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Alternative Models of the Sound Field in a Reverberant Room

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Alternative Models of the Sound Field in a Reverberant Room t.t'r¡ ALTERNAÎIVE MODELS OF lHE SOTJND FIEIÐ IN A REVERBERAI{T ROOM by Thomas J. Munro B. Sc. A thesÍs presented to the Faeulty of Engineering of the Unfverslty of Adelalde for the Degree of Master of Englneering Sclence Department of Mechanfcal Englneering January 1982 University of Adelaide 1'l TABLE OF CONTENTS Summary 1V Statement of 0ri gì naì i ty vi Acknowl edgements vìi chapter I AC0USTIC MEASUREMENTS IN REVERBERATI0N R00MS I l.l Introduct'ion I 1.2 His tory of Reverberant Sound Measurements 3 1.2 Ear'ly Ì¡lork 'in Room Acoustics 4 1.2 2 The Statistical Variation of Measured Quantì t'ies 9 1 .2.3 Determ'i nat'ion of Sound Power I niecti on i n Reverberant F'iel ds 12 1 .2.4 Methods of Obtainingan Accurate Space Average of l4easured Quantì b'ies l4 | .2.5 Stati sti cal Vari a'b'ion ì n Decayi ng Sound Fi el ds l5 1 .2,'o Comparison of Stat'ist'ical and Modal Models of Room Behavi our 17 1 .2.7 Predict'ion of Rate of Decay by Aìternative Model s l8 1.3 Standard l4easurenent Methods 20 1.3.1 Sou nd Power Level 21 r.3.2 Sou nd Absorption Coeff ici ent 25 I .3.3 Sou nd Transmi ss'ion Loss 27 Chapter 2 ALTERNATIVE THEORETICAL DEVELOPMENTS 30 2.1 I ntroduct i on 30 2.2 The hlave Equation and Soìutions 32 2.3 Steady State Behaviour of the Modal Sound Field 37 2 .3 .1 Equal Energy Di s'lri buti on 37 2.3 .2 Equaì Power F I ow 40 ?.4 Modal Descri pt'ion of the Decayi ng Sound F'iel d 41 2.5 Modal Damping by the Acoust'ic Ray Approxitnalion 43 2.5.1 Predi ction of Modaì Decay Rate 43 2.5.2 Norris-Eyrinq Equation as a Spec'ial Case 48 2.6 M'iscel I aneous Formul as 49 Chapter 3 MATERIALS AND METHODS 5l 3.1 Phys'icaì Fac'i I it'ies 5l 3.2 Instrumentat'ion, Sì gnaì Generat'ion and Measurement 53 3.2.1 Generation of Sound 54 3.2.2 Measurement and Process'ing of Sounci Pressure Level s 56 3.3 Experìmental Methods 57 3.3..l Measurement of Modal Decay Rate 57 3.3.2 Measurement of Steady State Modal Energy Level 60 lll Page Chapter 4 RESULTS: MODAL BEHAVIOUR IN THE EMPTY ROOM 62 4.1 Veri f i cati on of l4odal Behav'iour 62 4.1.1 Modal Behaviour in Steady State Conditions 62 4.1 .2 Modal Behavi our ì n a Decay'i ng Sound Fi el d 63 4.2 The llature of Moda'l DecaY 68 4.3 Anaìysì s of the Modal Rate of DecaY 73 4.3. I The Tradìtional ltlodal Approach 77 4.3.2 Modal Damping by the RaY Tracing Approxi rnatì on 79 4.3.3 Probable Reasons for the Fa'ilure of the Modeìs to Pred'ict Actual Behaviour 83 4.4 Initial Energy Level of Room Modes 84 Chapter 5 THE ROTATING DIFFUSER, ITS EFFECT ON MODAL STRUCTURE 90 5.1 Previous l,Jork lvith Rotat'ing Diffusers 9t 5.2 Experimental Results 92 Chapter 6 CONCLUS IONS l0l References 103 ]V SUMMARY Current standard test methods for acoustic measurements nade in reverberant rooms have been found to y'ieìd inconsìstent results at lolv frequencies. The theoret'ical basis of the standard test methods relìes upon the assumption that the reverberant sound field can be modeìed as compìeteìy d'iffuse ab al I f requencies. Thus, 'inconsi stenc'ies have been attributed to a lack of diffusivity at low frequencies. Th'is thesis exami nes the appropri ateness of the di ffuse fi el d assumpti on at I ow frequencìes and shows that the strong modal characteristics of a low frequency reverberant sound field precìudes the possibi'lìty of mode'ling such a field as dìffuse. A nlodel of the reverberant sound field which includes the observed modal characteri st'ics and i s based upon wel I known sol ut'ions to the wave equa'ti on i s presented . Th j s model 'i s dependent on the accurate predi cti on of the steady state ampì 'itude and the decay rate of i nd'ivi dual modes . I n pred'icti ng the steady state ampì 'itude of the modes contri bui'ing to a reverberant sound f iel d, the assumpt'ion i s usual ly made e'ither i mpì i c'itìy or expììcìt1y that there 'is equal energy distributioç between modes. In th'i s thes'i s, an al ternati ve approach based on the assumpti on that there 'is equal power flow to alì modes vlill also be consìdered. These assumpt'ions are di scussed and the'ir i nf