Application of the Herschel-Quincke Tube Concept to Higher-Order Acoustic Modes in Two-Dimensional Ducts

Transcription

1 Applicatio of the Herschel-Quicke Tube Cocept to Higher-Order Acoustic Modes i Two-Dimesioal Ducts Lori A. Brady Thesis submitted to the Faculty of the Virgiia Polytechic Istitute ad State Uiversity i partial fulfillmet of the requiremets for the degree of Master of Sciece i Mechaical Egieerig Ricardo A. Burdisso, Chair Alfred W. Wicks J. Keeth Shaw March Blacksburg, Virgiia Keywords: Acoustics, Noise Cotrol, Duct, Herschel-Quicke Tubes, Higher-Order Modes, Gree's Fuctio Copyright, Lori A. Brady

2 Applicatio of the Herschel-Quicke Tube Cocept to Higher-Order Acoustic Modes i Two-Dimesioal Ducts Lori A. Brady (ABSTRACT) The applicatio of the Hershcel-Quicke (HQ) tube as a oise reio device for oedimesioal plae-wave soud fields has bee studied i great detail i previous years. I this thesis, a aalytical techique is developed to ivestigate the potetial of the HQ tube cocept to cotrol higher-order modes. This aalytical method ivolves modelig the tube- iterfaces as fiite pisto sources, which couple the acoustic field iside the mai with the acoustic field withi the HQ tube(s). The acoustic field withi the HQ tube is modeled as plaewaves ad the acoustic field withi the mai is modeled by epadig the soud field i terms of the higher-order modes. This model is the used to ivestigate the oise reio mechaisms behid the atteuatio of higher-order modes. These mechaisms ivolve both the reflectio of the icidet wave as well as the recostructio ad recombiatio of the modal cotet of the icidet disturbace ito other modes. The effects of the modal cotet of the disturbace alog with the HQ tube geometric parameters, such as tube aial positio, legth, distace betwee iterfaces, ad cross-sectioal area, are studied with respect to the frequecies of atteuatio ad the reio obtaied. These results show the potetial of the Herschel- Quicke tube cocept to reduce higher-order modes i s..

3 Author s Ackowledgemets First, I would like to thak my advisor Ricardo Burdisso for all of his guidace ad support throughout this research ad the writig of this thesis. May thaks are also eteded to Al Wicks ad Ke Shaw for their support ad cotributios to this work ad as graduate committee members. I would like to ackowledge all of the admiistrative help I received from Daw Beett, Cathy Hill, Lyda Kig, ad Eloise McCoy. Without them I would have bee lost. I am also thakful to all the members of the Vibratio ad Acoustics Laboratories (VAL). The support amog its members ad the friedships i the lab proved a ivaluable asset i work preseted here. I am also grateful to the support of my family. I thak them for their love, ecouragemet, ad patiece. Fially, I would like to thak the NASA Lagley Research Ceter for their support of this research uder grat NAG-98 ad my graduate studies. It is greatly appreciated. iii

4 Table of Cotets Ackowledgemets...iii Chapter Itroio.... Prologue.... Review of Literature o Hershcel-Quicke Tube Cocept Objectives Outlie of Thesis...8 Chapter Aalytical Modelig...9. Herschel-Quicke Tube System Cocept...9 HQ Tube Model...3 Duct Model with Fiite Pisto Sources ad Disturbace...5 HQ Tube-Duct Couplig...9. Herschel-Quicke Tubes i Series....3 Modelig Simplificatios Sigle HQ Tube Pair...6 Compariso with Plae-Wave Modelig Approach...9 Frequecies of Maimum Atteuatio...3 Atteuatio at Mode Cut-Off Frequecies...33 Chapter 3 Numerical Aalysis Modal Aalysis with Icreasig Disturbace Compleity...35 Compariso with Plae-Wave Modelig Approach...35 Sigle-Mode Aalysis...38 Odd Disturbace Modes...4 Eve Disturbace Modes...45 All Disturbace Modes Parametric Studies...54 Tube Geometric Parameters: Aial Locatio, Legth, Iterface Distace, ad Area...54 Effect of Aial Positio...55 iv

5 Effect of Tube Legth...57 Effect of Iterface Distace...59 Effect of Tube area...6 Tubes i Series...63 Chapter 4 Coclusios Summary Future Research...67 Bibliography...69 Appedi A Duct Acoustics...7 Appedi B Plae-Wave Aalysis...76 Appedi C The Gree s Fuctio...8 Vita...86 v

6 List of Figures Figure.. Passive acoustic lier i rigid-walled... Figure.. Passive oise cotrol devices reactive type...3 Figure.3. Herschel-Quicke tube cocept...4 Figure.4. Herschel-Quicke tube with epasio chamber...6 Figure.. Simplified model two-dimesioal ifiite...9 Figure.. Eigefuctios for ( y) Φ for µ=,,, 3, ad 4... κ µ Figure.3. Herschel-Quicke tube modelig cocept... Figure.4. Ucoupled Herschel-Quicke system with fiite pisto source radiators...3 Figure.5. Ucoupled Herschel-Quicke tube with fiite pisto source radiators; (a) represetatio ad (b) simplified model...4 Figure.6. Ucoupled mai- with fiite pisto source radiators...7 Figure.7. Samples of the itegrated Gree s fuctios...8 Figure.8. Coupled Herschel-Quicke system...9 Figure.9. Herschel-Quicke tubes i series...3 Figure.. Sigle-mode frequecies of maimum atteuatio (h=5.4 cm, S=.7 cm, L=.7 cm, ad l=.6 cm)...3 Figure 3.. Phase relatioship betwee recombied waves...36 Figure 3.. Compariso of plae-wave ad higher-order mode models icludig; frequecies of maimum atteuatio, phase differece of recombied waves (rad), ad trasmissio loss (db)37 Figure 3.3. Sigle-mode frequecies of maimum atteuatio, phase differece of recombied waves, ad trasmissio loss for modes; (a) =, (b) =, (c) =, ad (d) = Figure 3.4. Sigle-mode aalysis of system characteristics with varyig HQ tube area, S=.3,.64,.7,.54, ad 5.8 cm, for modes; (a) =, (b) =, (c) =, ad (d) =3...4 Figure 3.5. Trasmissio loss (db) versus frequecy (Hz) for HQ system with disturbace comprised of the odd modes µ= ad 3; (a) together ad (b) idepedetly...4 Figure 3.6. Modal amplitude vectors at 495 Hz for disturbace comprised of µ= ad 3 modes; (a) µ= modal amplitudes ad (b) µ=3 modal amplitudes...4 vi

7 Figure 3.7. Modal amplitude vectors at 495 Hz; (a) ad (b) disturbace cosists of µ= mode, (c) ad (d) disturbace cosists of µ=3 mode oly...44 Figure 3.8. Trasmissio loss (db) versus frequecy (Hz) for HQ system with disturbace cosistig of; (a) all eve modes µ=,, ad 4, ad (b) eve modes idepedetly...46 Figure 3.9. Modal amplitude vectors at 495 Hz for disturbace comprised of µ=,, ad 4 modes; (a) µ= modal amplitudes ad (b) µ= modal amplitudes...47 Figure 3.. Modal amplitude vectors at 495 Hz; (a) ad (b) disturbace cosists of µ= mode, (c) ad (d) disturbace cosists of µ= mode oly...48 Figure 3.. Trasmissio loss (db) versus frequecy (Hz) for HQ system comprised of all si cut-off disturbace modes µ=,,, 3, ad Figure 3.. Trasmitted modal amplitude vectors at 4 Hz for disturbace comprised of µ=,,, 3, ad 4 modes...5 Figure 3.3. Trasmitted modal amplitude vectors at 4 Hz with disturbace cosistig of mode; (a) µ=, (b) µ=, (c) µ=, ad (d) µ= Figure 3.4. Modal suppressio of trasmitted mode µ=3 at 4 Hz, disturbace cosists of µ=,,, 3, ad 4 modes...54 Figure 3.5. Trasmissio loss (db) versus tube aial locatio, c (m), with disturbace comprised of µ=,,, 3, ad 4 modes at (a) 4 Hz ad (b) -3 Hz...56 Figure 3.6. Trasmitted modal amplitude vectors at 4 Hz for c = -.9 m ad disturbace comprised of µ=,,, 3, ad 4 modes...57 Figure 3.7. Trasmissio loss (db) versus tube legth, L (m), with disturbace comprised of µ=,,, 3, ad 4 modes at (a) 4 Hz ad (b) -3 Hz...58 Figure 3.8. Trasmitted modal amplitude vectors at 4 Hz for L= m ad disturbace cosistig of modes; (a) µ=, (b) µ=, (c) µ=, ad (d) µ= Figure 3.9. Trasmissio loss (db) versus iterface distace, l (m), with disturbace comprised of µ=,,, 3, ad 4 modes at (a) 4 Hz ad (b) -3 Hz...6 Figure 3.. Trasmitted modal amplitude vectors at 4 Hz for l= m ad disturbace cosistig of modes; (a) µ=, (b) µ=, (c) µ=, ad (d) µ=3...6 Figure 3.. Trasmissio loss (db) versus tube area ratio, AR (%), with disturbace comprised of µ=,,, 3, ad 4 modes at (a) 4 Hz ad (b) -3 Hz...6 vii

8 Figure 3.. Trasmitted modal amplitude vectors at 4 Hz for S=.. m ad disturbace cosistig of modes; (a) µ=, (b) µ=, (c) µ=, ad (d) µ= Figure 3.3. Trasmissio loss (db) versus frequecy (Hz) for both sigle ad double tube-array systems with disturbace comprised of µ=,,, 3, ad 4 modes ad c =.654 m...64 Figure 3.4. Trasmitted modal amplitude vectors at 43 Hz for double array HQ system with disturbace comprised of µ=,,, 3, ad 4 modes ad c = m...65 Figure 4.. Trasmissio loss (db) versus frequecy (Hz) with optimized HQ tube legth, L (cm), at each frequecy ad disturbace comprised of µ=,,, 3, ad 4 modes...68 Figure A.. Two-dimesioal ifiite...7 Figure B.. Geometry ad omeclature for Quicke tube derivatios Figure C.. Itegratio of Gree s fuctios; (a) pisto source local coordiate system ad (b) eamples of itegrated Gree s fuctio otatio...85 List of Tables Table 3.. Power ad amplitudes of modal compoets at 495Hz for odd modes µ= ad Table 3.. Power ad amplitudes of modal compoets at 495Hz for eve modes µ= ad..47 Table 3.3. Power ad amplitudes of modal compoets at 4Hz for modes µ=,,,ad 3..5 viii

9 Chapter - Itroio. Prologue Duct acoustics is the study of the propagatio of soud withi a cofied space. Give a certai oise source, the emitted soud will propagate with characteristics primarily depedet upo the surroudig geometry, acoustic properties of the boudary, ad the frequecy of ecitatio. A stadard geometry is the ifiite rigid-walled. The soud field withi such a is composed of both stadig waves i the trasverse directio ad travelig or evaescet waves i the logitudial directio. At low frequecies, those below the first cut-off frequecy of the system, the soud field is oe-dimesioal ad oly the plae-wave mode will propagate withi the. At higher frequecies, above the first cut-off frequecy, the soud field becomes more comple due to the presece of higher-order modes. These higher-order modes iclude both propagatig ad evaescet compoets []. There are may oise cotrol strategies, which ca be utilized to reduce oise i s. I geeral, oise cotrol methods ca be applied to the oise source, the trasmissio path, ad/or the receiver. However, oise problems are ofte times oly cosidered i pre-eistig systems where may of these modificatios ca be difficult to implemet due to iheret limitatios []. I additio, oise cotrol strategies ca be characterized as active, passive, or a combiatio of both passive ad active approaches. Active oise cotrol techiques i s cosist of the itroio of secodary oise sources to either cacel, suppress, or absorb part of the origial soud field [3]. This method typically ivolves usig a digital processor which drives a set of secodary or cotrol oise sources based o iformatio gathered about the soud field from a set of sesors, i.e. microphoes. Usually, the secodary sources will produce a soud field that is out of phase with the origial field ad oise cacellatio occurs due to destructive iterferece. The umber of secodary sources ad sesors required is determied by the compleity of the origial soud field ad the magitude of cacellatio eeded [4].

