PROJEKTKURS I ADAPTIV SIGNALBEHANDLING

Transcription

1 PROJEKTKURS I ADAPTIV SIGNALBEHANDLING Room Aouss wh assoaed fundamenals of aouss PURPOSE.... INTRODUCTION.... FUNDAMENTALS OF ACOUSTICS VIBRATIONS AND SOUND (WAVES)...4. WAVE EQUATION...6 A. The equaon of onnuy...6 B. The equaon of moon (Euler s equaon)...6 C. The equaon of sae...6 D. Equaons of lnear aouss...7 E. Wave equaon of lnear aouss...7 F. Veloy poenal SIMPLE SOLUTIONS OF THE WAVE EQUATION...8 A. Plane ravelng waves...9 B. Aous mpedane and haraers aous mpedane...9 C. Spheral waves... D. Energy densy, aous nensy, and aous power... E. Sound levels and debel sales....4 REFLECTION AND REFRACTION AT BOUNDARIES STANDING WAVES... 5 A. Refleon of plane wave a a fla rgd boundary... 5 B. Sandng waves n he reangular avy RAY ACOUSTICS (GEOMETRIC ACOUSTICS) ROOM ACOUSTICS SOUND IN ROOMS... 8 A. Dre sound and reverberan sound... 8 B. Aneho hambers, reverberan hambers and reverberan rooms... 8 C. Impulse response of a reverberan room... 9 D. A smple model for he growh of sound n a room... E. Reverberaon me... F. Radus of reverberaon STANDING WAVES AND NORMAL MODES IN ROOMS... 5 A. Normal modes and egenfrequenes n a reangular room... 5 B. Frequeny dsrbuon of room resonaes, and hgh frequeny approxmaon... 6

2 Purpose The purpose of he lessons s o nrodue he fundamenal knowledge and prnples on aouss and room aouss ha are needed n ondung he projes n ourse PROJEKTKURS I ADAPTIV SIGNALBEHANDLING. The knowledge and prnples are a bas ool for us o undersand how rooms affe sound wave propagaon, o desgn expermens for orrely measurng he aous properes of rooms, loudspeakers and mrophones, o nerpre he sgnals reorded by mrophones or oher sound reevers n rooms, and so on. Fg... Aous measuremen n a room. Inroduon Le us sar wh wo praal examples ha are losely relaed wh hs proje ourse of adapve sgnal proessng. These examples show wha he room aouss s used for. Example : Measuremen of mpulse response of a room The measuremen s made usng he seup and he reangular room n Fg... Suppose ha he loudspeaker and he mrophone boh have deal frequeny haraerss so ha he loudspeaker an generae mpulse oupus (sound) and he mrophone an reeve hs mpulse sound undsoredly. Plang he loudspeaker and mrophone as a eran dsane as shown n Fg.., he measured resul (he oupu from he mrophone) may look lke ha shown n Fg... In he fgure, we an see () ha for a sngle mpulse oupu from he loudspeaker so many mpulses are reorded by he mrophone; () ha he mpulses beome faded wh me; (3) ha he suessve mpulses ge loser n spang as me nreases, and hen beome dffuse afer a eran perod (e.g., ms). If he loudspeaker reaes anoher mpulse a a laer nsan, say ms laer han he frs generaon, he mrophone wll reord boh he seond mpulse response and he frs one ha s ms earler. Obvously, he seond mpulse sound response s dsurbed by he reverberan par of he frs one. Can we do somehng o suppress hs dsurbane by means of adapve sgnal proessng?

3 Fg... Impulse response of a room a a eran poson. Example : Measuremen of he frequeny response of a reangular room Now we plae he loudspeaker n one orner of he reangular room, and he mrophone n a dagonally oppose orner. By slowly nreasng he frequeny suppled o he loudspeaker from o Hz and smulaneously reordng he oupu of he mrophone, he reorded resul s n sold urve (see Fg..3). In he fgure, he dashed lne orresponds o he oupu of he loudspeaker measured n an aneho (eho-free) hamber. Ths example shows ha he frequeny response of he room s uneven. To oban a rue oupu of he loudspeaker, people may use an nverse fler o ompensae for unevenness n he frequeny response. Ths s he so-alled sound equalzaon. Fg..3. Frequeny response of a reangular room a low frequenes. The problems n hese wo examples are o be nvesgaed n he lessons on room aouss. Beause of he me lm of four leure hours for suh a bg subje, however, he room aouss o be presened here wll be onfned o he sope very lose o he above-menoned purpose and some relevan fundamenals of aouss requred by room aouss. I wll happen very ofen ha he resuls are drely gven whou dealed dervaons. 3

4 . Fundamenals of aouss Fg... A onverson proess. Le us look a a onversaon proess beween wo persons (Fg..). In he fgure he person on he lef speaks and he one on he rgh lsens. The speaker generaes sound, he sound propagaes hrough he ar o he lsener, and he lsener reeves he sound. Ths s a very ommon proess of aouss. From hs proess we an gve he defnon of aouss: ha s aouss s he sene of sound ha s onerned wh he generaon, propagaon, and reepon of energy n he form of vbraonal waves n maer. However, hs s a narrow-sense defnon. Aouss s an anen sene. Up o now has hghly developed and has been spl no a varey of branhes, suh as room aouss or arheural aouss, musal aouss, envronmenal aouss, amospher aouss, areoaouss, nose onrol, ulrasons, underwaer aouss, psyhoaouss, boaouss, and he lke. In hs ourse we are manly neresed n room aouss ha s mosly relevan. Room aouss s one of he bg branhes of aouss. To well undersand he room aouss, we frs need o nrodue some fundamenals of aouss ha are requred by he room aouss. Aouss s dsngushed from ops n ha sound s a mehanal, raher han an eleromagne, wave moon. The mehanal wave moon (sound) an only our n maer no n vauum... Vbraons and sound (waves) For a ompleely res medum, s slen, and here s no sound. When people alk, vbraons our o voal ords. As a resul of hese vbraonal dsurbanes, sounds are reaed. Ths means ha he vbraons are he soure of sound. Oher soures of sound are loudspeakers, unng forks, all knds of mus nsrumens, and he lke. The vbraons may undergo n a large range of magnude and frequeny. Fg... Sounds n dfferen ranges of frequeny and pressure. In presen usage, he erm sound mples no only he phenomena n ar responsble for he sensaon of hearng bu also whaever else s governed by analogous physal prnples; namely, nludes audble 4

