1918 bull indian assoc cultiv sci v15 p1-158 - 2

7/27/2019 1918 Bull Indian Assoc Cultiv Sci V15 p1-158 - 2

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Section X-Construction of the velocity-diagram when thebow is applied at a node

The discussion of the vibrational modes given in sections VI to IX proceeded onthe assumption that the bow is applied at a point of irrational division of thestring an d th at the p ressure of bowing is sufficiently large in relation to its velocityto ensure that the motion at the bowed point alternates between two an d onlytwo rigorously constant velocities, once or oftener in each period of vibration.When these conditions are satisfied, the resulting mode of vibration should ingeneral include the complete series of harmonics. We shall now pass on toconside r the cases in which owing to the p osition of the bow coinciding exactlywith a p oint of rational division of the string, the series of harmo nics having anod e at th at po int is completely absent in the resulting vibration. The absence ofthese harm onics is evidently necessary if the m ode of vibration of the string is tobe fully determin ate in terms of the motion at the bowed point, an d in section IV ithas been show n to follow from this an d the known characteristics of the motion atthe bowed point th at the form ofth e velocity waves must then be that ofa numberof straight lines parallel to one another with intervening discontinuities. Thevelocity-diagram of the string at a ny epoch of the vibration m ust therefore alsoconsist of parallel straight lines with intervening discontinuities. It will now beshow n tha t these straight lines are all parallel to th e x-axis, i.e. to the position ofequilibrium of the string.Let the velocity-diagram of the string at a certain epoch in the correspon dingirrational type of vibration consist of parallel straight lines inclined to thex-axis at an an gle a, with discontinuities of mag nitude d l, d,, d,, etc. interven ingat the points x = c,, c,, c,, etc. respectively . Th is velocity-diagram m ay be readilyanalyzed into its Fou rier components. Let the velocity at any point o n the stringat the epoch referred to, be represented by Il/(x). Then,

where the value of A, is determined by the equationnnxAn = J I)(*) sin- x.I

+(x) is equal to x tan a between the limits x = 0 nd x = c,. From x = c, up tox =c, , it is equal to (x tan ci - l) and then changes to (x tan a - l - ,) retainingthis value up to x = c,, and s o on. Integrating by parts, we haveSince tan a is a constant, the second term reduces to zero and the equation may be


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278 c v R A M A N : A C O U S T I C Swritten in the form

When n = 1, we have

When n = s, A, may be written in the form

The summation of the series 7A, sin (nnx ll) of which A, sin (7~x11) s thelead ing term gives us the original veloc ity-diagram $(x) which consists of parallelstraight lines inclined to the x-axis at an angle a and has a discontinuity dl atthe point x = c,, a discontinuity d 2 at the point x = c,, an d so on. From this, itfollows that the series = y A,, sin (nnx/l/s) of which A, sin (snx ll) is the lead ingterm would similarly give us w hen summ ed, a diagram also consisting of straightlines inclined to the x-axis at the same angle a, the magnitudes of thediscontinuities in it being d,/s, d,/s, d,/s, etc. an d the series being periodic forsuccessive increments of x by the length 211s instead of 21 as with the originalseries. Subtracting the ord inates of the diagram thus derived from those of theoriginal diagram $(x), the resulting figure in which the sth , sth,3sthharmonics, etc.are all absent, is seen to consist of straight lines parallel to the x-axis withintervening discontinuities. We have alread y seen that with the irrational types ofvibration, d l, d,, d,, etc. are all numerically equal to (v, - ,), som e belo ngin g tothe positive wave and so me to the negative. In the particular cases in which thepositions c , , c,, c,, etc. of the discontinuities are such that the motion at thebowed point is a simple two-step zig-zag, the summation of the series ofharmonics to be left out gives, when graphically represented, a diagram of thesimplest possible form, that is, one straigh t line terminated by on e discontinuityin eac h length of abscissa 21/s, the sthharm onic being the first of the series. This isvery readily proved. Tak ing the first type of vibration in which there is only onediscontinuity in the v elocity-diagram (figure 1 in section VI), we see that at theepo ch chosen a s the origin of time, c, = I. Fro m the exp ression for A, given above,it follows tha t the series A,,sin(nnx/l/s) has a discontinu ity equ al tov, - ,/s at the sam e poin t c1= . The next discontinuity would be at x = (s- )l/s,the line of the diagram passing throug h the intervening node at x = (s - )l/s. Fo rthe second , third, fourth types of vibration, etc. discussed in sections VII, VIII a ndIX, a precisely similar construction holds good when the motion at the bowedpoin t is a simple two-step zig-zag. For , the positions of the discontinuities c,, c,,c,, etc. in the initial velocity-diagram are such that ifth e bow is applied a t a no de


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of a given harmonic, the discontinuities are also situated a t certain other nodes ofthe same harmonic. A sufficient illustration of this fact and of the method ofcon struc tion is given in figure 5(i).Thi s shows the initial velocity-diagram for thethird type of vibration elicited by bowing at the no de 5118. The thin inclined linesshow the velocity-diagram with the complete series of harmonics, the thickhoriz onta l lines show the velocity-diagram obta ined by d rop ping ou t the 81h, 161hharmonics, etc. and the d otted vertical lines show the intervening discontinuities.From the subsequent movem ent of the discontinuities in the diagram thu s drawn,the vibration-curve at a ny desired point on the string is readily ob tained.Th e types of vibration set up by bowing the string at the nodal p oints 1/2,21/3,31/4,41/5,etc. are of special interest. The initial velocity-diagram in these cases isreadily derived from the first irrational type of vibration by the method ofconstruction described above. We shall now consider these cases a little morefully.Fig ure 6 illustrates the first case. Th e initial velocity-diagram is a straigh t lineparallel to the string with a discontinuity a t each end. These discontinuities movein towa rds the cen tre of the string, and when they meet at the expiry of a qu art er-period, the velocity at every po int on the string, except the c entre itself, is zero. Atthis epoch, the form of the string consists of two straight lines meeting at thecentre, being the same as at the corresponding epoch in figure 1. Incidentally, itwill be noticed that the mode of vibration of a string bowed at the centre is thesame as that-ob tained when it is plucked a t the centre.''Figure 7 illustrates the case of a string bowed a t a point of trisection. At the tw opoints of trisection, the vibration-curve is a simple two-step zig-zag, and at twocorresponding epochs, the configuration of the string consists of two straightlines meeting at the point of trisection. The nature of the motion as observed atthe centre of the string is of special interest in this case. Commencing from theposition of extreme displacement in one direction, the complete period ofvibration is seen to be made up of six phases of equal duration, in the first andthird of which the velocity of the centre of the string is the same as tha t of the bowapplied a t the point of trisection. In the fourth an d sixth phases the velocity is alsothe same num erically b ut in the opp osite direction. In the second an d fifth phases,the velocity is numerically double that of the bow, having the same sign in thesecond phase, an d th e o pposite sign in the fifth phase. On ly the first, fifth, seven th,eleventh harmonics, etc. contribute tow ards making u p the motion at the centreof the string in this case.