10 Lori A. Brady Chapter - Itroio Passive oise cotrol techiques i s geerally cosist of either the absorptio or the reflectio of the origial soud field. Absorptive techiques cosist of liig a with soudabsorbig materials, also called liers, as show i Figure.. These liers, which typically cosist of porous materials like fiberglass or foam, react locally with the soud field ad through frictio atteuate soud eergy through heat. Specific properties for these materials, which will determie its effectiveess i absorbig soud, iclude the thickess of the treatmet, its desity, ad its acoustic impedace [5]. Geerally speakig, the thickess of the soud absorbig material is related to the rage of wavelegths able to be atteuated. With icreasig thickess, a lier will be able to atteuate loger wavelegths, i.e. lower frequecies. For this reaso, liers are ormally used for high frequecy broad-bad disturbaces, which cosist of maily higherorder modes. Rigid-wall boudaries Protective Facig Liig Figure.. Passive acoustic lier i rigid-walled Reactive techiques cosist of chages i the geometry, which result i a reflectio of part of the origial soud field back toward the oise source. These reactive devices iclude, but are ot limited to, epasio chambers, Helmholtz resoators, ad Herschel Quicke tubes. All of these techiques are typically used for lower frequecy disturbace applicatios, which cosist of the plae-wave mode [6].

11 Lori A. Brady Chapter - Itroio 3 Both the epasio chamber ad the Helmholtz resoator are show i Figure.. The epasio chamber, i Figure.(a), is qualified as a acoustic low-pass filter, which is characterized by a elarged sectio of. At low frequecies, the soud reio is a fuctio of the relative differece betwee the ad epasio areas. The Helmholtz resoator, as show i Figure.(b), is used as a acoustic bad-pass filter. Thus resoator creates a bad of atteuatio cetered o a frequecy which is depeded upo the resoator volume, eck area, ad eck legth [7]. Duct Area Epasio Area Duct Area Volume Neck Legth Neck Area Epasio Legth (a) Epasio chamber (b) Helmholtz resoator Figure.. Passive oise cotrol devices reactive type A Herschel-Quicke (HQ) tube is essetially a hollow side-tube that travels alog a mai- ais ad attaches to the mai- at each of the two eds of the tube, as show i Figure.3. I geeral, a icidet plae-wave acoustic wave, travelig to the right, ecouters a brach i the path at the first itersectio of the side-tube ad mai-, amed the ilet of the HQ tube. The icidet wave divides ad will later recombie at the secod itersectio of the side-tube ad mai-, similarly amed the outlet of the HQ tube. A differece i path legth will create a phase shift betwee the recombied sigals ad cosequetly atteuatio of soud at a umber of discrete frequecies. Chagig tube parameters such as legth (L), area (S), ad the distace betwee ilet ad outlet opeigs, termed the iterface distace (l), the frequecies of cacellatio ca be adjusted.

12 Lori A. Brady Chapter - Itroio 4 This thesis will ivestigate the HQ tube approach for the case of higher-order acoustic mode disturbaces. This should be cotrasted to previous studies of the HQ approach where oly plae-waves were cosidered i the system. Mai- L HQ Tube Icidet Disturbace Soud y c S h Recombied Trasmitted Soud l Figure.3. Herschel-Quicke tube cocept. Review of Literature o Hershcel-Quicke Tube Cocept Herschel [8] first discussed the idea of acoustic iterferece of musical toes i such a system i 833. He predicted that cacellatio of toes would occur whe the path legth differece betwee the recombied sigals was (m+)(λ/), where λ is the wavelegth of the acoustic wave ad m is ay iteger. Later, i 866, Quicke [9] eperimetally validated that Herschel s system did effectively cacel soud. Stewart [], i the th cetury, foud Herschel s theoretical eplaatio to be isufficiet to iterpret eperimetal data he had observed. Usig a plae-wave aalysis, Stewart derived a aalytical model describig the ratio of trasmitted to icidet soud itesity i the system. He verified that cacellatio does occur whe the path legth differece is (m+)(λ/), as predicted by Herschel [8]. However, it was foud that atteuatio also occurs whe the path legth differece is mλ with limited atteuatio at other trasitioal frequecies.

13 Lori A. Brady Chapter - Itroio 5 Stewart s model assumed that the HQ tube system was composed of three sectios, each with costat cross-sectioal area. These three sectios described; the etrace portio of the mai- (up to the side-tube ilet), the HQ tube ad sectio of mai- betwee the HQ tube itersectios, ad the eit portio of the mai (followig the HQ tube outlet). The cross-sectioal areas of the HQ tube ad the sectio of mai- betwee the HQ tube itersectios were assumed to be equal. Although, Stewart s work was a valuable cotributio i the aalysis of HQ tubes, the model could oly be used i the absece of flow ad was still limited to plae wave acoustic fields. Oly recetly did Selamet, et. al. [, ] eted Stewart s work by derivig a aalytical model without the limitatios o cross-sectioal geometry. However this model still does ot iclude flow or higher-order modes. It is iterestig to metio that approimately 65 years passed betwee Stewart s ad Selamet s publicatios. I fact, durig this time period very little work was reported i the literature o the subject of HQ tubes. Beyod geeralizig Stewart s work, Selamet, et. al. [3-6] cotiued to develop the aalysis of the HQ systems to iclude alteratives i both modelig approach ad system geometry. I 994, Selamet, et. al. [3] studied multi-dimesioal effects, see at mai- ad HQ tube iterfaces, usig oe-dimesioal fiite differece ad three-dimesioal boudary elemet umerical approaches. It was foud that both umerical schemes compared well with each other ad with eperimetal data, idicatig the oe-dimesioal behavior of the soud fields well below the first mai- higher-order mode cut-off frequecy. However, some oplaar or higher-order mode behavior was observed at the HQ tube ilet. Still, Selamet, et. al. [3] cocluded that the mai- side-tube iterfaces did ot cotribute to ay deviatio from the oe-dimesioal plae-wave behavior. I 995, Selamet, et. al. [4] modeled several acoustic silecers, icludig the HQ tube system, usig a oe-dimesioal time-domai umerical approach. This model icorporated oliear behavior by simulatig usteady compressible flow. The objective of the work was to compare the well kow liearized acoustic theory with the umerical solutio, assumig zero mea flow. I the case of the HQ tube, it was foud that the umerical model correlated well

14 Lori A. Brady Chapter - Itroio 6 with acoustic theory i the lower half of the frequecy rage studied, but with icreasig frequecy some variatio could be see. However, this discrepacy was ot eplaied ad further validatio of the model was suggested for a future paper with eperimetal data. They speculated that if prove, the potetial for this modelig approach is substatial for the reaso that ow oliear flow disturbaces, as see i iteral combustio egies, could be cosidered. However, o publicatio was foud for the model validatio. I 996, Selamet, et. al. [5] looked at the ifluece of a epasio chamber i the HQ side-tube usig oce agai a plae-wave modelig approach, as show i Figure.4. Typically, i a HQ tube system without a epasio sectio, it is possible to cotrol the resoat frequecies simply by chagig the legth of the HQ tube. However, with the additio of a epasio chamber, it was foud that substatial shiftig of the resoaces ca occur by chagig the distace of the epasio chamber to the HQ tubes. This ca be useful i the desig of a HQ tube system for a oise cotrol applicatio i which the side-brached tubes are costraied to a certai legth. l l S l 3 S S 3 l h Figure.4. Herschel-Quicke tube with epasio chamber I 997, agai Selamet, et. al. [6] published a paper cosiderig yet aother geometrical variatio o the stadard HQ tube system. This work eteded the classical HQ tube system modelig to a multiple HQ tube cofiguratio. It was foud that with a icreasig umber of

15 Lori A. Brady Chapter - Itroio 7 HQ tubes the soud atteuatio characteristics become more ad more complicated. For this reaso, umerical results were limited to two ad three HQ tube arragemets. I geeral, it was foud that the HQ tube diameter iflueces the width of trasmissio loss bads ad HQ tube legth ifluece system resoat frequecies. Also, it was show that trasmissio loss characteristics for two ad three HQ tube cofiguratios differ greatly. More importatly, the itetio of the paper was to simply preset the modelig method for the multiple HQ tube system. This model ca be valuable i determiig the cost or beefit of a icreasig umber of HQ tubes for a specific oise cotrol applicatio. Although these previous works produced a wealth of theoretical eplaatios of HQ tube systems for various geometric ad modelig cofiguratios, still the basic uderstadig is limited to systems with oe-dimesioal plae-wave soud fields. The ifluece of the HQ tube system o more comple acoustic fields has ot as of yet bee addressed..3 Objectives There are two mai objectives i this work. The first, is to develop a aalytical model to address the effect of the HQ tubes whe applied to a i the presece of a comple acoustic field, i.e. higher-order modes. The secod is to determie the potetial of the HQ tube system to suppress higher-order mode disturbaces ad the physical mechaisms resposible for this atteuatio. Specifically, the developmet of the theoretical model will attempt to accomplish several tasks. The first is to model the ifluece of the two-dimesioal HQ system o higher-order mode acoustic fields. The secod is to reevaluate the relevacy of classical oe-dimesioal HQ theory. Fially, this model will be used to gai isight ito the physics behid the soud atteuatio of higher-order modes. The realizatio of these objectives will beefit future work i the desig of HQ tubes for practical oise reio applicatios.

16 Lori A. Brady Chapter - Itroio 8.4 Outlie of Thesis The mai goal of this thesis is to ivestigate the potetial of the HQ tube system to suppress higher-order mode acoustic disturbaces i two-dimesioal s. Chapter presets the aalytical modelig techique developed to address the effect of the HQ tubes whe applied to a i the presece of higher-order modes. This chapter also presets the etesio of the basic system aalysis to iclude HQ tubes i series, a compariso with the traditioal plae wave acoustic theory, ad various useful modelig simplificatios. I Chapter 3 umerical results are preseted, i the form of parametric studies, to idetify ad eplai the physical mechaisms behid the atteuatio of higher-order modes. Various chages to the modal compoets of the disturbace ad HQ tube geometry will be ivestigated. Chages i the modal compoets of the disturbace will cosist of icreasig the modal compleity of the disturbace while eamiig the system trasmissio loss characteristics. Modificatios to the tube geometry will iclude chages i tube aial locatios, legths, iterface distaces, cross-sectioal areas, ad the additio of a secod pair of HQ tubes. Fially, i Chapter 4, the coclusios for the thesis are preseted. The theory of the HQ tube systems, icorporatig higher-order mode acoustic disturbaces, is reviewed. The mai coclusios are described ad recommedatios are made for future research ad applicatios of this modelig techique ad the HQ tube for oise cotrol applicatios.

