Rayleigh, John William Strutt (1842-1919 ; 3rd baron ). The theory of sound . Volume I. 1995.

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THE

THEORY OF SOUND.

MARS~a~

THEORY OF SOUND.

JOHN WILLIAM STRUTT, BARON RAYLEIOH, M.A..F.R.8.FORMEK.YFELMWOFTRINITVCOLLEGE,CAMH)UnOE.

MACMILLAN AND 00.

1877

THF

&onlton:+

[~~ ~t~~ r~trt'ed.]J

HY

VOLUME 1.

OEambtitgc:r)t)M):Y'J'

ATT)tKUS)VRM)TV'')t'

PREFACE.

IN th work, of winch th prsent volume is an instal-lent, my endeavour bas been to lay before the readera connected exposition of the theory of sound, whichshould include ~he more important of th adv~nces madein modem times y Matliematicians and Physicists.The importance of the object winch 1 have had in viewwill not, I tinnk, be disputed Ly those competent tojudge. At th prsent time many of th most valuablecontributions to science are to be found onjy in scatteredpenodicala and traiMactions of socletles, pubMied invanous parts of th worid and in several languies, andarc often practically inaccessible to those vvho do nothappen to hvc in th neighbourhood of large publichbraues. In sucli a state of things the mechanicalimpedimonts to study entail an amount of unremunera-tive labour and consquent hindrance to th advanco-ment of science which it would be dimcult to over-estimate.

Since the wcH-known Article on Sound in the .E~-c'~)~ ~)~M~(~ by Sir John Herschel (1845),no complte work lias been publislied in wilich tliosuhject is trcatcd jnatl-cmatically. Ly th promaturedeath of Prof. Donkin tlie scientific worid was deprivedof onc w!)oso mathcmatical attainments in combmationwith a {n-ctic.-d kr~Judge of mua:c quahned hifn in a

VI PREFACE.

special manner to write on Sound. The first part of hisAcoustics (1870), though little jnm~ tbsm a fragment. issumcient to shew that my labours would l~ave been un-necessary had Prof Donkin lived to complte his work.

In tlie choice of topics to be dealt with in a. workon Sound, 1 have for th most part foUoAvedth exempleof my predecessors. To a great extent th theory ofSound, as commonly understood, covers th same groundas th theory of Vibrations in gnral but, unless somelimitation were admitted, th consideration of such sub-jects as the Tides, not to speak of Optics, would have tobe included. As a gnral ruie we shall confine ourselvesto those classes of vibrations for which our ears afford aready made and wonderfully sensitive instrument of in-vestigation. Without ears we should hardiy care muchmore about vibrations than without eyes we should careabout light.

The present volume includes chapters on th vibra-tions of systems in gnerai, in which, 1 hope, will berecognised some novelty of treatment and results, fol.lowed by a more detailed consideration of special systems,such as stretclied strings, bars, membranes, and plates.The second volume, of which a considrable portion isalready written, will commence with acrial vibrations.

My best thanks are due to Mr H. M. Taylor ofTrinity Collge, Cambridge, who bas been good enoughto read th proofs. By his kind assistance several errorsand obscurities have been eliminated, and th volumegenerally has been rendered less imperfect than it wouldotherwise have been.

Any corrections, or suggestions for improvements, witliwhich my rcaders may faveur me will be highly apprc-ciated.

TEttUNO PLACE, WmiAi.,

.~n7, 1877.

CONTENTS.

C'HAPTER I.MOE

1-27 i

Sound duo to Vibrations. Finite volocity of Propagatiou. Yelooity indo-pondent of Pitcb. Depmult'a oxporimonts. Sound propagated in water.'Witoatatono'aexperiment. Enfoeblomontof Sound by distance. Notesand Noisos. Musicalmoteaduo to poriodiovibrations. Siren of Cagniarddo la Tour. Pitch dopendont upon Poriod. Eelationahip betweenmusical notes. Tho samo ratio of perioda corresponds to the samointorval in all parts of tho scale. Harmonie sca-tes. Diatonio soaloa.Absolute Pitoh. Neoossity of Temperament. Equal Tomperament.Table of FroqnonoioB. Analyais of Notes. Notes and Tones. Qualitydepandont upon harmonie overtonos. Resolution of Notes by efu;un.certain. Simpletcnoa correspond to aimpla pondidona vibra.tiona.

CHAPTER II.

28-4.2 18

Composition of harmonio motions of like period. Harmonie Curvo. Com.position of two vibrations of nearly equal period. Bats. Fourier'aTheorem. Vibrations in porpendionlar directions. Lissajous' Cylindor.Lissajous' Figures, Bin.ckburu's poudulum. Kaieidophone. Opticalmethods of composition and analysis. Th vibration microscope. In.termittont Illuminntion.

viil CONTENTS.

~-c.s

Systumswithon~d~rouoffrt~dom. Indopcndottcu of ampiitudu~ndpfriod. Frictiomdfurccpruportimudtovotouit.y. l''))t'cnd\i)))'atiCHAPTER V.

CONTENTS. Ix

11814S. 127

Lnw of extension of n, titritig. Transvurso vihrntiona. Solution of tho pro.bJcm for n string whoso masH is concontrutcd in cquidistant points. D-rivation of no)ution for continuouo string. Pfu'tittI diffcrentitti quation.Hxpt't.'snious for nnd y. Hoat gonorat form of simpio harmonie mo-tion. StritiRS with xoJ extronitios. GcnoDt) motion of n stnng pcri-odio. Mcrsonuc'H L)tws. Sonomotur. Kurtnn.1 ntotlua of vibration.Dctunninat.ion of eonattuits to suit nrbitt'M'y initit ciroumfitttncus. Ca.Moof pluoknd HtrinR. Expression.-) for ' and y in torms of noriun] co-ordi-n

CONTENTS.

ordinatos. Normal equatious of motion. Dtermination of constant))l'AUI~

to suit initial conditions. Caso of rod etartod bv a b)ow. I~odstart~from rest as dofloctodby ~t~I ~~c. lu CL.~m ~os tho BorioHofnormal funoticus coanostu coj.vcro. Form of H~ norma! hmettonsli(:u-fiuebur. Lfnv9of Jopotj.Icuooof frofjuoucyou )ongt)i und tLiok-noM. Caso whou both oudH ~rc clampod. Normal fuuctions for ,)clampod.Ireo bar. Caleulaticu o puriud.s. CompnrinonHof pitch. Dis-oussiou of tho gravost modo of vibration of a freo-freo bar. Threunodos. Four ~oJoa. Gravost mojo for clampod-froobar. Position utnodos. Supportcdbar. Calculation of poriod for clamped-froobar fromLypothetleal typo. Solution of problom for n bar with a loaded ond.Euuct. uf adtUtious to a bar. lufluonco of irrogularitios of donsity.CorreottOMfor rotatory iuortia. L:ootsof functioua dorivediinoarly fromnormaJ fuuetioM. Formation of ~uatiou of motiou ~heu thoro is por.Mauout tousiou. Spoeial trmiual couditioua. Itosultaut of two trainsof wavcs of iic-iirlycqua) poriod. Fourior'Hsolution of problomfor ini.nito har.

CHAPTER IX.

1~213.Tension of a motubrano. Equation of motion. Fixod reotangular lonud-

ary. Expression for ~aud iu tenus of normal co-ordiuatos. Normalouations of vibratiou. Examplos of improssed forces. Frequoncy foran olongatod rectangle dpends maitiJy ou tho shortor sido. Casoo iuwhiohdifTerout modes of vibration havo tho samo poriod. Dorivodmodes thence arising. Effeet of 6li(;ht irrcgulanties. An irregu!aritymay rontovo nidoturmiuatonosa of normal modoa. Solutionsapplicableto a triaugle. Espro~ion of tho Honoraidiilorontml eqnation by polarco-ordm~tes. Of tho two functions, w),idt oceur in tho solution, ono iacxcluded by tho condition at tlio polo. Expressions for Bossel'a func-tions. Formutm rdating theroto. Tublo of tho first two functionsFixod eiroilar boundary. Conjugato proporty of tho normal functionswithout restriction of boandary. Values of integrntod squares. Ex-proHMounfor T nnd F in tcnua of normal functions. Normal oqua-tions of vibration for ciroular mombraue. Spoci.d easo of froo vibra-tions. Yibratioua duo to a harmonie force uniformty distributedUtohos of tho varionsshnpto tonoH. Tabio of tlio rootsof Bosscl'ofunc-tions. Nodal Fiur~. Circular mombrano with ono radius fixed.Bessel's Bonctionsof frnctional ordcr. Ejloct of sma'I lond. Vibrationsof a mombrano whoso boundary is approximatoiy ciroutar. In manycasos th pitch of a mombrano mny bu calculated from tlioaroa alonoOf aU .nombraues of equal aroa t)Mt of oireular form l.M tlio gravostpitch. l'itch of a mcmbrano whoso boundary ia au eDipso of smaltceeHntricity. Motliodof obtai)iii)g limits in casos that oumot bo dealtwitli rigorouf3ly. Comp~rison of fruqueucioaiu varions ca.sc.sof mcm-braues of eqna) arc.a. Histury of tho probion.. Bourh'ot'8oxperi.aonta) invostigfttiouB.

XICONTEN-TS.

