Edge diffraction implementation by semi-transparent surfaces in geometrical acoustics

Transcription

1 Edge diffractio implemetatio by semi-trasparet surfaces i geometrical acoustics Aders Isebae Bree & Strad austi AS, P.O. Box 1024 Hoff, 0218, Norway, aders.isebae@breestrad.o I various commercial acoustic software, geometrical acoustics (GA) is the basis for predictio of the soud field calculatios. However, a great disadvatage of this method is that GA is uable to hadle diffractio effects. Further, some exteded methods, still based o GA assumptios, have bee developed to successfully iclude diffractio. However, the subsequet calculatios have bee foud hard to implemet i existig acoustic software. This paper presets a alterative method for diffractio modellig, based o a idea that taes advatage of a defied semi-trasparet coefficiet assiged to the surface of the obstructig object of a give edge diffractio case. 1 Itroductio Whe dealig with complex geometric situatios geometrical acoustics (GA) is a efficiet method for solvig the wave equatio. However, the disadvatage of this method is that it has certai limitatios amog them ot beig able to hadle diffractio pheomea i a proper way. Because soud waves are regarded as rays the phase quality of the wave ature is igored, which ofte results i low frequecy errors as iterferece patters ofte occur ad affect this frequecy regio. Previous studies [1] have bee published regardig how edge diffractio successfully ca be implemeted i GA calculatios. However, a ofte itroduced complexity i the cosecutive calculatio algorithms has made it somewhat icoveiet to implemet these previously proposed modellig techiques ito commercially available GA software. More recetly, i Dammerud s [2] PhD thesis a smart ad easily implemeted tric was employed i the GA software CATT-Acoustic i order to mimic diffractio pheomea of soud propagatio through a symphoy orchestra. Briefly, diffractio was implemeted by claimig that the soud eergy propagates through the obstructig objects rather tha aroud them. I priciple, this meas that diffractio is hadled as a property of the obstructig object rather tha a property of the wave ature. Eve though such a mimetic diffractio implemetatio will itroduce some errors compared to the physical ature of wave propagatio, the method is still of further iterest both sice it is very efficiet, ad because highly accurate results ofte is outside the scope of iterest i practical egieerig acoustics. Further, a research by Isebae [3] supported the same applied mimetic modellig techique i a study that aimed to simulate the locally perceived acoustic coditios i a public hall audiece seatig area. Also Isebae s research revealed somewhat covicig results, ad cosequetly a geeral potetial i the semi-trasparet diffractio modellig techique was esured. However, the drawbac i both Dammerud [2] ad Isebae [3] is that their results were achieved i a somewhat tetatively proceedig form of research, where available parameters i the simulatios were utilized i order to optimally calibrate the model relative to a referece result. Accordigly, at this poit it would be of iterest to obtai a more i-depth mathematical liage betwee the true physical diffractio behavior ad the semi-trasparet modellig techique.

2 2 Theory 2.1 Geometrical acoustics (GA) I short, ray-tracig [4] regards soud propagatio as rays travellig ormal to the wave frot. This is implemeted by a defied poit source that seds out a large umber of rays distributed i all directios. Each ray the represets a certai agle of a spherical wave. Subsequetly, all rays are traced through their travel util all eergy has died out. Eergy is lost due to spherical propagatio dampig, or whe rays hit obstructig objects that are dedicated a absorptio coefficiet α set betwee 0 (o absorptio) ad 1 (total absorptio). No-rigid surfaces are implemeted by a dedicated scatterig coefficiet s set betwee 0 ad 1 with a correspodig umber of rays specularly ad radomly reflected. Fially, a receiver of a give spherical expasio is employed, beig hit by a stochastic umber of rays. Ideed, oe simplified way to describe how the ray-tracig method builds a impulse respose is by employig the sigal processig multipath trasmissio equatio: h K c 1 where K is the umber of rays set out by the source, c is the eergy cotributio of ray as it stries the receiver, ad is the the sample umber for whe ray stries the receiver. 2.2 Ifiite oise barrier edge diffractio I this paper a hard rigid ifiite oise barrier o a totally absorptive floor (otherwise free-field coditios) is studied. A simple geometrical cross-sectio of this give case ca be see i Figure 1. The preseted variables ad their otatios will be frequetly referred to trough this paper. (1) Figure 1: The ifiite oise barrier geometry A typical impulse- ad frequecy respose of such a oise barrier case ca be see i Figure 2. As ca be see, the impulse respose appears as a slightly time-smeared impulse. The frequecy respose slope teds to decrease for a icreased frequecy, similar to a low pass filter, which ideed agrees with our daily experiece. Figure 2: Typical impulse- ad frequecy respose of a oise barrier case. The receiver is located i the shadow zoe