'OIt is also readily seen that the velocity-diagram of a string plucked a t any o ther point and releasedshould consist of a straight line parallel to the x-axis and bounde d by two discontinuities which areinitially coincident at the point of plucking and start off with the velocity of wave propagation inoppo site directions. Th e form of the vibration-curves in this and o ther ana logou s cases is much m orereadily fou nd by the aid of the velocity-diagram than by tracing the configuration of the string.


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C V R A M A N : A C O U S T I C S

Velocity - diagrams Vibration -curve s and displacementdiagramsFigure 6. The string !owed at the centre.

Figure 8 illustrates the case in which the bow is applied at a point exactly aqu arte r of the length of the string from on e end, the initial velocity-diagram of thestring being obtained by the geometrical construction from the first type ofvibration. T he cases in which the bow is applied at a point one-fifth or on e-sixthor one-seventh, etc., of the length from one end may be similarly dealt with.Passing on to consider the modifications of the second, third and higherirration al types of vibration caused by the coincidence of the bow with a point ofrational division of the string, we find no difficulty in constructing the initialvelocity-diagram of the string in a ny specified case, and as we have already seen,the construction becomes further simplified when the motion at the bowed p ointis representable by a two-step zig-zag. Certain interesting relations the n becom eevident. The velocity-diagrams derived from the first and second types of


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II II IJe loc i ty d iogroms Vib ratio n curves and displacementd i a g ra m s

Figure 7. The string bowed at a point of trisection.

vibration are found to be identical when the bow is applied at a point of trisection(cj: section VII in which it was shown that these two types then becomecompletely identical). Similarly, the first and third types become identical whenthe bow is applied at a distance 114from one end, the first and the fourth when it isapplied at 115 from one end and so on. Further, the second and third types give thesame results when the bow is applied at a distance 2115 from the end, the ratio 0being 115 in both cases. The third and fourth types both give the ratio 0 = 117when the bow is applied at 2117, and so on. These relations may be stated andproved in the following general form.

W e n he motion at the bowed point is given by a simple two-step zig-zag, the pZh


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C V RAMAN: ACOUSTICS

Velocity-d iagram s Vibration -curves and displacementd i agr am sFigure 8. String bowed at point of quadrisection.

and qZhypes of vibration give identical vibration-curve s, provided the bowed pointcoincides with an intervening node of the (p + qYh harmonic.The proof of the relation stated above follows very simply from the generalrelation given in equatio n (17).Le t x, define the position of the bowed po int a ndlet the nearest nodes of the prhand qrh armonics be the ( r+ I Y h and the (s +respectively, counting from one end. Fo r the prh type, the ratio w = ( x , - llp)-+ l /p . For the qrh ype, the ratio o = (sllq- ,) - /q, it being assumed that rl/p< x , < sllq. T he pth and qr h ypes give the same vibration-cu rves when the value ofw is the same for both a t the bowed point. Equating w for the two cases, we deducex , = (r + s) l /(p+ q), in other words that the bowed point coincides with the


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(r + s + lYhnode of the (p+ qYh harm onic . Sim ilarly, if x, lies outside the limitsrlls and sllq, the p fh nd qfh ypes become identical when th e bow coincides withthe (r - + lYh nod e of the (p- Yh harm onic.As an illustration of the results, we may consider the case in which the bow isapplied at the poin t 3111 1. The fou rth an d seventh types of vibration both give theratio o = 111 1 and a re thu s identical in this case, but the third type gives the r atioo = 211 1, the first type gives the rat io o = 311 1 and the second type gives thelargest ratio of all, viz. o = 511 1.

Section XI-Some examples of the theoretical deter min ation ofthe vibration-curvesWe shall now proceed to figure a numb er of vibration-curves whose forms can becalculate d by the aid of the theoretical principles set ou t in the preceding sections.T o trace the vibration-curve at any given point o n the string, we require dat aregarding the position of the bowed point a nd also as to the particular type ofvibration (accord ing to th e meth od of classification explained in section V) which.may be assumed to be actually elicited. In the next section we shall consider atsome length the general relation connecting the position of the bowed poin t an dthe pressure, velocity and o ther features of the bowing ad opted, with the mo de ofvibration elicited. Meanw hile, as a work ing rule, it will be assum ed th at if the bowis applied at a p oint n ot far from a node of the nfh armo nic, n being one of the firstseven or e ight na tura l nu mbers, th e nth type of vibration is elicited. It will befurther assumed tha t the m otion a t the bowed point is a simple two-step zig-zagexcept in cases in which, as shown in section IX, it is necessarily of a morecom plicated type. These assum ptions ar e justifiable, provided the p ressure ofbowing is sufficient and other circumstances are favourable.The detailed method of tracing the vibration-curves is as follows: the initialvelocity-diagram of the string is first set down a ccordin g to the principles alreadyexplained. The ord inate a t the point w hose vibration-curves is to be drawn, givesits initial velocity. The positions and magnitudes of the discontinuities and thedirection of their motion, positive or negative, being known, the successive^chan ges of velocity at th e point of observation and th e intervals of time at whichthey take place are obvious to inspection. The successive intervals and theresulting displacements may then be pricked off on a time-displacement diag ramand when joined up, give us the desired vibration-curve.In figures 1 to 8 we have already had a num ber of illustrations of the graphicalprocess worked out in detail. The succeeding figures 9, 10, 1 1 and 12 presentadditional examples of vibration-curves determined by the entirely a priorime tho d set forth above. They were specially draw n for com parison w ith the first


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Point ofbowing Point ofobservationNear theend

2

3#-Figure 9. Examples of vibration-curves of a bowed string, found a priori.


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forty of the curves found experimentally by Krigar-Menzel and Raps andpublished with their paper.

N.B.-In these and the succeeding figures, the symbol e represents a smallfraction whose value was variously assumed.

The first curve in figure9 belongs to the first type of vibration. The second andthird curves belong to the fifth type of vibration, the bow being applied onopposite sides of the node of the fifth harmonic in the two cases.12 The formercurve is modified owing to the absence of the eleventh harmonic. The fifth andsixth types of vibration both give the ratio o = 111 1 when the bow is applied atthe point 21111 . (At the same point, the first type of vibration would give the ratiow = 211 1) . The fourth curve in figure 9 is a combination of the fourth and seventhtypes which become identical when the bow is applied at the point 31111 , the ratioo at the bowed point being 1/11 for both types. The fifth curve belongs to theseventh irrational type of vibration and the sixth curve to the third irrational type.It will be noticed that these two forms are closely analogous and it can be readilyshown that they merge into one another when the bow is applied exactly at thepoint 31/10,where they both give the ratio w = 1/10.The close approximation ofthe form of the seventh curve to that of a pure sine-wave will be noticed. Theeighth curve belongs to the third irrational type of vibration which is pictured ingreater detail in figure3. The ninth curve belongs to the fifth irrational type ofvibration. The tenth is a combination of the fifth and second types of vibrationwhich become identical when the bow is applied at the point 3117, both typesgiving the ratio 0 117.

