Chapter I. Concerning the Nature and Propagation of Sound.

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1 Euler's Disserttion De Sono : E002. Trnslted & Annotted by In Bruce 1. Chter I. Concerning the Nture nd Progtion of Sound. 1. The exlntion of sound by the old hilosohers ws very obscure nd confused, so much cn be understood from their writings tht hve come down to us. Some were of the oinion, like Eicures [ B. C.], tht sound emnted from ulsting body rther like the flow of river; while others with the foremost of the Ltin writers, believed with Aristotle, [ B. C.] tht sounds were formed from the breking of the ir which rose from the more violent collisions of bodies. Among the more recent commentries, Honoré Fbri [ ], nd Descrtes [ ], discovered tht sound consisted of tremors or vibrtions of the ir, but their resoning concerning these vibrtions were eqully confused. Newton [ ], with the shrest of minds, considered the mtter with more cre, nd undertook to set forth n exlntion esecilly for the rogtion of sound, truly with much more success. A determined effort hs been mde [by me] to grs the difficult mtters involved in n understnding of the nture of sound, which re set out in the two chters of this disserttion. In the first chter it becomes rent, fter some creful thought, wht the nture of sound relly is, nd how it is rogted from one lce to nother. Moreover, in the following chter, three sources of sound re to be considered. 2. However, before this work on sound is undertken, certin fcts relting to ir in the genertion of sound re first to be relted. I regrd ir s consisting of smll globules, in stte of comression from the incumbent tmosheric weight, nd this comression is relieved to gret extent with elevtion, s the force of comression diminishes with height, so tht the rticles cn restore themselves to their nturl stte. Thus, the weight of the ir bove comresses the ir below, nd the ir globules re not llowed to be extended. The elstic force of comression on the ir globules is equl to the weight of the tmoshere; on ccount of which one cn mesure this force by exeriment, which truly is equl to the mximum weight of the tmoshere resent. This weight is equl to column of mercury of height 2460 scrules or thousndths of Rhenish feet [One Rhenish foot is equl to mm], nd I will lwys dhere to these mesurements in the following text; if the tmoshere hs smller weight, equl to column of mercury of height 2260 scrules, then this too cn be tken s equivlent to the elstic force of the ir [t greter ltitude]. Indeed, the weight of the ir hs been determined with the id of neumtic ums; nd the rtio of the secific grvity of quicksilver to the secific grvity of the wrmest ir hs been observed to be in the rtio to 1; while for the coldest ir the rtio is round to If we consider one of series of ir globules to be comressed more thn the rest, then tht globule will dilte ccording to the lw discussed bove, while the surrounding globules become comressed by the force cting on them from the diltion of the single globule, which in turn comress others further wy, s the globules scttered t distnce exerience little of the [originl] comression. And by this line of resoning the sound is trnsferred to other lces. But, concerning the motion by which the globule considered exndes out, fter coming to rest reltive to the others, it then returns suddenly nd is unble to be confined, s it hs been extended excessively; hence it is gin comressed with resect to the other globules, yet gin excessively. Thus ech one of the not too distnt globules is itself dilted in this wy by the trembling motion of tht first globule considered, nd in this mnner ech globule is constrined to move. [Thus, the hysicl ide of centrl source consisting of n ir globule executing n S.H.M. is resented, with time dely or hse shift for neighbouring globules ; there is no hysicl rgument resented for the relity of such globules, which re convenient figment of the imgintion.] But such vibrtion of the globules of ir nerby ought not to occur for globules of very smll size, nd which hence deend on n indefinitely short time for single oscilltion; therefore innumerble oscilltions or undultions with finite eriod re to be given out by globule in the mnner rescribed, since truly the motion of ny such globule of continully decresing size cnnot hen. Moreover, finite time is required for ercetion by our senses, nd it is not ossible for sound to consist of vibrtory motion of tht kind in the ir. 4. Then t lst the sound is roduced by the sme globule, from the force exerted on other globules, with finite intervls lced between those llowed to hve denser comressions. It is of course required in order to roduce the sound, tht the sme globule is lterntely contrcted nd relxed, nd indeed the time for