214-235

Vibrations of PIatos. Potontial Enorgy of Bending. Transformation of 5~.Superuoial diiorontial equation. Dou.ndary conditions. Conjugatoproporty of normal functions. Transformation to polar co-ordinates.Form of gonorni solution continuons through polo, Eqnations doter-mining tho poriods for a froo ciroular p!nto. EirohhoC'a catouhtions.Comparison with observation, mdii of nodal cirolos. Irreguln.ntiesKivo riso to boats. Gonoralizution of solution. Cnso of cJampod, orHupportod,cdgo. Disturbn.uceof Chiadni's figures. Hifitoryof proUom.Mn.tl.iou's critieiamo. DoetfmguiM phtto with aupportoJ edgo. Itoct-nnguhn-plato with freo edgo. Boundary couditionH. Ono Hpocialcnso(~ =0) iHfunonabluto mfttttomaticfdtro~tmont. Investigation of codaifigures. WItcntatoue'Happlication of tho mothod of HnporpositionCompariMUof Whoat~tono'f. liguros wit]. thoso reaUy n.pp)io~!o to npMo in tho cnso = 0. Gravost modo of a squnroplate. Caiouhttionof poriod on hypothotica! type. Nodal ~igurcHinferrod from considor.atlona of symmetry. Hoxngon. Comparison hotweoncircle nnd squnre.Lnw connooting pitch and thicknoas. In tho cnso of a elfunpododgonny contraction of tho boundary raisos tLo pitch. No gravest form fora free plato of givon aron. In similar plates tho poriod is as tho linoardimension. Whoat.stono'a exprimenta on wooden plates. Knig'aoxperimontN. Vibrations of cylindor, or ring. Motion tangentinl aswoll as normal. Bolation betwoon tangoitial and normal motiona. Ex-prossinna for Mnetio and potontial nergies. Estions of vibration.L'requonciosof tonos. Comparison with Chiadni. Tangential frictionexcites tanguntiat motion. Exprimental vrification. Bats duo toirregularities.

CHAPTERX.RAOP

CIIAPTER I.

INTRODUCTION.

1. Tim sensation of sound is a thing s:M~e~eW~,not com-parab]e with any of our other sensations. No one can expressth relation between a sound and a colour or a smcil. Directlyor indirectiy, ail questions connected with this subject mustcomc for decision to thc car, as t!tG organ of hcaring; andfrom it thct'c can be no appea!. But wc are not thcrefore toinfci' that ail acoustical investigations arc conducted with thcunassistcd car. Whcn once wc have discovercd thc physicalphenomena which constitute th foundation of sound, our ex-plorations arc in great mcasurc transferred to another nc!d lyingwithin thc dominion of th pi-mciples of Mcchanics. ImportMitlaws arc in this way ai'rivcd at, to which the sensations of th carcanuot but conform.

2. Very cursory obscrvatioo. often succs to shew thatsounding bodics arc in a statc of vibration, and tha.t thc p)ic-nomena of sound and vibration are closcly connected. WIicn a,vibrating bell or string is touched by the finger, th sound cea~csat th same moment tha.t thc vibration is damped. But, in orderto affect th sens of hearing, it is not enough to have a vibratinginstrument t!icre must also be an uninterrupted communicationbetween thc instrument and thc car. A bcll rung in ~ac!

2 INTRODUCTION. [3.report from the flash. This rcpresents the time occupied bysouud in traveUIng from th gun to th observer, the rotardatinnof the nash duo to th finite velocity of light bcing altogethernegligible. Th first accuratc experiments wero mado by somomembers of the French Academy, in 1738. Cannons were nrc-d,and th rctardationof th reports at different distances ohscrvcd.Th principal prcaution -necessaryis to revo-se alternatcdy tliedirection along which the sound travels, in order to cllminatc thoinfluence of tlie motion of th air in mass. Down t!ic wind, forinstance, sound travels reJativeJy to th carth faster than itsproper rate, for the velocity of thc wind is added to that properto the propagation of sound in still air. For still dry air n.t atemprature of 0"0., thc French observerafound a velocity of 337metres per second. Observations of tho samo character wercmade by Arago and others in 1822 by th Dutch physicists Moll,van Beek and Kuytcnbrouwer at Amsterdam by Bravais andMartins between thc top of the Faulhorn and a station bclowand by others. Th gnerai result bas been to give a somcwhatlower value for tbc velocity of sound-about 332 mtres persecond. Thc effect of altration of temprature and pressure on thepropagation of sound will be best consideredin connectiouwithth mechanical theory.

4. It is a direct consequenceof observation,that within widelimits, th velocity of sound is independent,or at least very ncarlyindependent, of its intensity, and also of its pitel). Wcre thisotherwiso, a quick piece of music would be hcard at a littledistance hopelesslyconfused and discordant. But when the dis-turbances are vcry violent and abrupt, so that th altrations ofdensity concernedarc comparable with th whole density of theair, the simplicity of this law may be departed from.

5. An claborate sries of experiments on tlic propagation ofsound in long tubes (watcr-pipes) has been madc by Rcgnault\He adopted an automatic arrangement similar in principle to thatused for me~suring th speedof projectiles. At thc moment whena ptstol is fired at one end of tlie tube a wireconveying an electriccurrent is ruptnrcd by thc sliock. Tins causesth withdrawai of atracing point which was previonsly marking a line on a revolvingdrum. At tho furthcr end of thc pipe is a stretched membrano soarranged that whcn on th arrivai of the sound it yields to th

~MofrM

35.] VELOCITY OF SOUND.

impulse, the circuit, whichwas ruptured during tho passage of thsoun< i3 rccumpietfd. At thc sa.mcmoment tho tracing pointfaits back on tlic drum. Tho blank space loft uumarked corre-

sponds to thc thuc occupied by th Sound in t~aking the joumcy,and, wltcn th motion of th drum is known, givcsthe means ofdctcrmining it. Tho length ofthe journoy hctwccn th first wiroand the membrane is fouud by direct mcasurcmcnt. In thcsacxperimcnts the velocityof sound appcarcd to hc not quitc indc-pendent of th dl~meter of the pipe, whieh vn.)'Icdfrom 0'108to 1'100. Tho diso'cpancy is perhftps duo to friction, whose

j innucuco would hc greater in smallerpipes.G. AIthough, in practice, air is usually the vehicio of sound,

otiicr gases, liquids and solids are equally capable of conveyingit. In most cases,I)owever,th means ofmaking a direct mcasure-ment of the velocityof sound are wanting, and woM'enot yet ina position to consider tlie indirect methods. But in thc caso ofwater tho same diniculty does not occur. In th year 182G,

S Colladon ami Sturm investigated th propagation of sound in thcLake of Geneva. Tlie striking of a bell at one station wassimultaneous with a nash of gunpowder. The observer at a.

t, second station mcasured the interval between tho flash and thearriva! of th sound,applying Itis car to a tube carried beneathth surface. At a temprature of 8C.,th velocityof sound Inwater was thus found to bo 14-35metres per second.

7. Thc conveyanccof sound by solids may bc IHnstratedby a

pretty experiment due to Wheatstone. One end of a metallic wirois connecteclwith tho sound-board of a pianoforte, and th othertaken through th partitions or floors into anothcr part of thbuilding, where naturally nothing would be audible. If a reso-nancc-board (such as a violin) bc now placcd in contact with thewire, a tune p]ayed on th piano is easily heard, and th soundseems to cmanatc from th resonance-board.

8. In an open spacethc intensity of sound falls offwith grcatrapidity as tho distance from th source increases. Th saineamuunt ofmotion bas to do duty oversurfacesever Increa~ingas thesquares of the distance. Anything that confines the sound willtend to dimini.sh ttte falling off of intensity. Thus over th flatsurface of still watcr, a sound can'Ies furthcr than over brokenground thc cornerbetween a smoothpavement and a vertical wallis still botter; but the most crtcctive ofaU is a tubc-likc enclosure,

1S

[8.INTRODUCTION.

which prevents spreading altogether, Tlie use of speaking tubesto faciMtatecommunicationbetween thc dirent parts of abuHdir)

10.] piTcn. 5

spokcn of as always th samc for th same relationship. Thus,"horov.'r th~yMn.ynccm' lli th Hcale,a. note '~d ita.octave arcsep~u'atedby

6 INTRODUCTION, [12.

revolvc tho pitch will still be th same. Observation of othcrsources of sound, such as vibrating solids, leads to the samo con-clusions, though th difficulties arc often such as to rendernecessary rather rcnned exprimental mothods.

But in saying that pitch depends upon. period, therelurks an ambiguity, which dcscrves attentive consideration,as it will lead us to a point of grt importance. If avariable quantity is periodic in any time -r, it is also periodicin the timos 27-,3, &c. Conversely,a recurrence within a givenperiod r, docs not exclude a moro rapid reourrence withinperiods which are tho aliquot parts of r. It would appearaccording!y that a vibration really recurring in th time ~r (forexample)maybe regarded as having the pcriod -r,and therefore bytlie lawjust laid down as produciog a note of the pitch defined byT. Thc force of this consideration cannot be entircty evaded bydefining as tho pcriod th least time rcquired to bring about arptition. In tlie first place, th necessity of such a restriction isin itsc!f almost sufHcient to shcw that we have not got to th rootof the matter fur although a right to th period r may be dcuicdto a vibration rcpeating itself rigorousiywithin a time ~T, yet itmust bc auowcd to a vibration that may differ indefinitely littlethcrcfrom. In thc Siren cxperimcnt, suppose that in one of thcch'cles of holes containing an cvcn number, every alternate hole isdisp]accd along th arc of the circle by the same amount. Thedplacement may bo made sosmall that no change can be detectedin tlie resulting note but the periodic time on whieh th pitchdpends lias bccn doubled. And secoudly it is vident from thnature of pl'iodicity,tliat th superposition on a vibration of periodT,ofothurs having pcriods ~T,~T.&c., docs not disturb the period r,while yet it caniiot be supposed that th addition of th new cl-ments bas left thcqualityofthe sound unchangcd. Moreover.sincothc pitch is not affectcd hy their prsence, how do we kuow thatclcmcnts of the sliorter periodswere not tbercfromt)ie beginnin"'?