The first curve in figure 10 is a combination of the second and seventh typeswhich give a common ratioo = 119when the bow is applied at the point 4119.Thesecond curve represents the second irrational type of vibration (see also figure 2 inwhich this is worked out in greater detail). The third curve is a combination of thefifth and sixth types which give a common ratio o = 1 / 11 at the bowed point21/11. The fourth curve is similarly a combination of the fourth and fifth types(o 119 at the bowed point 2119).The fifth curve represents the third irrationaltype of vibration (see figure4 for a treatment in greater detail). The sixth curve is acombination of the third and eighth types,o being equal to 11 1 1 at the bowedpoint 4111 1 . The seventh curve in figure 10is representative of the fourth irrationaltype of vibration. The eighth curve is a combination of the fourth and seventhtypes (w= 1 /11 at the bowed point 31/11 for both). The ninth curve is acombination of the third and eighth types and the tenth is a combination of thethird and fifth types (o 11 1 1 and 118 respectively).

"The other experimental curves (twen tyfour n all) that appear with the paper of Krigar-Menzel andRaps are mostly simple tw o-step zig-zags. See also B arton's Text-Book of Sound, page 432.''The position of the bowed poin ts for the two correspond ing experimental curves have beenerroneou sly shown interchanged in the paper of Krigar-Menzel and Raps.


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Point o fbowing Point ofobservation

Figure 10. Examples of vibration-curves of a bowed string, found a priori.

The first curve in figure 11 belongs to the second irrational type of vibration(compare with the graphs in figure 2). The second and third curves in figure 11need no further remarks. The fourth curve is a combination of the sixth andseventh types of vibration,o eing equal to 1113 at the bowed point 21/13 for bothtypes of vibration. The fifth and seventh curves are examples of the well-known


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M E C H A N I C A L T H E O R Y O F TH E V I B R A T I O N S O F B O W E D S T R I N G S 287Point of Point ofbowing observation

Neor the ende-I S(2-eV8 8*

Figure 11. Examples of vibration-curves of a bowed string found a priori.

"staircase" figures discovered by Helmholtz. The sixth and eighth curvesrepresent two cases of the fifth irrational type of vibration. Th e ninth c urve is acom binatio n of the fourth an d fifth type of vibration, o eing equal to I/9at thebowed point 21/9 or both types. Th e last curve in figure 11 is of special interes t, asit is a case of the eighth type of vibration m aintained with a complicated m otion


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288 c v R A M A N : A C O U S T I C SPoint ofbowing Point ofobservation

3-- I7

Figure 12. Examples of vibration-curves of a bowed string, found a priori.

at the bowed point. In section IX, it was shown that the motion a t the bowedpo int c ann ot be a simple two-step zig-zag, if the eighth type of vibration is elicitedby applying the bow near the point 1/4.In the initial velocity-diagram from whichthis vibration-curve was drawn, two of the discontinuities were situated at thetwo en ds of the string respectively, and the o ther six were taken to be situa ted in


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pairs at the points (2 + 3e)/8, (4 - e)/8 and (6 + e)/8 respectively, correspondingto the position of the bowed point (2- )/8.

The first vibration-curve shown in figure 12 is of the type dealt with in greaterdetail in figure 8. The second curve is of special interest, as the eighth irrationaltype of vibration which it represents, is elicited by applying the bow near a node ofthe fourth harmonic and the motion at the bowed point for this case is thusnecessarily of a "complicated" type, viz. a four-step zig-zag instead of a two-step.The position of the bowed point being (2+ e)/8, the discontinuities in the initialvelocity-diagram were taken to be situated in pairs at the positions (1- )/8,(3- e)/8, (5+e)/8 and (7- e)/8 respectively. The third curve in figure 12represents the eleventh irrational type of vibration. The velocity-diagram of theeleventh type from which this was drawn gives a simple two-step zig-zag as thevibration-curve at the point (1+ e) /l l. An entirely different curve (in which the4th and 1 lth harmonics are dominant) is however obtained from the velocity-diagram which gives a two-step zig-zag at the point (3- )/ll, and actualexperimental trial, I find, gives a result in agreement with this, and not the curveshown in figure 12. Krigar-Menzel and Raps however describe one of theirexperimental curves which is very similar to the third in figure 12 as thatproduced by bowing at the point (3- )/ ll . If this description were correct, themotion at the bowed point could not have been a simple two-step zig-zag.

The other vibration-curves obtained theoretically and shown in figure 12 donot require any special remarks. The fourth curve is a combination of the fourthand seventh types which give identical motions ( o= 1/11) at the bowed point3/11 1. The fifth curve is the third irrational type of vibration modified by theabsence of the 13th harmonic with the node of which the bowed point coincides.The sixth curve is of the well-known staircase type. The seventh curve is acombination of the third and the eighth types given by the ratio w= 111 1. Theeighth curve is similarly a combination of the fifth and the eighth irrational typeswhich become identical when the bow is applied at point 51/13. The ninth curveillustrates the seventh irrational type of vibration, and the tenth curve is acombination of the second and fifth types of vibration which give a common ratio( o= 117) at the bowed point in the position 3/17.


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29 0 c v RAMAN : ACOUST IC SSection XII-The effect of the variation of the pressure and

velocity of bowing

Preliminary discussionIn the previous sections of the paper, we have considered the kinematics of all thepossible modes of vibration of a bowed string in which the motion at the bowedpoint consists of ascents and descents with strictly constant velocities. It nowremains to consider various subsidiary questions that arise. What are themodifications in the kinematical theory necessary, when, as foreshadowed in thediscussion on the modus operandi of the bow, the motion at the bowed point is notrigorously of the kind postulated? In other words, what is the effect produced ifthe velocity with which the bowed point slips past the hairs of the bow is notexactly constant in each period of vibration? Then again, what is the effectproduced by the finiteness of the region with which the bow is in contact, a regionwhich for the purpose of discussion we have so far taken as equivalent to amathematical point? Does any slipping occur when the string is being carriedforward by the bow? Is it possible in practice that by simply removing the bowfrom a nodal point to another closely contiguous to it, the missing harmonics inany given type of vibration are suddenly restored to their full strength as ourkinematical discussion tacitly assumed? Finally we have the all-importantquestion, what are the conditions of excitation, e.g. pressure and velocity ofbowing and so on, required for any given type of vibration to be elicited? Whatpart does the instrument on which the string is mounted and the handling andproperties of the hairs of the bow play in determining these conditions? What, forinstance, is the effect on the motion of the string produced by loading the bridgeover which it passes with a mute o r otherwise? This is a most formidable array ofquestions, but it does not seem entirely hopeless to find answers to them all,provided we proceed step by step, seeking answers to the questions one at a time.

We may commence by retaining the assumption that the region of contact ofthe bow may be treated as a mathematical point and that the velocities of ascentand descent of this point are rigorously constant. We have already seen that onthis assumption all the possible modes of vibration can be classified in a series ofwhich the ordinal number n is the same as the total number of discontinuities inthe velocity-diagram of the string, the bowed point being taken to divide thestring in an irrational ratio. If the point of bowing divides the string in a rationalratio, the mode of vibration may be derived from one of the irrational types bydropping out the series of harmonics having a node at that point (see section X).Another important point elicited by the discussion is that if n be not a primenumber, a two-step zig-zag motion at the bowed point is kinematicallyimpossible for that type of vibration if the bowed point lies within a distance of1/2n from any node of the nthharmonic which is also a node of some harmonic of


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lower frequency. Further, if the position of the bowed point be specified and itsmotion is representable by a simple two-step zig-zag, the entire mode of vibrationis uniquely determined by the number n of the discontinuities. But if the motionBtthe bowed point consists of several ascents and descents in each period, thenumber n does not uniquely determine the motion. One or more additionalconstants expressing the initial position of the discontinuities in the velocity-diagram must then be specified for the motion to be completely determinate.