3 3 Method 3.1 Semi-trasparet divisio ad the th sample barrier itersectio area Dammerud [2] too advatage of a semi-trasparet quality that ca be dedicated to surfaces i GA. This quality ca be implemeted by a trasparecy coefficiet τ defied specifically for each octave bad. It s rage goes from τ = 0 (zero trasmissio, oly reflected ad absorbed eergy) to τ = 1 (full trasparecy, o reflected eergy) [5, 6]. Now, i order to mime oise barrier edge diffractio it turs out essetial to develop a somewhat fuctioal desig algorithm of this semi-trasparet quality. Oe possible actio will the be to divide the total area of the oise barrier surface S tot ito a umber of equally large subareas ad the dedicate each subarea a suitable trasmissio coefficiet. For simplicity, it is ow assumed that each subarea ds will be hit by oe ray ad cotribute with oe certai amout of eergy c at the receiver. Figure 3 gives a illustratio of this tessellatio. Further, the arrival time of each ray will deped o the sum of the distace betwee the source S ad the barrier hit P ad the distace betwee the barrier hit P ad the receiver R. As all rays have reached the receiver a impulse respose has bee build up by addig all eergy cotributios, ad this impulse respose will ideed correspod to (1). For each ray, the dedicated eergy c i (1) will be give by the followig formula: c 1 SP RP s where τ is the trasmissio coefficiet of the subarea ds that ray propagates through, ad s is the scatterig coefficiet out from the oise barrier. These parameters τ ad s will be further discussed i subchapter 3.2. SP is the legth of the propagatio path betwee the source ad the curret subarea ds, ad RP is the legth of the propagatio path betwee the curret subarea ds ad the receiver. The value i (1) will be give by the followig formula: ds (2) where f s is the sample frequecy, ad c air is the speed of soud. f s SP RP (3) cair Now, give that the umber of subareas is large eough, it should i theory be possible to create ay desired shape of the impulse respose. As the oise barrier is divided ito a icreased umber of subareas, the travel legth betwee two succeedig icomig rays will aturally decrease. Therefore, as the umber of subareas grows towards ifiity, every relevat sample of the impulse respose h() will be the host for a umber of eergy cotributios c. Accordigly, the impulse respose shape ca be calibrated by assigig optimal trasmissio coefficiets τ. I order to assig the parameter τ optimal values, it will be a vital tas to map all rays/subareas that cotribute withi the same sample. These will be bouded by a certai barrier projectio, cetred by the barrier itersectio of the direct soud propagatio path SR. Figure 3 illustrates the occurrig th sample s barrier relatioship. Figure 3: Ray paths of the proposed modellig, ad the th sample barrier itersectio area

4 I theory, the exact shape ad size of each th sample barrier itersectio area will be bouded by a elliptic radiatio patter quite similar to a Fresel zoe, which i this particular case will be give by the source positio {S x, S y, S z } ad the receiver positio {R x, R y, R z }, as well as the sample frequecy f s ad the speed of soud c air. A cross-sectio of this elliptic zoe ca be see i Figure 4. Figure 4: Cross-sectio of the th sample barrier itersectio area For simplicity, it is ow assumed that the th sample barrier itersectio area is a circular des. The th sample barrier itersectio area ca be foud by the followig expressio: 1 2 Ph i S S (4) where Ph is the th sample barrier itersectio radius that ca be expressed by the followig formula: 1 i Ph SP RP cair 2 (5) SP f s 3.2 Trasmissio- ad scatterig coefficiets Further, a major challege of the preset research will be to dedicate optimal trasmissio coefficiets τ to each subarea ds. Let s assume for ow that the impulse respose h() is ow. The global trasmissio factor of the th barrier itersectio area will the be give by the followig formula: T SP RP h() (6) S Cosequetly, if all subareas that cotribute withi the sample is dedicated this global trasmissio coefficiet T a ideally correct impulse respose will be obtaied: ds S T (7) Further, oe highly desirable feature i GA is that it should fuctio for ay radom receiver positio simultaeously. Ideed, this should therefore also be tae ito accout whe regardig the semi-trasparet desig of the oise barrier. Oe way to implemet this feature is by itroducig a agle-depedet eergy trasmissio through the wall. For simplicity, it seems favourable to assume a simple cosius-related scatterig patter: where φ is the ray s agle out from the subarea ds. s cos (8) Ufortuately, ew challeges arise i the search for a semi-trasparet modellig that icludes this global receiver feature. At preset time, the optimal semi-trasparet desig algorithm remais uow. Still, oe proposed modellig method is preseted i subchapter 3.3.