13. Thse considrationslead us to expectrcmarkable rcJationsbetween th notes whose periods are as th reciprocals of thnatural numbers. Nothing can bc easicr than to invcstigate th

13.] MUSICAL INTERVALS. 7

be th double of that belonging to th second. On making theexperiment the two notes are found to stand to cach other inth relation of octaves; and we conclude that in passing fromany?M

INTRODUCTION. [15.8

we obtain th minor sixth. Again, by subtraction of a majorthird from a fifth we obtain th minor third; aud from this byinversion tho major sixth. The following table exhibits side byside th names of the intervals and the corrcsponding ratios offrcqucncies

Octave 2Fifth. 3Fourth. 4M~jorThird. 5MiuorSixth. 8Miner Third. GM~jorSixth. 5

Thcfjo are ail thc consonant intervals comprised witttin thclimits of th octave. It willbe remarked tliat tite correspondingratios are ail expressed hy means of ~M~t~whole numhers, andt!tat tliis is more particularly th case for th moro consonantintervals.

The notes whosc frequencics arc multiples of that of a givenune, are called its AM~M~M,and the whole scries constitutesa /M'7/io?!cscctle. As is well known to violinists, they may ailbo obtaiued from the samc string by touching it lightiy with theimgcr at certain points, whiloth bow is drawn.

Tlie establishment of th conncctionbetween musical intervalsand dfunte ratios of frequcncya fuudamcutal point in Acoustics-is duo to Mersennc (J63C). It was indeed known to thGrceks iu what ratios tlie Iougtlis of strings must bc chaagcdin ordcr to obtain tlie octave and rifth; but Mcrsenne duntou-strated tlie Jawconnecting thc length of a string with the ponodof its vibration, and madc thc first dtermination of the actualrate of vibration of a known musical note.

16. On any note takcn as a kcy-notc, or

16.] NOTATION. 9

lying immediately above th tonic, wo obtain notes whose fre-quenciesarranged in order ofmagnitude are

Do Re Mi Fa Sol La. SI Do

1,9 5 4 3 5 la

9 2.1, 8' 4' 3' 2' 3' 8' 2.

Hcro the common cbord on Do is Do-Mi-Sol, with thc5 3

ratios 1 th chord on Sol is SolSiRe, with th ratiosT~

~2x~=l:and thc chord on Fa is Fa-LaDo,0 0 T Xtlie c 101' on Fa 18i a- a- 0,

still with tlie samc ratios. Thc scale is completed by rcpeatingthcsc notes above and bebw at intervals of octaves.

If we take as our Do, or key-note, the lower c of a tcnorvoice,th diatonic scale will be

c d e f g a h c'.

Usage diffcrs slight~y p.s to th mode of distinguishing thedifferentoctaves; iu wllat followsI adopt th notation of Helm-hoitz. TIic octave below th one just referred to is written withcapital letters-C, D,

10 INTRODUCTION, f~g.18. Solong as wekeep to th diatonic scaleofc, th notes abovewritten are ail that are required in a musical composition. But itis frequentiy desired to change th key-note. Under thse circum-stances a singer with a good natural car, accustomed to performwitliout accompanimcnt, takes an entirely fresh departure, con-

structing a new diatonic scale on th new key-note. In tbis wayafter a fewchanges of key, tho original scalewill be quite departedfrom.and an immense varicty of noteshe used. On an instrumentwith fixednotes like tho piano and organ such a multiplication isimpracticahle,and some compromiseis necessary in order to allowth same note to perform different functions. This is not thplace to discuss the question at any length, wc will thcrefore takeas an illustration th simplest, as wcn as th commonest case-modulation into th key of th dominant.

By donation, th diatonic scale of c consists of th commonchords foundcd on c, g and f. Jn like manner th scale of g con-sists of tlie chords founded on g, d and c. Th chords of c and garc then commn'to th two sca!cs; but th third and fifth of dintrodnce new notes. Th thu-d of d written f# has a frcquency3 5 4a t J8 4 32 removed from any note in th scale of c.

But th fifth of d, with afrequc.cy differs but

little from a, whose frcqucncy is In ordinarykeyed instrumentsth interval betwecn the two, represented by and called ac~ is ncglectcd and th two note. by a suitable compromiseor ~?~-Hwc~ M-eidentined.

19. Various systems of tomperament have been used thsimplest and tliat nowmost generally used, or at least aimed at, isth equal temprament. On referring to the table of frequenciesfortlie diatonic sealc, it will be secn that the intervals from Do to Refrom Re to Mi, from Fa to Sol, from Sol to La, and fromLa to Si,are nearly th same, being rcpresented by or while tjjintervals fromMi to Fa and from Si to Do, represented by arela~1~' ?~ equal ~mperament treats ~cs~'ap.proximate relations as exact, dividing the octave into twelve eqnal

19.] EQUAL TEMPERAMENT. 11

parts called mean semitones. From thse twelvonotes th diatonicscalc belonging to any key may be selected accordingto tho fol-lowing rule. Taking the key-note as the first, fill up the serieswith th third, fifth, sixth, eighth, tcnth, twelfth and thirteenth.notes, counting upwards. In this wayail dKScultIesof modulationarc avoided, as th twolve notes serve as weUfor one key as foranothcr. But this advantagc is obtained at a sacrificeof true in-tonation. Th equal temprament third, being th third part ofan octave, is rcprescnted by th ratio ~2 :1, or approximately].'2a99, wliile th true third is 1-25. The tempercd third is thushigher than th truc by th interval 126 125. The ratio of thtempered fifth may be obtained from th consideration that sevenficmitoncsmakc a fifth, wliile twelve go to an octave. Th ratio is

thcrforc 2 1, which = 1-4.983. The tempered fifth is thus too]ow in th ratio 1'4!)83 1-5, or approximately 881 883. Thiscn'or is msignificaut; and even th error of th third is not ofmuch conse

INTRODUCTION. [20.

Th ratios of tho intcrvals of the equal teinpra.mentscalearegtvcn bclow (Zaunuluer)

21. Rcturning now for a moment to thc pbysical aspect of t!iequestion, we will assume, what wc shall af'terwards prove to bctruc within widelim its,that, whcu two or more sourcesof soundagitate th air sunultaneousiy, th resulting disturbance at anypoint ni the external air, or In th car-passage, is th simple sum(ni the extendeJ gcomotncal scuse) of what would be caused bycach source acd~g-separately. Lot us consider the disturbancoduc to a simultancous sounding of a note and any or ail of its]iarmouis. By durmition, th eompiexwholo formsa note havingt)ic same pcriod (and thcrefore pitch) as its gravcst element. Wc0Iiavoat present no criterion hywhichth two can bc distmguishcdor thc prsenceof th highcr harmonies recognised. And'yetinth case, at any rate, where th componcnt sounds have ail inde-pendent origin-it is usually not difncult to detect them hy thcar, so as to cnect an analysis of the mixture. This is as much asto say tliat a strictly periodic vibration may give risc to a sensa-tion which is not simple, but susceptible offurthcranalysis Inpoint of fact, it Ims lon~ been hnowu to musicians that undercertain circumstancusthe harmonies cf a note may Leheard alongw.t!t it, uven w!~n thc note is due to a single source, such as avibrato strier, but tl.e sig.lincancc of th fact was not undcr-stood. Since attention ]~asbccn

22.] NOTES AND TONES. 13

22. That kind of note which th car cannot furthcr resolve isc:i))ed hvHehnhoitz in Ccrmn.na ')!o?t.' Tyndall and other recentwritcrs on Acousticshave adoptcd 'tone' as an Enghsh quivalent,a practice which will bc followed in th prsent work. Thcthing is so important, that a. convenient word is almost a matterof nccessity. ~

14 INTRODUCTION, f'23.

have hardly any choice but to regard it as dpendent ccc~?'~~a.rt'&M~on the magnitude of th vibrations concerned.

24;. Wu Luve seoi that a musical note, as such, is due to avibration which is necessarily pcriodic but thc converse, it isevident, cannot be truc without limitation. A periodic repetitioMof a noise at intervals of a secondfor instance, tlie ticking of fi.clock-would not result in a musical note, be th repetition everso perfect. In such a case we may say tliat th fundamentai tonelies outside the.limits of hcaring, and although some of thharmonie overtoues would fall within them, thse would not ~iveriso to a musical note or ovcn to a chord, but to a noisy mass ofsound likc that produced by striking simultaneousiy tbe twelvenotes of thc chromatic scale. The experiment may be jnadc wititth Siren by distributing tho holes quite irregularly round thecircumferenco of a circle, and turning tho dise with a moJcratovelocity. By tho construction of tho instrument, everything re-curs after each complote revolution,

25. The principal remaining dimculty in tlie theory of notesand tones, is to explain why notes are sometimes analysed by thcear into toncs, and sometimes not. If a note is reallv comulcxwhy is not the fact immediately and certainly perccived,and t)tecomponontsdisentang!ed? The feebleness of th harmonie over-tones is not th reason, for, as ~veshall sec at a later staf-c of ourinquiry, titcy are often of surprisiug loudness,an(.1play a promiucnttpart in music. On th other hand, if a note is sometimesperccivedas a wholo,why does not this happen always? Thse questionshve been carefully considered by Hcimboitz', with a tolcrabiysatisfactory result. The difHculty,such as it is, is not peculiar toAcoustics,but may be paralleled in tlie cognate science of Pitysio-logicalOptics.

