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 reso- nance 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 cav- ity 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 ori- fice 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 transmis- sion 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 num- ber 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.