17 Chapter Aalytical Modelig I this chapter, the theoretical aalysis for the two-dimesioal HQ tube system is preseted. This aalytical model is desired i order to eplore the physical mechaisms behid the atteuatio of soud ad to ivestigate how chages i geometric parameters ifluece this atteuatio. Of primary cosideratio is developig a set of equatios that describes the soud field withi the due to a kow higher-order mode disturbace ad modified by the presece of the HQ tubes. This is accomplished usig a Gree s fuctio techique assumig that the iterfaces of the ad tubes ca be modeled as fiite pisto sources. This aalysis is preseted for both a sigle pair of HQ tubes ad for two pairs of HQ tubes i series. It should be kept i mid that the sigle-pair system is cosidered the primary system of iterest ad is modeled i detail. The system icludig two HQ tube pairs i series is simply a etesio of the basic sigle-pair system model. I additio, the aalysis will be eteded to iclude several closed form theoretical results which allows for a direct compariso with the traditioal plae-wave aalysis. Also, it will be show how to predict frequecies at which maimum atteuatio will occur icludig the predictio of atteuatio at the mode cutoff frequecies.. Hershcel-Quicke Tube System Cocept A disturbace oise field is assumed to propagate withi a two-dimesioal of height h i the positive -directio. The pressure due to the disturbace ca be foud assumig that the system ca be modeled as a ifiite rigid-walled as show below i Figure.. y=h y Disturbace Field r,t p D ( ) h Rigid-wall boudaries y= Figure.. Simplified model - two-dimesioal ifiite 9

18 Lori A. Brady Chapter Aalytical Modelig The pressure due to the disturbace, p D, at ay poit i the, r = (, y) r, ca be epressed as the sum of a set of N D modes of order µ propagatig i the positive directio. That is: p D r (,t ) i ( ) ( ωt k y e ) N = D Aµ Φ µ κ µ µ = (.) where A µ is the kow comple amplitude of the µ th mode (with µ= represetig the plaewave mode), Φ µ (.) is the eigefuctio or mode, ad κ µ is the eigevalue give as: ( κ y) cos( κ y) µπ = µ ; κ (.a,b) h Φ µ µ µ = I Figure., the eigefuctios are plotted for modes µ=,,, 3, ad 4, respectively. The positive ad egative sigs represet the istataeous positive ad egative acoustic pressure fluctuatios. _ + _ + y=h _ + + _ + µ= µ= µ= µ=3 µ= y= Figure.. Eigefuctios Φ(κ µ y) for µ=,,, 3, ad 4

19 Lori A. Brady Chapter Aalytical Modelig The propagatio characteristics of the modes are give by the aial waveumber k, which is give by: k k κ µ, = i κ µ k, k > κ µ k < κ µ (.3a,b) where k=ω/c is the free-field waveumber ad c is the speed of soud. A mode will propagate for k>κ µ ad will decay whe k<κ µ. The frequecy at which this chage i propagatio characteristic occurs is termed the cut-off frequecy for mode µ. It is give for k = κ µ yieldig: cπ ω µ ( rad s) = µ, µ =,,, 3... (.4) h I Equatio (.), oly propagatig modes that satisfy the coditio i Equatio (.3a) are assumed i the disturbace at each frequecy. The derivatio of the disturbace pressure distributio, as described above, is preseted i more detail i Appedi A. The Herschel-Quicke tube system cosists of a two-dimesioal ifiite rigid-walled which is modified to icorporate two side-brached tubes, symmetrically located about the ais. These tubes are referred there as a pair of Herschel Quicke (HQ) tubes. Figure.3 shows the system with a sigle pair of HQ tubes mouted at both the top ad bottom of the. For the sake of clarity ad uderstadig the dyamics of the system, the HQ tubes are described i two dimesios. Two HQ tubes are used to take advatage of simplificatios that ca be made due to symmetries whe icorporatig higher-order modes. The tubes are assumed to be of costat cross-sectioal area S, per uit width, ad have ceterlie legth L. The tubes aial positio is defied by the coordiate c at the first tube- iterface. The distace betwee the tube opeigs alog the ais is deoted as l.

20 Lori A. Brady Chapter Aalytical Modelig Mai- L HQ Tube c l y S h Figure.3. Herschel-Quicke tube modelig cocept The applicatio of these HQ tubes as a oise reio device has bee previously studied i great detail, but for oe-dimesioal soud fields oly, i.e. plae waves. The most recet theoretical treatmet, for the basic HQ system with a plae-wave soud field, was derived by Selamet, et. al. []-[] ad for the sake of completeess is reproduced i Appedi B. However, these plae-wave models ca ot be used to ivestigate the performace of HQ tubes i the presece of higher-order modes. Thus, the mai goal i this chapter is to develop models to study the oise reio mechaisms of HQ tubes for higher-order modes. The modelig of the HQ tube- system is carried out by first separatig the HQ tubes from the at the iterfaces, as show i Figure.4. The models of the soud fields i the HQ tubes ad i the mai are first developed idepedetly. The, these models are fully coupled by matchig the acoustic pressure ad particle velocity at the tube- iterfaces. The effect of the tubes o the is modeled by cosiderig the tube- iterfaces as fiite pisto sources radiatig ito the. Each pisto is assumed to have a ukow velocity v s (s=,, 3, ad 4) which represets the particle velocities at the eds of the tubes. It is importat to remark that, these pisto sources ca geerate both plae-waves ad higher-order modes i the.

21 Lori A. Brady Chapter Aalytical Modelig 3 ˆ HQ Tube Mai- p tube, vtube pisto source radiators p tube, vtube p, v p, v y p p 3 3, v 3 3 tube, vtube pisto source radiators p p 4 4, v 4 4 tube, vtube ˆ Figure.4. Ucoupled Herschel-Quicke system with fiite pisto source radiators HQ Tube Model I this study, the HQ tubes have bee depicted as semi-circular i shape. However, for modelig purposes they are cosidered as straight tubes with uiform cross-sectios, as show i Figure.5. The soud fields iside the separated tubes are assumed to cosist of plae-waves, i.e. there are o higher-order modes. This assumptio is valid as log as the frequecy rage of iterest is well below the first cut-off frequecy of the HQ tubes. This upper limit frequecy is defied as f upper = c S.

22 Lori A. Brady Chapter Aalytical Modelig 4 HQ Tube Legth L (a) i( t kˆ ) B + e ϖ p tube ( ) ptube ( L) v ( ) v ( L) tube i( t kˆ B e ϖ + ) tube ˆ (b) Figure.5. Ucoupled Herschel-Quicke tube with fiite pisto source radiators; (a) represetatio ad (b) simplified model The soud field iside the tube is give by: p tube ( ˆ,t ) = B + i( ωt kˆ ) i( ωt+ kˆ e + B e ) (.5) where B + ad B - represet the comple modal amplitudes of the plae-waves i the tube i the positive ad egative propagatig directios, respectively. The aial positio withi the HQ tube is deoted by the local coordiate ˆ. The acoustic pressures ad particle velocities at the eds of the tubes, i.e. ˆ = ad ˆ =L, are epressed as: p v tube tube + iωt + ikl ikl (,t) = ( B + B ) e ptube ( L,t) = ( B e + B e ) + iωt + ikl ikl ( B B ) e ( B e B e ) (,t) = v ( L,t) = ρc tube ρc e i e iωt ωt (.6a,b) (.6c,d) where ρ is the fluid desity, e.g. ρ=.ns /m 4 i air at C. The acoustic pressure ad particle velocity at the ed of the tube ca be epressed i matri form as:

23 Lori A. Brady Chapter Aalytical Modelig 5 p v tube tube ( L) ( L) cos( kl ) = i si ρc ( kl) ( kl) iρc si p cos( kl ) v tube tube ( ) ( ) (.7) Rearragig Equatio (.7), the acoustic pressure at the ed of the tube ca be rewritte i terms of the velocity as: p p tube tube ( ) ( L) ( ) ( L) ( ) ( L) cot( kl ) csc( kl ) vtube vtube = iρ c = [ Z tube ] (.8) csc( kl ) cot( kl ) vtube vtube where Z tube represets the impedace matri for the tube that relates the particle velocity to the pressure at the two eds of the tube. Equatio (.8) is valid for both of the HQ tubes i Figure.4. Thus, the relatioship betwee the acoustic pressures ad the particle velocities for both HQ tubes is coveietly writte i matri form as: p p p p tube tube 3 tube 4 tube = [ Z ] tube [ Z ] tube v v v v tube tube 3 tube 4 tube (.9) where the superscripts,, 3, ad 4 deote the four eds of the two tubes as idicated i Figure.4. Duct Model with Fiite Pisto Sources ad Disturbace As depicted i Figure.6, the soud field i the separated mai is obtaied as the liear cotributio of the soud due to the four pisto sources ad the disturbace. The pressure at a arbitrary poit r r i the mai ca be epressed as: p r 4 ( r ) p ( r ) + p ( r ) = s= s r D r (.)

24 Lori A. Brady Chapter Aalytical Modelig 6 r where ( ) p s is the pressure due to the s th r pisto source ad ( ) p D is the pressure due to the disturbace. The pressure due to the disturbace is give i Equatio (.). O the other had, the pisto sources are assumed to move with ukow comple velocities, i.e. v s is the velocity of the s th pisto. Thus, the soud field geerated by the pisto sources is required ad it is obtaied by itegratig the rigid-walled Gree s fuctio for a ifiite over the crosssectio of the source [7]. That is: s s + S s G( s S r r r p ( ) = iωρ v ) d (.) where s is the ceter coordiate of the s th r r pisto source. The Gree s fuctio ( ) G is defied as the pressure at ay poit, r, due to a poit source located at r. This is derived i Appedi C ad is give as: r G( ( κ y ) Φ ( κ N g i ik iωt ) = e e = Λk r Φ y ) (.) I Equatio (.) the aial waveumber k, eigefuctios Φ (.), ad eigevalues κ are the same as give i Equatios (.)-(.3), but distiguished by the subscript represetig the modal order of the Gree s fuctios. The orthogoalizatio costat, Λ, is h if = (the plae-wave mode) ad h/ if > (the higher-order modes) ad N g is the umber of modes icluded i the Gree s fuctio. Note that i the Gree s fuctio i Equatio (.) both propagatig ad evaescet modes are icluded. Thus, the effect of the pisto source earfields are icluded by the evaescet modes.