Th knowledgo of external things which wo derivo from thindications of our sensos,is for th most part thc result of inference.When an object is beforc us, certain nerves in our rctin arcexcited, and certain sensations arc produced, which wo areaccustomcdto associatewith th objcct, and we forthwith infer itspresence. In the case of an unknown object th process is muchthe samc. We interpret th sensations to which we arc subjcct soas to form a pretty good idea of their exciting cause. From thsliglitly dincrcnt perspective views reccived by titc two cycs weinfer, oftcn by a liglily claboratc process, th actual relief and

~

25.] ANALYSIS 0F NOTES. 15

distance of th object, to which we might otherwise have had no~np. Thcse inferences are madc with extrmerapidity a.~dquiteUitCunsciousiy. Tbu 'it~l& life ui' baciiono of us is a continuedlusson in intcrpreting tho signa presented to us, and in drawingconclusionsas to the actualitics outside. Oulyso far as we succeedin doing tins, arc our sensations of any use to us in th ordinaryaffairsof hfe. TI)is being so, it is no wonderthat the study of oursensationsthemselvesfalls into th background,andthat subjectivephenomena, as they are called, becomc exceedingly difficult ofobservation. As an instance of this, it is suNdeiil to mention the.'blifid spot' on thc retina, which might a ~'K))-~ have beenexpectcd to manifest itself as a conspicuousphenomenon, thoughas a fact prohahly-not one person in a hundredmillion would nndit out for themselvcs. Th application of these i-emar'ksto thcquestion in hand is tolerably obvious. In tho daily use of our earsour object is to disentangle from the wholemass of sound thatmay rfach us, thc parts c&mlngfrom sourceswhich may interestus at th moment. 'Whcnwelisten to th conversationof a friend,wc fix our attention on th sound procecding from him andcndcavour to grasp that as a whole, while wc ignore, as far aspossible, any other sounds, regarding them as an interruption.Therc arc usually sufilcient indications to assist us in making thispartial analysis. Whcn a man spcaks, th whoJe sound of hisvoice rises and falls together, and wc have no dirnculty in recog-nlsiug its uoity. It would bc no avantage, but on thc eontrarya grcat source of confusion,if we werc to carrythe analysis furthcr,and rsolve thc whole mass of sound prsent into its componenttones. A] though, as regards sensation, a resolution into toncsmight be expectcd, tho necessities of our positionand th practicoof our lives lead us to stop tho analysis at thc point, beyondwhich it would ccase to bc of service in deciphering our sensa-tions, considcrcd as sigus of extcrnal objccts\

But it may sometimes liappcn. that however much wc maywish to formajudgment, th materials for doing so arc absolutelywanting. When a note and its octave are sounding close togetherand with perfect uniformity, there is nothing in our sensations tocnahic us.to distinguish, whctiicr th notes have a double or asingle origin. In thc mixture stop of tlie organ,the pressing downof each keyadmits th wind to a group ofpipes,giving a note and

Matprohubtytho poworof nttondingto thoinipt-tantnndignoringthonnimportantpartofourseusationaisto ftgreatoxtontiuhcntodtQhowgreatairoxtontwoshftt)perha.paBovcrknow.

1C INTRODUCTION. [25.its first three or four harmonies. Tho pipes of each group aiwayssound together, and th result is usually pei-ceived as a singlenotfj n!though .h~'H

27.] PENDULOUS VIBRATIONS.

I

17

weare led to inquirc what is thc physical characteristic of tones,to winch they owetheir pecuHarity? What sort of periodic vibra-tton it, whicii prod~ces a. simple tone ? According to wha.tmatl)cmatical function of t)ic time does tlie pressure vary inth passage of thc car ? No cluestion in Acoustics can be moreimportant.

The simpicst periodicfunctions with which mathcmaticians arcacquainted are the circular functions, expressed by a sine orcosine; indecd t!)cre are no otJiers at aU approaclung them ia.simphcity. TIiey may bc of any penod, aud a

CHAPTER II.

IIARMONIC MOTIONS.

28. TllE vibrations expressed by a circular function of thetime and variously designated as simple,~w~t~M~OM~or /mr?)M)n'c,are so important in Acoustics thatwc cannot dobotter thaii (levotea cha.pter tu thcir consideration, Leforecntcring on tlic dynamicalpart of our subject. Thc quantity, whose variation constitutcsth 'vibration,' ma-ybc tlie displacement of a particle mcasuredin a given direction, th pressure at a fixed point in a iluidmdium, and Buon. In any case denoting it by M,wo have

in which a dnotes tho MHp~

38. j COMPOSITION. 19

so tliat if K'=~, ~fvanishcs. In tliis case th vibrations arc oftens:ud to t'~cr/b-e, but the expression is rather misleading. Twosounds may vcry propeny bc said to interfre, whenthcytogethcrcause silence; but th mere superposition of two vibrations(whcthcr rest is the consequence,or not) cannot properly bc socalled. At Icast if tbis bc Iiitei-furence,it is difficult to say whatnon-intcrforenco can bc. It will appcar in th course of thiswork that whcn vibrations exccetl a, certain intensity tucy nolonger compound by more addition;

20 HARMONICMOTIONS [29.

whcre called the wavc-!cngt]i, is written in place of r, bothquantities dcnoting tho range of tlic indcpendcnt varia.biccorre-sponding to a complte rcurrence of thc fonction. The harmoniecurvc is tlius thc locus of a, point subject at once to a uni-form motion, and to a ha-rmonic vibration in a perpcndicuta.rdirection. In th next chapter we shall sec tha.t the vibrationofn. tuning fork is simple harmonie; so that if an excited tuningfork is movcd with uniform velocity parallcl to th lino of itshandio, fLtracing point attached to th end of onc of its prongsdcscribesa harmonie curve, which ma.ybc obtained in a permanentfonn by allowing the tracing point to bcar gently on a piece ofsmokcdpaper. In Fig. 2 the continuons linos arc two harmoniecurves of thc same wavc-lcngth a,ndamplitude, but of diSercnt

phases th dotted curve represents haf thcir rcsu~tant, bcing

30. J OF NEARLY EQUAL PERIOD. 21 1.

One case of the compositionof harmonie vibrations of dinereutperiods is worth special discussion, na.me!y,when the dinerencoci' the periods is small. Ii' we nx our attention on the courseof thiugs during an interval of time including mcrcly a fcwporiods, wc sec that the two vibrations are nearly t!ie same asif their periods were absolutely equa!, in whic]t case they would,as wcknow, bc cquiva!cnt to another simple harmonie vibration01 tho samc poriod. For a fcw periods thcu tho rsultantmution is approximatcly simple harmonie, but tho samc har-moniewill not continue to rcprescnt it for long. Th vibrationhaving th stiorter period continuaDy gains on its icilowthm'cby altering th dittcrcncc of phase on which th lmentsof th rsultant dpend. For simplicity of statement let ussuppose that tho two components Iiave oqual amplitudes, fre-quencies rcpresentcd by ??~and ?!, wlicre ??t?!. is small, andthat when first obsorvod their pitases agre. At this momentthuir cn'ccts conspire,and th rsultant ha.san amplitude doubleof that of the components. But after a time 12 (M~) thcvibration ?~ will hve gaincd ha)f a period rclatively to thothcr; and thc two, boing now in comptete disagreemcnt, ncu-trahze cach other. After a furtiicr intcrval of time equal tothat abuve named, Mtwill hve gained altogether a who!evibra-tion, and complte aceordanccis once more rc-establishod. T!)ersultant motion is thcrcfore approximately simple harmonie,wiLhan amplitude not constant, but varying from zero to twiccthat of thc componcuts, thc frcqnency of thse altrations beingM-M. If two tuniug f

2222 HARMONIC MOTIONS. [30.

pipes adjust thcmselvcs to complete opposition, so tliat at alittle distance nothing can be heard, except th hissing of thcwind. If by a rigid w:dt bctwecn th two pip~s one souudcould bc eut off,th othcr would bc Instautly restored. Or tbobalance, on which silence dpends, may bc upscb by connectingth car with a tube, whose other end lies close to tlie mouth ofeue of the pipes.

By meaus of bats two notes may be tuned to unison withgrt cxactncss. Tlie object is to make th bats as slow aspossible, siuce th numbor of be~ts in a second is oqual to thdiScrcnce of t]te frcqucnei.os of thc notes. Under favourablecircumstanccs bats so slow a.3 onc in 30 seconds mn,y be re-cognised, and would indica.te th~t th highcr note gains onlytwo vibrations a ?~M:M

31.] FOURIER'STHEOREM. 23

31. Anothcr case of grt importance is the composition ofvibrations correspondingto a tone and its harmonies. It is knownthat thc most gericml single-valued nuito periodic function canbc expressedby a sriesof simple harmonics-

a theorem usually quotcd as Fourier's. Analytical proofswill befouud in Todhuuter's J~~e~ra~Calculus aud Thomsou and Tait's~~M)Y~r/~7oso~/ty and a line of argument almost if not quiteamounting to a dmonstration will bo given later in this work.A fcwremarks arc ail tliat will bo required bore.

Fourier's thoorem is not obvious. A vague notion is not un-common that tlie innnitudc of arbitrary constants in tho sriesof necessity endows it witli the capacity of ropresenting an arbi-trary pcriodic function. Tha,t tbis is an error will be apparent,wlicn it is observed tliat the samoargument would apply equally,if one term of tbe serieswere omitted in which case th expan-sion would not in general be possible.

Another point worth notice is that simple harmonies are notthc orily functions, in a series of which it is possible to expandone arbitrarily given. Instead of the simple elementary tcrm

formed by adding a similar one in th samc phase of half theamplitude and period. It is vident that thse terms wouldserve as wcUas tlie others for

a~t?!

so that eacli term in Fourier's sories, and thereforc the sum oftho sries, can be expressed by means of the double elementary

24 HARMONIC MOTIONS ~31.

';,t.t:r.v.

terms now auggcstcd. This is mentioncd hero, becausc students,not, b~in? nc't"aintcd wit~' ~thf) expansions, m~y imagine that

simpic h~rmoniofunctions arc 1bynature tiio only oncs (tu:;Jitiedto bo thc clements in t!ic dcvclopmont of a periodic function.

Thc rcason of th prccmincnt iinport.a.nceof youncr's scries in

Acoustics is thc mccha.uic:Uonc rcfcrrcd to in thc procedingch~pter, and to bc cxp~incd more fuHy))cre:U'tcr,namciy, th:).t,in guncrfd,simple harmonievibrations are th oniy kind titat arc

propagatcd through a vibrating systcm without sun'eringdecom-

position.

