Physics in Entertainment and the Arts

Transcription

1 Physics in Entrtainmnt and th Arts Chaptr VI Arithmtic o Wavs Two or mor wavs can coxist in a mdium without having any ct on ach othr Th amplitud o th combind wav at any point in th mdium is just th sum o th wavs displacmnts at that point This is why whn you listn to an orchstra playing you can sparat out th individual instrumnt s s sounds You can har th clarints vs th trumpts vs th violins th sounds ar simply addd togthr not jumbld togthr! Th suprposition principl is also why w can snd TV, FM, AM, cll phon, tc radio signals through th sam air and sparat thm out at th rcivr Thy don t gt mixd togthr; thy just gt addd tmporarily Lt s start by studying two pulss with positiv amplituds travling along a long slinky On travls to th lt; on to th right Th pulss will mt tmporarily in th cntr o th slinky Sinc th pulss both hav positiv amplituds thy both add in th cntr but only tmporarily Th pulss pass through ach othr with no lasting ct Now lt s s what happns whn th pulss hav opposit amplituds On with a positiv amplitud; on with a ngativ amplitud Now th pulss both subtract in th cntr but only tmporarily Th pulss onc again pass through ach othr with no lasting ct Figur rom Physics, Cutnll & Johnson, 7 th d. Figur rom Physics, Cutnll & Johnson, 7 th d.

2 Not that in this subtraction cas vn though th slinky is tmporarily lat at th momnt o intraction th wavs hav not bn dstroyd! Th wavs just tmporarily add to giv a zro displacmnt at that instant Continuous Wavs - Lt s now mov on to th suprposition o two continuous wavs Assum that th wavs hav th sam rquncy I th wavs ar o dirnt rquncy, it s a lot mor complicatd! Suppos w gnrat two wavs with th sam rquncy moving in th sam dirction down th sam string How w do this is airly asy, but not important or this discussion I th two wavs vibrat up and down at th sam tim w say that th wavs ar in phas with ach othr In phas mans that ach wav rachs a crst togthr, and thn a trough togthr as thy mov down th string Not that whn continuous wavs travl down a string th rsulting wav is th sum (suprposition) o both wavs A A A total Th total wav amplitud is th sum o th individual wav amplituds A A = + A total Altrnativly, i on wav vibrats up whil th othr wav vibrats down at th sam tim w say that th wavs ar 80 o out o phas with ach othr 80 o out o phas mans that on wav rachs a crst and th othr a trough togthr as thy mov down th string Not that whn continuous wavs travl down a string th rsulting wav is still th sum (suprposition) o both wavs, vn though thy ar 80 o out o phas A A A total Th total wav amplitud is th sum Th wavs o th individual wav dstroy ach amplituds othr! A A = A total In phas and 80 o out o phas ar th two xtrms th wavs can b any numbr o dgrs out o phas with ach othr (th math just gts hardr to do thn) Th rd wav is 0 o out o phas with th blu wav. Th yllow wav is th suprposition (sum) o th two wavs Not that th combind yllow wav has an amplitud much gratr than zro, but much lss than th sum o th rd and blu wavs Ths wavs ar rrrd to as simply out o phas

3 Why is th phas dirnc important? Lt s look at a typical xampl Suppos you hav a stro systm that t you wish to wir up or hom us I you wir up th spakrs corrctly th sound rom ach spakr will rinorc ach othr at your ar S t r o I you wir up th spakrs incorrctly th sound rom ach spakr will oppos ach othr at your ar S t r o In phas spakrs: wavs add at your ars 80 o out o phas spakrs: wavs cancl at your ars Howvr, vn with corrct spakr wiring thr could still b potntial problms with wav cancllation Suppos you don t stand qual distancs rom ach spakr Dspit th spakrs producing sound in phas th distanc travld by ach sound wav is dirnt! S t r o d d d > d d Evn though ach wav starts out in phas, thy ar out o phas whn thy rach your ar bcaus thy travld dirnt distancs! d > d Th wavs crsts and troughs no longr match up at your ar I d > d, w can din th path lngth dirnc or th two wavs as PLD = d d I th PLD quals a hal a wavlngth th wavs will rach your ar 80 o out o phas and hnc cancl ach othr out! This ct is calld dstructiv intrrnc and th wavs ar said to b dstructivly intrring with ach othr Mathmatically: PLD Dstructiv Intrrnc = d d = Not that complt wav cancllation only occurs i both wavs hav th sam rquncy both wavs hav th sam amplitud th wavs start out in phas th wavs maintain th sam phas rlationship or at last a short priod o tim This is calld cohrnc, and th wavs ar said to b cohrnt Any othr conditions will only giv partial cancllation