2 Euler's Disserttion De Sono : E002. Trnslted & Annotted by In Bruce 2. these oscilltions should not to be indefinitely smll, but finite, in order tht the number of these vibrtions or oscilltions for given time cn be determined. [Note: Mersenne, in his Hrmonie (1635) hd lredy set out tbles of frequencies ssocited with musicl scles, nd determined the seed of sound exerimentlly.] Of course the number of ulses rriving on the er from n orgn note in given finite time cn be exressed numericlly. 5. With the time now noted for which the sound is resent, it is esy to exlin the differences of sounds; here I will only distinguish between the rincil kinds. Generlly there re loud nd soft sounds. A sound is loud or violent when the comressions of the ir globules re stronger, nd sound is soft or smll when these comressions re weker. When the sound mde by the oscillting globule is rogted by the communiction of the comressions with ech of the globules lced round it, the number of these increses in the rtio of the squre of the distnces from the lce of origin, nd the strength of the sound decreses in the inverse squre rtio of the distnces, unless erhs the sound is ugmented from elsewhere. The distinction between notes of low nd high tones lies in the nture of the mximum durtion of the movement. The cse for low notes occurs when the vibrtions of the ir globules follow ech other in turn more slowly, or for given time the undultions re sent out less frequently. Moreover, the note is of higher tone when the vibrtions hve shorter delys lced between them, in order tht more oscilltions re crried out in the sme time. Hence the notes, with resect to low nd high notes, re in the rtio of the number of oscilltions mde in given time intervl. 7. A sound is lso either simle or comosite. A simle sound [or note] is one in which the vibrtions hve equl distnces between ech other, nd they re of equl strength. A comosite sound is constructed from mny simle sounds roduced t the sme time, nd this sets u either concordnt or discordnt sounds. Concordnt notes [or chords] re erceived s being roduced by simle sounds or notes mintining the simlest rtio between the comonents, such s two s in the octve, or s one nd hlf s in the musicl fifth, etc. On the other hnd, the dissonnt [or discordnt] sounds hve their comonents in more bstruse rtio, such s two suerimosed frctions s in three tone. [The interested but uninformed reder my wish to hone u on the hysics of music scles; good introduction is found in rther dted book : Anlyticl Exermentl Physics, by Ference, Lemon, & Stevenson (1956) U. Chicgo, Ch. 33] 8. Now we my observe the rogtion of sound with little more ttention, tht cn be done with some consistency. For the distnce which give sound cn trvel cross in given time cn be found from the theory set out bove : for the minutes nd seconds of the hour re found by observtion to be the sme for ll sounds, either loud or soft, low or high tones, to be crried through given distnce; nd in fct these sounds lwys move forwrds with the sme seed. In order tht this shll be so, this question my be sked : during the time the globule of ir is comressed, wht distnce does it get thrust forwrds? This question cn be nswered without difficulty from the rules governing the communiction of the motion nd from the nture of ir. Indeed wy cn be found, but I omit doing this, s I refer not work with imgined quntities. I ut in its lce wht results re found from hysicl mesurments. 9 In order tht I cn consider the roblem in generl wy, the secific grvity of mercury to ir is ut in the rtio n to 1; the height of mercury in the brometer is equl to k, the length of endulum is f, from which it is lesnt tsk to mesure the time tken for the sound to trvel distnce from the endulum's oscilltions. From these denominted fctors, I cn find the rtio of the time for one oscilltion of the endulum f to the time for the sound to trvel distnce to be s 1 to 4 nkf [Following Newton, we ssume the endulum is one tht follows the rc of cycloid, in which cse it executes simle hrmonic motion (shm) whtever the mlitude, nd the eriod T of this oscilltion is f relted to the length f of simle endulum by T = 2π. On the other hnd, the seed of sound g P ccording to Newton v = = ga, where A is the the height of the homogeneous tmoshere. Hence, ρ