From the preceding, it is obvious that the first step for establishing the mode ofvibration in any given case is to find the number n of the discontinuities. Primafacie, we may exclude the consideration of all types of vibration of which theordinal number n is very large. This is evident, for the larger the number ofdiscontinuities, the more important is the part which harmonics of high ordercontribute to the motion,13and it is known from various considerations that theseharmonics of high order are difficult to elicit and maintain as part of a perfectlyperiodic motion with the frequency of the gravest mode. As has been well shownby Prof. E H Barton,14 the higher the frequency of any given component in themotion, the more sensitive it is to any departure from the exact adjustment forresonance. Under ordinary circumstances, the free periods of vibration onlyapproximate to, and do not actually form a harmonic series. We are therefore ledto conclude that any type of vibration in which the number of discontinuities isvery large and in which therefore harmonics of high order are dominant wouldnot, in practice, be elicited by the bow.

Two other considerations also point to the same result as that noted above. It isvery probable that the higher harmonics are subject to a greater degree ofdamping and the bow would therefore tend to elicit types of vibration in whichthey are relatively subordinate. Then again, we see that the finiteness of the regionwith which the bow is in contact would operate in the same direction. For, fromequation (17)

which expresses the ratio of the time during which the string moves past the hairsof the bow to the whole period of the vibration, we see that the larger n is, the morerapidly does the value of o change with any alteration in the position of thebowed point, x, being the distance of this point from the nearest node of the n'*

"This becomes almost selfev iden t on attempting to draw a velocity-diagram for the string con sistingof parallel straight lines inclined at a constant angle a to the x-ax is and separatedbya large number ofequal discontinuities situated at various intervals along the string.14E H Barton, "On the Range and Sharpness of Resonance to Sustained Forcing,"Bull. Indian Assoc.Cultiu. Sci., No . 13.


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292 c v R AMAN : ACOUST IC Sharm onic . The distu rbing effect of applying th e bow over a finite region instead ofat a mathem atical po int wo uld therefore be much greater when the value of n islarge, and would thus have a n unfavourab le effect on the pr odu ction of types ofvibration involving a large number of discontinuities.

We thu s see tha t if the motio n at the bowed po int adm its only of discontinuouschanges of velocity from one value to anoth er an d vice versa, the possible modesof vibra tion would be confined to a restricted num ber of types and their variants.Each type would, according to eq uation (17), have a ch aracteristic value of o ifthe position of the bowed po int were specified, and a passage from one type t oanoth er would involve a sud den change from one value ofo at the bowed point toanother.Takin g the results summarised above, together with those arrived a t in thediscussion on the modus operandi of the bow in section 111, it is now possible totrac e the general effects of varying the pressure or velocity with which the bow isapplied. If there is a change of type, the value of w undergoes the concom itantdiscontinuous change given by equation (17). If the pressure of the bow isincreased o r its velocity decreased, the value of o in the concom itant change oftype decreases as a rule, so tha t the bow and string move with a com mo n velocityfor an increased fraction of the period. The change would be in the oppositedirectio n if the press ure of the bow were decreased o r its velocity increased. T hechan ge of type may o r may no t, according to the circumstances, involve a changein the m otion at the bowed point from a simple two-step zig-zag to one withseveral ascents and descents, or vice versa. Another variety of change tha t m ay becaused by th e alteration of the pressure or velocity of the bow is a readjustment ofthe positions of the discontinuities with th e corresponding changes in the relativeamplitudes of the harmonics, the value of wwhich is determined by the number ofdiscontinuities in the motion remaining unaltered. This species of change, asalready show n, would only be possible if the motion at the bowed po int involvedseveral ascents and descents in the period.W hen the pressure with which the bow is applied is considerable and the o the rcircumstances are favourable, the maintained motion would be the one whichgives, as nearly as possible, the sm allest value of w at the bowed point. When thebowed point is in the neighbourhood of a node of some fairly importantharmon ic, say the nth, t is readily seen from equ ation (17) that the nth ype wouldgive a very small value of w and is therefore the type that w ould be maintained.This is the justification of the result which was assumed in section XI fortheoretically determining the vibration-curves and comparing them with thoseobtained experimentally by Krigar-Menzel and Raps. From the publishedaccount of these experimenters it appears that the style of bowing actuallyadop ted by them was the on e referred to above. F or sm aller pressures of bowingthan those adopted by them, it might be expected that the modes of vibrationfigured therein would be incapable of being maintained and would yield place totypes giving larger values of w.


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Graphical treatmentA clearer comprehensio n of the foregoing results may be obtained by discussingthe specific cases in which the bow is applied at some poin t between th e extremeend of the string and the node of the fifth harmonic distant 1/5 from the same end.This includes the "musical range" of bowing. When t he bow is applied very closeto th e end of the string, only the first type of vibration is kinematically cap able ofgiving a two-step zig-zag motion at th e bowed point. This also gives the smallestvalue of w and is accordingly th e type th at is elicited by firm bowing. The highertypes give four, six, or eigh t-step zig-zags, etc. as the case may be at the bo wedpoint, with correspondingly larger and larger values of o, and can thus beobtained only by increasing the velocity or reducing the pressure of bowing. Asthe position of the discontinuities in these cases is susceptible of variation, each ofthese types is itself capab le of mod ification by a ltering the p ressure o r velocity ofbowing, and the relative amplitudes of the harmonics may be profoundlymodified without any change in the value of w.Wh en the bow is removed from the extreme end to a point at some distancefrom it, som e of the higher types also a re capable of giving a two-step zig-zagmotion at the bow ed point. Let us now assume for simplicity tha t the bow doe snot elicit vibrations involving more than nine discontinuities, so that the ninthtype is the highest that n eed be considered. Within th e range between 0 nd 1/5,the bow may elicit the first type anywhere. The table IV shows for the other typesthe maxim um ranges within which the bow must be applied for a two-step motionto be even kinematically possible.

Table I V0< x, < 115

Ninth type 1/18< x, < 116Eighth type 1/16


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Ninth type [ / l a < xb


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inadmissible, we find that at the nodal points 1/10,1/9,1/8, etc. the value of w is1/10, 1/9,1/8, etc. respectively. The lines for the higher types intersect in pairs atthe points whose abscissae are 21/19,21/17,21/15,21/13 and 2111 1 respectively. Atthese points, the first type gives the value of w to be 2/19, 2/17, 1/15, etc.respectively. But the higher types give the value of w at these same points to be1/19, 1/17, 1/15, etc. if, therefore, these types are to be elicited, considerablepressure and small velocity of bowing must be adopted, and at the same time, theregion with which the bow is in contact must be very narrow. With a short,heavily-damped string such as that of a violin, these conditions are not readilyobtainable and, at any rate, they differ from those actually adopted in the wholerange of violin practice. We are therefore led to conclude that in the musicalapplications of the subject, we have to deal only with the first type of vibrationand its modifications. On an ordinary monochord, however, by suitable bowingthe higher types of vibration may be obtained within the ranges considered. Thediagram shows that except at the nodal points already mentioned, 3 or even 4different values of o re possible.