5 5 sound whose frequeny les n he range of abou o, Hz wh pressure range of ~ Pa, nfrasound wh oo low frequenes (< Hz) and ulrasound wh oo hgh frequenes (> khz) o be heard by a normal person (see Fg..). Sound ravels n an elas medum n a form of wave moon. Thus, s also alled sound wave. The medum an be gaseous, lqud, and sold. Waves an be of varous ypes n erms of he dreons of parle vbraon and propagaon. The erm parle means a volume elemen large enough o onan mllons of moleules so ha he medum may be hough of as a onnuous medum, ye small enough so ha all physal varables may be onsdered nearly onsan hroughou he volume elemen. Of he mos ommon ypes are longudnal wave (also alled ompresson wave) for whh he dreons of parle vbraon and wave propagaon are parallel, and ransverse wave (or alled shear wave) for whh he dreons are perpendular (see Fg..3). Sound waves propagang n ar are longudnal waves. Transverse waves an no propagae n ar beause ar does no susan ransverse sress. Longudnal wave Transverse wave Fg..3. Sound waves of wo ommon ypes: longudnal and ransverse waves. Sound wave s a physal phenomenon and proess ha s relaed wh varaons of pressure, parle veloy, densy, and emperaure. For example, for a medum n an equlbrum sae n whh here exss no sound, has sa pressure p (equlbrum pressure), and onsan densy ρ (equlbrum densy). When a sound wave ravels n he medum, he pressure and densy n he medum are hanged o p ' and ρ ', respevely. The dfferene of he pressures p = p' p, (.) s alled aous (sound) pressure, and he dfferene of he denses ρ = ρ' ρ, (.) s relaed wh ondensaon 5

6 ( ρ' ρ ) ρ s =. (.3) Boh p and s (or ρ ) vary wh me. Speal noes: () I should be noed ha he only hng ha ravels n he wave s s sae, n he ase of longudnal wave he sae of ompresson and rarefaon. How fas he wave ravels s measured by sound veloy or sound speed. () Noe ha ravelng of a wave does no mean ha he wave ranspors he parles. The parles perform elas osllaons only abou her posons of res, and reman n plae. The rae of moon of he parles s measured by parle veloy. () Sound veloy and parle veloy are dfferen parameers desrbng he moons of wo dfferen hngs.. Wave equaon To alulae sound waves or felds of sound waves, we need mahemaal equaons desrbng he waves. As menoned above, sound waves ause varaons of pressure, parle veloy, densy, and emperaure n he medum where he waves propagae. I mples ha he wave equaon may be formulaed usng he relaons beween hese physal quanes n erms of eran physal laws and mehansm. A. The equaon of onnuy The equaon of onnuy s expressed as where ρ' = ( ρ' v), (.4) s he dvergene operaor so ha A = A x + A y + A z ha s salar. The equaon s obaned from he onservaon of mass. I relaes parle veloy wh he nsananeous densy. Noe ha s nonlnear. B. The equaon of moon (Euler s equaon) The equaon of moon s wren as dv v p = ρ', equvalenly p = ρ ' + ( v ) v, (.5) d where v s he parle veloy, and s he graden operaor so ha f = xˆ f x + yˆ f y + zˆ f z (where xˆ, yˆ, zˆ are he un veors n he x-, y- and z-dreon, respevely). The equaon of moon s obaned based on Newon s seond and hrd laws. I onnes pressure wh parle veloy and densy. I s sll nonlnear. C. The equaon of sae The equaon of sae, generally, relaes pressure wh densy n a ompressble flud n he followng way, ( ρ) p ' = p'. (.6) x y z Applyng a Taylor-seres expanson o he above equaon, we have 6

7 p' p' p ' = p + ( ' ) + ( ' ) +... ' ρ ρ ' ρ ρ, (.7) ρ ρ ρ whh s obvously a nonlnear equaon. ρ D. Equaons of lnear aouss Sne he above equaons (he equaons of onnuy, moon and sae) above derved are nonlnear, hey are nraable n general. In many suaons n prae, sound waves have small amplude. These aous waves an, hus, be regarded as small-perurbaons o an equlbrum sae. Ths means ha p and ρ and v are small quanes, ha s p << p, ρ << ρ. By negleng he seond- and hgher-order erms of he small quanes n he nonlnear equaons, we an lnearze hese equaons, and hus oban he equaons for lnear aouss. Sne ρ << ρ, ondensaon s s small. Negleng he seond- and hgher-order erms, we ge he lnearzed onnuy equaon, s = v. (.8) Makng suh approxmaons ha lnearzed moon equaon, dv / d = v / + ( v ) v v /, and ρ' = ρ ρ ρ, we oban he v p = ρ, (.9) Reanng he lowes order erm n he equaon of sae (Eq. (.7)), we have he lnearzed sae equaon, p = s, (.) ρ where ( '/ ρ' ) ρ = p (.) s referred o as he sound veloy. E. Wave equaon of lnear aouss Combnng he lnearzed equaons, Eqs. (.8)-(.), and sorng ou a dfferenal equaon wh sngle dependen varable, for example, pressure, we oban he wave equaon n erms of pressure n a lossless flud, p p =, (.) where = + + s he hree dmensonal Laplaan operaor. The dervaon of he wave x y z equaon was made n erms of pressure as he dependen feld varable. The same form n Eq. (.) also holds for ρ, v, and s. 7