3 Euler's Disserttion De Sono : E002. Trnslted & Annotted by In Bruce 3. the time T ir for the sound to trvel distnce is given by T ir = / v = / ga. Hence, f T / Tir = 2π ga / = 2π fa / = 2π nkf /. Here A is the height of the homogeneous g tmoshere which is ρ ir g A, or ρ Hg g k, from which A = nk. Newton ws the first erson who hd grs of the mechnicl nture of wve motion to the extend tht he ws ble to roduce formul for the seed of sound c in ir; tht this formul does not redict exctly the correct vlue for c does not detrct in the lest from the method, but only reflects the lck of understnding t the time of how het ws involved in the gs lws. Thus, the rising nd lowering of the temerture in the ir due to comressions nd rrefctions re considered dibtic s they hen so quickly. These effects were ccounted for by Llce hundred yers lter by inroducing n extr fctor γ into Newton's formul for c (γ = c /c v, the rtio of the secific hets of ir t constnt ressure nd temerture). Thus, in Euler's dy, these temerture relted effects were still unknown. We now look t Euler's formul.] 10. If nd k re mesured in scrules, but in lce of f is ut 3166 [ scrule is ~ 0.34mm; hence f ~ 1076mm, giving time of 2 seconds for comlete swing, or 1 second for swing from one side to the other], will give the vlue, the number of seconds in which the sound should be rogted nk distnce. For the length of the endulum with time of one oscilltion [i. e. hlf-swing] is indeed 3166 scru. Thus, with the distnce solved for the time, [v = /t =] /, the distnce tht sound nk trvels out in time of one second will be nk scru. [Thus v = 4 ( ) = scru. or 415 m/s. If we ut π in lce of 4 in the formul for the hlf-swing, we get 326 m/s, which is ner the correct vlue.] 11. Thus these conclusions follow on. The seed of sound remins the sme if nk remins the sme too, if the density of the ir nd the ressure re in roortion, the sounds re crried with the sme seed. For one my know tht there is no difference to the senses nor seed for sound in ir of the mximum comression thn for sound moving in ir with the mximum rrefction. Hence sound on the summits of mountins ought to trvel with the sme seed s sound within the vlleys, excet for other cuses to be exlined soon tht might be dded. 12. By incresing the fctor nk, the seed of sound should be incresed. Hence with the density of ir remining the sme or decresing little, but with the ressure incresed, then the seed of sound will be greter ; but truly from the contrry with the density of the ir incresing, nd with the ressure remining the sme or decresing little, the sound is slowed down. And hence collecting these things together, since both the density or weight nd ressure of the ir surrounding the erth re subject to vrious chnges, the seed of sound is constntly chnging lso. Hence the mximum seed of sound will be [found] on the hottest dys with cler sky, or with the most creful mesurements mde from brometers nd thermometers t the highest levels of elevtion. Truly with the hrshest cold nd the fiercest storm, the seed of sound should be minimum, tht which my come bout by mesurements from the liquids in brometers nd thermometers resent in the lowest lces. 13. Hence the mximum seed of sound is found, if is ut in lce of n nd 2460 scru., in lce of k s thus the distnce trveled by sound in time of one second is = , i. e. the mximum seed of sound following my theorem should be 1222 Rhenish feet er second. Indeed the minimum seed of sound is obtined by utting for n nd 2260 for k, s thus the distnce ssed through in one second is = , or Hence the distnce which sound sses through in one second ought to be contined between the limits 1222 nd If those numbers re brought together with exerimentl trils, they re found to be in excellent greement with those, tht confirms my theory. For FLAMSTED mde observtions [J. Flmsteed ( ) circ 1675 hd determined the seed of sound in ir in Greenwich with Ed. Hlley ( ).] nd DERHAM [W. Derhm ( ), Exeriments nd observtions concerning the motion of sound