32. As in other cases of a similar character, c.g. Tay~or'sthcorcm, if th possibility of thc expansion be known,th co-

cfncicnts maybc determined by a. comparativcty simpio process.\Vc may writc (1) of 31

Multip)ying by ces or sinand Intcgratmg over

;).complte period from

33.] IN PERPENDICULARDIRECTIONS. 25

33. Another ~MS of compotin(led vibrations, intcrestingfrom

tlic facility with which they Icnd themsel-~csto optical observa-

bion, ~cur wt~i harino~c vib~doi~U."

r~-ticlc arc exccutcd ~e;~e;~tCM~r ~rcc~ons,

more cspecialty

whcn th pcriods are not oniy commensur~bic,but in the ratio

oi' two SM~tM whoc uumbcrs. Th motiorL is thcn complter

pcriudic, with pcnod not manytimcs grcatcr th~u tliosc cf th

co.nponents, and thc curve dcscribcdis re-cutrant. If M and v

ho thc co-oi-dmatcs, wc may takc

reprcscnting in gnral an ellipse,whose positionand dimensions

dpend upon tlie amplitudes of th original vibrationsand upon

tlie dincrcncc of thcir ph~es. If th phases ~er by a quarter

poriod, co3=0, and th quation becomes,

In this CMCthe axes of th ellipse coinci'dc with't~osc of

co-ordinatcs. If furthcr th two cfjmpMieutsha.vcecju~!ampli-tudes, th locus (JcgenGmtcsinto thc cirete..

which is described with uniform velocity. This shows how a

uniform circuiM' motion may bo analyscd into two rcctilmca.r

hn-monicmotions,whosc directionsarc pori~enJicula.r.

If thc phases of thc components agre, E=0, and the cl!Ipsc

dc"'cncrates iuto the coiticident stmight liucs "r"j

When the unison of the two vibrations is exact, thc cUiptic

path remains pcrfeetly stcady, but in practicc it willahnnst

:dways happcn th:i.t there is n sli~it ditTo-cnecbctwocuthc

periods. TI~oconsequeuccie timt though a f~xcJeHipsercprcscnts

IIARMONIC MOTIONS.26 [33.thc curve described with sufHcient accuracy for a fcw perlods,tho ellipse! itsc)f iyradually cttangos in -jon'cspondencewith t)io'cr,noB m t,hujua~uiLudoof e. It becomcs thcroiorc a matterof interest to cojisider thc system of ellipses rcprescntcd by (2),supposing a and b constants, but j variable.

SInec tho extreme values of u and are i a, t b respcctivcly,thc cHipse is iti all cases insct-ibcd in thc rectangle whose sidcsarc 2(,26. Sterling with tlio pitascs in agrcemcnt, or 6=0, wchavo tlic cHipsc concident with tlie dia"'ona.l

= 0 Asemcrcascs fromC to ~-n-,thc ellipse opcns out until its equationLeon)es

From tins point it closesup ngam,ultimutely comcidingwith thcother

diagonal += 0, eon-cspondingto thc incrcMcofefrom ~Tr

to 7r. Aftcr t!iis, as e mngcs from vr to 2~ th dHpsc retracesIts course untU it again coincidcs with t!ie first diagonal. TIiosequoice of changes is exhihitcd in Fig. 3.

Thc ellipse, having a,lrca.dyfour given tangents, is compictclydctcrmiucd by its point of contact P (Fig. 4) with thc linc ~=&.

33.] ] LISSAJOTJS'CYLINDER. 27

In order to connect this with e, it is Ruflcient to observe tha.t

when ~=6. cos27r?:

28 HARMONIC MOTIONS. [34.

with uniform velocity, which insurcs a harmonie motion for .P,wc obtain a complte rcprcsuutation of thc varying orbitdcscribcd by thc point wh~n Lhcperiods uf thc two compunentsdiffer slightiy, eacli complte revolution answoring to a gain orloss of a single vibration'. Th rvolutions of th cyliuder arcthus synchrouous vlt)i th bats which woutd rcsult f)'om thccompositionof thc two vibru-tious,if they wcrc to act in thc s.uucdirection.

35. Vibrations of thc Mnd hre considercd arc very easilyrcn.Hxt'dexpcrimeutn.))y. A Ii(j:Lvypondulum-bob, hung from aiixud point by a long wirc or string, descrihes cliipscHundcr t))caction of gravity, which may in particular cases, according to thcircumstunc'e.sof projection, pass into straight lincs or circles.But in order to sec th orhits to thc best advantagc, it is necessarythat thcy sliould be described so quic)dy tl~at th 'Itnprcssio!iou th retina madc by th moving point at any part of its coursebas not time tofade materially, heforc tl)epoint cornes round againto its action. This condition is fulfilled by th vibrationof a silvered bead (giving by reflection a luminous point), winch isatt~ched to a straight mctaUic wire (such as a knitting-necdie),firmiyclamped in a vice at the lower end. When tiie systemis setinto vibration, the luminous point dcseribcs ellipses,whichappearas fine lines of light. Thse ellipses wouldgradually contract indimensions under th influence of friction until t!iey subsidcdinto a stationary bright point, without undergoing any othcrchange, wcre it not that in ail probability, owing to somc wantof symmetry, the wire lias s)ightly ditiering puriods accordingtothc plane in which th vibration is cxceutcd. Undcr thse cir-cumstances th orbit is sceu to undcrgo t!io cycle of changesalready cxplaincd.

3G. So far we Itavc supposcd tho periods of th componentvibrations to be equal, or nearly cqual; thc next case in ordcr ofsitnpiicity is when one is the double of tho othcr. Wc have

M=acos(4~7!-

3G.] CONSONANTINTERVALS. 29

which for ~1 values of e reprsenta a curvc inscribed in the rect-

angle 2ct,2&. If e = 0, or 7r,woha.vc

reprsenta p~bolas. FIg. 7shcws thc various curves for U~c

iutcrvals of tLc octave, twuifth, aud fth.

To aU thse systems Lissajous' method of represontation by

th transparent cylinder is applicable, and whcnthe relative

phase is altcrcd, whctber from th diffrentcirc~mstanccs of

projection in diiferent cases, or continuously owingto a sbght d-

viatior. from exMtness in tho ratio of tbe poriods, th cylinder will

app~r to turn, so as to prsent to the eyedigrent aspects of th

sa.DiOline traced on its surface.

37. There is no dinicutty in arranging a vibrating systemso

that th motion ofa point shall consist of twoharmonie vibrations

in perpendicular planes, with their periodsin any assigued ratio.

The simplest is that known as Blackhurn's pendnlum.A wire

~tC-Bis fastcncd at ~1and two nxcd points at tbe samc Icvel.

Tbe bob P is attached to its middle point by another wirc CP.

For vibrations in th plane ofthe diagram, thc point of suspensioniHpractically C,provided that th wires are sunIcicQtIy

stretched

30 IIARMONIC MOTIONS. J37.

but for a motion porpendiculin-to this plane, th bob turns aboutD, can-ying tho wire ~O'j9 witli it. TIic pc.ri~s of vibration in

thc principal planes arc in the ratio of t!.c square roots of CPandDP. Thus if ~C=36'~ the bob describc.s th figures of thcoctave. To obtain tito squence of curvc.s correspondin~ to~pproxnnatc unison, Y~ must bc so ncarly tight, tiiat isrdativeJy small.

3S. Another contriv~ncocalled thc kalcidophonc was origin-ally invented by Whcatstoiie. Astraight tllin bar of steel carry'i~a bcad at its uppcr crid is fastcncd in vice, as cxpMncd in aprevious p~ragraph. If the section of th bar is square,or circule-th poriod of vibration is indepeudcnt of thc plane in which it ispcrformcd. But let us suppose that the section is a rectalewith unequal sidcs. Tlie stress of tl.c bar-tho force withwhich it rcsists Lcndin~-is thcu grcater in t!te plane of mc.aterHuc~nc.ss,aud tlie vibrations in this phuie have th shortcr pcriodBy a suitable adjustmcnt of tho thickncsses, the two poriods ofvibration may bc brought into any required ratio, aud th eor-responding curve cx]iibitd.

Thc defeet in this arrangement is that thc samc bar will r.Ivconly one set of figures. In ordcr to ovurcome tins objectionth fullowlng modification lias bccn deviscd. A slip of steci istakcn whosc rectangular section is very ciongated, so tliat asregards bcnding in onc plane the stiHhcssis so gr~t as to amountpractically to rigidity. Thc bar is divided into two parts, and the

38.] OPTIOAL METIIODS. 31

broken ends reunited, the two pices bcing turned oa one anotherthroush a rigtit angle, so that tho plane, which contains th smallLitickucfiMoi'ojf:, ~

33 UARMONICMOTIONS, [40.But it often happonsthat the application of this mcthod would

ho dimcult or inconvnient. Jn such cases we may substituts forthc uniform n'

41.] INTERMITTENT ILLUMINATION. 33

verified; though for this latter purpose a more thoroughlyacoustical mthodeto be described in a future chapter, may beprcfcncd.

42. Another method of examining thc motion of a vibratingbody dpendsupon thc use of intermittent illumination. Suppose,for exampic, that by mcans of suitable apparatus a series ofcleetric sparks are obtained at regnfar intcrvals T. A vibratingbody,whoseperiod is also T, cxamined by thc light of thc sparksmust appear at l'est, because it can be sccn only in one position.If, Itowcvcr, th period of th vibration differ from T cvcr solittle, the iHuminatcd position varies, and the body will appearto vibrato slowly~ith a frequcncy which is thc diffcrcncc of thatof the spark and tliat of the body. Th type of vibration canthon be observed with facility.