8 I has found expermenally ha aous proesses are nearly adaba (whh means ha here s nsgnfan exhange of hermal energy from one parle of flud o anoher). Therefore, a good approxmae equaon of sae for ar s he adaba equaon of sae for a perfe gas expressed by p' p ρ' = ρ γ, (.3) where γ s he rao of he spef heas a onsan pressure and volume, respevely. Subsung Eq. (.3) no Eq. (.), we have he sound veloy n gas, γp =. (.4) ρ Makng use of he deal gas equaon (Boyle s law), Eq. (.4) an alernavely be expressed as, = Tk 73 = + / 73, (.5) / T where s speed a beomes C, k T s he absolue emperaure n kelvns and T s he emperaure n C. In ar, = T / 73. (.6) Ths reveals ha emperaure has effe on sound speed. A C, he sound speed n ar s m/s. F. Veloy poenal An alernave formulaon ha leads o he wave equaon s n erms of veloy poenal. Sne he url of he graden of a funon mus vansh, f =, from he equaon of moon (Eq. (.9)) he parle veloy mus be rroaonal, v =. Thus, we an express he parle veloy as he graden of a salar funon Φ, v = Φ. (.7) From Eq. (.9), we an easly oban he relaon of veloy poenal wh pressure, p Φ ρ. (.8) =.3 Smple soluons of he wave equaon The hypohess ha sound s a wave phenomenon s suppored by he fa ha he lnear aouss equaon and herefore he wave equaon have soluons onformng o he noon of a wave as a dsurbane ravelng hrough a medum wh lle or ne ranspor of maer. Smple soluons exhbng hs feaure ha plays a enral role n many aous oneps are plane ravelng waves and spheral waves. 8

9 A. Plane ravelng waves For a plane wave, all aous feld quanes vary wh me and wh some Caresan oordnae x bu are ndependen of poson along planes normal he x-dreon. In hs ase, he wave equaon redues o p = x p, (.9) where p = p(x, ). One has been a general soluon o hs equaon, expressed as p ( x, ) = p+ ( x) + p ( + x), (.) where p + and p are wo arbrary funons of argumens ( - x) and ( + x), respevely. Inserng p + or p or p are he soluon of he wave equaon. I an p n he wave equaon, we an onfrm ha eher + be shown ha p + and p represen he plane waves ravelng forward and bakward wh sound speed, respevely. The sum of he wo soluons makes he omplee general soluon of he wave equaon. For a j( ω + ϕ ) harmon soure ha s relaed wh snusodal vbraon n he form e (where ω s he angular frequeny), he omplex form of he harmon soluon for he pressure of a plane wave s jk ( x) jk ( + x) p = Ae + Be, or (.a) where p = Ae j( ω kx) j( ω + kx) + Be, (.b) k = ω / = π / λ s alled wave number, λ = / f s he wave lengh, and he assoaed parle veloy s obaned from he moon equaon n Eq. (.9), A v = e ρ j( ω kx) B ρ e j( ω+ kx) where xˆ s an un veor n he x-dreon. xˆ, (.) B. Aous mpedane and haraers aous mpedane The rao of aous pressure n a medum o he assoaed parle veloy s defned as aous 3 mpedane n Pa s /m or N s /m, p Z =. (.3) v For plane waves ravelng forwards or bakwards hs rao s Z = ±, (.4) ρ where ρ ha only depends on he maeral properes s alled haraers aous mpedane (smply haraers mpedane). In general, aous mpedane wll be found o be omplex,.e., Z = R + j X, where R s aous ressane and X s aous reaane. For ar a C, he sound veloy s 343 m/s, he densy s. C 3 kg / m, and hus s haraers mpedane s 45 Pa s /m. For dslled waer a, he sound veloy s 483 m/s, he densy s Pa s /m. 3 kg / m, and hus s haraers mpedane s 9

10 C. Spheral waves In he oordnaes (r, θ, φ ) he Laplaan operaor s of he form, p p p p p = + + snθ +. (.5) r r r r snθ θ θ r snθ φ If he waves have spheral symmery, he aous pressure s a funon of radal dsane r and me bu no of he angular oordnaes θ and φ. Then he equaon (Eq. (.5)) beomes p p ( rp) p = + =. (.6) r r r r r and he wave equaon n Eq. (.) redues o ( rp) = r ( rp). (.7) Treang rp as a sngle varable, he equaon s of he same form as he plane wave equaon wh general soluon rp ( r, ) = p+ ( r) + p ( + r), or (.8a) p ( r, ) = p+ ( r) + p ( + r). (.8b) r r The frs erm represens a spheral wave dvergng from a pon soure a he orgn wh speed ; he seond erm represens a spheral wave onvergng ono he orgn. The mos ommonly used spheral wave n prae s he dvergng one n he harmon ase, expressed as p( r, ) = A e r j( ω kr), (.9) and from he moon equaon he parle veloy beomes, j p v( r, ) = rˆ. (.3) kr ρ In he ase of pressure spheral wave, he amplude s A/r, and obvously, s no onsan, bu dereases wh he dsane from he soure, even f he wave propagaes n a lossless medum. The aous mpedane s no ρ, bu or kr jζ Z = ρ e, (.3a) + ( kr) jζ Z = ρ osζ e, (.3b) where ζ = aran( / kr). (.33) The aous mpedane Z s dependen on he dsane from he soure.

11 D. Energy densy, aous nensy, and aous power The energy ranspored by aous waves hrough a flud medum s of wo forms: he kne energy of he movng parles and he poenal energy of he ompressed flud. The nsananeous energy densy s he oal aous energy (he sum of he kne and poenal energy) n a un volume E = p ρ + v, (.34) ρ 3 whh s measured n joules per ub meer (J/ m ) and s boh poson and me dependen beause he parle veloy and pressure are funons of boh poson and me. The me average of E gves he energy densy E a any pon n he flud E = E = T T o E d, (.35) where T s he perod of a harmon wave. For a harmon plane wave, / v = ρ so ha he nsananeous energy densy beomes p E pl p = = ρ v. (.36) ρ and f P A and V A are he ampludes of he pressure and parle veloy, from Eq. (.35) he energy densy of he plane wave s E pl A PAV A P = = = ρv A. (.37) ρ To be analogous o he eleromagne waves, we use he so-alled effeve ampludes, P = / and V e = V A /, and Eq. (.37) beomes e P A E PV e e Pe pl = = = ρ Ve. (.38) ρ The aous nensy I of a sound wave s defned as he average rae of energy hrough a un area normal o he dreon of propagaon. From he defnon, follows I = pv = T T o pvd. (.39) The fundamenal un of aous nensy s was per square meer (W/ m ). For a harmon plane wave, s aous nensy s, I pl e P = PeV e = = ρ Ve. (.4) ρ