P o s it i on o f b o w e d p o i n tFigure 14. Show ing the chief types of vibration, their ranges and charac teristics and the effect of thevariation of the pressure of bowing.

Figure 14 shows the different types of vibration with discontinuous changes ofvelocity that may be obtained by applying the bow anywhere within the regionlying between 115 and 113. The heavy line, as before, is the graph for the first type.The other types are shown by lines passing through the nodes of the


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296 c v R AMA N : ACOUST IC Scorresponding harmonics, thin continuous lines representing types which maygive a simple two-step motion at the bowed point, and broken lines (----)representing types which necessarily involve more complicated motions at thebowed point. F or instance, through the point 113, we have one continuou s linerepresenting the third type an d two broken lines representing the sixth an d ninthtypes respective ly. Similarly , thr ou gh th e point 114, we have two co nti nu ou s lines,on e on each side, representing th e fourth type an d two bo rken lines, one on eachside, representing the eighth type. The diagra m enables us t o find at a glance, thegeneral effect of applying the bow at a ny given point w ithin the range a nd of anyconsiderab le variation of the pressure o r velocity with which it is applied. At thetwo nodal points 115 and 113, for a two-step zig-zag motion, the diagram showsthat the value of o an o nly be 115 an d 113 respectively. At th e poi nt 114, o asgenerally t he value 114, but an oth er v alue, i.e. 112 belonging t o th e second type, isbarely on t he limit of possibility. Elsewhere, three, four, five o r even six differentvalues of o re possible, the number being determined by the position of thebowed point, and the particular value out of those possible which is actuallyobtained being determined by the pressure, velocity and other features of thebowing adopted in any given case.Figure 15 similarly represents the types of vibration th at m ay be obtain ed bybowing at points lying between 113 and 112. It is noteworthy that all the threediag ram s give the values of o t the bowed p oint, both in the cases in which thispoint divides the string in a rational ratio as well as in those in which it divides the

Position o f bowed po i n tFigure 15. Showing the chief types of vibration, their ranges an d ch aracteristics and the effect of the

variation of the pressure of bowing.


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string in an irrational ratio. In cases of the latter kind, the diagrams may also beused to find the value of o elsewhere than at the bowed point.

There is another and a very instructive point ofview from which the effect of thevariation of pressure may be regarded. In section I1 of the paper, it was shownthat if the bow is applied at a point of rational division of the string, the harmonicshaving a node at that point are not excited. It is obvious also that if the bowedpoint is situated at some little distance from the node instead of actuallycoinciding with it, the forces required to maintain the harmonics in question withappreciable amplitudes would be of considerable magnitude relatively to thoserequired for the maintenance of the other components in the motion. We are ledtherefore to conclude that the types of motion maintained would tend to be thosein which harmonics having a node near the bowed point are relatively large orsmall in amplitude according as the pressure with which the bow is applied ismuch or little. We are also led to conclude that as an increase of the velocity of thebow involves an increase of the amplitude of vibration, its effect on the characterof the motion would be analogous to that of a decrease in the pressure of the bow.The preceding results are in agreement with those already arrived at fromsomewhat different considerations.

Mathem atical theoryWe now proceed to consider the mathematical theory of the subject. For thepresent, we shall adhere to the simplifying assumption already made that theregion of contact may be.regarded as a mathematical point. Let the motion of thestring in any actual case be represented by the Fourier series

or more briefly

From the formulae found in section I1 of the paper it is seen that the forcerequired to maintain this motion may be represented by

sin(2n = m + en+ e:C k n B nn = l n7cx0sin-

when k n and ek are quantities independent of the position of the bowed point and


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298 c v RAMAN : ACOUSTICS8

xo is the distanc e of this po int from the fixed end of the string. If for a particularvalue o r values of n the denom inator sin nnx,/l of the term in the series is verysmall but not actually zero, and if at the same time the amplitude B, of thecorresponding harmonic or harmonics remains finite, the magnitude of therespective term or terms in the expression for the ma intaining force is necessarilygreat. But if sin nnxo/l is actually zero for certain values of n, the correspo ndingterms B, in the maintaine d motion m ust necessarily vanish as we must ob viouslyexclude the possibility of infinitely large values for the mainta ining force. Theseries given above represe nts the variable part of the frictional force at the poin t ofcontact. T his force is a function of the pressu re P with which the bow is appliedan d the relative velocity (v- ,) at the point of contact. It may be written in theform F(P, - ,), the function remain ing determ inate and always of the same signso lo ng a s the relative velocity (v - ,) does not become zero or change sign. Wemay therefore write

n = 1 r r r ~ r r osin-where P o is the non-periodic part of the frictioA1 force.Eq uatio n (22) may be re garded as the dyn amica l relation from which th emotio n a t the bowed point an d therefore also the entire mode of vibration of thestring is to be determin ed. For, if the velocity v of the bowed point beexpre ssed asa function of the time, the amplitud esB, of the harm onics can be found therefromby integration over the complete period. The various types of vibration whosekinem atical theory was discussed in the first pa rt of the paper may be regarded asform ing a series of limiting solutions of this dynam ical equation to one o r oth er ofwhich the actual results approximate more or less closely according tocircumstances. How these limiting solutions are obtained has already beenindicated in th e discussion on the modus operandi of the bow. W e now proceed t oexamine more closely the degree of approximation within which the limitingsolutions represent the actual results.Ou r knowledge of the form of the function F(P, v-vd is not at present verydefinite for surfaces of the kind we are concerned with in this paper. It mayhowever be accepted as demontrably correct that when the relative velocity(v - ,) is actually zero, the friction is not g reate r than a certain maximum static alvalue P,which increases continuou sly for increasing values of the pressure P withwhich the bow is applied. We may a lso suppo se that w hen the relative velocity(v-vd is greater than zero, the magnitude of the friction decreases with increas ingvalues ofthe relative velocity, at first somewhat rapidly and later on perhaps no t sorapidly. Arguing from these premises, the relation given in (22) enables us toarrive at certa in imp ortan t conclusions. In the first place, since the quantities k,,ek depend on the construction of the sounding box on which the string is


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stretched, the summation of the series on the right-hand side of (22) cannot, ingeneral, result in ou r finding a cons tant value for the relative velocity (v- ,) inany o r all of the epochs in which it is greater tha n zero. The relative velocity mayhowever approach or attain practical constancy in any or all of the epochsreferred to, if the variation of the sum of the series on the right-hand side ofeq ua tio n (22) within those e poc hs is negligibly small. It is seen tha t such pra cticalconstaqcy would be attained if the pressure P with which the bow is applied issufficiently arge to ens ure that the necessary variation in the value of the functionF(P,v - ,) ca n be secured by negligib ly small chan ges in the relative velocity(v- ,). Th e practical cons tancy in the relative velocity durin g the epochs referredto m ay also be secured even for mo dera te values of the pressure P, provided thatin the series on th e righ t-han d side of (22) there a re no large, rapidly-varyingterms. F or instance, if the string were bowed a t imp ortan t nodes such as 113 o r 114even with very moderate pressure, the absence of the corresponding series ofharmonics in the resulting motion removes terms which would otherwiseintro duc e large and rapidly varying fluctuations in th e value of the series for themaintaining force. The constancy of the velocity of the bowed point in slippingpast the hairs of the bow is thus remarkably perfect in such cases.Assuming the solution of (22) to be one of the limiting types in which therelative velocity (v - ,) has a co nstan t value in all the epoch s in which it is notactually zero, we may easily find a n inferior limit to the pressu re with which thebow must be applied for the given type of vibration to be possible. Thus, let(v, - ,) be the c onstant value of the relative velocity during any such epoch an dlet PA e the sum of the series o n the right-ha nd side of (22) for tha t epoch . Thenwe must haveFurther, let PA e the max imum value of the series durin g any of the epochs inwhich the relative velocity is zero. Then, we must have