The sries of sparks can bc obtained from an Induction-coihwhose primnry circuit is periodicauybroken by a vibrating fork,or by somcothcr intcrruptcr of snrRcientregularity. But a bette)'rcsult is afforcledby sunlight rendered intermittent with tlie aid ofa fork,whosc prongs carry two small plates of meta], parallel tothe plane of vibration and close togethcr. In each plate is a slitpM'aIIclto thc prongs of th fork, and so placed as to aAbrd afj'ccpassagethroug)i th plates whcn th fork is at rcst, or passingthrough th middte point of its vibrations. On th opening soformed,a beam ofsunHght is concentrated by meansof a burning-glass,and thc object undcr examination is placed in th cne ofrays diverging on thc furthcr sidc'. When tlic fork is made tovibratoby an cicetro-magnetic arrangement, thc illumination is eutoff exccptwhen the fork is passing through Us position of equi-librium,or nearly so. The nashcs of light obtained by this methodarc not so instn.nta.nouusas clectric sparks (especially when ajar is connected with thc sccondarywire of th coil), but in myexprience th rcguhu'ity is more perfect. Carc shoultl bc takcnto eut on' extrancous ]ight as far as possible, and thu cnect is thonvery striking.

A similar result may bc arrived at by looking at th vibratinghodythrough a sries of holes arranged in a circlc on a-revolving(tisc. Several sries of holes ma.y be providcd on the same

HARMONIC MOTIONS. [43.34

Except with respect to the sharpness of definition, the result isthe samf when the pcriod of th light is any multiple of tt~t of.th vibmtin~ ~c'y. Tiiis pouit. ~HHt bu att,ciided tu ~i)eu threvolving wheel is used to determine an unknown frequency..

When the frequency of intermittence is an exact multiple ofthat of th vibration, t!te object is seen without apparent motion,but generally in more than one position. Titis condition of thingsis sometimes advautageous.

Similar effects arisc when th frcquencies of th vibrationsand of th flashesare in th ratio of two smaUwholenumbers. If,for example, th number of vibrations in a given time be halfas grt again as the number of flashes, th body will appearstationary, and in general double.

CHAPTER Iir.

SYSTEMSIIAVINGONEDEGREE0F FREEDOM.

43. THEmatcrial systems, with whosc vibrations Acoustics isconcerned, are usually of considrable complication, and are sus-ccptible of very varions modes of vibration, any or a!l of whichmay cocxist at any particular moment. Indeed in some of thmost important musical instruments, aa strings and organ-pipes,th number of independent modes is theoretically infinite, andthe consideration of several of tliem is essential to the most prac-tical questions relating to the nature of tho consonant chords.Cases, however, often present thcmselvcs, in which one mode isof paramount importance and cvcn if this were not so, it wouldstill be proper to commenceth consideration of thc general pro-blem with th simplest case-that of one degrce of frcedom. Itneed not be supposed that th mode treated of is th only onepossible,because so long as vibrations of other modes do not occurtheir possibility under other circumstances is of no moment.

44, TIte condition of a system possessing one degree of frec-dom is denncd by th value of a single co-ordinate M,whoseoriginmay be taken to correspond to thc position of cquilibrium. TIieMnetic and potential nergies ofthc system for any given positionarc proportional respectively to and

r=~~ F=~(i),whcre w and are in general functionsof M. But if we Hmit our-selves to tlie consideration of positions M!.the ?'y~:e~'

ONE DEGREEOF FREEDOM. [4~.36

now proceed. If there he no forces,cither rcaulting frominternaifriction or viscosity,or imprcss'~don the systcm from without, the\vholeenergy remains constant. Thus

y+ 1~=constant.

Substituting for T and Vtheir values, and differentiating with

respect to tho time, wc obtain tlie e~ua-tionofmotion

~m + /tW= 0 (2)of which th complte integral is

~=(tcos(?)< a) (3),

whcrc ?~=/7):, rcprcscnti))~ a ~Muc vibration. It will boscc that thc pcriod alone is detemuned by th nature of thesystem itself; the amplitude and phnse dpend on cothttcral cir-cumstances. If tlie difrercutial equation wcrc exact, that is to

say, if T werc strictly proportional to and F to thon, without

any restriction, th vibrations of th system ahont its conDgurationof equilibrium would bc accuratc)yharmonie. But in th majorityof cases tlic propoi'tionaHtyis only approximate, dcpending on an

assumption that tlie displacemeut ?

44.] DISSIPATIVEFORCES. 37

u. th force of th quivalent spring. Thus an augmentation of

mass, or a rc!f).xationof spring, incrcas

38 ONEDEGREE0F FREEDOM. [45.If thc friction be so grt that > thc solution changes itsfonn, and no lorger f'orrRsp.ndsto nn os

3946.] FORCEDVIBRATIONS.

to ~thc magnitude of tlie force, and the period is the sameas that ofthe force.

Let us now supp

ONE DEGREE 0F FREEDOM.40 [46.If p bc Jcss tha.n ?;,the rctardation of phase relatively to thoforcelies betwech xeruand a qu:u-te!-pcriod, aud whcn is~reater

tit:m}.[.,butwcchi(.~U!u'~t'~(.'i:m!n.i,.bntfuut~d.In t!)c cuscof a systcln devoidof i'riction, tlie solution is

When is amaller ttian ?~thc pl.ase uf tiiu vibration agres withtliat of thc force,but whcn~ Is th grever, the sign of th vibra-tion is clianged. Th change of phase from complte agreementto complote disagrcemeut, which is graduai wlien friction acts,hre take~ place abruptty as pa.ssesthrough t!ic value 7t. At thsamc tune thc expressionfor th amplitude bccomes inanit. Ofcourse this oniymeans that, iu thc case of cqual periods, friction7~

46.] PRINCIPLE0F SUPERPOSITION. 41

If a force of nearly equal period with th free vibrationsvnry s1o\v1yto a maximum and then slowly dcrte, thc dis-pjacement docs not rcach its maximum untd aftcr th forceliasbcgun to diminish. Under thc opration of the force at itsmaximum, thc vibration continues to increasountil a certain limitis approachcd,and this incrcase continues for a time cven att))ouglttlie force, having passed its maximum, begin.sto diminis)). Outtds principic tlie t'utardation of spring tidcs bchiud tlie da.ysofucw and full luooulias bccn cxp]ained'.1.

47. From tlie linearity of the cquations it follows that themotion rcsulting from thc simuItanGOusaction of any numbcr offorces is thc simple sum of tlie motions duc to the forcesta~enscparate!y. Each force c:uises tlie vibration proper to itself,wthout regard to tlie presoicc or absence of any othos. Thcpeculia-ritiesof a force arc thus in a manner transmitted into thmotion of tho system. For example, if thc force be periodic intimc T, so will be th resulting vibraLion. Each ])armonic ele-ment of tlie forcewill call forth a correspondingharmonie vibrationin tl system. But since tlie rctardation of phase e, and the ratioof amplitudes M is not the samc for th different components,the resulting vibration, though periodic in th same time, is dif-frent in c/t

ONEDEGREEOFFREHDOM.42 [48.important, and must bu olearly understood. Thc pcrioJ of t))cformer is detenniucd solcly by thc force whicli is supposed to act.u Umh~ lic-ni~rdiQut:,~hHu Lii:t ut' thc htttcr dpends un)yon the constitution ofthe system itself. Anothcr point of din'er-cnce is that so long as the extcrnal influence continues to opcratc,a forced vibration is permanent, being rcpresentcd strictly by aharmnic function; buta frec vibration graduallydies away, be-coming ncghgibic aftcr a timo. Suppose, fur cxample, that thesystcniis :),trcst when thc force 7~cos~j{bcgins to operate. Su.chrinitc vaincs must bc givcn to th constants jd and a iti (1) of 47,that buth and ii arc initiatty zro. At first tllen tiiere is a ffrec vibration not less important than its rival, but after a timefriction rednces it to insignificanee,and the forced vibration is leftill complte possessionof the nc!d. Tins condition of things willcontinueso longas the force oprtes. Wlien thc force is removed,thcrc is, of course, no discontimuty in the valucs of Mor !

40.] VARIOUS DEGREES OF DAMPING. 43

2If = ~T,wc havea;= log la a, system subject to on)yaxTnnjucru-Lu du~reu ut' dampmg, \vc tua.y tn.kc upprox.UTmtcly,

Thi.sgives thc number of vibrations which arc performed,heforcthea.)nplituduf!iih)to0.Thc inuucncc of damping is aiso powcrfu~y Mt in a, forccd

'Ibra.tion, wlicu thcre is a. :uear approach to isochronism. In thecase ci' an exact equality betwcen a.nd?~it is thc damping alonewitich prcvcnts thc motion becommgm~nite. We might casilyauticipate thatwheu tUc damping is small, a. compara.tivelyslightdcvia-tionfrom perfect isoein'onismwuld cause a large fa.Hmgoffin thc Magnitude of thc vibration, but that with a larger damping,thc s:uuc precisionof adjustmcut would not bc rcquired. Fromtlie enuatious

so that if bc sm~l!, must bc verynearly equa.1to 7)jlu.ordcr toproducc a,motion ]iot grea.tly Icssthan th maximum.

Th two principal eScets of damping may be compared byclimijiating betwecu (1) and (2). Th result is

where th sign of tlie square root must be so cliosenas to makethe right-hand sidcngative.

If, when a systemvibrtes frcely,tlie ampUtude be reduced inthe ratio after x vibrations then, wben it is acted on by a force(p), thc energy of the resulting motion will bc less than in thcaseof perfect isochronism in the ratio T T~. It is a mattcr ofmdiifcreucc whcthcr th forced or tlie free vibration bc th higher;all dpends on theM/erua~.

In most casesof interest thc intcrval is small; and then, puttingp = ~+8~ tlie formula may be written,

44 ONEDEGREE0F FREEDOM. ['49.The following table c~culatcd from thcso formulte haa been

givcn by Hcimlioltx'

Ijttcrvfd con-cspon.Ung to a rduction v'Lmtif.nM nfter whiuh thoof Uto ruM.ttauco to ouc-touth. i'itcnHity of a frco vibrntiou is ro-

y y ~Q ducudtoono.tunth.

~=A.

tonc. ~.oo'~

19'00

9-50

? G-33Whuif! tonc. 4-75tuno. 3.~0

y tono = minor thu-d. g. jy7 toile. 2-71

Twu whuit' tonea~ major third. ~'37

Formula (4) shcws that, w!ien i.s small, it varies c~~M'1. asMtt)~~ as a:

50. From observations of forced vibrations due to knownforces, tlie natural period and dampiug of a system may Ledeter-tniGd. TIio formuhuare

~ifH~/ntJf~fyc~ p. 221.