25 Lori A. Brady Chapter Aalytical Modelig 7 Mai- Pisto Source c l Disturbace Soud Field y p p, v 3 3, v S p p, v 4 4, v Figure.6. Ucoupled mai- with fiite pisto source radiators h Oce the pisto velocities are foud, the acoustic pressure aywhere i the mai- ca ow be computed usig the epressios i Equatios (.), (.), ad (.). To this ed; (i) the pressure ad particle velocity at the iterface betwee the HQ tubes ad the mai- eed to be matched. Thus, the average pressure over the pisto sources will be matched to the uiform pressure ad (ii) the pisto velocity will be matched to the particle velocity at the eds of the tubes, respectively. Usig Equatio (.), the average acoustic pressure over the pistos ca be epressed i matri form as: p p p p 3 4 = iωρ ~ G ~ G ~ G v v v v 3 4 p p + p p D D 3 D 4 D = [ Z ] s v v v v 3 4 p p + p p D D 3 D 4 D (.3)

26 Lori A. Brady Chapter Aalytical Modelig 8 where s p is the average pressure over the s th pisto due to the disturbace ad all four pisto sources ad Z s is a impedace matri that relates the pisto source velocities to the average pressure over the sources. Thus, the average pressure over the s th pisto ca be writte as: 4 p = iωρ v + p (.4) s s' = rs s s D The rs elemet of the impedace matri Z s, i Equatio (.4), represets the average pressure over the r th pisto source due to a uit velocity of the s th pisto source. To derive this fuctio, the pressure from Equatio (.) due to the s th pisto eeds to be itegrated over the surface of the r th pisto. From the symmetry of the problem i Figure.7, there are oly two cases to cosider, which are preseted i Appedi C. The elemets of the Z s matri are give as: rs N ( yr ) ( ys ) g Φ κ Φ κ rs = e = Λk Ng Φ ( κ yr ) Φ( κ ys ) = i e = Λ k s ik ikl, si k r s, = r s s (.5) ad are illustrated i Figure.7. Mai- c l y 3 4 S Figure.7. Samples of the itegrated Gree s fuctios

27 Lori A. Brady Chapter Aalytical Modelig 9 HQ Tube-Duct Couplig Figure.8 shows the coupled system. The models for the HQ tubes ad the mai- are coupled by matchig (i) the pressure o the surface of the pistos to the costat pressure at the ed of the HQ tube ad (ii) the pisto velocity to the particle velocity at the ed of the HQ tube, i.e. s tube s p = p ad s tube s v = v, for s=,, 3, ad 4 (see Figure.4). Replacig s v tube by s v i Equatio (.9) ad substitutig this ito Equatio (.3) leads to a liear system of equatios that ca be solved for the ukow pisto source velocities, s v. That is: v v v v 3 4 = [ Z ] tube [ Z ] tube [ Z ] s p p p p D D 3 D 4 D (.6) Mai- L HQ Tube p R p D r ( r,t) r,t y ( ) v c 3 v l v 4 v h p T r ( r,t) S Figure.8. Coupled Hershcel-Quicke system Oce the source velocities are kow, the pressure at ay locatio withi the ca be foud usig Equatio (.). It is coveiet to epress the pressure withi the i terms of the disturbace, trasmitted, ad reflected modal amplitudes as depicted i Figure.8. The

28 Lori A. Brady Chapter Aalytical Modelig modal amplitude of the disturbace is give i Equatio (.), i.e. A µ. The modal amplitudes of both the trasmitted ad reflected waves are foud usig Equatio (.). The trasmitted pressure is defied at c +l ad is give as: p T 4 T (, y) p (, y) + p (, y) = s= s D, c + l y h (.7) where T p s is the trasmitted acoustic pressure due to the s th pisto. Addig the cotributios of the four pisto sources ad the disturbace, the trasmitted soud field is give as: p T (, y) N g si = ωρ = Λ ND + A µ = µ s ( k ) k cos e ik c ( ) i( ω t k ) κ y e µ ik 3 4 ik i [( ) ( ) ( )] ( ) ( t k ) l l ω v + v e cos κ h + v + v e cos κ y e (.8) where the velocity terms are obtaied from Equatio (.6). Thus, the modal amplitude of the trasmitted th modes is give as: A T = ωρ si Λ s ( k ) k e ikc ikl 3 4 ikl [( v + v e ) cos( κ h) + ( v + v e )] + A, =,...,N > c + l g (.9) Similarly, the reflected pressure distributio is defied at < c ad is give as: p R 4 R (, y) p (, y) = s= s, < c y h (.) that leads to: p R (, y) N g si = ωρ = Λ s ( k ) k e ikc ik 3 4 ik [( ) ( ) ( )] ( ) i( t k ) l l ω + v + v e cos κ h + v + v e cos κ y e (.)

29 Lori A. Brady Chapter Aalytical Modelig where the velocity terms are agai obtaied from Equatio (.6). Thus, the modal amplitude of the reflected modes is give as: A R = ωρ si Λ s ( k ) k e ikc ikl 3 4 ikl [( v + v e ) cos( κ h) + ( v + v e )], =,...,N < c g (.) Usig the modal amplitudes of the disturbace, trasmitted, ad reflected pressure distributio, the performace of the HQ tube system will be studied. The effectiveess of the HQ tubes as a oise cotrol device is determied usig the trasmissio loss (TL) as a metric. The TL is defied as a ratio of the disturbace to trasmitted power as: TL log Π Π = D (.3) T Power is the rate at which eergy flows through the aially ad is give as the acoustic itesity i the -directio itegrated across the cross-sectio: r Π I ( ) dy (.4) = h where the itesity is defied as: I r r r = (.5) ( r ) real[ p ( r ) v ( r )*] ad the otatio * idicates the comple cojugate. The particle velocity is defied usig Euler s equatio ad i the aial directio is give as: r v( ) = iωρ p r ( r ) (.6)

30 Lori A. Brady Chapter Aalytical Modelig Thus, the epressio for trasmitted power i Equatio (.4) ca be rewritte i terms of the disturbace, trasmitted, ad reflected modal amplitudes: Π D = real ωρ * ( k ) Aµ Λµ * ( k ) AT Λ Π T = real (. 7a,b,c) ωρ Π R = real ωρ * ( k ) AR Λ where the modal amplitudes A µ, ad Equatio (.) respectively. A T, ad A R are defied i Equatio (.), Equatio (.9),. Herschel-Quicke Tubes i Series The previously described model for the sigle pair of HQ tubes served the purpose of describig the modelig approach i detail. To eted the model to multiple pairs of HQ tubes, the same approach is take. This aalysis is udertake to study whether multiple pairs have sigificat atteuatio beefits over a sigle pair system. The ifiite is modified to icorporate two sets of HQ tube pairs, placed i series as show i Figure.9. The first pair of tubes has the same cofiguratio as the sigle-pair system described i Sectio.. The secod pair of HQ tubes is agai assumed to be of costat crosssectioal area S ad ceterlie legth L. The tubes are positioed aially at a distace c from the outlets of the first pair. The distace betwee the tube opeigs alog the ais is deoted as l.

31 Lori A. Brady Chapter Aalytical Modelig 3 Mai- L st HQ Tube Pair d HQ Tube Pair L v v 5 v 6 v c l c l h y 3 v S 4 v 7 v S 8 v Figure.9. Herschel-Quicke tubes i series The modelig approach is idetical to the method used for the sigle-pair system. The modelig is carried out by first separatig the HQ tubes from the at the iterfaces. The models for the soud fields i the HQ tubes ad i the mai are first developed idepedetly. The, these models are fully coupled by matchig both the pressure ad particle velocity at the HQ tube ad mai- iterfaces. The effect of the tubes o the is modeled by cosiderig the tube- iterfaces as fiite pisto sources radiatig ito the. Each pisto is assumed to have a ukow velocity v s (s = through 8) which represets the particle velocities at the eds of the tubes. These pisto sources ca geerate both plae-waves ad higher-order modes iside the. The pressure distributio withi the HQ tubes is assumed to cosist of oly plae-waves as epressed by Equatio (.5). Thus, oce agai, the soud field withi the mai is give as the summatio of the pressure from the eight pisto sources ad the disturbace. That is: p s 8 ( r ) p ( r ) + p ( r ) = s= s r D r (.8)

32 Lori A. Brady Chapter Aalytical Modelig 4 where the pressure due to the disturbace p D is evaluated usig Equatio (.) ad the pressure due to a fiite pisto source is agai obtaied by itegratig the rigid-walled Gree s fuctio for a ifiite, as described i both sectio. ad Appedi C. Oce the soud fields withi the separated HQ tubes ad mai are defied, the couplig of these soud fields is accomplished by matchig both pressure ad particle velocity at the iterfaces. The pisto velocities are ow attaied as: v ~ pd TD I TDII I,I I,II = 8 II I G M iρ c k II,I M (.9) TD TD II,II 8 v pd where the sub-matrices for the HQ tube dyamics are give by: [ ψ ] [ ] [ ] [ ψ ] [ ζ ] [ ] [ ] [ ζ ] TDI = TDII = (.3) ad the matrices Ψ ad ζ are give by: ψ cot = csc( kl ) ( kl) csc( kl ) cot( kl ) csc( kl ) ζ = csc( kl ) csc( kl csc( kl ) ) (.3) The secod matri i Equatio (.9) is defied as:

33 Lori A. Brady Chapter Aalytical Modelig 5 I,I II,I = = ~ G ~ G ~ G ~ G ~ G ~ G I,II II,II = = ~ G ~ G ~ G ~ G ~ G ~ G (.3) where the sub-idices I ad II deote associatios with the first ad secod HQ tube pairs respectively. Therefore, as a eample the elemets of matri I, II represet the average pressure over the pisto sources of the first HQ tube pair due to a uit velocity of pistos o the secod HQ tube pair. Oce the source velocities are kow, the pressure at ay locatio withi the ca be foud usig Equatio (.8). Agai, it is coveiet to epress the pressure withi the i terms of the disturbace, trasmitted, ad reflected modal amplitudes. The amplitude of the trasmitted ad reflected modes are give as: A A T R N ik c e = ωρ Λ k si + si ikc g e = ωρ = Λ k s ikl ikl ( k )( [ ) ( ) ( )] v + ve cos κ h + v3 + v4e + A s ik ( l+ c ( ) ) ikl ikl k [( + ) ( ) + ( + )] e v5 v6 e cos κ h v7 v8e s ikl ikl si( k )( [ v + ve ) cos( κ h) + ( v3 + v4e )] s ik ( l+ c ( ) ) ikl ikl + si k e [( v5 + v6 e ) cos( κ h) + ( v7 + v8e )] (.33a,b) where the equatios are valid as log as > c +l+ c +l ad < c for the trasmitted ad reflected wave amplitudes respectively.