P, (the statical friction) 4 Po + P A . (24)Sub tractin g the q uan tities in (23) from the two sides of the inequality in (24),wefind that the pressure with which the bow must be applied is such tha t

P, - (P , vA - ,) 4 P >- P A .If p be the coefficient of sta'tical friction for the pressure P, and pA be thecoefficient of dynam ical friction for t he pressure P an d relative velocity (v, - ,),the inequality may be written thus,If the pressure of the bow is less tha n the critical value given by (25), the vibration

would no longer be maintained and would therefore alter in character. We may


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300 C V R A M A N : ACOUSTICS

now consider the effect of increasing the pressure a bov e this critical value. It isobv ious from eq uatio n (22) tha t if the relative velocity ( v - ,) is assum ed to havea co nstan t value in all the epoch s in which it is not actua lly zero, the increase inthe frictional force F(P ,v - ,) cau sed by an increase of the pressure of the bowmight m erely result in raising the non-p eriodic con stant Po on the right-hand sideof the equ ation and thus leave the periodic part of the forces acting on th e stringunaltered. W hen, therefore, any on e of such types of vibration is once thorou ghlyestablished, considerable latitu de is permissible in the pressure of the bow so longas i t does not fall below the critical value. It must not however be understoodfrom this that the pressure can be increased indefinitely. For, if there be someother possible type of vibration whose critical pressure be higher than the first,the mo tion originally maintained would be relatively less stable and wo uld yieldplace to the othe r, when the pressure of the bow exceeds the critical value of thelatter. The ma nne r in which an increase of pressure sets up a n instability of theoriginal type of vibration is best understood by analogy with that of theequilibrium of a system with one degree of freedom. The equation of motionabo ut the po sition of equilibrium of such a system under the ac tion of the bow is

where k is the dam ping coefficient and 1 s a positive quantity proportional to thevelocity-rate of change of the frictional force which may be increased byincreasing the pressure of the bow. When the velocity as given by 2 is initiallyzero, there is apparently nothing to cause the system to depart from the positionof equilibrium. If, however, by increasing the pressure of the bow, I is increasedtill it exceeds k, the equilibrium becomes unstable and any small motion ismagnified continuou sly till it reaches the limit set by the velocity of the bow. Th emechanical disturbance caused by laying on the bow or by increasing its pressureis suficient to give effect to the instability. Similarly in the case of the bowedstring, a steady state of vibratio n in any given type is obviously ou t of the qu estionif any slight disturban ce results in a progressive chang e and readjustme nt of theam plitud es of the harmonics. Referring to equa tion (22), it is seen that a n increasein the non-periodic part Po of the frictional force caused by an increase in thepressure P of the bow wo uld tend to set up such instability if an altern ative mod eof vibration with a higher critical pressure were possible.On comparing equations (22) and (25), it is seen that, in general, the type ofvibration which has a higher critical pressure than another would also containthe harmonics having nodes near the bowed point with relatively largeramplitudes.

Analysis of the motion


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Differentiating, we have

But in section X, we found th at the analysis of the velocity-diagram consisting ofparallel stra ight lines givesdy nnx-=EA,sin--d t 1where

d l , d,, d,, etc. being the magnitudes of the discontinuities in the velocity-diagramsitu ate d at poin ts whose abscissae a re c,, c,, c,, etc. respectively. Since som e of thediscontinuities move towa rds the origin and others aw ay from it, it is convenientto use instead of c,, c,, c,, etc. the qua ntiti es C,, C,, C,, etc. where c,= C, + at ifthe discontinuity d, belongs to the positive wave, and c, = 21- C,+at ) if itbelongs to the negative. We then have, as a relation true for the whole period ofvibration,

nn(C, + at ) nn(C2+ at)+ d 2cos 1n 1Com paring (26) and (27), we m ay w rite

1 nnCl . nnC2b n = --[d,sin-2 n 2 T 1 + d2 in---+etc.1This gives the amplitudes of the series of harmonics in terms of the discontinuitiesand their initial positions. In the case of the bowed string we have

Fro m (28) and (29), it is easy to verify the remark already ma de th at w hen thenum ber of discontinuities is large, the higher harmonics gain considerably in theirrelative importance.Subs tituting the values of the amplitudes B, of the harmonics found from thepreceding a nalysis in the expression for the m aintaining force on the righ t-handside of equa tion (22), the interesting question arises whether this expression forthe m aintain ing force is really a convergent F ourier series. If k, varies as nz wherez > I , the expression for the fcrce on the right-hand side of (22) is a non-


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302 c v R A M A N : A C O U S T I C Sconvergent series and it follows that under such circumstances, the bow would beeven theoretically incapable of eliciting an infinite series of harmonics. Even whenkn varies as nz where z is not greater than unity, the existence of the factorsin nnxo/ lin the denominatorof each term in the series brings in a very interestingdifficultyas to its convergency. If we assume xo/l to be an irrational fraction, it isobviously possible by assuming a suitable and sufficiently large value of n to makethe quantity sin nnxo/l smaller than any specified fractional number, and, further,it is evidently possible to find an infinite number of such values of n which wouldreduce sin nnx,/l to a value below the specified limit. Under the circumstances itbecomes rather a delicate mathematical question whether the expression for theforce given in (22) would then be a convergent series, and, if so, for what values ofz. This difficulty does not arise if x,/ l be a rational fraction, for the values ofsin nnxo/l then form a recurring series and the expression for the force is obviouslyconvergent for values of z not greater than unity. It is of great interest todetermine and represent the form of the expression for the force at the bowedpoint graphically and compare it with the form of the vibration-curve at thebowed point; if the values of k, be arbitrarily assumed and the amplitudes Bnaredetermined from the formulae given in (28) and (29), it is to be expected that, ingeneral, the equation for the maintaining forces given in (22) is not rigorouslysatisfied even if the value of the constant Po and the initial positions C,, C,, C,,etc. of the discontinuities be suitably assumed. In other words, the motion at thebowed point cannot in general be rigorously of the type on which our kinematicaldiscussion was based, that is, having forward motions which involve no slippingand backward motions with uniform velocity of slipping. But in a special class ofcases, as we shall presently see, the motion at the bowed point may be exactly ofsuch type consistently with the dynamical conditions expressed in equation (22).