4 Euler's Disserttion De Sono : E002. Trnslted & Annotted by In Bruce 4...., Philosohicl Trnsctions (London) 26, 1708, no. 313,. 1] with the most ccurte exeriments set u found tht sound trveled through 1108 feet in one second, which is number lced lmost midwy between the limits found. If now we consider wht NEWTON hs to sy in [I. Newton, Philosohie nturlis rincii mthemtic Book II ro. 50, scholium; 2nd Ed, Cmbridge 1713, ; see lso this website on Newton] in the Phil. Book. II, Section VIII, he found tht sound sses through distnce in one second (with his resoning reduced to our wy of tlking) of 3166nk Rhenish scrules with d denotining the rtio of the dimeter to the erihery of circle, i. e. tht is roximtely 7 : 22. And thus his exression is less thn our one, if indeed NEWTON introduced the fctor 3 to 3166 nk, however, I will dhere to 4 in lce of this number. [Newton ctully comred the times with endulum nd so did not introduce π.] 15. Hence this is not so wonderful, since the most cute NEWTON found the exceedingly smll distnce tht sound will rech in second ; tht he could not determine greter thn 947 feet, which certinly is huge discrency [relly only ~ 11% out] from tht distnce which is found by exeriment ; but he reorts in order to confirm his method, tht the descrency is shred by the imurities mixed with ure ir tht slow it down. Indeed ir is corruted by vours, but the force of the ressure is lwys equl to the tmosheric weight, nd the weight of the ir to the senses too does not chnge. It is not ossible to be lwys mintining the chnge in the seed of sound in tems of these fctors, nd neither does the size of the molecules of the ir mke ny difference to these things. d 1 7 CHAPTER II. CONCERNING THE PRODUCTION OF SOUND. 16. It is well-known requirement in the roduction of sound, for the theory set forth in the revious chter, tht the vibrtions hve to be re-lied in some mnner, in order tht the globules of ir cn hve serte lternte contrctions nd exnsions for short time intervls. I hve been ble to infer three kinds of vibrtory motion from the three wys in which sound is generted in the first lce. Whereby few words should be sid here concerning the three diverse wys in which sounds cn be roduced. Moreover, I refer to s the first kind of sound, the sound of stringed instruments, drums, bells, musicl instruments under the control of the tongue, etc., ll well-known sounds, which hve their origin in the vibrtion of solid body. To be referred to s the second kind of sound, re sounds such s thunder, bombrdment nd the sning of twigs, nd ny in which body is set in some more violent stte of motion, ll nevertheless re sound rising from the sudden restitution of comressed ir, nd s stronger ercussion of the ir. Finlly, the third kind of sound I dd re the sounds of wind instruments such s the flute, nd I will send some time in exlining the nture of these with cre, since no one u to the resent hs given nything of substnce concerning these instruments. 17. As fr s ll the sounds of the first kind re concerned, so much I know, tht clerly ll sounds initilly were referred to nd considered to be generted by the vibrtions of some solid body, nd no other kind of sound considered ossible; but the flseness of this ide will soon be shown, when I set forth the two remining wys in which sound cn be roduced. But now, concerning the mnner in which sounds re roduced, the first hs been exmined with greter cre. Indeed for the resent I need only consider how sounds of different kinds rise from strings, s the other sounds of this tye cn esily be deduced from this exmle. I exmine closely how the exct tension in the string cn be obtined by being extended round the column of the instrument in order tht the force in the string is llowed to be mesured ccurtely. 18. Before ll else, it is to be noted tht strings give rise to sounds with the sme equl rtio of low or high itch, whtever the lied ulsting force, lthough it is ossible to hve huge difference in the rtio of the strength to wekness of the force; indeed the intensity of the sound is roortionl to the seed with which the string bets the ir, nd sounds re eqully intense if the ir is struck with the sme force.