34 Lori A. Brady Chapter Aalytical Modelig 6.3 Modelig Simplificatios Sigle HQ Tube Pair The models developed i previous sectios allow for the predictio of the reio of oise i a for a arbitrary cofiguratio of HQ tube pairs. The accuracy of the model depeds o the umber of modes to be icluded i the Gree s fuctio. However, it is iterestig to derive simplified epressios that would allow for easier uderstadig of the physics as well as guide i prelimiary desig of the HQ tube system. I this sectio, simplified epressios are preseted to accomplish these goals. The first step is to take advatage of the reciprocity property of the impedace fuctios i the case of o fluid flow. That is: rs = sr (.34) which implies that the average pressure at pisto r due to a pisto source at s is equivalet to the pressure at pisto s do to the pisto source at r. I other words, it is the distace betwee the observatio pisto ad source pisto which characterizes the impedace. This results from the epoetial term foud i the Gree s fuctio, which is depedat upo the absolute value of the differece betwee the observatio poit ad source locatio i the -directio. I additio, it is clear that: = = (.35) = Also, due to the symmetric placemet of the HQ tubes about the ais, the followig simplificatios ca be made (see figure.8): = 34 3 = 4 4 = 3 (.36) Cosiderig all of the above simplificatios, Equatio (.6) ca be reduced to:

35 Lori A. Brady Chapter Aalytical Modelig 7 v v v v 3 4 cot( kl ) csc( kl ) = iρc csc( kl ) cot( kl ) cot( kl ) csc( kl ) k csc( kl ) cot( kl ) ~ G ~ G ~ G 4 3 p p p p D D 3 D 4 D (.37) where it is ow oly ecessary to evaluate four impedace terms. It is possible to further simply the previous equatio by cosiderig that the disturbace iput cosists of eve ad odd modes separately. Due to the symmetric locatio of the HQ tubes, if the disturbace is comprised of oly eve (odd) modes, the pisto velocities at the same aial locatio will be i phase (out of phase) ad of equal magitude; i.e. 3 v = v ad 4 v = v ( v 3 = v ad v 4 = v ). Therfore, the impedace elemets ca be further simplified as = 3 ad = 4 ( =- 3 ad =- 4 ). Usig this iformatio, it is ow oly ecessary to fid two pisto source velocities, either to specify the etire system. Equatio (.37) ca ow be reduced to: v ad v or 3 v ad v 4 v v ~ cot( kl ) csc( kl ) PD = G iρ c k csc( kl ) cot( kl ) (.38) PD where: N Φ g = = Λ k ik ( κ ) s h e N ik g l s Φ ( κ h) e si( k ) = i = Λ k (.39a,b) The system of equatios i Equatio (.38) ca be solved i closed form ad the epressios for the modal amplitudes of the disturbace, trasmitted, ad reflected waves ca be obtaied. They are:

36 Lori A. Brady Chapter Aalytical Modelig 8 A D = A µ (.4) for the disturbace, A T cos s ( κ h) si( k ) ikl ( v + v e ) + A, > + l ikc = ωρ e c Λk (.4) for the trasmitted, ad A R s ( κ h) si( k ) ikl ( v + ve ), c cos ik c = ωρ e < (.4) Λ k for the reflected modes, where the two pisto source velocities, v ad v, are give as: v v = = P P D ( α iωρ ) PD ( β iωρ ) ( α iωρ ) ( β iωρ ) D ( α iωρ ) PD ( β iωρ ) ( α iωρ ) ( β iωρ ) (.43a,b) where: iρc iρc α = ; β = (.44a,b) ta ( kl) si( kl) ad the pressure due to the disturbace is defied usig Equatio (.). This simplified model will be used i a compariso with the plae-wave modelig approach, to predict the frequecies of maimum atteuatio for idividual modes, ad to predict atteuatio at the cut-off frequecy for a idividual mode.

37 Lori A. Brady Chapter Aalytical Modelig 9 Compariso with Plae-Wave Modelig Approach The model developed here for the HQ tubes ca be used to ivestigate ay disturbace modal distributio, i.e. plae-wave ad higher-order modes. Thus, the model should also be capable of predictig the respose at frequecies below the first cut-off frequecy where oly the plae wave mode is preset. The simplified model developed above will be used to predict, i closed form, the atteuatio for the case of a plae wave mode i the disturbace. I additio, this epressio will be compared to the traditioal plae wave model. The plae-wave aalysis method for ivestigatig the HQ tube is show i Appedi B. For the sake of compariso, the higher-order mode model as derived i Sectio. ca be simplified for direct compariso with the plae-wave model. Assumig o fluid flow, the ratio of the disturbace to trasmitted wave amplitude for a sigle mode ca be foud usig Equatio (.4) ad Equatio (.4) as: A A D T = 4ωρ cos ( ) ( Λk α iωρ β iωρ ) s ( h) si( k )( [ i ) ( i κ α ωρ β ωρ ) cos( k l) ] + (.45) This epressio ca be further simplified for the plae wave mode, by settig =, k =k, ad Λ =h givig: A A D T hk = 4ωρ si ( ) ( α iωρ β iωρ ) s ( k )( [ i ) ( i α ωρ β ωρ ) cos( kl) ] + (.46) Additioally, it is assumed that the acoustic wavelegth is much larger tha that of the source dimesio, λ>>s. The harmoic fuctio ca the be approimated as: si k s k s ; cos k s (.47a,b)

38 Lori A. Brady Chapter Aalytical Modelig 3 Takig ito accout these approimatios ad evaluatig ad usig Equatio (.39a,b) the ratio of the disturbace to trasmitted wave amplitude for the plae wave mode is give by: A A D T cs h ρ α h = ρcs ρcs α h ρcs β e h ρcs β e h ik ikl l cos ( kl) + (.48) ad rearragig for compariso with the plae-wave model, as depicted i Appedi B, gives: A A D T e 4 = ikl ikl ikl ikl + AR( + e e e ) AR ikl ikll ( e )( e ) + ikl e 4 e ikl ikl e + AR e ikl ikl e (.49) where: S AR = (.5) h is kow as the system area ratio. Usig Equatio (.3), the TL for the plae wave mode ca be writte as: TL = log A A D T = e 4 ikl ikl ikl ikl + AR( + e e e ) ikl ikll ( e )( e ) e 4 e ikl ikl e + AR e ikl ikl + AR (.5)

39 Lori A. Brady Chapter Aalytical Modelig 3 Comparig this epressio for TL to the epressio derived i Appedi B usig the traditioal plae-wave aalysis shows that the epressios are idetical. Therefore, it is prove that the higher-order mode modelig approach is equivalet to the plae-wave approach whe cosiderig oly the plae-wave mode i both the disturbace ad Gree s fuctios. Thus, the traditioal plae-wave model is a special case of the higher-order model developed here. Frequecies of Maimum Atteuatio Aother useful applicatio of the simplified model developed i Sectio.3 is that it ca be used to predict the frequecy of maimum atteuatio for idividual modes. Maimum atteuatio is obtaied whe the umerator of the ratio of the trasmitted to disturbace wave amplitudes is set equal to zero. This ratio is defied, usig Equatio (.4) ad Equatio (.4) as: A A T D cos = 4ωρ s ( κ ) ( )( [ ) ( h si k α iωρ β iωρ ) cos( k l) ] k ( i ) ( i Λ α ωρ β ωρ ) + (.5) Usig agai the approimatio i Equatio (.47a,b) ad settig the above equatio to zero, the frequecies of optimum atteuatio for the selected mode is obtaied. This leads to the followig trascedetal equatio: si Sk = Λ k l (.53) ( kl) si( k ) The sigificace of this epressio is that for a fied set of HQ tube parameters, e.g. L, l, ad S, Equatio (.53) shows that each mode has a set of optimum frequecies of atteuatio, which icreases with icreasig modal orders. The left had side of the equatio depeds oly o the tube legth. The right had side of this equatio is a fuctio of the tube cross-sectioal area, distace betwee iterfaces, ad the modal order. Thus, the optimal frequecy of atteuatio is

40 Lori A. Brady Chapter Aalytical Modelig 3 differet for each mode preset i the. This is depicted i Figure. where the left ad right had side of equatio (.53) are plotted as a fuctio of the frequecy. For a give tube legth, the left had side fuctio is a sigle curve show i Figure. (black lie). The right had side of the equatio depeds o the mode aial waveumber k. The curves for modes,,, ad 3 are show i the figure. The frequecy of optimum atteuatio occurs whe the left ad right had side curves itersects as show i the figure. For the same mode, there are may frequecies where the curves itersects, i.e. roots of equatio (.53). It is also iterestig to show that as the area of the HQ tube is reduced (S ), the optimum frequecy of atteuatio approaches the atural frequecies of the HQ tube assumig pressure release boudary coditios, i.e. kl=. It is also iterestig to ote that for the plae-wave mode, i.e. = ad k =k, Equatio (.53) is the same epressio foud by Selamet et. al. [, ] usig the traditioal plae-wave aalysis. Numerical results usig this equatio will be preseted i Chapter 3 of this thesis. si( kl) HQ - Tube st Resoace S h si( kl) Sk Λ k si ( k ) l HQ - Tube d Resoace Sk Λ k si ( k ) l Sk Λ k 3 si ( k ) l Freq. Optimum Atteuatio Figure.. Sigle-mode frequecies of maimum atteuatio (h=5.4 cm, S=.7 cm, L=.7 cm, ad l=.6 cm)

41 Lori A. Brady Chapter Aalytical Modelig 33 Atteuatio at Mode Cut-Off Frequecies Additioally, it is also of iterest to ote that the simplified model ca also be used to predict atteuatio at the cut-off frequecy for a idividual mode. This ca be show mathematically by lookig at the limit of the trasmitted to disturbace wave amplitude as the mode waveumber approaches zero: A T Lim = k AD (.54) where the ratio of trasmitted to disturbace modal amplitude is give by Equatio (.5). Agai assumig that that the acoustic wavelegth is much larger tha that of the source dimesio, i.e. λ>>s, ad usig the resultig approimatios as stated i Equatio (.47a,b), this ratio is reduced to: Lim k A A T D = Lim C k α k Cαk β k αk βk + Cβk iclk icβlk + ic lk + C l k + (.55) where C is defied as: ωρs C = (.56) Λ The limit i Equatio (.55) is actually udefied because of the first term i the epressio, i.e.. This udefied term ca be maipulated to obtai its limit usig L Hopital s rule [8]. L Hopital s rule states that for a give ratio f ( ) g( ) ca be show that:, for which both f( )= ad g( )=, it f Limit g ( ) ( ) ( ) ( ) f' = Limit (.57) g'

42 Lori A. Brady Chapter Aalytical Modelig 34 as log as f() ad g() are cotiuous ad have derivatives that are cotiuous o some ope iterval that cotais. Equatio (.55) ca ow be reduced to: Lim k A A T D = C α k Cα β α β icl = + = α + β + icl k α β icl + Cβ 4iCβlk + ic l + C l k + (.58) where the ratio of trasmitted to disturbace wave amplitude approaches zero i the limit as the waveumber approaches zero. Thus, at the cut-off frequecy the atteuatio produced by the HQ tubes is perfect idepedet of the tube dimesios.