4 Also not that th PLD or dstructiv intrrnc is dirnt or dirnt rquncis Exampl #: sound wavs at khz dstructivly intrr at a PLD = 0.55 t = 6.6 in v / ( 00 t / s) /000 Hz PLD = = = = t Exampl #: sound wavs at 5 khz dstructivly intrr at a PLD = 0. t =.3 in v / ( 00 t / s ) / 5000 Hz PLD = = = = 0. t Incrasing th rquncy by a actor o 5 dcrasd th PLD or dstructiv intrrnc by th sam amount Tim or a rality chck Th conditions or complt dstructiv intrrnc (complt wav cancllation) ar diicult to mt in a ral situation or svral rasons In a ral stro systm Th spakrs don t produc idntical wavs That would b monaural sound, not stro! Th wavs rlct o th walls, th loor, th ciling, urnitur, tc complicating th PLD gomtry You hav two listning locations (ars) sparatd by 6 7 inchs o dad spac Your had is about hal a wavlngth thick at khz Dirnt PLD s or ach ar! O cours, i th PLD is a whol wavlngth rathr than a hal wavlngth th two wavs will arriv at your ar in phas This is calld constructiv ti intrrnc and th two wavs ar said to b constructivly intrring with ach othr Mathmatically: PLD Constructiv Intrrnc = d d = Finally, wav intrrnc occurs or all typs o wavs not just sound wavs as w v bn discussing I you listn to AM radio whil driving, you v probably hard intrrnc cts It s that ading in and out o th signal as you pass in and out o th partial cancllation points (causd by multipl rlctions o th radio wav) Th constructiv and dstructiv points ar actually mor gnral than this Thr ar multipl points in rality: PLD PLD Constructi Dstructiv v Intrrnc Intrrnc = d n 0,,,3,K d = n whr = = d d = n + whr n = 0,,,3,K Bat Frquncy On othr intrrnc ct is calld bats or th bat rquncy btwn two wavs I th two intrring wav hav slightly dirnt rquncis thir phas dirnc is not constant Bat Frquncy Bcaus o this, th wavs altrnat btwn constructiv and dstructiv intrrnc as thy pass a point Your ars will har both wavs plus a third wav having a rquncy qual to th dirnc btwn th original wavs rquncis

5 Bat Frquncy This dirnc rquncy is calld th bat rquncy Bat Frquncy = usually xprssd as a positiv numbr Bat Frquncy Th Hz bat rquncy will b hard as a pulsation in th loudnss o th avrag o th two rquncis Th pulsations having a rquncy o pulss pr scond at a 44 Hz rquncy 440 Hz Standing Wavs I a puls on a string rlcts o a ixd ndpoint (or a chang in mdium) th puls is invrtd It undrgos a 80 o chang o phas Exampl: Pur tons o 440 Hz and 44 Hz produc a bat rquncy o Hz CH6_bat_dmo.ds 44 Hz 44 Hz w/bats Th sam is tru or a continuous wav Figur rom Physics, Cutnll & Johnson, 7 th d. Figur rom Physics, Cutnll & Johnson, 7 th d. Standing Wavs Standing Wavs Standing Wavs For continuous wav rlction, you nd up with two wavs travling in opposit dirctions with opposit phas Th suprposition o ths two wavs is calld a standing wav bcaus th addd wav appars to b standing still Th standing wav has rgions o constructiv and dstructiv intrrnc dstructiv intrrnc crats a nod constructiv intrrnc crats an antinod A distanc o hal a wavlngth sparats on nod rom th nxt as wll as on antinod rom th nxt Standing Wavs An important point about standing wavs: Unlik ral wavs, standing wavs carry no nrgy Standing wavs stor nrgy howvr Standing Wavs Th prvious discussion assums only on nd o th string is ixd and thror a standing wav o any rquncy can b gnratd But i w ix both nds (such as in a guitar) only vry spciic standing wav rquncis ar gnratd Lt s discuss thos spciic rquncis now Som musical instrumnts ar basd on standing wavs on strings guitars violins cllos harps pianos banjos Lt s xamin how wavs ar producd on stringd instrumnts