Frictional-force curves and motion at the bowed pointAs indicated above, it is a most important matter to determine and representgraphically the nature of the variation of the frictional force at the bowed point. Ifthe maintained motion, the position ofthe bowed point and the magnitudes of theseries of constants kn and en are all known, the harmonic components in thefrictional force are all determinate, except those, the frequencies of which are thesame as that of the missing harmonics having a node at the bowed point. If suchcomponents be assumed to be zero, or at any rate be assigned specificmagnitudes, the frictional-force curve at the bowed point may be drawn by theaid of a machine for harmonic synthesis. Such curves would be very useful forcomparison with the form of the vibration-curve at the bowed point, speciallyduring these epochs at which the bowed point slips past the hairs of the bow; theycould also be used for determining the critical value of the pressure of the bow forthe possibility of the type of vibration considered.

The form of the frictional-force curves may be found analytically and drawn


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witho ut the aid of a harmo nic machine in an im portan t class of cases which maybe regarded a s representative of the phenom ena actually met with in experiment.We shall assume th at the values of e; can all be put e qual to n/2. This would be avery close appr oxim ation to the tr uth , as may be seen from the formu lae given insection I, if the natural frequencies of vibration of the string form a strictlyharm onic series and the yielding at its ends is negligible. Furth er, we shall assumethat t he values of knare propo rtiona l to n. This assumption cann ot be very wide ofthe mark, for, if k, were proportional to a power of n higher than unity, theexpression for the frictional force would no t be a convergent F ourie r series an d itwould n o longer be possible to consider the motion of the string as comprising a ninfinite series of harmonics. In practice, moreover, it seems very unlikely that thedam pin g coefficients of the higher harmonics increase at a m uch lower rate th antha t given by the formula knu n, as the higher harmo nics would otherwise be mo reconspicuous in the maintained vibration than they usually are.'O n these assum ptions, the ex pression for the frictional force assumes the form

L-rn= 1 ~ K X ~sin-T o effect the summ atio n of this series, we have to assu me tha t x,/l is a ratio nalfraction an d the values of the den om inators therefore form a recurring series. Fo rthose values of n for which the deno mina tor is zero, B, is also equal t o zero, an dthe series therefore con tains a series of indeterminate terms. In the most generalcase in which the vibration-curve at the bowed point can be represented by anum ber of straight lines of which alternate on es are not necessarily parallel to o neanother, we may write the expression for the force in the form

n = 30

Cn = l ~ K X ~sin-the values of an an d b, being given by the formulaea = -- ~ K C ,[ d l c o s Tn 2 n 2 T + d, cos-+ etc. ,nnC2 Ibn= -- ~ K C ,[d,s inT + d 2sin-n 2 n 2 T 1 + etc. .nnC2 I

"It may be noted that the assumptions e; = ( 4 2 ) nd k, zn are quite correct when the dissipation ofthe energy of the string is solely due to a frictional force proportional to the velocity resisting eachelement of the string. Cf. Andrew Stephenson, January 191 1, Philos. Mag.


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304 C V RAMAN: ACOUSTICSTh e summa tion of the expression for the force is thus seen to depend u pon thesummation of a number of separate sine and cosine series of the form

nnC nnCsin-2nntCO S- 1 I 2nntcos- and I-- sin -.Ennnx , T n nnx, Tsin- sin-Since x,/l is, by assum ption , a rati on al fraction, we have to exclude theinde termin ate terms in which the deno min ator sin nnx,/l is zero, in order t o effectthe summation of these series.Th e form of the expression suggests that the g raph rep resenting the sum of eachof the series given above shou ld consist of a nu mb er of straight lines parallel to theaxis of time, separated by intervening discontinuities. The summation of theseries then reduces itself to finding the positions and magnitudes of thesediscontinuities. Taking, for instance, any one of the cosine series, we may assu methat its graph has a discontinuity 6, at either of the points t = f a, T/2n), a

discontinuity 6, at either of the points t = f a, T/2n), and s o on. This graph isrepresentable by the expression

whereLf,= -[dl sin nu, + 6, sin nu, +.etc.]nn

....o find the values of the quantities dl , 6, a,, a,... etc., we have the set ofequations,n cos nC/l6, s ina , + 6,s ina2 + ...=...-------2 sin nxO/1n cos 2nC/l6, sin 2a, + 6, sin 2a2+ ... = ...-2 sin 2nxo/ln cos nnC/l6, sin nu, + 6, sin na, + . . .= ...-2 sin nnx,/l

It is obvious that if C/l is an irrational fraction, the quantity(n/2)[(cos nnC/f)/(sin nnxo/l)] will never recur in value, howeve r much n be incre-ased. The number of independent equations is then infinite and the method ofevaluation proposed appears to fail altogether. If however C/l be a rationalfraction, and the two fractions C/l and x,/l be reduced to their lowest com mon


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denominator, the expression (n/2)[(cosnnC/[M(2sinnnxo/l)] will recur when n isincreased by any multiple of twice this common denominator. To enable theequa tions to be satisfied, the qu antities sin nu,, sin nu,, etc. should similarly recur,and the angles a,, a,, etc. must therefore all be multiple s of n divided by th e lowestcom mo n denom inator of the two fractions C/1 and x,/l. The num ber of unknow nquan tities (a,, 6,, etc.) to be evaluated is the same as the num ber of independ entequations available, and it is thus possible to determine 6,, d2 , etc. completely.The several series contained in the expression for the frictional force may thu sbe added up and their sum represented graphically. It is obvious that thefrictional-force curve assumes the least complicated form, that is, has fewestdiscontinuities when the initial positions C,, C,, C,, etc. of the discontinuities inthe velocity-diagram of the string coincide with the nodes of the principalmembe r of the missing series of harm onics . For , the fraction s x,/l and C,/l, C,/l,C,/l, etc. would then h ave the smallest possible comm on deno min ator. It may benoted that the method given above for drawing the frictional-force curve isapplicable in the general case when the vibration-curve at the bowed pointconsists of any num ber of straight lines forming a contin uou s "curve," and is thu snot restricted to the cases in which the velocities in the forward and backwardmovements are both constant and uniform. If these velocities are constant anduniform, the discontinuities d,, d,, etc. in the velocity-diagram of the string are allequal t o on e ano ther (see section IV).When the vibration-curve at some one point on the string (not necessarily a tthe bowed point) is a simple two-step zig-zag, the calculation of the form of th efrictional-force curve becomes particularly simple. When the motion at thebowed point is itself of this type, the expression reduces to the form

K(v, - ,) (- 1)" sin n n o 2nnt1 nnx, CO S-2T n sin2- T '1

Whe n the motio n at a point x , w hich is not the bowed point is of the two-stepzig-zag form, the expression for the frictional force may be written a sK(vL - ',) (- 1)" sin nn o' 2nntcos-2T nnx, nnx, T 'n sin- in-1

the quan tities v>,v', and w' having reference to the mo tion at the point x,. The twoformulae given abo ve may be readily verified from (22) by analysing the m otionpostulated at the bowed point and substituting the values thus obtained for thequantities B,. Figures 16, 17, 18, 19,20 and 21 in the text represent n o fewer tha n45 curves for the frictional force calculated for various cases from the expressionsgiven above, the vibration-curve a t the bowed p oint which forms the basis of thecalculation being shown alongside for comparison. Figures 16 to 19 represent