If tlie equilibrium thcory Le known, tlie comparisonof ampli-tudes tells us tlie value of sav

4550.]] STRINGWITH LOAD.

and e is also kuown, whence

51. As bas been ah-cadystated, the distinction of forccd andfrcc vibra.tionsis important but it may be remarked that most ofth forced vibrations which we shall Laveto consider as affectinga system, take tbcir origin ultimately in the motion of a second

system, which influences thc first, and is innuenccd by it. Avibration may thus have to be reekoned as forced in its relationto a system whose limits are fixed arbitrarity, cvcn when that

pystem lias a share in dctcrmining thc period of the force whichacts upon it. On a wider view of thc matter embracing both th

Systems,thc vibration in.question will be recognizcdas free. An

example ma.yma~c tliis clca.rer. A tuning-fork vibrating in airis part of a compound system including th air and itself, andin respect of this compound system the vibration is free. But

although thc fork is influenccd by thc raction of th air, yet thcamount of such innuence is smaU. For practical purposes it isconvenieat to eonsidcr the motion of the fork as givcn, and Lhatofth air as forced. No crror will be committed if th f

4G ONE DEGREE0F FREEDOM. [52.

position the tension is ~reatcr but we limit ourscivcs to the caseof vibrations so small that tt~eadditiona! strotching is a ncgJigibJefraction of the who)c. On th~ (~ncHii~n thc ~

53.J METHOD 0F DIMENSIONS. 47

rfrence to thc three fundamental unitsof length, time, andmnssure a, and T. Let th solution of the problem bownLLeu-

This equation must rcta.in its form unchanged, whatever mayl)ethe fundamcntal units by means of which th four quantitiesarc nnmerically expressed, as is vident, when. it is consideredthat in deriving it no assumptions wouldbe made as to th mag-nitudes of those units. New of all tlie quantities on which fdpends, Tis the oniy onc involving time and since its dimen-sions arc (Mass)(Length) (Ti.me)' it followstl~at whe!i ? and ~farc constant, oc.T' otherwise a change in th unit of timewould necessarilydisturb the equation (3). Tins being admittcd,it is ca~y to see that in order that (3) may be independent of theunit of Icngth,wemust Imver ceT"~ n~,when Is constant andfinally, in order to secure indcpcndence of the unit of mass,

Therc must be no mistake as to what this argument doesanddocs not prove. Wc Iiave nMKMte~that thcrc is a deHnitcperiodic time dcpeuding on no other quantities, having climen-sions in spacc, time, and mass, t!ia.nthosc aLovemcntioncd. Forexample, we hve not proved that r is indpendant of thc ampli-tude of vibration. That, so far as it is truc at ail, is a cons-quence cf thc linearity ofthe approximate dinercntial quation.

From the neccssity of a complte cnumeration of all thequantifies on which th required rcsult may dpend, thc methodof dimensions is somewhat dangerous but when used with properCt~reit is unqucstionably ofgreat powerand value.

ONE L.MCREE 0F FREHDOM. F~..'3 1 lie solution of th prsent problemmight bo made thcfoun~tion of a ,nethcd for Lheabsolute n~asurerncnt of pitch.pnncip~J impedunc-nt to accuracy would prubabjy.be t!todifBculty oi u~ku~ suf!ictcutfyi~ iu relation to th m~ oftlie ~u.c, without at tlie samc timc lo~crin~ th note toomuch inth musical scalc.

T)ic wirc may bo strctc)ied by wcight ~t~chcd te itsfur hcr en

54.] COMPARISON WITH UNIFORM STRING. 49

mated with a.variable velocity, increasing from either end towardsthc centre. If we attribute to th whole mass tho vclocity of thocentre, it la vident that th kinetic cnergy will bG cousidcrab!yovcr-estimated. Again, at the moment ci' maximum excursion,th uniform string i.smore stretched than its suhstitutc, winchfoltows thc straight courses ~1~ and accordingly the poteu-tial cncrgy is dumnished y t)to substitution. TIiGconcentration"i th mass at the middie point at once increascs tho kincticcncrgy whcn a;= 0, and decreascs ttte potential energy when ~-= 0,and thercforc, accordingto the principle explained m 44, prolon'~the pcriodic timo. For a string thon the period is less than thatcatcuiatud from the formula of the last section, on the suppositionthat ~1/dnotes th mass of the string. It will afterwards appeart)jat in order to obtain a correct result we should !)avc to takc in-

4 4stcadof.Von!y-~V. Of thefactor-~ hy far thc more import-TT TT

ant part, viz. is duc tu (,hedifTcrcnceof tlie kinetic nergies.

55. As another example of a Systempossessingpracticu.Hybutone dcgree of freeclom,let us considcr tlie vibration of a spring, oneend of which is clamped in a vice or otherwise held fast, wliile thcotiter carries a heavy mass.

In strictncss, this System !iko tho last liasan innnite numbcr of Indcpcndent modes of vi-bration but, whcn thc mass of t!tc spi'mg is)-e!ativc!y sn-i:d),ttiat vibration which is ncarlyindcpcndont of its inci'tla. buconics so much thmost important t!)at tho othcrs may bo ignored.Pusinng this idca, to it.s limit, we may regard thespring merety as tite origin of a force urging thattaehed mass towards th position of equilibrium,and, if a certain point be not excecded, in simpleproportion to thc disp!acement. Thc result is aharmonie vibration, with a period dpendent onth stinhess of tho spring and the mass of thetoad.

56. In consquenceof tho oscillation oi' the centre of inertia,H~ci-eis a,constant tendency towards the communicationofmotionto tlie supports, to resist which a.dequate!y th latter must bevery ni'm and massive. In ordcr to obviate this inconvenience,

R. 4

~0 ONE DEGREE0F FREEDOM. [5G.two prccisely similn-r springs and lo~ds m~y bc mountcd outlie same frame-work in a symmctrical manncr.If thc two loads pcrformvibrations of cqual amp)i-tude in such a,manner that the motionsarc a.Iwn.ysopposite,or, {mit may otherwisebc e.xprcsscd,witha phasc-tiiHcrcucc of !m]f a period, thc centre ofinertia of thc whole system rcmains at rcst, andthcro is no tendency to set thc fra.mc-workintovibra.tion. We shaU sec in a future chapter thatthis peculiar relation of phases will quiddy esta-b)ish itself, wt~tever may be tho original disturb-a.nce. In fact, any part of tho motion winch doesnot conform to the condition of Icaving thc centreof inertia unmoved is soon extinguished by damp-ing, unless indccd thc supports of tbe system arcmore than usually nrm.

57. As in our first exemple wc found a rough illustration oftho fundamental vibration of a musical string, so hre with tliespring and attachcd load wc may compare a uniform slip, or bar,of elastic material, one end of which is securejy fastencd, such forinstance as the ~:

S~J TUNJXGFORKS. 5jI

madc, thctwo parts opcned ont to form the prongs of the fork,and th whole workcd by tho iiammer and n!u into thc rcquircdshape. T)ic two prongs must bc cxactiy symmctricat with respectto a plane passing through the axis of thc liandie, in ordcr thatduring t!ie vibration th centre of incrtia may remain unmovedunmoved, tiiat is, in thc direction in which thc prongsvibrato.

Thc tuning is cnected t)ms. To make th note higher, thquivalent incrtia of thc Systemmust bc rcduccd. This is donchy nling away t)ie ends of thc prongs, cithor diminishing theirthickncss, or actuaiy sliortening thcm. On the other hand, toJowcrthe pitch, tlic substance of the prongs ncar thc bcnd maybe rcduced, the effect of which is to diminish th force of the.spring,Icaving t)te inertia pmctically unchangcd or the inertiamay be increased (a mcthod whichwould be prfrable for tcm-porary pm-poscs) by loading thc ends of thc prongs with wax, orothcr material. Large forks arc somctimusprovided with movc-able weights, which slide along thc prongs, and can be nxcd inany position by screws. As thse approach thc ends (whcro thcvetoeity is greatcst) the quivalent incrtia of thc Systemincrcascs.In this way a considrable range of pitch may bo obtained fromone fork. TJ)c number of vibrations per second for any position

ofthe weights may be markcd on th prongs.Tite relation bctwcen the pitch and thc''size of tnnin~ forks is

rcmarkably simple. In a future chapter it will be provcd thatprovided the material remains thc samc and tho shape constant'tt.c period of vibration varies, dircctty as t)te linear dimensionTIrns, if t!ic linear dimensions of a tuning fork be doubicd, itsnote falls an octave.

58. Thc note of a tuning fork is a ncarly pure tone. Imme-diateJy after a fork is struck, high tones may indccd be hcard,con-esponding

to modes of vibration,whosc nature will bc subse-qucnHy considered; but thse rapidiy die away, and cven whilc

s they exist, they do not b!cnd with th propcr tone of the forkpart~y on account of thcir very high pitch, and partly bccauseihey do not bchng to its harmonie scale. In the forks examincd

~.byHelmhoitz the first of thse overtoneshad a frequcncy from 5-8to n-Gtimcs titat of the proper tone.Tunmg forks are nowgeneraUysupplied with rsonance cases,whosc effect is greatly tu augment the volume and pnrity of the

42

53 ONEDECREEOF FREEDOM. [58.

sound, according to principles to be hcreaftcr dcve!opcd. In

oiJer to excite thon, a -viotinor ccHobow, wcll supp)icd with

')~ :~dr~t.t .Cr~~ rh< prongs'u ~'u dit'

(;0.'j SCHEIBLER'STONOMETER. 53

point at th bottom of tho sries. Th pitch of th other forkscouldbc dcduccd.