43 Chapter 3 Numerical Aalysis I this chapter, the umerical aalysis for the two-dimesioal HQ tube system is preseted. These umerical results are desired i order to eplore the physical mechaisms behid the atteuatio of soud, i particular the atteuatio of higher-order acoustic modes. To accomplish this, it will be ecessary to ivestigate how varyig both the modal compoets of the disturbace ad the system geometric parameters ifluece this atteuatio. The HQ system trasmissio loss characteristics will be eamied first with oly a plae-wave disturbace, but this disturbace will be icreased i compleity to iclude both eve ad odd higher-order modes. Also, the ifluece of geometric parameters such as tube aial positio, legth, distace betwee iterfaces, cross-sectioal area, ad the umber of tube arrays will be eamied. 3. Modal Aalysis with Icreasig Disturbace Compleity The followig sectio describes the oise reio mechaisms ivolved i the atteuatio of high-order mode acoustic disturbaces. This is accomplished by eamiig several aspects of the system characteristics by gradually icreasig the modal compleity of the disturbace. The umerical results were computed for the two symmetrically located HQ tubes as show i Figure.3. It was assumed that there was o fluid flow i the system, but that the fluid is air, i.e. c=343 m/s ad ρ=. Ns /m 4. Compariso with Plae-Wave Modelig Approach The first case ivestigated compares the simplified higher-order model developed i Sectio.3 for a plae-wave disturbace to the traditioal plae-wave aalysis by Selamet et. al. [-], which was reproduced i Appedi B. The dimesios of the system are h=4.9 cm, S=.45/ cm, L=.8 m, ad l=.4 m for the higher-order mode model ad the parameters for the plae-wave model are S d =4.9 cm, S t =.45 cm, l 3 =.8 m, ad l =.4 m. The first cut-off frequecy, i.e. µ=, is 35 Hz. For this compariso, several system characteristics will be ivestigated icludig, the frequecies of maimum atteuatio, the phase relatioship betwee the recombied waves at the outlet of the HQ tube, ad the trasmissio loss. 35

44 Lori A. Brady Chapter 3 Numerical Aalysis 36 Equatio (.53) is used to predict the frequecies of maimum atteuatio assumig a sigle mode both i the disturbace ad i the Gree s fuctio. This equatio is best depicted by plottig the left ad right-had sides o the same plot, as is show i Figure.. The frequecies of maimum atteuatio occur at the itersectios of these curves. The sigificat of these results is that for a give set of system geometric parameters (h, S, L, ad l) each mode has a set optimum frequecies of atteuatio, which icreases with modal order. This aalysis ca be take a step further by ivestigatig the phase relatioship betwee the recombied waves. The phase relatioship is easily eplaied usig Figure 3.. I this figure, the icidet soud field, travelig to the right, ecouters a brach i path at the ilet of the HQ tube. The icidet wave the divides ito the wave travellig through the HQ tube ad the wave remaiig i the mai-. At this first itersectio, it is assumed that all of the waves are i phase. Both of these divided waves travel ad recombie at the outlet of the HQ tube. The waves i the HQ tube ad mai- travel a distace of L ad l respectively. Therefore, the relative phase betwee the recombied waves is defied as: kl k l (3.) Assume waves are i phase kl y l The relative phase betwee the two waves is kl-k l k l L Figure 3.. Phase relatioship betwee recombied waves

45 Lori A. Brady Chapter 3 Numerical Aalysis 37 Figure 3. shows these system characteristics over a frequecy rage below the first cutoff frequecy, i.e. 35 Hz. The top subplot is the left ad right-had side of Equatio (.53), the ceter subplot is the phase differece betwee the recombied waves, ad the bottom subplot is the trasmissio loss compariso betwee modelig approaches. The higher-order mode results were calculated assumig that oly the plae-wave is preset i the disturbace, but that the fiite pisto sources are capable of geeratig higher-order modes i the, i.e., the evaescet modes were icluded i the Gree s fuctios or N g =9. Plae-Wave HOM Figure 3.. Compariso of plae-wave ad higher-order mode models icludig; frequecies of maimum atteuatio, phase differece of recombied waves (rad), ad trasmissio loss (db) As see i the top subplot of the previous figure, the frequecies of maimum atteuatio occur at the itersectios of the two siusoidal curves. Two such itersectios have bee highlighted ad occur at 4 ad 6 Hz. At these frequecies the phase differece betwee the recombied waves are 3. ad 9.4 rad, correspodig to path legth differeces of λ/ ad 3λ/, respectively. Also of ote is the icreasig discrepacy with frequecy betwee the plaewave ad higher-order mode trasmissio loss curves i the bottom subplot. The differeces betwee the two curves are due to the iclusio of the ear-field effects of the evaescet modes i the higher-order mode model. The effect of the evaescet modes becomes more sigificat as the frequecy icreases ad approaches the cut-off frequecy of the first higher-order mode. Thus, the traditioal plae-wave approach is oly valid at very low frequecies, well below the cut-off frequecy of the µ= mode. Whe the evaescet modes, created by the fiite pisto

46 Lori A. Brady Chapter 3 Numerical Aalysis 38 sources, are removed the plae-wave ad the higher-order mode results are idetical. I fact, it has bee show aalytically that the traditioal plae-wave model is a particular case of the higher-order model, see Sectio.3 for details. Sigle-Mode Aalysis I order to address the performace of the HQ tube to suppress higher-order modes, a aalysis was performed i which the disturbace is comprised of a sigle mode. I this aalysis, the fiite pisto sources geerate the same sigle mode as icluded i the disturbace ad the system dimesios are as follows; h=5.4 cm, S=.7 cm, L=.7 cm, l=.6 cm, ad c =5. cm. Figure 3.3 depicts this aalysis for the first four modes, i.e. µ=,,,ad 3. Each subplot i this Figure cotais three graphs. The first is the left ad right-had side of Equatio (.53), the secod is the phase differece betwee the two waves at the secod tube- iterface, ad the third is the trasmissio loss. For eample, Figure 3.a shows that for the µ= mode maimum atteuatio of 9 ad 36 db occurrig at 377 ad 747 Hz, respectively. This subplot also idicates a phase differece of.64 ad.3 rad at these frequecies. The sigificace of these phase differeces is that for a small area ratio they do ot correspod to multiples of λ/. Similar results are preseted for the first three higher-order modes, i.e. µ=, ad 3, i Figures 3.(b), 3.(c) ad 3.(d), respectively. What is obvious is that the frequecies of maimum atteuatio are differet for each mode ad that with icreasig modal order these frequecies also icrease.

47 Lori A. Brady Chapter 3 Numerical Aalysis (a) µ= (b) µ= Hz (c) µ= (d) µ=3 Figure 3.3. Sigle-mode frequecies of maimum atteuatio, phase differece of recombied waves, ad trasmissio loss for modes; (a) µ=, (b) µ=, (c) µ=, ad (d) µ=3 This same type of sigle-mode aalysis ca be used to ivestigate how system geometric parameters ifluece the frequecies of maimum atteuatio. This is preseted i Figure 3.4, for several values of the HQ tube cross-sectioal area, S=.3,.64,.7,.54, ad 5.8cm ad for the first four system modes. These areas correspod to area ratios ragig from.5% - 4%, see Equatio (.5). It should be kept i mid however that i practice implemetatio of a HQ tube i a egie would be limited to area ratios of % or less. For this reaso the baselie system has a area ratio of %.

48 Lori A. Brady Chapter 3 Numerical Aalysis 4 (a) µ= (b) µ= (c) µ= (d) µ=3 Figure 3.4. Sigle-mode aalysis of system characteristics with varyig HQ tube area, S=.3,.64,.7,.54, ad 5.8 cm, for modes; (a) µ=, (b) µ=, (c) µ=, ad (d) µ=3 I the previous figure, it is iterestig to see graphically that as the area of the HQ tube is reduced (S ), the optimum frequecies of atteuatio correspod to the atural frequecies of the HQ tube, i.e. 35 ad 7 Hz. Also, the highest-order mode preseted, the µ=3, is affected more tha the lower modes by the cross-sectioal area. Although this type of aalysis is show oly for variatios i HQ tube cross-sectioal area, oce the tube legth is selected it is useful for ivestigatig ay of the other system geometric parameters.

49 Lori A. Brady Chapter 3 Numerical Aalysis 4 Odd Disturbace Modes To better uderstad the performace of the HQ tubes to suppress multiple higher-order modes, a disturbace cotaiig the first two odd modes was et cosidered. I this aalysis, the fiite pisto sources are capable of geeratig higher-order modes i the, i.e. N g =9. The dimesios of this system remai at h=5.4 cm, S=.7 cm, L=.7 cm, l=.6 cm, ad c =5. cm. For this system, the cut-o frequecies for the µ=,,, 3, ad 4 modes are, 675., 35, 6, ad 7 Hz, respectively. The two odd modes, µ= ad 3, are assumed to be preset i the icidet disturbace field with equal amplitude, A D =A D3 =. Note that due to the symmetric cofiguratio of the HQ tubes, the eve modes will ot be ecited i this system. Thus, the µ= is the oly propagatig mode i the below 6 Hz while at higher frequecies both the µ= ad 3 are propagatig. The resultig trasmissio loss characteristic are show i Figure 3.5 for a disturbace comprised of both odd modes, Figure 3.5(a), ad for a disturbace comprised of each odd mode idividually, Figure 3.5(b). µ= µ=3 (a) (b) Figure 3.5. Trasmissio loss (db) versus frequecy (Hz) for HQ system with disturbace comprised of the odd modes µ= ad 3; (a) together ad (b) idepedetly Substatial atteuatio is clearly achieved at 5 Hz due to the reflectio of the oly mode preset, i.e. µ= at this frequecy. This result is ot uepected sice the HQ tubes are very effective at suppressig sigle modes as i the case of plae-wave mode i Figure 3.. O the other had, at 495 Hz the trasmissio loss is 4.5 db due to the atteuatio of both odd

50 Lori A. Brady Chapter 3 Numerical Aalysis 4 higher-order modes, i.e. µ= ad 3. This is evidet by plottig the comple amplitude of the icidet, trasmitted, ad reflected waves for each mode at the frequecy of 495 Hz. This is show i Figure 3.6. µ= µ=3 (a) (b) Figure 3.6. Modal amplitude vectors at 495 Hz for disturbace comprised of µ= ad 3 modes; (a) µ= modal amplitudes ad (b) µ=3 modal amplitudes Figure 3.6(a) shows that the µ= icidet mode, A D =, was reflected back with comple amplitude A R, yieldig a et trasmitted wave with amplitude A T. Figure 3.6(b) shows the same aalysis for the µ=3 mode. From the modal amplitudes, the soud power reio of the µ= ad 3 modes is 3. ad 8. db, respectively. Thus, these results demostrate that the HQ tubes are also effective i reducig multiple higher-order modes i a. It is also very iterestig to ote that the amplitude of the reflected µ=3 mode is larger tha the amplitude of the icidet µ=3 mode, i.e. A R3 > A D3, see Table 3. which lists the power ad the amplitude of each of the of the modal compoets. This clearly implies that some of the acoustic eergy from the µ= mode has to be spilled over ito the µ=3 mode. Thus, the atteuatio due to the HQ tubes ca ot simply be eplaied as reflectio of the mode back to the source.