12 For a harmon spheral wave wh he effeve ampludes of pressure and parle veloy, he aous nensy P e and V e, Pe I sph = PeV e os ζ = = ρ Ve os ζ, (.4) ρ where ζ s deermned by ζ = aran( / kr) (see Eq. (.33)). The aous power of a soure s defned as he average rae a whh oal energy radaed by he soure flows hrough a losed surfae. Is un s wa. For a pon soure produng a spheral wave ha s expressed by Eq. (.9), he aous power an be obaned by alulang he average rae of energy flow hrough a losed spheral surfae of radus r surroundng he soure n he manner, W sph PA πa = 4π r I sph = 4πr =, (.4) ρ ρ where he relaon P A = A / r s used n he hrd equaly. Ths equaon shows ha he aous power of a spheral wave from a pon soure s ndependen of he radus of he surfae, a onluson ha s onssen wh onservaon of energy n a lossless medum. E. Sound levels and debel sales I s usomary o desrbe sound pressures and nenses usng logarhm sale known as sound levels. One reason for dong hs s ha he logarhm sale ompresses he very wde range of sound pressures and nenses enounered n our aous envronmen, e.g., audble nenses range from approxmaely o W / m. The seond reason s ha humans judge he relave loudnesses of wo sounds by he rao of her nenses, a logarhm behavor. The mos generally used logarhm sale for desrbng sound levels s he debel sale (db). The nensy level L s defned by I ( I ) L = log / (.43) I I ref where I ref s a referene nensy, L I s expressed n db referened o I ref, and log represens logarhm o he base. Sound levels are also measured usng sound pressure level I L p ha an be derved from he nensy level L. In he ases of plane and spheral waves, he nensy and effeve pressure are relaed by I = Pe ρ ). Thus, he sound pressure level s p where ( ( P P ) L = log /, (.44) e ref L p s expressed n db referened o measured effeve pressure of he sound wave. The referene sandard for nensy level s P ref, P ref s a referene effeve pressure, and P e s he I = W / m, (.45) ref

13 whh s approxmaely he nensy of a Hz pure one ha s jus barely audble o an unmparedhearng person. The orrespondng referene effeve pressure s 5 P ref = Pa = µ Pa. (.46) In underwaer aouss, he dfferen referene pressures are used o spefy sound levels..4 Refleon and refraon a boundares When an aous wave ravelng n one medum enouners he boundary of anoher medum, refleon and refraon of he wave our. General dsusson of hs phenomenon s dfful. Here he dsusson s onfned o a smples, bu mporan ase ha boh he nden wave and he boundary beween he meda are planar and ha all he meda are fluds. Le he plane a x = be he boundary beween flud I of haraers mpedane Z = ρ and flud II of haraers mpedane Z = ρ. Assume ha a plane wave s oblquely nden ono he boundary a angle θ, he wave s refleed no flud I a he boundary a angle θ r, and ransmed no flud II a angle θ, as shown n Fg..4. Then he nden, refleed, and ransmed waves are expressed as p p r j ω k x os θ k y snθ ) ( = A e, (.47) r j ω + k x os θ k y sn θ ) ( r r = A e, (.48) j ω k x os θ k y sn θ ) ( p = A e. (.49) The ransmed and nden waves have he same frequeny, bu dfferen sound speeds, and, and hus dfferen wave numbers, k = ω / and k = ω /. There are wo boundary ondons ha mus be sasfed: () he aous pressure on boh sdes of he boundary are equal () he parle veloy (or dsplaemen) normal o he boundary are equal. The frs ondon, onnuy of pressure, means ha here an Fg..4. Refleon and refraon be no ne fore on he plane separang he fluds. The seond of plane waves a he boundary. ondon, onnuy of normal veloy, guaranees ha he fluds are always n ona. Applyng he ondon for onnuy of pressure a he boundary x =,.e., p + p r = p, yelds jk y snθ jk y sn θ jk y sn θ + Are = A e, (.5) A e Sne hs expresson mus be rue for all y, he exponens mus all be equal. Ths means ha k snθ k snθ r = k = snθ, and hus we have θ = θ r, (.5) whh ndaes ha he angle of ndene equals he angle of refleon, and 3

14 snθ snθ =, (.5) a saemen of Snell s law. Usng he ondon for onnuy of he normal parle veloy a he boundary, v x + v rx = v x, gves v os θ + v osθ = v osθ. (.53) r r Consderng he relaons v = p Z, vr = pr Z, v = p Z, and θ = θr, Eq. (.53) beomes osθ A. (.54) ρ ρ ( A Ar ) = osθ Combnng Eqs. (.5) and (.54), we oban R A ρ osθ ρ osθ r = =, (.55) A ρ osθ + ρ osθ whh s defned as pressure refleon oeffen and T A ρ osθ = =, (.56) A ρ osθ + ρ osθ whh s defned as pressure ransmsson oeffen. From Eqs. (.55) and (.56), s easy o fnd +R = T. From Snell s law, we have osθ = sn θ = sn θ. (.57) I s mporan o noe hree mplaons of Eq. (.57) () If >, θ s real and less han he angle of ndene. A ransmed beam exss n he seond medum and hs beam s ben oward he normal o he boundary for all angles of ndene. () If < and θ < θ where θ s he ral angle gven by θ = arsn, (.58) θ s agan real bu greaer han he angle of ndene. A ransmed beam exss bu he beam s ben away from he normal o he boundary for all angles of ndene ha are less han he ral angle. () If < and θ > θ, os θ s now pure magnary. The nden wave s oally refleed. Inensy refleon and ransmsson oeffens are also ofen used n prae, and hey are defned as R I = I r I and T I = I I, respevely, where I, I r and I are he nenses of he nden, refleed, and ransmed waves. The nensy refleon and ransmsson oeffens an be derved from he orrespondng pressure oeffens, 4

