Aoustis 08 Paris Soud diratio i periodi suraes i aiet aritetural strutures J.A. Cruz Calleja Istituto Póliteio Naioal, Av Sata Aa No. 1000 ol. Sa Fraiso Culuaa, Del Coyoaa, 04430 Méio, Meio jorgearuz@otmail.om 091
Aoustis 08 Paris I tis work is aalyzed te soud diratio i aritetural strutures aused by periodi suraes. It is studied te soud iteratio wit aritetural strutures usig dieret types o soures, materials ad sizes. Te apparetly satterig eet o te stairases i aiet ostrutios (Kukula s pyramid stairase i Cie Itza, te spee ommuiatio to large distaes ad te spatial ilter eet usig te glide repetitio pit teory are also aalyzed. Is te Cirp eo o te aiet preispai pyramids a itetioal eet? Computatioal models o te peomeo ad real measuremets were used to establis te peomea pysis priiples. 1. Itrodutio Te aousti diratio i te pyramid o Kukula as bee aalyzed by Lubma, Delerq, Beristai ad Bilse. I 007 N. Tsigos eposed a geometrial model o te satterig i te stairase Kukula pyramid stairase. Te model allows veriy te spatial ilter eet o te stairase ad te diratio urves ormatio. I te same orm te model is apable to geerate a soud rederig o te eo, but do ot eplai te peomea. Some teories ave bee eposed to eplai te peomeo ad some questios ave bee eposed. New i situ eperimets ad matematial model o te diratio eet i aiet strutures as allowed veriy te validity o te models. Te distae o te soud soure to te stairase modiies te arrival time eo ad diratio requeies. We te soud soure is joi stairase te diratio is ot preset at all. Te possible trajetories tat a ross te soud to be reeived by te observer are idiated i igure 1. 1 Te soud tat travels diretly to te pyramid ad is reeived diretly. Te soud tat travels diretly to te pyramid ad is reeived ater to ave releted i te groud. 3 Te soud tat is releted i te groud beore travelig to te pyramid ad tat is reeived diretly. 4 Te soud tat as bee releted i te groud beore travelig to te pyramid ad tat is reeived ater to ave releted i te same oe.. Eo pysis priiple A eo aused by a lat surae we a soud impulse is geerated i rot, it is a repetitio o te origial evet delayed i time ad wit mior itesity. I te lutter eo te otiuous repetitio o te impulse is preset. I a periodi surae te eo geerated i rot is reated by otiuous repetitio o te origial impulse geerated by te reletios o te stairase steps. Te iitial reletio is aused by te irst step ad te ial by te last step. Tese soud reletios reate ostrutive or destrutive itereree aused by te pase ages o te soud sigals, origiated by time delay preset i ea reletio. Fig.1 Iteratio o te reletios i a periodi surae. 3. Cirp eo parameters Diverse parameters are ivolved i te eo ormatio wi modiies te eo legt, te preset diratio requeies ad te upper - lower requey diratio limits. 3.1. Soud soure positio Some irp eo arateristis deped o soud soure positio. Te soud soure eigt modiy te arrival time o te eo ad te lies diratio urvature. Fig. Pyramid s stairase ad soud pats desriptio. Usig a modiiatio o te glide repetitio pit teory, te utio tat allows to alulate te orret trajetories S( tat origiates te releted rays i te stairase or te osideratio umber oe is: S( y For steps wit dieret wide ad eigt te epressio is: Beig q = Wide o te step q y = Heig o te step (1 d q ((( 1 q ( S( For tis ase, we use te et epressio: S d q ((( 1 q (3 ( Beig te step umber ad its irst value is ero. Te matematial equatio tat allows alulatig te preset requeies i tis ladslide it is dedued by te ollowig way. Te irease i distae betwee trajetories o two suessive reletios is give by: S S( S( (4 S ( 1 ( 1 S( S( (5 Te irease o te distae is equal to te wavelegt or a give reletio. S( (6 y 09