6 Whn a string ixd at both nds is pluckd many dirnt spciic standing wav rquncis ar producd and thy ar all rlatd! I w pluck th string just right th pluck will produc a standing wav o rquncy, th lowst possibl rquncy This is what our string would look lik: It has nods and antinod / Frquncy is calld th st harmonic or undamntal rquncy o th string Figur rom Physics, Cutnll & Johnson, 7 th d. Th st harmonic rquncy,, is rlatd to th lngth o th string (L) and th spd o wav on th string (v): And sinc v = : = v L = L or L = I w pluck th string dirntly, w would gt a dirnt spciic standing wav pattrn: / Whn xactly is masurd, it is ound to b = is calld th scond harmonic or st ovrton o th string In this scond cas, th lngth o th string is L = Continuing to th nxt spciic standing wav rquncy: 3 Whn xactly 3 3 / is masurd, it is ound to b 3 = 3 - Summary - Summary 3 is calld th third harmonic or nd ovrton o th string In this scond cas, th lngth o th string is 3 L = 3 3 / 3 / / L = L = 3 L = 3 So, th lngth o th string is always an intgr numbr o hal wavlngths Th lngth o th string dtrmins th spciic wavlngths allowd Th tnsion and mass o th string dtrmin th spd o th wav on th string and hnc th spciic wavlngths v =

7 - Summary st harmonic: undamntal v L nd harmonic: = 3 rd harmonic: = 3 = 3 Th undamntal rquncy dtrmins th pitch o th wav producd - Summary Th pitch o th wav can b changd in two ways changing by changing th string lngth As whn you hold th string lat against th rts on a guitar changing v by changing th string tnsion This is how musicians tun thir instrumnts! Pip Harmonics Som musical instrumnts ar basd on standing wavs in pips pip organs clarints saxophons luts trumpts wind chims Lt s xamin how wavs ar producd in pipd instrumnts Pip Harmonics Thr ar two typs o pipd instrumnts pips with both nds opn Exampl: a lut pips with on nd opn Exampl: a clarint Each typ o pip producs wavs with dirnt charactristics Pip Harmonics Opn Endd For a pip opn at both nds, th standing wavs hav antinods at ach nd This is a slight approximation, but a good on A = antinod = = N = nod Figur rom Physics, Cutnll & Johnson, 7 th d. Pip Harmonics Opn Endd Th standing wav rquncis or an opn ndd pip hav th sam mathmatical orm as thos or a string ixd at both nds Th wav s physical orm is dirnt howvr sound wav vrsus mchanical wav Opn ndd pips produc all harmonics st, nd, 3 rd, 4 th, tc Pip Harmonics Closd End For a pip closd at on nd, th standing wavs hav an antinod at th opn nd and a nod at th closd nd 4 A = antinod = = N = nod Figur rom Physics, Cutnll & Johnson, 7 th d. Pip Harmonics Closd End Th standing wav rquncis or a pip closd at on nd ar dirnt! Du to wav rlctions o o th closd nd, th vn harmonics undrgo complt dstructiv intrrnc and do not surviv! Only th odd harmonics ar producd st, 3 rd, 5 th, 7 th, tc Pip Harmonics Summary Th undamntal rquncis or ach typ o pip ar dtrmind by th lngth o th pip opn ndd pip closd nd pip L = = L L = = 4 L 4 v v v v = = = = L 4 L

8 Pip Harmonics Summary Pip Harmonics Summary Pip Harmonics Exampls Th highr harmonic rquncis ar opn ndd pip v v n = = n,,3,4,k or n = n L closd nd pip v v n = = n =,3,5,7,K 4 or n n L Th undamntal wavlngths ar opn ndd pip: = L closd nd pip: = 4 L For an opn ndd pip, th undamntal wavlngth is twic th lngth o th pip; or a closd nd pip, its our tims Exampl # A t long opn pip vrsus a t long closd pip Opn: v 00 t / s = = = 75 Hz L t v 00 t / s Closd: = = = Hz 4 L 4 t Pip Harmonics Exampls Pip Harmonics Exampls Pip Harmonics Exampls Exampl# So an opn ndd pip producs a undamntal rquncy twic as larg as a closd nd pip o th sam lngth What i th lngths arn t th sam? Exampl # A t long opn ndd pip vrsus a t long closd nd pip Opn: v 00 t / s = = = 75 Hz L t v 00 t / s Closd: = = = 75 Hz 4 L 4 t Exampl# So an opn ndd pip twic as long as a closd nd pip producs th sam undamntal rquncy! Pip Harmonics Exampls Exampl#3 Sound rom an opn ndd pip twic as long as a closd nd pip with th sam undamntal rquncy (75 Hz), but dirnt highr harmonics Standing Wavs Final? O cours, ral musical instrumnts ar much mor complicatd than ths asy xampls Ral instrumnts hav ingr hols, valvs, slids, dirnt dnsity strings, rsonant cavitis tc CH6_opn_pip.ds CH6_closd_pip.ds Whil w can usually gt a rough stimat o th instrumnt s opration with what w v larnd so ar it s not narly nough to gt it xactly corrct!