If conscutiveforks givc four bats per second, C.5in a.llwill

bc rnquirod to bridge over thc intcrval frora c' (2.')G)to c' (5L2).Un thisaceountthc mctitod is laborious; but it is probably th

most accm-atcfor tl)Coriginal dctcrmina.tion of pitch, as it Is liabtc

to no ct-rors but such as care and repetitioti will clhninatc. It

mn.ybc obscrvcd tliat thc cssctYtiatthing is tho mcasurcmcnt of

t)tc ~er~ce of frcqucncics for two notes, whosc ~o of frcqucn-cics is UKlcpcndcnt]yknown. If wocould be sure of its accm-a-cy,thc Intci-v:dof th nfth, fourt)i, or cvcn.major third, might bc suh-

stitutcd for th octave,with th advantagc of rcducmgttie numberof th ncccssary interpolations. It is proba.b!c tttat with thc aid

of optic!d mcthods t))i.s course might bc succcs.s(ut!yadoptcd, as

thc con'csponding Lissajous' ngurcs a.rc casily rccognised, and

thcit- stcadinoss is a vcry sovcrc test of t!ie accm'acywith whicb

tt'e ratio isattainud.Thc frcqnency of large tuning forks may bc detcrmincd by

aHowiugthem to trace a harmonie curve on smokudpapcr, which

tnnycouvcnicnHybc mountcd on thc circumicrenco of a rcvo)vingdrmn. Thu muubci' of wavcs cxccutcd in a second of thnu givcsthcfrcqucncy.

In many cases tbc nsc of Ittterniittcnt Hturnination duscribcd

in 4-2givcs a convcniunt )net))odof dctcrmining an nnknown

frcqucncy.

(il. A scrics of forks ranging at snndi int.crv:usoveran octave

is vcry uscf\d for th dcturtnination of thc frcqucncy of any)nusic:dnote, and is caUcd Schuibtcr's Tonomctur. It may a~obc nscd for tnuing a note tu any desirct).pitch. In cilber case

thu f')-u(Utcncyof th note is dctermincd hy tl)e nuinher of beats\vhic)i it givcs with thc i'orks, which lie aearest to it (on cach

sidu)in pitch.For tuning pianofortcs or organs,a. set of twelvc forks may be

uscdgiving thc notes of thc cbromatic sealc 0)1tho equal temp-rament, or any dcsircd system. Tbc corrcsponding notes are

adjusted to unison, and t])Cotbcrs tuned hy octaves. It is betto-,I~owevcr,to prpare th forks so as to givc four vibrations persecond k-ss than is above proposed. Eacli note is thcn tuncd

little higher than tlie corresponding fork, until thcy givc whensounded togcthor cxactiy four bats in thc second. It will be

54 C~K DEGREE 0F FREEDOM. [61.

ubservcd that tho addition (or subtraction) of a constant numberto thc frcfptencics is not the samc thing as a more displaccnicutofthescatcinabsolutcpitch.

In th ordinary practicc of tuners a' is takcn froma fork, andtlie other notes dctermiued by cstinm.tionof (Iftt)s. It will bcrcmcmbcrcd that twefve truc ~fths arc slightiy in excess of sevenuct.ivcs, so thitt on tho equal tcmpcranicut System cn.chf)ft)~is :).little fit. Thc tuner procccds upw:irds from Ly succci-isivufifths, coining down au octave aftcr about every altet'ttate stop, mordor tu reimutt in nearly the same part of the scfdo. TwcivcHfths should britig ])itn back to . If this Le not thc case, thewurk must bc ruadjustud,unt,iJ all the twe)vc ftfths arc too fhtt by,as nearly as can bcjndgcd, thu samo sma!) amount. Thu incvita-biu o-ror is thcn hnpartiaUy di.stributed, an

C2."] COMPOUND PENDTJLUM. 55

New oecos~

5G ONH DEGREE OF FRKEDOM. [62.

of a. pcriod, by winch amount th uppcr pcndulum is in a.dvancc.If the two pcriods hf vnry dHTo'cnt, thc vibrations ~either a.g)'ecor arc compictcly opposed in p!)asc, accordin~ to quations (4)and (5) of 4C.

63. A vo'y good cxa)np)c of {tfurccd vihruttcn i~iabrdcdbyn. ibrk under thc iunucnec of tui intermittent ctcctric cui'rcnt,

~hoso period is ncarly cqual to its own. ~).CZ?is the fork; 7?asma!) c)ectro-magnct,formedby winding insula.tcdwire on nn ironcorc of tho shape shcwu ni E (simila.rto titat known as 'Sioncn.s'armature'), ~nd supportcd betwccu tho prongs ofthc fork. Whcuan intermittent current i.ssent through th wire, a periodic forceacts upon the fork. This force is not cxprcssibic by a simpic cir-cular fonction; but mn-ybc cxpandcd by Fouricr'.s theorcm lu ascrics of sucli functions, ha.vlngporiods T, T, T, &c. If nny ofthcsc, of not too small amplitude, bc ncarly isochronouswith thefurk, thc latter will be canscd to vibrato othcrwisc t]tc effect isinsigninca.nt. In wbat followswcwill suppose that it is the com-plete pcrioclT whicb ncarly ngrcc.switlt tliat of the furk, and cou-scqucntly rega.rdthc sries expressing th pcriodic forceas reduccdto its first term.

lu order to obtain t))Cmaxitnum vibration, thc fork must becai'cfuHytuncdhy a small siiding pice orby w:LX',uutit its j~turatpcriod (without friction) is cfpud to that ot'th force. Dus is bcstcloueby actual trial. Witen tho desired c~uidity is approacticd,and th fork is a)!owcd to start from rc'st, thc ibrccd and com-ptctncntary frce -vibration arc of nearly cqual amplitudes a.ndfrequencics,and therefore (4-8) in thc bcginning of thc motionproduce~ef< whose stowncss is a measuro of the accm'acy of

y"rUu~j'urposc\\nxmny

G3.] RELATION 0F AMPLITUDE ANDPHASE. 57

tho adjustmcnt. It is not until a-ftertl)c froc vibration lias badtime to subside,that th motion assumes its peru.anent ch~'acter.T))Cvibrations ofa tuning fork properly constructed and inountedarc subject to very little damping; consequcntly a vcry slightdviation from perfect isoclironism occasionsa markcd falittig off

in th intcnsity of the rsonance.The nmpHtudoof thc forccd vibration can bc obsci-vedwith

suficicntaccuracyby thc car or cyc but th exprimenta! verifi-cation of thc relations pointed out by thcory bctwccn its phaseand that of th force which ca.uscsit, re

ONEDEGREE0FFREEDOM.58 [C3.

It appears tbat a. considrable altration of phase ni eitherdirection may be obtaincd without very materialty reducin"' thamplitude. Whcn one furk is vibrating at its maximum, thcothcr may be made to dinfcrfrom it on either sido by asmuc)t asCO"in phase, without lo.singmoro than t)alf its amplitudu, or by asmuch a.s-I

G4.] FODK INTEBRUPTER. 59

or m a, second, whose periodic interruption is effectedby anotherrider dipping luto mci'cury cups'.

Tho ??~(~

60 ONE DEGREE OF FREEDO~f. [G4.

taxation apoken ofrcdnccs this advance. If th phasc-diu'crcncebe rcdueed to 90",th force acts in th most favourable manner,~tU.'t.x~t p'-K-~bh.~Yibmuon.i.sptudtiL-cfJ.

It is important to notice t))at (cxccpt in thc case just, rcfcrrcdto) the actual pitch of ttie mterruptcr dKFcrsto some cxtent fromtbat natur.'d to thc fork according to th hnv cxprcsscdin (5) of 4G, e being in thc prsent case a. prescribcd pbase-difcrencodepcnding on t!)c na.turo of thc contacts :ind

C5.] RESONANCE. 611

required in th adjustment of th pcriods in order to bring outth effect. dpends un tlic degrcc of damping to winch th systcmIs subjcctcd.

Amongbodics winch resound without an extrme prcision oftuning, may be mentioned strctched membranes, and strings asso-ciated with sounding-boards, as in tho pianoforte and thc violin.\Vhcn th propcr note Is sounded in thcir neighbourhood,thcscbodies arc caused to vibrato in a very perceptible nianner. Thccxperimcnt may bo made by singing into a pianoforte tho notegiveu by any of ils ttrings, Iiaving nrst raised tlie con'cspondingdampcr. Or if onu of tbo Mtringsbeionging to any note bc plu.ckcd()ikc a Itarp string) with tlie nnger, its feHowswill be set Intovibration, as may immediatcly bc proved by stopping thc nrst.

T)tCpiienotncnon of rsonance is, howover, mo.st striking incases ~'hero n. vo'y accm'ato c([uality of pcriods is nccessary inorder to cHeit t))c full cfrcct. Of tins class tuning forks, 'muuntcdon rsonanceboxes, are a.conspicucusexample. Witen thc UMisonis perfect thc vibration of ono fork wIHbe taJ~cn up by anothcracross thc width of a room, but thc slightcst dviation of pitchis sumcicnt to l'cnder thc phenomcnon almost insensible. Forksof 25C vibrations pcr second arc commonlyused for thc purposc,and it is found that a dviation from unison giving oniy one bcatin a. second makcs ail thc dincrencc. Whcn thc forks arc '\vcUtuncd and ciose togcthcr, th vibration may be transferred back-wards and furwards bctwcen thcm scvcral times, by damping thcma!ternatc!y, with a toucit of th nngcr.

IMustrutions of tho powerfui c~ccts of isochronism must bcvitinn t))e exprience of every onc. Tticy are often of importancein very dinerent neld.sfrom any with which acoustics isconccrned.For cxample, few things are more dangerous to a ship than to liein th trough of th sea.undcr thc innucncc ofwavcs whose pcriodis ncarly that of its own natural ro)Hng.

(iG. Th solution of thc quation for frcc vibration, viz.M+/C!t+)'M =0 (1)

may be put into another form by cxprcssing tlie arbitrary con-stants of intgration J- and a in ternis of tlie initial values of !