5 Euler's Disserttion De Sono : E002. Trnslted & Annotted by In Bruce 5. Wherefore, since musicl sounds of either low or high in itch should be of equl strength, in order tht lesnt hrmony is roduced, this is given ernest ttention in the mking of musicl instruments, in order tht equll sounds with the sme rtio of intensity or robustness re roduced, which should be obtined following the rules, which indeed hve been found now from much gross tril nd error rctise by the most recent crftsmen, the truth of which indeed is erfectly rent from wht follows, which hve been diligently observed. I. The lengths of the strings re in the recirocl rtio of the notes, i. e. the number of vibrtions sent out in given time. II. The thickness of the strings or the trnsverse cross-sections re lso in the recirocl rtio of the notes, if the sme mteril of course my be sid to be used; if indeed this is not so, then the rtio of the densities hs to be included with the rtio of the widths. These rules cn be lied for the construction of wind instruments such s the flutes, where in lce of the length of the strings is to be tken the length or height of the flute nd in lce of the thickness of the strings the internl width of the flute. 19. When the string is vibrting, it strikes the ir globules which re nerby nd since they unble to recede, they re comressed ; but with the enduring oscilltory motion the ir globules suffer continul new comressions, thus the note rises. And thus the ir strikes the er or the tymnic membrne of the er s often s the string returns. Indeed the number of ercussions of ny note crried to the er for given time cn be found of course by finding the numberof oscilltions of the string emitting the note in the sme time. Moreover, my solution, which with the solutions of Cl. Cl. D. D. JOH. BERNOULLI [JOH. BERNOULLI ( ), Medittiones de chordis vibrntious, Commt. cd. sc.petro. 8 (1728), 1732,. 13; Oer omni, Lusnne et Geneve 1742, t. III,. 198] nd BROOK TAYLOR [( ), Methodus incrementorum direct et invers, Londini 1715,. 26 ] re in exct greement, is the following: 20. Let the weight holding the string =, the weight of the string = q, nd the length of the string =, from which three given quntities the number of vibrtions to be found in given time cn be roosed. I find the number of oscilltions set forth in second to be : , 7 q where should be mesured in scrules. [Recll tht the eriod of the endulum mesuring single seconds f for single swing from left to right of vice vers is given by T = π, where f is 3166 scrules. Hence, g 3166 g the time of one second corresonds to T = π. If the tension in the string is s force, the density of the string is q/ s mss er unit length, then the seed of trnsverse wve in the string is given by q v =, while the time to trvel length is given by t = / v = / =. If the recirocl is the q q frequency of the ssocited sound wve, then the frequency is q. If is given s mss, then the ssocited force is g, nd hence the frequency is given by q g ist =. This eriod comred with the endulum is sound g q while the ssocited eriod T q / / q g 1 q sound T =. g π = g π g 3166 = The frequency of the sound is hence π π s Euler hs shown.] Since the number of vibrtions of sound is in roortion to this number, q sounds sent forth by different strings re mongst themselves in the rtio q, i. e. the frequencies of the sounds re in squre root rtio comosed in direct roortion to the weight of the tension, nd indirectly to the length nd weight of the string. I do not deduce more rticulr consequences from this, but I will

6 Euler's Disserttion De Sono : E002. Trnslted & Annotted by In Bruce 6. inquire into the nture of known sounds, nd the number of vibrtions nswering to these formule from the exeriment tht is set u by me in the end. 21. If I tke coer string with the density of tht kind of substnce, which is indicted in 18, of length 980 scrules, which hs weight of ounds, nd I hve stretched this with weight of 11 4 ounds, the sound ulses from which re to be tken in greement with tht instrument in the chorl mode, s they sy, to be fitted [i. e., by someone singing the sme tone], which to the musicl is herd s ds. Hence therefore, one is ermitted to count u, how often tht sound is herd, nd hence how my vibrtions of the other sound in the given time re herd from the mechnicl device being struck ; indeed in the given generl formul, if 980 is substituted in lce of, in lce of 11 4 ounds, in lce of q 49, then the sound ds is found to hve 559 vibrtions er second nd since ds shll be to c s 6 to 5 [ minor third], the sound c hs 466 vibrtions [er second] nd thus t the minimum C is 116 vibrtions [er second]. 