15 R I r = R, (.59) I I = and T I I ρ osθ = = T. (.6) I ρ osθ.5 Sandng waves A. Refleon of plane wave a a fla rgd boundary Consder he ase n Fg..4 where he boundary s rgd. In hs ase, he plane wave s ompleely refleed, and no energy s ransmed no medum II. Therefore, we have A = A r, (.6) and he resulng wave n he x spae s he sum of he nden and refleed waves, p = p + p r = A os( k x osθ )sn( ω k y sn θ ). (.6) For he normal ndene ( θ = ) he equaon beomes p = A os( kx)sn( ω). (.63) The soluon an be nerpreed n wo dfferen bu equvalen ways: () he nerferene of wo waves of equal amplude and wavelengh ravelng n oppose dreons, and () a waveform does no propagae, nsead remans saonary. Therefore, suh a wave s alled a sandng wave and s mahemaally haraerzed by an amplude ha depends on he poson along he propagaon dreon. A represenave sandng wave s shown n Fg..5 where he pressure magnude a varous posons s ploed. The posons of zero pressure are alled nodes and he posons of maxmum pressure are alled annodes. The dsane beween he nodes s half wavelengh λ /. Fg..5. Sandng wave ploed n erms of he varaon range of pressure magnude as a funon of poson. 5

16 B. Sandng waves n he reangular avy Consder a reangular avy of dmensons L x, L y, L z, as shown n Fg..6. Ths box ould represen a reverberan room or audorum, a smple model of a oner hall, or any oher reangular spae ha has few wndows or oher openngs and farly rgd walls. Fg..6. The reangular avy Assume ha all surfaes of he avy are perfely rgd so ha n ˆ v = a all boundares. Then n ˆ p = and hus p x p = x x= x= Lx p p p p = ; = = ; = = y y z z z z= L y= y= Ly = z (.64) Sne aous energy an no esape from a losed avy wh rgd boundares, approprae soluons of he wave equaon are sandng waves. Subsuon of p jω ( x, y, z, ) = X ( x) Y( y) Z( z) e (.65) no he wave equaon and separaon of varables resuls n he ses of equaons X x Y Z + k X x = ; + k Y = y ; = y + k z Z (.66) z where he separaon onsans mus be relaed by x y z k = k + k + k. (.67) Applaon of he boundary ondons n Eq. (.64) shows ha osnes are approprae soluons, and Eq. (.65) beomes p lmn lmn jω lmn ( k x) os( k y) os( k z) e ( x, y, z, ) = A os, (.68) xl ym zn where k k k = lπ, l =,,, xl L x = mπ, m =,,, (.69) ym L y = nπ, n =,,, zn L z 6

17 Thus, he allowed frequenes of vbraon, ω k or f = ω π, are quanzed, lmn = lmn lmn lπ mπ nπ ω lmn = k = + +, or Lx Ly Lz whh are deermned by he avy's dmensons l m n f lmn = + + (.7) Lx Ly Lz L x, L y, L z wh ombnaons of l, m, n. Eq. (.68) shows a seres of dsree funons. These funons are alled egenfunons, or normal modes, or resonane modes. Assoaed wh eah of he soluons s a unque frequeny known as egenfrequeny, or normal mode frequeny, or resonane frequeny, deermned by Eq. (.7). The form n Eq. (.68) gves hree-dmensonal sandng waves n he avy wh nodal planes parallel o he walls. Beween hese nodal planes he pressure vares snusodally, wh pressure whn a gven loop n phase, wh adjaen loops 8 ou of phase..6. Ray aouss (geomer aouss) Ray heory, alhough approxmae, an gve farly good predon o expermenal measuremens n reverberan rooms. Here he ray heory s very brefly nrodued. In he real world, nsead of plane waves, we fnd sound beams whose ross-seonal area and dreons of propagaon may hange as he beams raverse he medum. In suh rumsanes, we frequenly fnd useful o hnk of rays raher han waves. A ray an be defned as a lne everywhere perpendular o he surfaes of onsan phase (Fg..7). Is usefulness les n he nuve feelng, mahemaally jusfed under eran ondons, ha energy s arred along a ray. In many ases, espeally where s a funon of spae or where he wave s resred o a lmed sold angle (suh as he beams of sound from a hghly dreonal soure), desrpon n erms of rays s muh easer han wave frons. However, rays are no exa replaemens for waves bu only approxmaons ha are vald under eran raher resrve ondons. The mehods of reang refleon and refraon (Snell s law) of plane waves apply o he propagaon of rays. Fg..8 shows an example of how he rays from a pon soure propagae n a reangular room. Fg..7. Rays a a surfae of equal phase. Fg..8. Rays from a pon soure and her propagaon. 7

18 3. Room aouss Up o now que a lle knowledge has been presened abou fundamenals of aouss whh s nended o esablsh a foundaon ha s enough for servng room aouss. Ths seon s gong o presen he room aouss ha s drely relaed wh hs proje ourse of adapve sgnal proessng. 3.. Sound n rooms Rooms ha are eher bg or small are he neessary spae for our lvng, workng, eneranng, e. Los and los of aves lke onferenes, oners, spors evens, and so on, happen n rooms. Perhaps we have suh experene ha he same speehes an feel very dfferen when hey are gven n a oner hall, n a bg onferene room, n a lassroom, n he rooms wh a same sze bu dfferen walls. Therefore, he shapes, dmensons and wall's surfae sruure of rooms have effe on sounds. How hese affe sound s he subje of room aouss. Fg. 3.. A proess of sound generaon and perepon n a reverberan room. A. Dre sound and reverberan sound Usng ray model we an nuvely and approxmaely explan he sound proess happenng n a room. Assume ha he speaker s approxmaed as a pon soure ha generaes spheral waves. The rays from he soure ravel ouward n he dvergng dreon. A eah enouner wh he boundares of he room, he rays are parly absorbed and parly refleed. Anoher assumpon s ha he lsener s hough of as a small reever (mrophone). When he speaker makes a sound n a room, he sound hen pereved by a lsener or a mrophone onsss of he sound omng drely from he soure (e.g., Ray D n Fg. 3.) plus he sounds refleed or saered by he walls (e.g., Rays R, R and R 3 n Fg. 3.) and by objes n he room. The sound drely omng from he soure s alled dre sound, and he sound havng undergone one or more refleons s alled reverberan sound. For an mpulse soure he reverberan sound orresponds o a seres of ehoes. B. Aneho hambers, reverberan hambers and reverberan rooms If he dre sound wave predomnaes almos everywhere, he room s aneho (eho-free); rooms so desgned are aneho hambers. When he rooms are desgned so ha he reverberan wave predomnaes 8