51 Lori A. Brady Chapter 3 Numerical Aalysis 43 Power (db) Magitude of Modal Amplitude µ= µ=3 µ= µ=3 Disturbace Reflected Trasmitted Table 3.. Power ad amplitudes of modal compoets at 495Hz for odd modes µ= ad 3 I order to gai further isight ito the oise cotrol mechaisms ivolved i the suppressio of multiple higher-order modes, a modal aalysis is carried out where the amplitude of the reflected ad trasmitted waves are computed for each of the icidet modes applied idepedetly. This aalysis is preseted i Figures 3.7(a) through 3.7(d). For eample, Figure 3.7(a) shows the icidet µ= mode amplitude, A D =, ad the resultig trasmitted, reflected, A t, ad A r, wave amplitudes for mode µ=. Note that these amplitudes are differet tha the oes i Figure 3.6(a) which agai idicates that there are iteractio effects betwee the modes. Figure 3.7(b) shows the resultig trasmitted, A t3, ad reflected, A r3, wave amplitudes for mode µ=3 due to the disturbace mode µ=. The coclusio of this is that some of the eergy from the µ= mode is spilled over ito the µ=3 mode. Figures 3.7(c) ad 3.7(d) show the same aalysis for the case of the disturbace cosistig of oly the µ=3 mode with the same geeral coclusios. This sort of aalysis is valid because the system is liear.

52 Lori A. Brady Chapter 3 Numerical Aalysis 44 (a) (b) (c) (d) Figure 3.7. Modal amplitude vectors at 495 Hz; (a) ad (b) disturbace cosists of µ= mode, (c) ad (d) disturbace cosists of µ=3 mode oly These plots clearly demostrate that there are two mechaisms i the reio of the icidet modes. Firstly, the eergy i a icidet mode is i part reflected back to the source, the same as i the classical plae-wave aalysis. Secodly there is some eergy spilled from oe mode ito the other mode. The results i Figure 3.7 also suggest that there is more eergy spilled from the low-order mode µ= ito the higher-order mode µ=3 tha vice-versa, i.e. t3 3 t A > A. Note that the eergy spilled ito other modes will propagate both upstream ad dowstream as trasmitted ad reflected waves, respectively. This is evidet i the fact that t3 A = A r3 ad A 3 t= A 3 r. The upstream or trasmitted compoets, A t3 ad A 3 t, will the

53 Lori A. Brady Chapter 3 Numerical Aalysis 45 cotribute to the total trasmitted soud power. Thus, the suppressio of a particular mode is due to the combiatio of the spilled-over cotributios from the various icidet modes. This is demostrated i Figure 3.6(b) which illustrates the suppressio of the µ=3 mode. Usig the priciple of liear superpositio, the resultig trasmitted wave of the µ=3 mode, A T 3, is due to the vector sum of the spill of the µ= mode ito the µ=3 mode, A t3, ad the trasmitted 3 compoet due to the same µ=3 mode, A t3. Thus, the et trasmitted wave for the µ=3 mode is r r r 3 the vector sum A = A + A which is idicated i dotted lies i Figure 3.6(b). This behavior T 3 t3 t3 should be cotrasted to the traditioal plae-wave aalysis of the HQ tube where the oly oise reio mechaism that takes place is reflectio of the icidet plae-wave. Eve Disturbace Modes Usig the same method to illustrate the dyamics of the odd modes, the eve modes for the same system will be described. Agai, the system dimesios are h=5.4 cm, S=.7 cm, L=.7 cm, l=.6 cm, ad c =5. cm resultig i cut-o frequecies for the µ=,,, 3, ad 4 modes of, 675., 35, 6, ad 7 Hz, respectively. The three eve modes, µ=, ad 4, are assumed to be preset i the icidet disturbace field with equal amplitude, A D =A D = A D4=. Note that due to the symmetric cofiguratio of the HQ tubes, the odd modes will ot be ecited i this system. Thus, the µ= is the oly propagatig mode i the below 35 Hz, above 35 Hz but below 7 Hz both the µ= ad modes are propagatig, ad at higher frequecies, those above 7 Hz, all of the eve modes µ=, ad 4 are propagatig. The resultig trasmissio loss characteristics are show i Figure 3.8 for both a disturbace comprised of all three eve modes, Figure 3.8(a), ad for a disturbace comprised of each mode idepedetly, Figure 3.8(b).

54 Lori A. Brady Chapter 3 Numerical Aalysis 46 µ= µ= µ=4 (a) (b) Figure 3.8. Trasmissio loss (db) versus frequecy (Hz) for HQ system with disturbace cosistig of; (a) all eve modes, µ=, ad 4, ad (b) eve modes idepedetly Referrig to Figure 3.8(a), which depicts the trasmissio loss characteristics for a disturbace comprised of all three eve modes, substatial atteuatio agai is clearly achieved at Hz due to the reflectio of the oly mode preset, i.e. µ=, at this frequecy. This result is ot uepected sice the HQ tubes are very effective at suppressig sigle modes. However, what is also apparet is that, ulike the odd modes case, there is ot ay other substatial atteuatio at aother frequecy. Specifically, at 495 Hz there is less tha db i oise atteuatio. Show i Figure 3.8(b) are the trasmissio loss curves for the system with a disturbace comprised of each eve mode idividually. I this figure, it is clear that there is substatial power atteuatio, 5.3 db, of the µ= mode at 495 Hz. To gai further isight ito the performace for these modes, further studies are carried out by plottig the comple amplitude of the icidet, trasmitted, ad reflected waves for each mode at the frequecy of 495 Hz. This aalysis is show i Figure 3.9.

55 Lori A. Brady Chapter 3 Numerical Aalysis 47 µ= µ= (a) (b) Figure 3.9. Modal amplitude vectors at 495 Hz for disturbace comprised of µ=,, ad 4 modes; (a) µ= modal amplitudes ad (b) µ= modal amplitudes A Figure 3.9(a) shows for the µ= mode that oly a small amout of the icidet wave, D =, was reflected back with comple amplitude A R, yieldig a et trasmitted wave with amplitude A T. Figure 3.9(b) shows the same aalysis for the µ= mode, where agai most of the icidet wave was trasmitted with comple amplitude A T. From the modal amplitudes, the soud power reio of the µ= ad modes is.96 ad.6 db, respectively. This iformatio is summarized i Table 3. which lists the power ad the amplitude of each of the of the modal compoets. Thus, these results agai demostrate, that for a disturbace comprised of all of the eve modes, there is egligible reio at 495 Hz. Note that the µ=4 mode has a cut-o frequecy of 7 Hz ad is therefor ot preset at this frequecy. Power (db) Magitude of Modal Amplitude µ= µ= µ= µ= Disturbace Reflected Trasmitted Table 3.. Power ad amplitudes of modal compoets at 495Hz for eve modes µ= ad

56 Lori A. Brady Chapter 3 Numerical Aalysis 48 I order to agai gai further isight ito the oise cotrol mechaisms ivolved, a modal aalysis is carried out where the amplitude of the reflected ad trasmitted waves are computed for each of the icidet modes idepedetly. This aalysis is preseted i Figures 3.(a) through 3.(d). For eample, Figure 3.(a) shows the icidet µ= mode amplitude, A D =, ad the resultig trasmitted, A t, ad reflected, A r, wave amplitudes for mode µ=, while Figure 3.(b) shows the resultig trasmitted, A t, ad reflected, A r, wave amplitudes for mode µ= due to the disturbace mode µ=. Figures 3.(c) ad 3.(d) show the same aalysis for the case of the disturbace cosistig of oly the µ= mode. (a) (b) (c) (d) Figure 3.. Modal amplitude vectors at 495 Hz; (a) ad (b) disturbace cosists of µ= mode; (c) ad (d) disturbace cosists of µ= mode oly

57 Lori A. Brady Chapter 3 Numerical Aalysis 49 Cosiderig the case i which the disturbace is comprised of the µ= mode oly, Figure 3.(a)-(b), it is clear that most of the icidet wave eergy, A D, is trasmitted i the µ= compoet of the trasmitted wave, termed A t. However, there has bee a small amout of eergy reflected i the µ= compoet of the reflected wave, A R, ad some spill of eergy ito both the trasmitted ad reflected compoets of the µ= mode, At ad A r respectively. This behavior illustrates what is already kow about the soud atteuatio at this frequecy from Figure 3.8(b). Namely, that for a disturbace comprised of the µ= mode oly, there is very little trasmissio loss at 495 Hz. Figure 3.(c)-(d) depict the case i which the disturbace is comprised of the µ= mode oly, A D. From these plots, it is evidet that a small portio of the icidet wave eergy has spilled ito the trasmitted ad reflected compoets of the µ= mode, A t ad A r respectively. I additio, it ca be see that most of the icidet wave eergy is reflected i the µ= compoet of the reflected wave, A r. This eplais the reaso there is soud atteuatio at 495 Hz for a disturbace comprised of oly the µ= mode, as see i Figure 3.8(b). Like the odd modes case, these plots show that there are two mechaisms ivolved i the modal recostructio of the icidet modes. First, the icidet wave eergy is i part reflected back to the source, ad secod there is some eergy spilled ito the other higher-order modes. This spilled eergy will propagate both upstream ad dowstream as trasmitted ad reflected waves respectively. The upstream compoet cotributes to the total trasmitted soud power. These results also suggest that there is more eergy spilled from the low-order mode µ= ito the higher-order mode µ= tha vice-versa, i.e. t t A > A. As metioed before, the suppressio of a particular mode is due to the combiatio of the spilled-over cotributios from the various icidet modes. This is demostrated i Figure 3.9(b) as vector sums idicated as dashed lies. For eample, the resultig trasmitted µ= mode, A T is due to the vector sum of the spill of the µ= mode ito the µ= mode, A t, ad the

58 Lori A. Brady Chapter 3 Numerical Aalysis 5 trasmitted compoet due to the same µ= mode, case the two compoets A r t ad A r t A t, i.e. r A T r r = A + A. However, i this are early i phase ad their cotributios do ot cacel each other, as was the case for the odd mode disturbaces i Figure 3.6. Further ivestigatio i later sectios will show that various system parameters affect the spill-over / additio behavior. t t All Disturbace Modes I the two previous sectios, the suppressio of the eve ad odd disturbace modes has bee studied i detail, but idepedetly. For the sake of completeess, the aalysis of the HQ tube system is eteded to iclude all of the eve ad odd cut-o disturbace modes i the frequecy rage of iterest. The same system geometric parameters are used with h=5.4 cm, S=.7 cm, L=.7 cm, l=.6 cm, ad c =5. cm. Resultig i cut-o frequecies for the µ=,,, 3, ad 4 modes of, 675., 35, 6, ad 7 Hz, respectively. The first five modes, µ=,,, 3, ad 4, are assumed to be preset i the icidet disturbace field with equal amplitude, A D =A D = A D = A D3 = A D4 =. The resultig trasmissio loss characteristic is show i Figure 3.. Figure 3.. Trasmissio loss (db) versus frequecy (Hz) for HQ system comprised of all si cut-off disturbace modes µ=,,, 3, ad 4

59 Lori A. Brady Chapter 3 Numerical Aalysis 5 At Hz, the trasmissio loss is 6.7 db due to the atteuatio of both the plae wave, µ=, ad the first higher-order, µ=, modes. Cetered aroud 4 Hz, is a bad of atteuatio with a peak of. db. At this frequecy, this reio is due to the atteuatio of the first four modes, µ=,,,ad 3. This bad of atteuatio is studied by plottig the comple amplitudes of the trasmitted waves for each mode for the frequecy of 4 Hz, this is show i Figure 3.. Figure 3.. Trasmitted modal amplitude vectors at 4 Hz for disturbace comprised of µ=,,, 3, ad 4 modes I this figure, the comple modal amplitudes, i black, red, blue, ad gree colors are used to represet the µ=,,, ad 3 modes respectively ad should ot be cofused with the color scheme used i previous sectios to represet icidet, trasmitted, ad reflected wave amplitudes. It is clear that there is a reio i the magitude of the modal amplitudes, see Table 3.3. This reio i magitude leads to the atteuatio of power. I additio, it ca be see that there is a chage i phase of the comple amplitudes.