22. To this method of sound roduction, reference is mde to the sounds roduced by the tongue s well s by elstic ltes [vlves] inserted in tube in strong irflow, though both these methods re relted lso to the third mnner of sound roduction ; for indeed they re ech relted to the other. Mechnisms of this kind re to be seen in the vrious comressed ir ies such s trumets nd bugles, s in the imittion of the humn voice in song, ll of these instruments must hve ir blown into them in order tht they cn roduce sound. The comressed ir, by itself seeking to ss out, uncovers the tongue from the oening, like vlve, but holds tht oening to n excessive extent, so tht the tongue cts s vlve nd closes gin by stretching to its former stte, which is thus oened new, in order tht the ir ssing through cn be given vibrtory motion. Indeed it is unvoidble, with the ir flow roduced by uniform brething, tht this kind of vlve finlly comes to rest nd the sound ceses; but this wrning lies only to the wind instruments themselves, for if the ir is exelled from bellows [s in n orgn], it strikes the oenings of the mechnism in non-uniform mnner, nd with the hel of n inserted vlve the wind in the tube is crried long nd the vibrtion continues. 23. Clerly the humn voice is roduced in the sme wy; indeed the eiglottis holds in lce the set of the tongue in the orgn of seech, the vibrtion of which is mintined by the ssge of the ir scending through the rough windie. Besides, the vibrtory motion of the ir escing from the end of the rough windie is chnged in the cvity of the mouth in number of wys, by which the low nd high-itched tones of the voice cn be roduced, nd different vocl effects re formed, which with the hel of the tongue, lis, nd the hrynx rovide sounds with consonnts So with the nose, for it is rent tht the ir in continuous vibrtion coming from the eiglottis cn leve by the nose too; but with regrd to the vrious high nd low itched notes which cn be roduced in this wy, they re unble to give rise to either cler seech nd neither cn consonnts be reserved, which re thus different from sounds roduced by the mouth. 24. But these sounds sent forth by the vibrtions of the tongue, unless they re strengthened by ies or tubes, re soon to become very wek, s is to be seen in the vibrtions of thin ltes [in contct with the mouth, one imgines], where hrdly nything is erceived by the er. But the mircle is the wy in which these sounds re mlified in tubes, nd lso the humn voice by the mouth cvity, not in the lest being the mnner sounds of low nd high notes sent into ie undergo gret chnge. Truly this is not the lce to sow the seeds [for further study] of how sound wves re intensified nd reflected within ies; there would need to be work with secil title, in which tht mteril could be set out more crefully, nd in which the mirculous mlifiction of sound by meghones could lso be estblished, s well s the rincile of the echo nd mny other henomen; but the tsk of crefully considering tht mteril more crefully is not yet comlete, nd concerning wht is held in other writings, lot of things tht I hve exmined in these sources re very confused nd in the greter rt flse. 25. By the second clss of sounds I refer to these sounds which rise either from the sudden relese of noteworthy quntity of comressed ir or which strike the ir with greter force. The ir in the ltter mode is lso comressed, since it ttemts not to leve the lce to the vibrting body, from where the ir left hs gin exnded. Thus the cuse of the sounds belonging to the second clss is the restitution of the ir to its