I uence on the rel ati ve di ff erence 'i n the steady state sound pressure I evel of any two modes i s analyti ca'l ly 'i nvest'igated. Two models for predicting the decay rate of ind'iv'idual modes are also examined. The fjrst is based upon an approx'imate solution to the wave equation for a very ìightìy damped room. The second is based upon a com- 'ied b'ina'li on of the modal approach and the ray tracì ng approach as appì to the rectangu'lar room. Both models reìy upon the assumption that the 'loca'l wal I s can be accurateìy model ed as ly reacti ve and that the wal l V .impedance can be modeled as constant over the wall surfaces and throughout the frequency range investigated. Experintental results are presented for comparison of each model. Results ind'icate that the wall surfaces of the room cannot be modeled as being locally reac'bive as theìr response to excitation by the reverberant sound fjeld in the frequency range'investjgated is characterized by modaì vibration. In consequence ìt j s shown that neìther model accurate'ly pred'icts the relative rates of decay of the measured modes. The relative ampì i tudes of i nd'ivi dual modes are seen to be h'igh'ly dependent on the effective coupìing between each mode and the sound source. Th'is is shown to be highly unpredictable for commonly used sound sources. 'i A wi deìy recommended means of enhanc'ing the di ff usì on n a rever- berant room is the installation of a rotating diffuser. Th'is thes'is examines the effects of a rotatìng diffuser on the modal characteristics of a reverberant sound field and shows that jt substantially disrupts the nodal structure of the sound field. vl To the best of the candidate's knowledge and belief' thi s thesì s contai ns no materi al whi ch has been accepted for the award of any degree or dipìoma in any Unì versi ty, and conta'i ns no materi al previ ously pubìished or written by another person, except where due reference 'is made 'in the text. Thomas J. Munro vii ACKNOt.lLEDGEMENTS The work described in this thesis was carried out in the Departntent of Mechani cal Engi neerì ng of the Un'ivers'ity of Adel a'ide under the aeg'i s 'i of Prof essor R . E . Luxton. The autholi s ndebted to Professor Luxton f or the opportunity to carry out th'is research. To Dr D . A. B'i es , who supervi sed the work , the author i s deepìy 'indebted, both for his constant interest and encouragement throughout the course of the research and for his helpful suggest'ions during the prepara- ti on of th'i s thesi s . To the University staff, thanks are due for their assistance jn the preparati on of the f aci I 'iti es i n wh'ich the work was compì eted, and i n part'icular to Mr H. Bode and Mr R. Curt'in. The author gratefully acknowledges the Australian American Eciuca- ti onal Foundati on who provi ded the f i nanc'ial support wh'ich made the conrpletion of this work possible. F< Ejf ACOUSTIC MEASUREMENTS IN REVERBERATION ROOMS I . I INTRODUCTiON Acousti c measurements ì n reverberatì on rooms ane w'ide'ly used i n standard tests to determi ne the acoustì c propert'ies of vari ous bui I d'ing materials and of both industrial and domestjc equipment. The most commonly used tests are for the determinatjon of the sound power output of noise sources, the sound transmìssion loss through building partions, and the determìnation of the absorbing propertìes of various material surfaces. None of these properties can be measured directly, so test methods have been developed which allow their determination from those room variables which can be measured, such as the sound pressure level (SPL), the physical d'imens'ions of the test fac'ilities, and the time e'lapsed for a specified decrease in the mean sound pressure level of the reverberant field after the sound source has been shut off. Current standard methods for conduct'ing acoustic tests are pub'lished by the International Standards 0rganisation (lS0), the Standards Association of Australìa, and various other national and international bodies. tJh'ile the prescribed standard test procedures usuaì ly y'ieì d usef ul resul ts , si gnì f icant d'i screpanc'ies have been noted in results and these are the subject of contìnu'ing research in the field.
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