Circular concentric Helmholtz resonatorsa) A. Selamet and P. M. Radavich Department of Mechanical Engineering, The Ohio State University, Columbus, Ohio 43210 N. S. Dickey The University of Michigan, Ann Arbor, Michigan 48109 J. M. Novak Powertrain Operations, Ford Motor Company, Dearborn, Michigan 48121 ~Received 27 December 1995; accepted for publication 30 July 1996! The effect of specific cavity dimensions of circular concentric Helmholtz resonators is investigated theoretically, computationally, and experimentally. Three analytical models are employed in this study: ~1! A two-dimensional model developed to account for the nonplanar wave propagation in both the neck and the cavity; ~2! a one-dimensional solution developed for the limit of small cavity length-to-diameter ratio, l/d, representing a radial propagation in the cavity; and ~3! a one-dimensional closed-form solution for configurations with large l/d ratios which considers purely axial wave propagation in the neck and the cavity. For low and high l/d, the resonance frequencies determined from the two-dimensional approach are shown to match the one-dimensional predictions. For cavity volumes with l/d.0.1, the resonance frequencies predicted by combining Ingard’s end correction with one-dimensional axial wave propagation are also shown to agree closely with the results of the two-dimensional model. The results from the analytical methods are then compared with the numerical predictions from a three-dimensional boundary element method and with experiments. Finally, these approaches are employed to determine the wave suppression performance of circular Helmholtz resonators in the frequency domain. © 1997 Acoustical Society of America. @S0001-4966~97!05312-5# PACS numbers: 43.20.Ks, 43.20.Mv, 43.50.Gf @JEG# INTRODUCTION area discontinuity from the neck to the volume, but it is only effective for a limited range of geometries when used with Helmholtz resonators, which consist of a volume com- the classical model above. In their extensive work, Miles municating through an orifice or neck to some external exci- ~1971! and Miles and Lee ~1975! investigated the Helmholtz tation, produce narrow bands of high wave attenuation. The resonance behavior of harbors by employing the electrical classical approach in modeling these resonators is to neglect analogy; the former treats the simple shapes including circu- the spatial distribution leading to an equivalent spring–mass lar and rectangular harbors with constant depth, the latter system where the mass of air in the neck, m5r0Aclc ,is develops analytical approaches for irregular geometries with driven by an external force and the volume acts as a spring variable depth. Alster ~1972! extended the spring-mass anal- 2 2 with stiffness s5r0c0Ac /V, Ac and lc being the neck area ogy by including the mass of the spring and incorporating a and length, respectively, and V the resonator volume ~Ray- spring with varying stiffness. His experimental results for a leigh, 1945, Art. 303; Kinsler et al., 1982, Chap. 10!. For number of different resonator shapes showed a significant this one degree of freedom system, vr5As/m, leading to a improvement over the lumped model. Several other simple single resonance frequency of f r5(c0/2p)AAc /lcV, which end corrections are also listed by Chuka ~1973!. To predict is a function of the cavity volume, but independent of the the resonance frequency, Tang and Sirignano ~1973! as- volume dimensions. Experimental observations, however, sumed one-dimensional wave propagation in both the reso- have deviated from this frequency, which is attributed to the nator neck and cavity volume. Panton and Miller ~1975! de- motion of some additional mass on both sides of the neck. veloped a relationship for the resonance frequency which To improve the accuracy, the neck length is usually ‘‘cor- matched experimental results well when used with Ingard’s rected’’ by adding a term for each end in order to account for end correction. Their effort was based on a one-dimensional this fluid motion, thereby modifying lc in the foregoing ex- wave motion in the cavity alone and a spatially lumped short pression for f r by lc8 5 lc 1 dv 1 d p . A number of analytical neck length. Monkewitz and Nguyen-Vo ~1985! studied non- treatments based on somewhat simplified physics have been planar effects in two-dimensional resonators with a semicy- introduced to develop these end correction factors. Rayleigh lindrical cavity and three-dimensional resonators with a ~1945! derived a length correction for resonators mounted in hemispherical cavity. By asymptotically matching the solu- a baffle. Ingard ~1953! developed an end correction to ac- tions of the linearized inviscid equations in terms of low- count for multidimensional wave propagation excited at the frequency expansions in the exterior, neck, and cavity, they proposed volume and length corrections for these configura- a!A preliminary version of this study has been presented at the