19 overwhelmngly, hey are alled reverberan hambers. The rooms n whh we are lvng are neher aneho nor reverberan hambers, bu he rooms n beween and wh eran reverberan effes; hey are alled reverberan or lve rooms. C. Impulse response of a reverberan room Le us go bak o he suaon n Fg. 3.. Also le a loudspeaker subsue he speaker ha an make an mpulse sound, and a mrophone subsue he lsener ha pereves he sound n he reverberan room. A ypal, represenave measuremen an look lke ha n Fg. 3., whh was shown a he begnnng of he aouss sesson. The frs mpulse o he mrophone s he dre sound, he seond mpulse ha s smaller s he frs refleon from he surfae loses o he mrophone, he hrd, he forh and he ohers are all he refleed sounds from he frs or mulple refleons. The refleed sounds beome smaller and smaller beause hey have more enouners wh he surfaes and hus ge absorpon by he surfae. As he more me has elapsed, he more refleed rays reah he mrophone. Therefore, he mpulses ome loser and loser n me,.e., he nerval beween he adjaen mpulses ges smaller and smaller unl hey beome dffused so ha he sound hroughou he room s well blended. Ths sound proess may vary from room o room, dependng on he aous properes of he rooms. To spefy he aous propery of a room, one mporan aous parameer s always used ha s he reverberaon me, defned as he me requred for he sound pressure o drop 6 db from he nal level, whh wll be presened n deal n he followng seon. Fg. 3.. Impulse response of a reangular room. Wha wll happen f a soure s urned on and onnuously operaes n a room? Expermens have shown ha n hs ase he aous nensy a any pon n he room bulds up o hgher values ha would exs f he soure were operaed n open ar, he gan n nensy ofen beng greaer han enfold. In oher words, he sound grows up n he room. For any gven enlosure (or room) hs gan s nearly proporonal o he reverberaon me. Therefore, a long reverberaon me s desrable f a weak soure of sound s o be audble everywhere n he room. If he soure s shu off, he reepon of dre sound eases afer a shor me nerval = r/, where r s he dsane from he soure o he pon of observaon and s he speed of sound n ar. The refleed waves onnue o be reeved as suesson of arrvals of dereasng nensy. The presene of hs reverberan aous energy ends o mask he mmedae reognon of any new sound, unless suffen me has elapsed for he reverberaon o fall down some 5 o db below s nal level. Sne he reverberaon me s a dre measure of he perssene of suh sounds, s obvous ha a shor 9

20 reverberaon me s desrable o mnmze maskng effes. The hoe of he bes reverberaon me for a parular room mus, herefore, be a ompromse. D. A smple model for he growh of sound n a room When a soure of sound s sared n a reverberan room, refleons a he walls produe a sound energy dsrbuon ha beomes more and more unform wh nreasng me. Ulmaely, exep lose o he soure or o he absorbng surfaes, hs energy dsrbuon may be assumed o be ompleely unform and o have essenally random loal dreons of flow. Ths s well known Sabne heory. Under hs assumpon, le he aous energy densy E be unform hroughou he room wh a volume of V. The rae a whh he energy falls on a un area of he wall an be found o be de E =, (3.) d 4 where s he sound speed. If s assumed ha, a any pon whn he room, energy s arrvng and deparng along ndvdual ray pahs and ha he rays have random phases a he pon, hen he energy densy E s he sum over all rays of he energy denses E of he ndvdual rays. Supposng ha he mh ray has effeve pressure P em, we have Em = Pem /( ρ ), and hus E = m ρ P em m = ρ er P m, (3.) where P er = P em s he spaally averaged effeve pressure amplude of he reverberan sound feld. If he oal sound absorpon of he room s A, hen follows from Eq. (3.) ha he rae a whh energy s beng absorbed by all surfaes s A 4 E. (3.3) Noe ha A has a un of square meers and ofen gven n mer sabn ( m ) or Englsh sabn ( f ). Aordng o he energy onservaon, he rae of absorbed sound energy ( AE 4 ) plus he rae of growng sound energy ( V de d ) hroughou he neror of he room mus equal he rae of energy W produed by he sound soure(s). From hs, we oban a fundamenal dfferenal equaon ha governs he growh of sound energy n a lve room, A E + V 4 de d = W. (3.4) If he sound soure s swhed on a =, soluon of hs dfferenal equaon and use of Eq. (3.) yelds P where exp 4W = er ( ) ρ A E, (3.5) 4V E = (3.6) A s he me onsan governng he growh of he aous energy n he room. If A (he oal sound absorpon) s small and he E s large, a relave long me wll be requred for he effeve pressure amplude and energy densy E o approah he ulmae values of

21 Wρ ( ) = and A 4 P er 4W E( ) = (3.7) A The growh of sound n Eq. (3.5) an be expressed n erms of relave sound level n he followng manner, P er ( ) L = log = log exp Per ( ) E (3.8) whh s shown n Fg. 3.a. E. Reverberaon me Fgure 3.a. Growh of he sound energy n a room If he sound soure n a reverberan room wh unformly dffuse sound s urned off a =, he pressure a any laer me an be found from Eq. (3.4), as follows P er ( ) = P () exp( / ). (3.9) er E The equaon shows ha for eah me nerval E he energy densy dereases by /e. Eq. (3.9) an be wren, n erms of relave sound level, as P ( ) L = log P () er er logexp = E (3.9') The me requred for he sound level o drop by 6 db s defned as he reverberaon me T (Fg. 3.a),.e., solvng Eq. (3.9 ) for = T wh L = 6 db gves 55.V T = 3.8 E =. (3.) A Wh = 343 m/s ( C ) hs beomes.6 V T =. (3.) A If he surfae area of he room s S, he average Sabne absorpon a s defned by a = A / S, (3.) and he reverberaon me n Eq. (3.) beomes.6 V T =. (3.3) Sa The reverberaon me s an mporan parameer deermnng he aous performane of a room. To pred he reverberaon me of a room wh gven aous properes, one needs o know he oal sound absorpon ha depends on he areas and absorpve properes of all he maerals whn he room. Sabne adoped he plausble assumpon ha he oal sound absorpon s he sum of he absorpons A of he ndvdual surfaes, n