60 Lori A. Brady Chapter 3 Numerical Aalysis 5 Power (db) Magitude of Modal Amplitude µ= µ= µ= µ=3 µ= µ= µ= µ=3 Disturbace Reflected Trasmitted Table 3.3. Power ad amplitudes of modal compoets at 4Hz for modes µ=,,, ad 3 It is importat to study how the array of HQ tubes produces this reio i the disturbace modal amplitudes. I order to accomplish this, a modal aalysis is performed i which the trasmitted modes are computed for each of the icidet modes idepedetly. This aalysis is preseted i Figures 3.3(a) through 3.3(d). For eample, Figure 3.3(a) shows the icidet µ= mode (black solid lie) amplitude, A D =, ad the resultig trasmitted wave amplitudes, A t ad, A t, for mode µ= ad µ=, while Figure 3.3(b) shows the resultig trasmitted wave amplitudes, A t ad A t, for mode µ= ad µ= due to the disturbace mode µ= (blue solid lie). Figures 3.3(c) ad 3.3(d) show the same aalysis for the case of the disturbace cosistig of the µ= ad µ=3 mode respectively. It should be oted that at this frequecy the µ=4 mode is ot cut-o.

61 Lori A. Brady Chapter 3 Numerical Aalysis 53 (a) (b) (c) (d) Figure 3.3. Trasmitted modal amplitude vectors at 4 Hz with disturbace cosistig of mode; (a) µ=, (b) µ=, (c) µ=, ad (d) µ=3 Agai, from these plots it ca be see that there are two mechaisms i the reio of the icidet modes. A part of the icidet eergy is reflected back to the source, ad secodly there is some eergy spilled ito the other higher-order modes. The suppressio of a mode is due to the combiatio of the spilled-over cotributios from the various icidet modes. This is demostrated i Figure 3.4 which illustrates the suppressio of the µ=3 disturbace mode, A T3. The resultig trasmitted wave of the µ=3 mode is due to the vector sum of the spill of the µ= mode ito the µ=3 mode, A t3, ad the trasmitted compoet due to the same µ=3 mode, 3 A t3.

62 Lori A. Brady Chapter 3 Numerical Aalysis 54 Thus, the et trasmitted wave for the µ=3 mode is T 3 t3 3 t3 A = A + A which is idicated as a vector sum i a solid lie i Figure 3.4. Figure 3.4. Modal suppressio of trasmitted mode µ=3 at 4 Hz, disturbace cosists of µ=,,, 3, ad 4 modes 3. Parametric Studies This sectio cotiues to describe the oise reio mechaisms ivolved i the atteuatio of higher-order mode acoustic disturbaces. This is accomplished by eamiig the system trasmissio loss characteristics by chagig geometric parameters such as tube legth, distace betwee iterfaces, cross-sectioal area, aial positio, ad the umber of tube arrays. The umber of modes icluded i the Gree s fuctios was 3, icludig the plae-wave mode, or N g =9. Tube Geometric Parameters: Aial Locatio, Legth, Iterface Distace, ad Area The atteuatio of higher-order mode acoustic disturbaces has bee attributed to the reflectio of eergy, the spill of eergy betwee modes, ad the recostructio of compoets i each mode. Sice the phase of each mode chages as it propagates dow the, system geometric parameters eist which ifluece the modally restructured magitudes ad phases to optimally recombie, resultig i the miimum overall trasmitted amplitude. Such ifluetial

63 Lori A. Brady Chapter 3 Numerical Aalysis 55 system characteristics are tube aial positio ( c ), legth (L), distace betwee iterfaces (l), ad area (S). Numerical results were computed usig the two symmetrically located HQ tubes as show i Figure.3. The baselie dimesios of this system are h=5.4 cm, S=.7 cm, L=.7 cm, l=.6 cm, ad c =5. cm. Resultig i cut-o frequecies for the µ=,,, 3, ad 4 modes of, 675., 35, 6, ad 7 Hz, respectively. The first five modes, µ=,,, 3, ad 4, are assumed to be preset i the icidet disturbace field with equal amplitude, A D =A D = A D = A D3 = A D4 =. The umber of modes icluded i the Gree s fuctios were N g =9. Effect of Aial Positio I Figure 3.5(a), the trasmissio loss characteristic is show versus the aial positio c of the HQ tube for a frequecy of 4 Hz. The trasmissio loss for a rage of frequecies, - 3 Hz is preseted i Figure 3.5(b). Clearly, the distace c has a sigificat ifluece o the atteuatio of specific modes. For istace, there eists a locatio c,opt which is optimum for the atteuatio of all five of the disturbace modes. I additio, the optimal locatio repeats periodically dow the. Figure 3.5 (b) shows that the frequecy of maimum atteuatio is isesitive of the aial positio of the HQ tubes. The reaso for havig a optimal aial locatio for the HQ tubes is that the relative phase betwee the modes chages as they propagate alog the because of their differet aial waveumbers. This implies that at some positio alog the the phase of the modes is such that leads to the best recombiatio of the spill of eergy betwee the modes ad miimum et trasmitted power. Agai this behavior should be cotrasted to the case of cotrollig a sigle mode where the reio is idepedet of the aial positio of the HQ tubes.

64 Lori A. Brady Chapter 3 Numerical Aalysis 56 (a) (b) Figure 3.5. Trasmissio loss (db) versus tube aial locatio, c (m), with disturbace comprised of µ=,,, 3, ad 4 modes at (a) 4 Hz ad (b) -3 Hz Figure 3.6 shows the modal compoets of the trasmitted wave amplitudes at 4 Hz for a uit modal amplitude of the icidet modes, µ=,,, 3, ad 4, ad for the rage of values of the tube aial locatio, i.e. c = to.9 m. The rage of modal amplitudes correspodig to the rage of c values are idicated by the curve coectig the two vectors associated to the

65 Lori A. Brady Chapter 3 Numerical Aalysis 57 lower ad upper values of the rage. This figure shows that the magitude ad phase of the modes are greatly affected by the tube aial locatio. Figure 3.6. Trasmitted modal amplitude vectors at 4 Hz for c = -.9 m ad disturbace comprised of µ=,,, 3, ad 4 modes Effect of Tube Legth The ifluece of the legth of the HQ tube, L, for all five disturbace modes, is show i Figure 3.7 for a frequecy of 4 Hz i Figure 3.7(a) ad for the rage of frequecies -3 Hz i Figure 3.7(b). These results show that the HQ tube legth is also a very importat parameter o the atteuatio of multiple higher-order modes. I additio, this atteuatio repeats periodically, with a period of.43 m. It is iterestig to ote that cosiderig this periodicity i space as a wavelegth, it correspods to a frequecy of approimately 4 Hz. This result is quite logical because the atteuatio occurs ear the HQ tube resoace frequecies ad a chage i the legth of the tube results i a shift of the resoace frequecies. For eample, a tube legth of L=.3 m results i the secod tube resoace to be ear 4 Hz (kl=π) ad thus this resoace is resposible for the soud atteuatio. By chagig the tube legth to L+ where =π/k, the fourth resoace is ow ear 4 Hz ad thus is resposible for the atteuatio. Withi the rage of tube legths studied, maimum atteuatio of approimately. db is achieved with a tube legth of.84 m at 4 Hz. Figure 3.7(b)

66 Lori A. Brady Chapter 3 Numerical Aalysis 58 shows that chagig the tube legth results i a chage i the frequecy of maimum atteuatio. It is clear to idetify the bads of atteuatio for the st, d, etc., tube resoaces. (a) (b) d st resoace Figure 3.7. Trasmissio loss (db) versus tube legth, L (m), with disturbace comprised of µ=,,, 3, ad 4 modes at (a) 4 Hz ad (b) -3 Hz Figure 3.8 shows the modal breakdow aalysis at 4 Hz for a uit modal amplitude of the icidet modes ad for a rage of values of the tube legth, i.e. L= m.

67 Lori A. Brady Chapter 3 Numerical Aalysis 59 This figure shows that the magitude ad phase of the modes created by the spill of eergy mechaism is greatly affected by the tube legth. (a) (b) (c) (d) Figure 3.8. Trasmitted modal amplitude vectors at 4 Hz for L= m ad disturbace cosistig of modes; (a) µ=, (b) µ=, (c) µ=, ad (d) µ=3 Effect of Iterface Distace The ifluece of the iterface distace of the HQ tube, l, for the disturbace comprised of µ=,,, 3, ad 4 modes, is show i Figure 3.9 for a frequecy of 4 Hz i Figure 3.9(a) ad for the rage of frequecies -3 Hz i Figure 3.9(b). The HQ tube iterface distace, costraied by the system geometry as S< l >L, also iflueces the atteuatio of higher-order

68 Lori A. Brady Chapter 3 Numerical Aalysis 6 modes. Withi the rage of tube iterface legths studied, maimum atteuatio of approimately. db is achieved with a tube legth of.874 m at 4 Hz. Figure 3.8(b) shows that the frequecy of maimum atteuatio is isesitive of the iterface distace of the HQ tubes. (a) l (m) (m) (b) Figure 3.9. Trasmissio loss (db) versus iterface distace, l (m), with disturbace comprised of µ=,,, 3, ad 4 modes at (a) 4 Hz ad (b) -3 Hz

69 Lori A. Brady Chapter 3 Numerical Aalysis 6 Figure 3. shows the modal breakdow aalysis at 4 Hz for a uit modal amplitude of the icidet modes ad for a rage of values of the tube iterface distace, i.e. l= m. Agai, this type of aalysis, shows that the magitude ad phase of the modes created by the spill of eergy mechaism is greatly affected by the tube iterface distace. (a) (b) (c) (d) Figure 3.. Trasmitted modal amplitude vectors at 4 Hz for l= m ad disturbace cosistig of modes; (a) µ=, (b) µ=, (c) µ=, ad (d) µ=3 Effect of Tube Area The last parameter, for the sigle-array HQ tube system, to be studied is the crosssectioal area of the side-tubes, S, for the disturbace comprised of µ=,,, 3, ad 4 modes. This aalysis is show as a fuctio of area ratio (AR), where AR=S/h, i Figure 3. for a

70 Lori A. Brady Chapter 3 Numerical Aalysis 6 frequecy of 4 Hz i Figure 3.(a) ad for the rage of frequecies -3 Hz i Figure 3.(b). Withi the rage of tube cross-sectioal areas studied, maimum atteuatio of approimately. db is achieved with a AR of.8% at 4 Hz. It should be kept i mid, that i the modelig of the side-tube dyamics, the pressure distributio withi the tubes was assumed to cosist of plaes waves oly. This method is therefore limited to frequecies well below the first cut-off frequecy of the trasverse modes i the side-tubes, c S. (a) (b) Figure 3.. Trasmissio loss (db) versus tube area ratio, AR (%), with disturbace comprised of µ=,,, 3, ad 4 modes at (a) 4 Hz ad (b) -3 Hz