7 Euler's Disserttion De Sono : E002. Trnslted & Annotted by In Bruce 7. stte before being comressed. It is rent tht the sound rises from tht restorton, the ir which ws comressed itself exnds too nd hence gin contrcts nd so on, with which undultion of the ir shll be, s the smllest of the ir globules too, which of course mke u the mss of the ir, rticite in tht vibrtory motion nd in consequence led the sound forwrds; it is to be noted, tht if greter body of ir is comressed, deeer sound is roduced, nd if it is smller, the sound will be shrer. Truly sounds roduced in this wy re not ble to endure for long time, but from wht is left they must sto, becuse the vibrtory motion of the ir set u long wy off is lost t once. 26. Therefore ll the cuses revil for the roduction of suitble sound, whereby either the ir is comressed for the sound to be sent, or truly it is thus comressed gin in order tht the exnsion cn be stoed. Whereby ll the swifter motions of bodies in the ir ought to emit sound; indeed from the motions of bodies the ir on ccount of its own inerti is unble to go freely nd is comressed, nd gin by exnding itself the vibrtory motion of the ir is induced in the smllest of ir globules, to roduce suitble sound. Thus sounds flow from the stronger vibrtions of brnches, s from bodies moving more swiftly. Also the origin of the sounds of the breeze nd of the wind rise from this font; indeed the ir from the receeding is comressed s for tht following exmle, s from hrd body. 27. Esily the loudest of the sounds which rise from the relxtion of suddenly comressed ir re those ssocited with rtillery nd thunder. Moreover vrious exeriments set u with sulhur nd sltetre owder rove tht the cuse of these immense sounds is the restortion of comressed ir to its former stte; since it is found tht the ir there is in stte of mximum comression when the owder is set light nd wys re found for it to emerge, so tht it cn burst forth with the mximum force. Since moreover from the sltetre nd fiery dust, nd the mny constituent clouds nd vours tht re resent, is it not mircle tht fire rises from tht mteril, nd the stonishing sounds tht then reverberte. 28. The sounds of flutes constitute the third kind of sounds. The exlntion of the nture of these sounds hs been stonishingly twisted by scrutineers t ny time. Most hve thought tht with the infltion of the ies the smllest rticles of the inner surfce re struck nd the vibrtory motion is set u, thus the internl surfce of the ies restores the vibrtion by infltion nd enbles the oscilltions to be communicted with the ir; but how tht exlntion is consistnt with the lws of nture nd of motion, they themselves my exmine. I myself m certinly unble to conceive this, ll the sme it is ossible to exlin how ies of different heights cn give rise to differences of sounds of the sme mlitude ; for indeed if the internl rticles cn bring bout motion t ny time, then I m unble to see how, for tubes of different heights, it is ossible to hve different kinds of oscilltions. In next to no time I cn decide by setting u n exeriment even with single ies, how it is ossible to exlin the theory. 29. Moreover, since I will resolve this mtter, t first the structure of flutes is to be considered, nd then wht indeed is going on inside while ir is blowing through them hs to be considered with gret cre. For flutes re ies or tubes, for which beyond the junction of the tubes there is cvity for the mouth [the embrouchure or blow hole] in order to receive suitble ir blown in, which is lced towrds the end of the tube nd which chnges the ir flow by striking shr edge on the inner side of the hole on the side of the tube. This finlly llows ir from the mouth to be sent in through the hole in the tube, nd the ir is llowed to move slowly bck long the inner surfce of the tube [Euler thus seems to believe tht reflected ir moves bck long the inner surfce of the tube slowly, rther thn reflected sound]; if the ie is mde in this wy, sound is emitted on ir being blown in, tht is redily rent, for if some tube on its own is blown into by the mouth, so tht the ir in the tube cn move slowly long on the inner surfce, s it lso sends out the sme sound s the flute. The internl surfce of the tube should be hrd nd go to the left, so tht the ir cn neither intrude, nor be given lce in tht extended tube [to the right] where it is ossible to exnd [i. e., stnding wves re estblished in the left-hnd rt of the tube, the smll sce to the right cting s sort of buffer, evens out sounds of different frequencies], wherefore the ies should be rered from tubes with closed end nd smooth rigid sides. [Note however tht the flute cts lmost like tube oen t both ends, due to the resence of the blow hole; thus, if ll the note holes re closed, then the sound roduced will hve wvelength roximtely twice the length of the instrument. The closed end now hs stoer, nd its exct osition is imortnt for tuning.]

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