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A one-dimensional preliminary design impedance prediction model is derived for Helmholtz resonators constructed with circular ori ces. The model addresses only the effects of resonator geometry and incident sound-pressure amplitude. The effects of grazing ow are not included. An experimental program was conducted to calibrate unknown empirical parameters derived in the model. The model was shown to predict, reasonably accurately, impedance data published in the open literature for resonators constructed with single and multiple ori ces over a wide range of sound-pressure amplitudes and frequencies. The model provides a rapid means of calculating resonator impedance as a function of resonator geometry, incident sound-pressure amplitude, and frequency. Nomenclaturece inertia length de ned by Eq. (19) d N = multiple-ori ce diameter de ned by Eq. (34) d o = single-ori ce diameter F = sound frequency H = ori ce nonlinear inertial length de ned by Eq. (20) and Fig. 1 K ac = acoustic viscous loss parameter de ned by Eq. (22) K ss = steady-state viscous loss parameter de ned by Eq. (21) k = sound wave number !=c 0 L c = cavity depth N = number of ori ces P c = peak cavity sound pressure de ned by Eq. (7) P pk = p 2P 0 P 0 = peak driving sound pressure acting on control volume upper surface S o R = real part of resonator impedance R fd = steady-state fully developed laminar resistive loss de ned by Eq. (16) R L = linear resistive loss de ned by Eq. (15) r o = ori ce radius S BL = ori ce boundary-layer area, S BL D S o ¡ S inv de ned in Fig. 1 S c= cavity cross-sectionalarea S inv = ori ce inviscid area de ned in Fig. 1 S N = multiple ori ce center-to-center spacing S o = ori ce area S w = ori ce wetted area t = time u BL = acoustic velocity passing through S BL de ned in Fig. 1 u inv = acoustic velocity passing through S inv de ned in Fig. 1 u o = ori ce peak acoustic velocity hu o i rms = rms amplitude averaged over each half-cycle V non = nondimensional acoustic particle velocity de ned by Eq. (23) X = imaginary part of resonator impedance ± = ori ce instantaneous boundary-layer thickness µ p = phase angle between P 0 and u o de ned by Eq. (5) · vis = ori ce viscous loss parameter de ned by Eq. (10) = sound wavelength ¹ = uid viscosity º = uid kinematic viscosity ½ 0 = uid density ¾= ori ce open area ratio, S o =S c ¿ = ori ce thickness ¿ w = acoustic shear stress acting on ori ce wetted area de ned by Eq. (8) ! = sound radian frequency (D 2¼ F) Subscripts c = cavity L = linear NL = nonlinear res = resonance

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A fluid mechanical model of the acoustic behavior of small orifices is presented which predicts orifice resistance and reactance as a function of incident sound pressure level, frequency, and orifice geometry. Agreement between predicted and measured values (in both water and air) of orifice impedance is excellent. The model shows the following (1) The acoustic flow in the immediate neighborhood of the orifice (i. e. , in the near field) can be modeled a s a locally spherical flow. Within this near field, the flow is, to a first approximation, unsteady and incompressible. (2) At very low sound pressure levels, the orifice viscous resistance is directly related to the effect of boundary-layer displacement along the walls containing the orifice, and the orifice reactance is directly related to the inertia of the oscillating flow in the neighborhood of the orifice. Previously, orifice resistance and reactance were modeled by empirical end correction expressions. The model also shows that, at low to moderate sound pressure levels, the resistance can be dominated by weak nonlinear jet-like losses but that the overall impedance can still be constant (i. e., independent of incident sound pressure level) providing the orifice resistance is very small relative to the reactance. This is shown to occur when the amplitude of the incident acoustic pressure P is less than p [ w ( D + L)] ', where w is the sound radian frequency, D and L a r e the orifice diameter and thickness, respectively, and p is the fluid mean density. (3) When P/p[ w(D + L)] >> 1, the orifice impedance is dominated by nonlinear jet-like effects. This corresponds to very high sound pressure levels, at which the orifice behaves in a predominately quasi-steady manner. Thus, the model establishes explicitly the quasi-steady nature of the flow in orifices exposed to intense sound. of t h e a c o u s t i c b e h a v i o r o f small o r i f i c e s i s p r e s e n t e d w h i c h p r e d i c t s o r i f i c e r e s i s t a n c e a n d reactance a s a f u n c t i o n o f i n c i d e n t s o u n d p r e s s u r e l e v e l , f r e q u e n c y , a n d o r i f i c e g e o m e t r y . A g r e e m e n t b e t w e e n p r e d i c t e d a n d m e a s u r e d v a l u e s ( i n b o t h w a t e r a n da i r ) o f o r i f i c e i m p e d a n c e i s e x c e l l e n t . The m o d e l s h o w s t h a t ( 1 ) The a c o u s t i c f l o w i n t h e i m m e d i a t e n e i g h b o r h o o d o f t h e o r i f i c e ( i . e . , t h e n e a r f i e l d ) c a n b e m o d e l e d a s a l o c a l l y s p h e r i c a l f l o w . W i t h i n t h i s n e a r f i e l d , t h e f l o w i s , t o a f i r s t a p p r o x i m a t i o n , u n s t e a d y a n d i n c o m p r e s s i b l e . ( 2 ) A t v e r y l o w s o u n d p r e s s u r e l e v e l s , t h e o r i f i c e v i s c o u s r e s i s t a n c e i s d i r e c t l y r e l a t e d t o t h e e f f e c t o f b o u n d a r y -l a y e r d i s p l a c e m e n t a l o n g t h e w a l l s c o n t a i n i n g t h e o r i f i c e...