SAE Noise tions. Selamet et al. ~1993, 1995a! developed an expression and Vibration Conference as SAE 951263, Traverse City, MI. for the transmission loss of a Helmholtz resonator with wave 41 J. Acoust. Soc. Am. 101 (1), January 1997 0001-4966/97/101(1)/41/11/$10.00 © 1997 Acoustical Society of America 41 approach, Sec. III the boundary element method, and Sec. IV the experimental setup. The results from the analytical ap- proaches and the boundary element method are compared with experiments and used to evaluate the end corrections in Sec. V. The study is concluded with final remarks in Sec. VI. I. ONE-DIMENSIONAL ANALYTICAL APPROACHES The generation and propagation of multidimensional waves in the resonator volume is clearly dependent on the relative magnitudes of the wavelength, the neck and duct diameters and the volume dimensions. Provided that the in- cident wave from the neck to the volume is planar, there are two limiting configurations for which the wave propagation FIG. 1. Helmholtz resonator dimensions ~d p54.859 cm, dc54.044 cm, 3 in the volume can be considered one-dimensional, thereby lc58.5 cm, and V54500 cm !. allowing a relatively simple closed-form solution for trans- mission loss. This section presents the expressions for the motion. They also provided experimental results illustrating wave attenuation properties for these one-dimensional limits. that the relationship for resonance frequency given by Tang A. Radial propagation limit and Sirignano ~without end correction factors! worked well for volumes with large length-to-diameter ratio, while show- The discrepancy between the analytical results and the ing some deviation at low lv /dv ratios ~hereafter, l/d, for experiment is largest for l/d of the order 0.1 to 1.0. An brevity!. In a preliminary investigation, Selamet et al. accurate prediction of the transmission loss and resonance ~1994a! studied the effect of multidimensional propagation frequency in this region then requires a multidimensional and showed deviations in resonance frequency from the analysis. For smaller l/d ratios as the volume approaches a lumped parameter analysis, particularly at low l/d ratios. Re- ‘‘pancake’’ geometry, a one-dimensional solution in the ra- cently, Chanaud ~1994! developed a relationship for the reso- dial direction becomes possible as discussed next. nance frequency of configurations with a rectangular paral- Consider planar propagation in the main duct and con- lelepiped cavity volume. Finally, by curve fitting finite nector and purely radial wave motion in the cavity volume. element results, Sahasrabudhe et al. ~1995! provided polyno- For circularly symmetric propagation of acoustic waves in a mial expressions for the end correction as a function of ex- hollow disk of constant width, the linearized, inviscid wave pansion ratio and frequency. equation may be written as The present study considers the circular concentric 1 ] ]p 1 ]2p Helmholtz configurations with constant cavity volume and r 5 . ~1! r r r 2 t2 neck length, as shown in Fig. 1. The objective is to investi- ] S ] D c0 ] gate, as a function of the l/d ratio: ~1! the discrepancy in the Introducing the harmonic dependence resonance frequency from classical approaches, and ~2! the ivt acoustic attenuation behavior. To estimate the end correction p~r,t!5P~r!e , ~2! accurately at the neck–volume interface, a two-dimensional and using the separation of variables, the solution for com- axisymmetric analytical model is developed for these reso- plex pressure amplitude may be determined as nators. The study also employs two different one- P r 5C H~1! kr 1C H~2! kr , 3 dimensional analytical solutions based on ~1! radial propaga- ~ ! 1 0 ~ ! 2 0 ~ ! ~ ! tion for small l/d ratios, and ~2! axial propagation for large where C1 and C2 are complex constants related to the am- l/d ratios, to be referred hereafter as one-dimensional radial plitude of oscillation for the inward and outward traveling ~h! and axial solutions, respectively. Both closed-form solutions waves, respectively, and Hb is the Hankel function ~the are used in the study to examine their corresponding limits. Bessel function of the third kind! of order b and type h.As The pressure field inside the resonators is determined by the a function of the latter variable, h, Hankel functions may boundary element method and used to evaluate the degree of readily be expressed as H(1)5J1iY and H(2)5J2iY with J multidimensionality of the sound-pressure field at both ends and Y being the Bessel functions of the first and second kind, of the resonator neck. Experimental results are obtained by respectively. The momentum equation installing Helmholtz resonators in an impedance tube setup. ]u 1 Results from the boundary element method and experiments 52 “p ~4! are then compared with the resonance frequency relationship ]t r0 of the one-dimensional axial model modified by Ingard’s end combined with the harmonic relationship correction for multidimensional propagation.