5 3.3 Proposed method for semi-trasparet modellig The followig proposed desig algorithm is ow employed: 1. A sigle source ad a array of receivers are employed to dedicate trasmissios coefficiets. I additio to beig the source ad receivers, these objects, S G ad R G, will also fuctio as a buildig it for the semi-trasparet properties of the oise barrier. 2. The oise barrier is divided ito two differet trasmissio zoes. As ow from the quasi-aechoic recordig techiques [7, 8, 9], a Fourier trasform of the iitial samples of a time-smeared impulse respose will still give a complete eergy represetatio at high frequecies. Evidetly, this ca be associated with the ray cotributios that arrive earliest at the receiver. Cosequetly, it ca also be claimed that all ray cotributios that arrives somewhat further out i the impulse respose will oly perform a accomplishmet of boostig the lower frequecy eergy to a more ad more correct eergy level. Based o this, the utilized idea is that the mid/high pea zoe represeted by the earliest icomig rays, raises a somewhat flat frequecy respose, while the bass tail zoe represeted by the more time-delayed rays, helps to build the more typical shape of a low pass filter. To illustrate the proposed modellig techique, Figure 5 gives a setch of the zoe divisio, ad the cocept of a impulse respose beig shaped by the two zoes. Of course, i practice there will be a more blurred overlap i the trasitio betwee the two zoe s impulse respose cotributios, ad the fial curve shape will be expected somewhat less smooth. 3. The mid/high pea zoe: is bouded by the uio of the 1st barrier itersectio areas of all receivers i the array R G. Slightly simplified, all the subareas of the mid/high pea zoe will ow be dedicated a costat trasmissio coefficiets τ based o the expected mea pea value of the later achieved impulse resposes at R G. Such pea value owledge ca e.g. be obtaied by employig a library, although it would of course be beeficial to develop a more sophisticated mathematical relatioship for fidig these impulse respose pea values (typically based o the simple geometric parameters of Figure 1). However, for simplicity, through the rest of this paper the pea values will be obtaied by employig a Matlab toolbox EDBtoolbox [10] (developed by the Acoustics Group, NTNU). 4. The bass tail zoe: ivolves all subareas of the oise barrier that are ot occupied by the mid/high zoe. For simplicity, the bass tail zoe is dedicated a costat trasmissio coefficiet that equals to the trasmissio coefficiet of the mid/high pea zoe multiplied by a factor 0.1. The itetio is that the cotiuous damped tail of the impulse respose should appear due to the steadily icreased spherical propagatio dampig of succeedig rays, combied with a icreased scatterig ifluece. Figure 5: The proposed semi-trasparet modellig techique 4 Result Figure 6 gives a plot of the mea octave bad error at the receiver array R G whe S G is the employed source. The source- ad receiver positios were bouded withi the itervals x = [1.0, 30.0] m ad z = [0.25, 0.75] m, with a symmetric property regardig the y-axis. I order to preset reliable results for geeral source- ad receiver positios, the preseted octave bad error plots are based o the averaged outcome of a umber of 100 simulatios series where source- ad receiver positios where chose radomly. As the referece model the already metioed EDBtoolbox [10] was employed.