Aoustis 08 Paris Te time tat would take a wave travel tis irease o distae is give by: S( T ( (7 Te distae iremet is ot ostat so te dieree betwee two suessive ireases o distae is give by: S S( S( (8 ( 1 Te sum o te ireases is equal to te total variatio o te wavelegt. S ( (9 i i i (10 (11 I order to alulate te requey ( is used te ollowig proedure: ad T T ( S( ( S( S( 1 (1 (13 (14 (15 Te ial epressio wi allows alulatig te requey or a step is: ( m 171.5 d ( 1 q (( q d q (( 1 q Beig m is te armoi umber (16 We a alulate o S( usig te ollowig osideratios. Te releted agle o releted ray is give by: te Te ray pat is give by: y arta (( 1 q arta ( q d (17 (18 S( (19 os ( q d ( (( 1 q osarta ( q d S Tereore, beig =343 requey ( is alulated by: 171.5 ( m (( 1 q d ( q d (( q (( 1 q osarta osarta (( 1 q d ( q d (0 (1 3.1.1. Distae ad eigt Te resultig eo o te diratio proess is utio o te distae ad eigt wi te impulse is emitted. Te soorous soure positio ages te S( trajetories, tereore te diratio requeies. D=0 D=0 Fig.3 Variatio o S( ad t(. To distaes very ear stairase irease o S( teds very quikly to zero (see Fig. 3, reaso wy te diratio o te soud waves is little pereivable. We te distae o te soorous soure to te stairase is great, te irease o S( teds slowly to zero ad te age o ( is smoot; te diratio o te soud waves is learly appreiable. Fig.4 Variatio o diratio urves wit eigt. Te eigt o te soure also tis ivolved i te ormatio o te eo; we te eigt o te soorous soure is ireased, te trajetories o soorous rays or lower steeps are ireased, ad or te superior oes are dimiis ad te iitials values o ( are ireased. 3.. Impulse arateristis m=1 d=10 D=100 3..1.Impulse legt Te soud used to produe te eo must be a impulse wit smaller duratio to d/, i were d is te distae o te soud soure to te irst step ad te soud veloity. Tis oditio avoids a iteratio betwee te iidet wave ad te releted wave, ad allows to reeive te sequee o origiated steps reletios wit its respetive delay o time, i su a way it a be ormed ad be augt te requey slide. I te soud used is greater or otiuous, tere would be a iteratio betwee te iidet wave ad te releted wave, wi would prevet to otie te requey ladslide ad te eet o spae ilter aused by te diratio o te waves i te stairase is preset as a omb ilter. 093
Aoustis 08 Paris 3...Type soorous soure Commoly a adlap as bee used to geerate te eo i te pyramid o kukula. Usig oters eitatio soure types te eo geerated is dieret. Playig dieret Pre- Hispai istrumets i rot o te stairs o te kukula pyramid, was veriied te relatio eistet betwee te requeies preset i te eo ad te impulse geerated by oe soud soure []. Te soograms learly sow tat te requeies are determied by te type o soud soure. So te irp eo a be produed we te soud soure geerates a impulsive soud. 3.3. Periodi surae proprieties 3.3.1.Material proprieties Te limestoe properties are desribed by Delerq; te limestoe stairase o Kukula s pyramid as a desity o 000 kg/m 3, te veloity o te logitudial wave is 4100 m/s ad 300 m/s or te sear wave veloity. Tis data ave to be veriied usig a limestoe sample. Te soud itesity or a audible soud is mior tat a ultrasoi wave, so te iteratio betwee te soud beam ad solid periodi surae is small. 3.3..Iluee o te groud I tis setio it is aalyzed te possible iluee o te groud i rot o te pyramid i te eo ormatio. Te trajetories or te poits, 3 ad 4 o te setio 3.1 are alulated i tis part. Te groud i rot o te stairase allows a part o te soorous sigal be releted toward te stairase. Te agle tat would allow a ray releted i te groud goes to te last step o te stairase a be alulated wit te ollowig epressio: H ( r ma arta D d For smaller agles te soud wave is releted i te diretio o te stairase. 