22 An = A S a, (3.4) = n n n n where a n s he Sabne absorpvy of he nh surfae S n. Thus, he average Sabne absorpvy beomes a = S n S n a n. (3.5) Eah a n s o be evaluaed from sandardzed measuremens on a sample of he maeral n a reverberan hamber. Beause of he frequeny dependene of he absorpon of eah surfae, s neessary o spefy he reverberaon me for represenave frequenes overng he enre range ha s mporan o speeh and mus. The frequenes usually hosen are 5, 5, 5,,, and 4 Hz. Some represenave absorpon oeffens of surfaes are gven n Table 3.. Table 3. Represenave absorpon oeffens of surfaes. Absorpon oeffen a Maerals 5 Hz 5 Hz 5 Hz Hz Hz 4 Hz Brks, unglazed Plaser, gypsum or lme, on brk On onree blok Conree blok, oarse Paned Plywood, -m-hk Panelng Cork,.5 m hk wh arspae behnd Glass, ypal wndow Drapery, lghwegh, fla on wall Heavywegh, draped o half area Floor, onree Lnoleum on Heavy arpe on Floor, wood Celng, gypsum board Plasered Plywood, m hk Suspended aousal le, m hk Gravel, loose and mos, m hk Grass, 5 m hgh Rough sol Waer surfae, as n a pool Soure: A. D. Pere, Aouss an nroduon o s physal prnples and applaons, MGraw-Hll, 98, p. 56.

23 Convenonally, when he erm reverberaon me s used whou spefaon of any parular frequeny, s generally undersood o refer o he frequeny of 5 Hz. An example s gven n Fg. 3.3, showng measuremens of reverberaon mes a hree dfferen frequenes, 5, 5 and 5 Hz. Fg Measuremens of reverberaon mes a hree dfferen frequenes, 5, 5 and 5 Hz. Crera for wha onsues good aouss for rooms nended for spefed purposes have been exensvely developed sne a long ago. The reverberaon me plays a enral role n he quanave formulaon of some of he smpler rera, lke he example shown n Fg Fg Opmum mdfrequeny (5 o Hz) reverberaon mes for fully ouped rooms versus volume (from A. D. Pere, Aouss an nroduon o s physal prnples and applaons, MGraw-Hll, 98, p. 7). 3

24 F. Radus of reverberaon Whenever a onnuous soure s presen n a room, wo sound felds are reaed. One s he dre sound feld from he soure. The oher s he reverberan sound feld ha s produed by he refleons from he surfaes of he room. Assume he reverberan feld has no effe on dre sound feld or soure power. Ths s no exaly rue, espeally f he frequeny band of neres s narrow or f he soure s emng a pure one, bu may be regarded as approxmaely so f one hnks n erms of loal spaal rue. For a pon soure wh aous power W, he effeve pressure amplude P ed of he dre sound feld a a moderae dsane from he soure s desrbed by P ed ρ W =. (3.6) 4πr The reverberan feld s assumed o be ompleely dffuse so ha s haraerzed by Eq. (3.7)). The oal mean squared pressure s P er = 4Wρ A (see P 4 = ρ W +. (3.7) 4πr A Ths formula gves an ndaon of how far, or lose o, he soure one mus be o be assured ha he reverberan (or dre) feld predomnaes. A he radus of reverberaon, or ral radus, A r r = (3.8) 6π he wo felds are of equal onrbuon, and he sound pressure level s 3 db hgher han expeed from eher alone. Inserng Eq. (3.) no Eq. (3.8), we have he relaon beween he reverberaon radus and me n he followng manner V r r =. 56. (3.9) T In he radal regon of r r, he dre feld s larger han he reverberan feld, and ousde hs regon, he dre feld s smaller han he reverberan feld (Fg. 3.5). Fg Dre feld, reverberan feld and her superposon. 4

25 3.. Sandng waves and normal modes n rooms In he prevous seon 3., based on ray aouss, we have derved formulas for deermnng he reverberaon me, he average Sabne absorpon, reverberaon radus, and he ohers. Ray aouss, however, does no provde a omplee heory of he behavor of sound n an enlosure. A more adequae approah mus be based drely on wave heory. The wave equaon has been solved (a leas approxmaely) for smple enlosures (suh as reangular and hemspheral spaes) and new oneps have emerged from examnng he ransen and seady-sae behavors of sound n suh enlosures. Even n omplaed enlosures for whh he wave equaon anno be solved, he heory has been used o supplemen and exend resuls provded by ray-aous mehods. Fg Fve dfferen normal modes of pressure feld n a reangular room wh dmenson of 6x4 m. A. Normal modes and egenfrequenes n a reangular room As presened n Se..5, he soluon of he wave equaon n a lossless, rgd-wall, reangular avy of dmensons of L, L, L, resuls n he normal modes x y z p l m lmn ( x, y, z, ) Almn os πx os πy os πz L x L y L z n jπf lmn = e, (3.) whh are a seres of dsree funons, eah wh s own egenfrequeny expressed by 5

