫͩ n ͪ 2 1 ͑ ͒ ͗ C,n , A,s͘A A5 Ά a1 a2 a1 r 1 ␣ ␣ ͵ 2ͩ ␣ ͪ ϭ 2 2͑␣ ͒ ͩ n ϭ s ͪ rJ0 s dr a1J0 s , 0 a1 2 a2 a1

2 a2 r J ͑␤ ͒ r 1 1 ͗␺ ␺ ͘ ϭ ͵ ͭ ͩ ␤ ͪ Ϫ 1 s ͩ ␤ ͪͮ ϭ 2␺2 ͑ ͒Ϫ 2␺2 ͑ ͒ ͑ ͒ B,s , B,s B r J0 s ͑␤ ͒ Y 0 s dr a2 B,s a2 a1 B,s a1 , A6 a1 a2 Y 1 s a2 2 2 ␺ ␺ ͗ C,n , B,s͘B a a ␣ 1 ͩ ␣ 1 ͪ ␺ ͑ ͒ n J1 n B,s a1 a2 r r J ͑␤ ͒ r a a ͵ ͩ ␣ ͪͭ ͩ ␤ ͪ Ϫ 1 s ͩ ␤ ͪͮ ϭ 2 2 ͑␣ Þ␤ ͒ rJ0 n J0 s Y 0 s dr 2 2 n s a a a Y ͑␤ ͒ a ␤ ␣ ϭ 1 2 2 1 s 2 ͩ s ͪ Ϫͩ n ͪ Ά a2 a2 a2 r r J ͑␤ ͒ r 1 1 a ͵ ͩ ␣ ͪͭ ͩ ␤ ͪ Ϫ 1 s ͩ ␤ ͪͮ ϭ 2 ͑␣ ͒␺ ͑ ͒Ϫ 2 ͩ ␣ 1 ͪ ␺ ͑ ͒ ͑␣ ϭ␤ ͒ rJ0 n J0 s ͑␤ ͒ Y 0 s dr a2J0 n B,s a2 a1J0 n B,s a1 n s , a1 a2 a2 Y 1 s a2 2 2 a2 ͑A7͒

a a ␣ 1 ͑␣ ͒ ͩ ␣ 1 ͪ s J0 n J1 s a1 r r a a ␣ ␣ ͵ ͩ ␣ ͪ ͩ ␣ ͪ ϭ 2 2 ͩ s Þ n ͪ rJ0 n J0 s dr 2 2 0 a a ␣ ␣ a a ␺ ␺ ϭ 1 2 ͩ s ͪ Ϫͩ n ͪ 2 1 ͑ ͒ ͗ A,n , C,s͘A A8 Ά a2 a1 a1 r 1 ␣ ␣ ͵ 2ͩ ␣ ͪ ϭ 2 2͑␣ ͒ ͩ s ϭ n ͪ rJ0 n dr a1J0 n , 0 a1 2 a2 a1 ␺ ␺ ͗ B,n , C,s͘B a a ␣ 1 ͩ ␣ 1 ͪ ␺ ͑ ͒ s J1 s B,n a1 a2 r J ͑␤ ͒ r r a a ͵ ͭ ͩ ␤ ͪ Ϫ 1 n ͩ ␤ ͪͮ ͩ ␣ ͪ ϭ 2 2 ͑␣ Þ␤ ͒ r J0 n Y 0 n J0 s dr 2 2 s n a a Y ͑␤ ͒ a a ␤ ␣ ϭ 1 2 1 n 2 2 ͩ n ͪ Ϫͩ s ͪ Ά a2 a2 a2 r J ͑␤ ͒ r r 1 1 a ͵ ͭ ͩ ␤ ͪ Ϫ 1 n ͩ ␤ ͪͮ ͩ ␣ ͪ ϭ 2 ͑␣ ͒␺ ͑ ͒Ϫ 2 ͩ ␣ 1 ͪ ␺ ͑ ͒ ͑␣ ϭ␤ ͒ r J0 n ͑␤ ͒ Y 0 n J0 s dr a2J0 s B,n a2 a1J0 s B,n a1 s n , a1 a2 Y 1 n a2 a2 2 2 a2 ͑A9͒

2 a2 r a ͗␺ ␺ ͘ ϭ ͵ 2ͩ ␣ ͪ ϭ 2 2͑␣ ͒ ͑ ͒ C,s , C,s C rJ0 s dr J0 s . A10 0 a2 2

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J. Acoust. Soc. Am., Vol. 113, No. 4, Pt. 1, April 2003 A. Selamet and I. Lee: Helmholtz resonator with extended neck 1985