1 (3 ta or S r S r ( q d ( (( 1 q osarta ( q d (6 d q ((( 1 q (7 ( I order to alulate te trajetories tat ulill te oditios o poit ad 3 we ave: ( m (( 1 q d (( 1 q d (( q (( q os arta os arta ( q d ( q d ( q d ( q d (( 1 q (( 1 q os arta os arta ( q d ( q d For te oditios establised i poit 4: ( m (( 1 q d ( q d (( q (( 1 q os arta os arta (( 1 q d ( q d 3.3.3.Iluee o te z ais (8 (9 We a epad te glide repetitio pit teory to tree dimesios addig te possible displaemet o te soorous ray trougout te stairase (z-ais. Te epressio tat allows alulatig te trajetories or soorous rays is: Fig.7 Diagram to alulate de z ais S( y z (30 For our ase: S d q ((( 1 q ( z (31 Fig.5 Groud releted soud pats. Te trajetory or a ray releted i te groud ad releted i a step "" is give by: For a give reletio: te X1 S r ( (4 os (( 1 q arta ( q d (5 Te values o z are limited by te wide o te stairase (9m legt ad go rom 0 to 4,5 ad rom 0 to -4,5, takig te value zero te etral poit rom te same oe. Te variatio o S( i utio o z or te ase o te pyramid o Kukula is small, or tis reaso te variatio o ( is also small. I Fig. 6 te diratio urves or te dieret values rom "z" appear very et to tose o te etral value (z=0. Tis ould eplai so tat te urved presets i soograms o te real soud do ot appear as ti lies, else ave a determied tikess, or tis reaso te urves o diratio or te possible values o z are preset i it 094
Aoustis 08 Paris 4. Numerial results Fig.6 Eo i utio o te steps umber. (m=1 I order to demostrate te iluee o te parameters ivolved i te eo ormatio were realized some eperimets i situ usig Bruel & kjaer equipmet ad i quality reordigs; te reordigs were made i 3 bits resolutio ad were aalyzed usig omputatioal sotware. Te soograms were made usig a Blakma Harris Widow wit 104 size. 3.3.4.Number o steps Te umber o te steps otaied i te periodi surae ages te eo legt. I te stairase as a lot o steps te eo legt will be great. By oter side i te stairase as little steps te eo legt will be sort. Te eo legt is depedet o te umber o steps 3.3.5.Steps dimesio Te geerated eo is aused by te ombiatio o diratio, dispersio ad reletio o te soud. For dirated waves te limits are stablised by: Speular releted waves d q D (3 s (33 q beig q= surae periodiity ad D= distae betwee te eter o two steps Cosiderig te steps as a lie array o radiatig soures te miimum requey tat a be dirated by te stairase based o te dimesios o te steps, ad tis give by: d= distae betwee te soud soures d (34 d For smaller requeies to d te waves are added i oeret orm ad te diratio is ot possible absolutely, beause te wavelegt o te ivolved requeies is bigger ta requey slide (its equivalet wavelegt, so tis small distae does ot allow its aellatio by destrutive itereree, beig te produed reletio similar to te produed by a lat surae. For ig requeies, i wi te ivolved wavelegt is small i ompariso wit te dimesios o te step, te reletios produed are speular, reaso wy te soud waves are govered by te laws o te geometri aoustis. Besides, ig requeies suer atteuatio due to te air absorptio. 3.4. Medium o trasmissio Te air umidity ad te temperature ave a eet o te soud veloity. Te umid Yuata air ave a desity o =1.1466 kg/m 3 ad te soud veloity as bee take as =343 m/s [1]. I te alulus were take te previous values wi ave to be veriied i situ. Fig.9 Cirp eo i utio o te distae. Soograms o a eo reorded at 5 ad 50 meters respetively. Te iluee o te soud soure positio was demostrated i a teoretial way. Tis iluee a be demostrated i a pratial way aalyzig te eo reordigs toke i dieret poits i rot o te pyramid. Te groud eet is supported wit te tikess preset i diratio urves. Tis eplais wy te diratio urves are ot ti lies i te real eo. Usig Fig.9 it is possible to demostrate te urvature age o te diratio slide i utio o te distae. We te distae is great te requey slide is latter. Measured value Calulated value =1.5m Arrival Eo Arrival Eo time Legt time Legt Distae 0.09 