26 l m n f + + lmn =. (3.) Lx Ly Lz The egenfrequenes are ompleely deermned by he naure of he room. The modes are labeled by he neger se (l, m, n). If all he negers are no zero, he normal mode s ermed oblque. If one of he negers s zero, he mode s ermed angenal beause he propagaon veor of he wave s parallel o one par of he room surfaes. If wo of he negers s zero, he mode s ermed axal beause he propagaon veor of he wave s parallel o one of he surfae axes. Fg. 3.6 llusraes some normal modes n a reangular room 6x4 m. B. Frequeny dsrbuon of room resonaes, and hgh frequeny approxmaon Knowledge of he egenfrequenes of a room s essenal o a omplee undersandng of s aous properes. When a soure s presen, he room wll respond srongly o hose sounds havng frequenes n he mmedae vny of any of hese egenfrequenes. I s jus hs haraers ha affes he oupu of a loudspeaker as measured n a reverberan room and auses he dsored resuls of he loudspeaker s properes. Eah sandng wave has s own egenfrequeny and hus s own parular spaal paern of nodes and annodes. In effe, superposon of he haraerss of he room wh hose of any sound soure presen resuls n he fluuaons n sound pressure ha vary wh poson and frequeny. Therefore, as a mrophone s moved from pon o anoher, or as he frequeny of he soure s vared, he fluuaons may ompleely oneal he rue oupu haraerss of he soure. I s for hs reason ha measuremens of he response urves of loudspeakers should be arred ou eher n he open ar or n aneho hamber. If he absorpon oeffen s greaer han.99, he reverberan feld s neglgble ompared o he dre wave. Eah of he ndvdual sandng waves of a avy an be exed o s fulles exen by a sound soure loaed n regons where he parular sandng wave paern has a pressure annode (reall Se..5). Observng he normal modes n Eq. (3.), we see ha he pressure ampludes of all paerns of sandng waves n a reangular avy are maxmzed n he orners of he room. Therefore, f he soure s a he orner of suh a room, wll possble for o exe every allowed normal mode o s fulles exen. Correspondngly, f a mrophone s loaed n he orner of he room, wll measure he peak sound pressure of every normal mode ha has been exed. By onras, when a soure s loaed n a regon where a parular normal mode has a pressure node, ha mode wll be exed only weakly. For nsane, f a loudspeaker s loaed n he ener of a reangular room, only hose modes havng even numbers smulaneously for l, m, and n wll be exed ( abou mode n ) as he drvng frequeny s slowly vared from low o hgh frequenes. An example, whh s he seond example presened a he begnnng of he Inroduon, s o be looked no n more deal. A reangular room has a dmenson of 3.x4.69x6.4. The normal modes wh orrespondng normal mode frequenes below Hz are gven n Table 3.. A loudspeaker s posoned a one orner of he room, and he mrophone s loaed n a dagonally oppose orner. In hs rumsane, he measuremen was arred ou by slowly nreasng he frequeny suppled by he loudspeaker from o Hz and smulaneously reordng he oupu of he mrophone. The reorded resul s shown n Fg. 3.7 n sold lne. In he fgure he oupu of he loudspeaker measured n an aneho hamber s gven n dashed lne. In omparson, he nfluene of he room s apparen. When eher he loudspeaker or mrophone s posoned n he ener of he room, only hose peaks orrespondng o he (,, ), (,, ) and (,, ) modes would be observed below Hz. 6

27 Table 3.. Normal mode frequenes below Hz for a reangular room of 3.x4.69x6.4 wh = 345 m/s. l m n f (Hz) l m n f (Hz) Fg Normal mode response of a reangular room 3.x4.69 x 6.4 m a low frequenes. Modal densy s defned as he number of room normal modes per un frequeny bandwdh. Le N(f) denoe he number of room normal modes whose egenfrequenes are less han a gven value of f. In he ase of a reangular room wh dmensons of L, L, L, he oal number of modes N(f) an be alulaed usng Eq. (3.) wh f lmn f, x y z 4πV N( f ) 3 f 3 S + 4 f + L 8 f + 8, (3.) where V s he volume of he room, S= ( L L + L L + L L ) s he oal surfae area and 4 x y y z z x L= ( L + L + L ) s he oal lengh of all he edges n he room. The number of normal modes n a x y z frequeny bandwdh f and enered a frequeny f, denoed by Thus, he average number of modes per un frequeny bandwdh (modal densy) s, an be esmaed as [ dn f df ] f dn( f ) 4 V S L = π f + f +. (3.3) 3 df 4 8 N ( ). 7

28 When he room dmensons are large ompared wh wavelengh, he frs erm predomnaes. Takng he leadng erm yelds he form of modal densy ha s ommonly used n he hgh frequeny range, he number of normal modes n a frequeny bandwdh f beomes N dn( f ) 4πV f = 3 df f f. (3.4) Ths shows ha he number of normal modes n a gven frequeny band wdh f nreases rapdly as he ener frequeny of he band (or he sze of he room) s nreased. As a resul, he responses of he sandng waves wll overlap more and more a hgher frequenes so ha he response of he room wll beome que smooh wh nreasng frequeny. Sne he sandng wave paern orrespondng o eah egenfrequeny s n general assoaed wh some parular se of dreon osnes, any nrease n N ndaes an nrease n he randomness of dreons of he assoaed waves. Ths s suppored by he observaon ha reverberaon equaons based on a dffuse sound feld, suh as Eq. (3.), are n beer agreemen wh expermens a hgh frequenes. The response of a room s observed o beome less unform wh nreasng s symmery beause he number of degenerae modes havng dfferen (l, m, n) bu he same naural frequeny s nreased. From Eq. (3.4) we an esmae he average frequeny spang f mode (n Hz) of egenfrequenes n he frequeny bandwdh f n he followng manner, 3 f f mode = =. (3.5) N 4πVf Whenever he soure drvng frequeny f s suffenly lose o he egenfrequeny f(n), a resonane s apparen. The nh normal mode beomes domnan as f f (n). The bandwdh of he resonane peak an be found approxmaely o be 3.8 f res = =. (3.6) π πt If he average spang E beween he resonane peaks s of he order of or less han, say. 5, he f mode resonane peaks may be regarded as a smoohed-ou onnuum. Sne he average spang wh nreasng frequeny, here s a frequeny f mode f Sh f res f mode derease, alled Shroeder uoff frequeny, below whh <. 5 s no sasfed and above whh s. Ths frequeny s denfed, from Eqs. (3.5) and (3.6), as f res T f Sh =. (3.7) 55. V 5 3 Ths, wh = 34 m/s, beomes T f Sh 9. (3.8) V Sh The Shroeder rule says ha above he Shroeder uoff frequeny a sum over mode ndes an be approxmaed by an negral. In oher words above f he normal modes may be regarded as a smoohedou onnuum. In hs ase, room aouss should be reaed from he sasal aspes. Devaon of aous quanes from he averages preded by he Sabne model are frequenly gven a sasal nerpreaon. 8