6 Figure 6: Mea octave bad error at the receiver array R G 5 Discussio By first impressio, it must be argued that the error plot i Figure 6 reveals a fairly hopeful result. The mea errors are very small ad almost eglectable. Sadly, large errors are itroduced for the max/mi-values, as 10 db errors caot be tolerated. The mai reaso for these errors is probably caused by a geeral ucertaity i the applied modellig techique. O the other had, the performed simulatio sessio with radomized source- ad receiver positios will certaily iclude a large amout of critical positios, where it is obvious that the modellig techique will have a limited fuctioality. Errors are expected for positios very close to the barrier, as well as for positios close to the floor ad barrier height. These expected errors are e.g. caused partly due to some assumptios itroduced for the th sample barrier itersectio area S, ad partly due to a potetially ufavourable scatterig behaviour. Especially, a combiatio of a odd source ad a odd receiver positio must be expected to result i a misleadig impulse respose shape. Fortuately, these ufavourable source- ad receiver combiatios ted to be rare i ay practically relevat case. Nevertheless, it must also be restated that geometrical acoustics i ay case ofte itroduces some errors i the lower octave bads, as room modes, comb filterig effects, etc. are eglected. It could therefore be questioed whether the diffractio errors of Figure 6 at all should be questioed ahead of these already existig ucertaities. Moreover, it will perhaps also be advatageous to implemet a mimetic edge diffractio modellig that itroduces some ucertaities versus a modellig that yields o edge diffractio eergy cotributios at all. However, it must further be assumed that the proposed modellig techique may itroduce additioal errors that are ot revealed by the plot i Figure 6. E.g., oe obvious cocer arises as the applied symmetric y-axis coditio is erased. The mid/high pea zoe will the be more smeared out ad arrive further out i the impulse respose. I additio, the eergy cotributios c of the mid/high pea zoe rays will be further affected by the scatterig coefficiet s, due to a icreased agle out from the oise barrier. However, this scatterig dampig may actually ot be such a bad approach to how the higher frequecies decrease alog the o-symmetric lie. Hece, the major cocer should most liely be give to the delayed arrival time of the mid/high pea zoe rays. Iterestigly, the preseted modellig is based o a broadbad simulatio techique, which to some extet is advatageous, but which also will cause some issues regardig a potetial GA software implemetatio. Normally, to imitate a frequecy-depedet absorptio property of surfaces, simulatios are ru separately for each octave bad. From there, the echogram values of each octave bad are post-processed to form a iterpolated frequecy respose. Fially, this frequecy respose is combied with a DSP filter to geerate a cotiuous broadbad impulse respose curve. Accordigly, the questio is how a broadbad simulatio techique ca be implemeted i this applied octave bad simulatio techique? Oe possible solutio could perhaps be to ru a separate "edge diffractio" simulatio i additio to all the octave bad simulatios, ad the somehow merge the edge diffractio impulse respose to the fial output impulse respose by post-processig. By cotrast, oe obvious alterative modellig techique could be to develop a oise barrier desig that ivolves a octave bad-based trasmissio coefficiet. Such a solutio correspods to both the previous semi-trasparet edge diffractio modellig studies by Dammerud [2] ad Isebae [3].

7 Further studies should also be performed regardig the reverbat cosequeces of the preseted edge diffractio modellig. I this study oly early edge diffractio eergy cotributios are cosidered. How is the ofte preset reverberatio field affected by this modellig techique? Moreover, sice this is a broadbad simulatio method, what will happe as these rays hit a frequecy-depedet absorptive surface? 6 Summary To summarize, a geeral potetial i the proposed semi-trasparet edge diffractio modellig techique has bee revealed. The preseted results ad discussio idicate a somewhat successful modellig of the ifiite oise barrier edge diffractio case. However, some uwated errors are itroduced at rare source-receiver positios. Thus, possible future studies should be performed i search for a somewhat improved modellig algorithm. So far, o attempt has bee made to implemet the proposed edge diffractio modellig i commercially available GA software. Some possible complicatios are itroduced as the preseted modellig is a broadbad ray-tracig simulatio method, whereas the commo procedure is to perform idividual octave bad simulatios. I coclusio, further research should study to which extet the proposed modellig techique actually ca be implemeted i commercial GA software. A mai outcome of this report reveals that a fairly satisfyig model of the oise barrier edge diffractio case ca be developed based o owledge about the expected impulse respose pea values. Accordigly, there should be of further iterest to fid a somewhat simple relatioship betwee the impulse respose pea value ad the simple geometry of the give ifiite oise barrier case. Refereces [1] U. P. Svesso, R. I. Fred & J. Vaderooy, A aalytic secodary source model of edge diffractio impulse resposes, J. Acoust. Soc. Am. 106, , [2] J. J. Dammerud, Stage acoustics for symphoy orchestras i cocert halls, Ph.D. thesis, Uiversity of Bath, Eglad, [3] A. Isebae, Modellig audiece seatig area i geometrical room acoustic software, Faculty of iformatio techology, mathematics ad electrical egieerig, Norwegia uiversity of sciece ad techology, Norway, [4] A. Krostad, S. Strøm & S. Sørsdal, Calculatig the acoustical room respose by the use of ray tracig techique, J. Soud ad Vibratio 8, , [5] CATT-Acoustic, CATT-Acoustic v8.0 user s maual, [6] CATT-Acoustic, Addedum to CATT-Acoustic maual, [7] J. M. Berma & L. R. Ficham, The applicatio of digital techiques to measuremet of loudspeaers, J. Audio Eg. Soc. 25, , [8] L. R. Ficham, Refiemets i the impulse testig of loudspeaers, J. Audio Eg. Soc. 33, , [9] E. Bejami, Extedig quasi-aechoic electroacoustic measuremets to low frequecies, AES Covetio paper 6218, [10] U. P. Svesso,