0.15 0.09s 0.184 5m 0.057 0.179 0.058s 0.179 10m 0.91 0.159 0.91s 0.158 50m 0.348 0.156 0.349s 0.156 60m Table 1 Arrival ad legt times or a impulse geerated i rot o te Kukula s pyramid at dieret distaes Te time arrival ad eo legt i utio o soud soure distae to te stairase is veriied i Table1. Te iluee o te step size was veriied i a eperimetal way i dieret aiet strutures; or te Kukula ad Moo (Teotiuaa pyramids were alulated ad measured te limit values or dirated waves. I Fig. 10 it is possible to observe te requey limits or two aiet periodi suraes. Te umber o te steps aets te eo legt. Geeratig a impulse i rot o a Veus temple (Small pyramid 14 steps it was possible demostrate tis at. See Fig. 11. Te requeies otaied i te irp eo are depedet o te soud soure arateristis. Usig a adequate MLS sigal (0.34s legt te irp eo was geerated al 0 meters. I Fig. 11 it is possible to observe te requey limits alulated. 095
Aoustis 08 Paris q d D q(m D(m 608.36Hz Cie Itza 461.09 Hz 0.63 0.3719 30.01Hz Teotiuaa 406.94Hz 0.98 0.414 Table Frequey limits or dirated waves i two aiet pyramids Fig.1 Soograms o Lubma ad J. Cruz eo reordigs. I blak lies te alulated diratio urves (I rame te urves wit groud eet; uder te real eo soograms. Fig.10 Soograms o Moo ad kukula pyramids. Te lies elp us to mark te upper ad lower requey limits. At 10 meters te maimum duratio o te impulse will ave to be 59ms to avoid a iteratio betwee te iidet wave ad te releted wave. Usig a otiuous pik oise soure i rot to te stairase o pyramid o Kukula, te omb ilter is produed. We a pure requey or arrow bad oise is geerated i rot o te stairase, te geerated irp eo oly otais te orrespodet requeies. Akowledgmets Te autor is takul to INAH (Istituto Naioal de Atropología e Historia by te oered support, wit ollowig studets wo assisted te reordig o te studied souds: Agelia Villaueva Almaraz, Jessia Mora Alatar, to Proessor Gerardo Heradez Suilla ad te aoustis arrier studets (8EM8. Te autor is also takul to Nio F. Delerq by is ivaluable support ad advie. Reerees Fig.11 Eo geerated wit MLS sigal. To te rigt itereree aused by otiuous oise soure, i rot o te Kukula pyramid. Colusio Te validity o te proposed matematial model based o te repetitio pit glide was veriied. I te same orm te ivolved parameters i te eo geeratio was aalyzed ad its iluee i te eo ormatio was orroborated usig some i situ eperimets. Te model alulates te arrival time, eo legt ad requeies i a orret way. Usig tis model it is possible to determiate tat te irp eo Lubma s reordig [6] was made at 3 meters approimately ad ot at 10 meters. I Fig. 1 is preseted te real soograms ad te alulated diratio urves, we a observe te orrespodee betwee te alulated ad te real urves.te Autor s reordig was made at 10 meters o te Kukula s stairase. Tis at was veriied eperimetally. [1] Delerq Nio F., A teoretial study o speial aousti eets aused by te stairase o te El Castillo pyramid at te Maya ruis o Cie-Itza i Meio, Joural o te Aoustial Soiety o Ameria 116(6, 338-3335 (004. [] Cruz C. Jorge A., Aousti peomea i araeologial zoes i Meio, 19t Iteratioal Cogress o Aoustis, Madrid, España, -7 September 007. [3] Sergio Beristaí, Ceilia Coss, Gabriela Aquio, ad Jose Negrete, Toal respose o te stairway o te mai pyramid at La Ciudela, Teotiuaa araeologial site, Joural o te Aoustial Soiety o Ameria 11, 85 (00. [4] Fras A. Bilse, Repetitio Pit glide rom te step pyramid at Cie Itza, Joural o te Aoustial Soiety o Ameria 10(, 594-596 (006. [5] Tsigos, Niolas; Leebvre, Sylvai; Dasbaer, Carste; Dellepiae, Matteo, Etedig geometrial aoustis to igly detailed aritetural eviromets, 19t Iteratioal Cogress o Aoustis, Madrid, España, -7 September 007. [6] David Lubma: ttp://www.oasa.org/mayapyramid.tm ~site visited o 1/13/007. 096