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306 c v R A M A N : ACOUST I C SPosition ofbowed point

Value of u,at the bowedpoint

Figure 16. Frictional-force curves and motion at the bowed point.

cases in which the vibration-curve at the bowed point is assumed to be of thesimple two -step zig-zag type, and figure 20 represen ts cases in which it is take n tobe a four-step or a six-step zig-zag, though at so me oth er point on the string thevibration-curve is a two-step zig-zag.As examples of the method of calculation, we may consider a few of the casesillustrated in the figures. Let the string b owed a t a distance of one-seventh of itslength from one end. From figure 13 it is seen that in this case o = +. T he


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M E C H A N I C A L T H E O R Y O F T H E V I B R A T I O N S O F B O W E D S T R I N G S 307

expression for the force then reduces to

.. - n sin-Th e positions of the discontinuities in the frictional-force curve must thereforebeWe then get the set of equations

of which only the first six are in depe nden t (i.e. n = 1 , 2, 3,4, 5 or 6). On writingdow n the equations, it is found a t once that 6 , = 6-3= 65 = 0. Multiplying bothsides of the equ atio n by sin nn/7, simplifying and utilizing the re lation

we find that 2kvBd2:64:d6 as 1 : 2 : 3 , and 6, =- n T 'Th e frictional-force curve for this case is shown in figure 1 9 . As another example,we may take the position of the bowed point to be 21/11 and that w = 1 / 1 1 .(Referring to figure 1 3 , it will be seen tha t this gives a com bina tion of the fifth an dsixth types of vibration.) O n writing down t he equ atio ns as before, it is found t ha t6, = 6, = 6, = 6, = 6, = 0 and that the equations reduce to the form

. 2nn 4nn 6nn 8nn 1Onn62sln-+~4sin-+b6sin-+6,sin-+b,,sin-1 1 1 1 1 1 1 1 1nn1 1kv, sin-

= ( - 1 ) " - I 1 12nn '2nT. sinZ-1


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30 8 c v R AM A N : A C O U S T I C SMu ltiplying b oth sides by 2 sin2 2n7c/l I) , i.e. by (1 - os (4n lrlll)) and simpli-fying, the equ ation s may be easily solved by grou ping to gether all the terms w hichhave a com mo n coefficient sin 4 1 , all those which have the coefficient sin 2741 1and so on. The solution is found to be 6 2 : 6 4 : 6 6 : 6 8 : 6 1 0 as 3:- :9 : - 0 :15 and 6 ,=6kv,/.rrT. On plotting, these values, a very interesting figure is obtained inwhich the fifth and sixth harmonics a re very prom inent an d by their sup erpo-sition, give an app eara nce similar to tha t of "beats." (see figure 19) . All the curv esshown in figures 16 to 21 were calculated by metho ds closely analog ous to thoseemployed in these two exam ples. They were all drawn to exactly the same scale,the velocity of the bow being taken to h ave a fixed value, and the curves are thusall strictly comparable.

Posi t ion ofb o w e d point Value o fw 01 thebowe d point



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From a scrutiny of the 45 frictional-force curves shown in figures 16 to 21, anum ber of impo rtant generalisations can be arrived at, which we now proceed toconsider.Th e curves shown in figures 16 to 21 a re of special interest in many ways. Th e 29curves shown in figures 16 to 19 all deal with cases in which the bow is applied a t apoint of rational division and the mo tion a t this point is a two-step zig-zag, thevalue of w for which has either 1 or 2 as its numerato r. I t will be seen that, in allthese cases, the frictional force has a m inimum constant value during the epochsat which the bowed point slips past the hairs of the bow. It is also seen that insom e of these cases the frictional force has the maximum or a maximum value justbefore the slipping begins an d just after it is over. But this is by no means a generalrule, for in 12 ou t of the 29 cases, the frictional force before and after this epo ch isnot a maxim um at all. Furthe r, in all the cases in which the bow is applied at apo int of aliqu ot div ision, e.g. 1/2,1/3,1/4,1/5 or 116, etc. the fricti on is a max im um atthe middle of the stage during which the bowed point is carried forward by thebow. The nine curves of this kind shown in the series all bear a strong familyresemblance to one ano ther. Similarly the frictional-force curves for th e bowedpoints 2/15, 2117, 2119, 21/11, etc. are worthy of careful study and inter-comparison. When the value of w for these cases is 115, 117, 119, 1/11, etc.respectively, the curves have a distinctive form in which the two chief harmo nicshaving a node on either side of the bowed point are specially prominent. Butwhen w =2/5,2/7,2/9,2/1 1, etc. respectively, the form of the curve is more closelyanalogous to those for aliquot points of division referred to above. This isevidently because the fundamental is more prominent when w has the largervalue in such cases.Krigar-Menzel and Raps rem ark in their paper as follows: "It is clear from alltha t has been said that the idea of the mechan ical action of the bow which we formis in all the cases the sam ea s that which has already been described by Helmholtz.The bowed point adheres to the rosined hairs of the bow and is carried forwardwith a constant velocity equal to that of the bow. This situation accounts for theascending line of moderate slope in the vibration-curve at the bowed point.Finally, throug h the increasing tension of the string, the adhering po int b reaksloose and glides dow nward s against the bow under str ong friction with a constan tmaximum velocity till the cycle commences to repeat itself anew." If Krigar-Menzel and Rap s were quite correct in their explanation of the "b re ak ~n goose ofthe bowed point from the hairs of the bow" as due to the increasing tension of thestrin g, we sho uld hav e expected to find that the friction of the bow is a maximu mimmediately before the release takes place. As we have just seen, this is by nomeans generally the case, and we are therefore forced to conclude that the purekinematics of the motion is also a factor in determining the release oft he string bythe bow. As we shall see presently, Krigar-M enzel and Raps were also wro ng intakin g the velocity of slipping against the hairs of the bow t o be uniform . It is ingeneral only approximately uniform or even largely non-uniform, and such


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310 c v R AM A N : ACOUST IC SPosition ofthe bowed

pointValue ofw at thebowedpoint

Figures 18. Frictional-force curves and motion at the bowed point.

variations of the velocity have a n imp ortan t significance and result.Referring again to the curv es shown in figu res 16 to 19, it will be seen tha t in allcases, the harmonics which have a node near the bowed point are prominent inthe frictional-force curve. This is what might be naturally expected from the formof the expression for the force.Further, in all cases when w is a small fraction, the frictional-force curvesbecom e very steep, that is, the friction becom es grea ter. If this friction exceed s themaxim um statical value, the mo tion ceases to be possible. We thus see that if thevalue of w is to be small, the pressure of the bow must be considerably increased.For instance, from the last five cases shown in figure 19 it is clear that the


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P os i t i on ofthe bowedpoint

Volue of or ott h e bowedp o i n t


minimum pressure of the bow must be increased as the bowed point app roach esone en d of the string. Similar effects are als o noticeable in the cases in which th ebow closely approaches importa nt nodes such as 112, 2/15, 113 or 114.Fo r any given position of the bowed po int, the larger value ofo means smallerfriction and therefore a reduced minimum pressure. When the bow coincidesexactly with an imp ortant nod e such as 112 or 113, the friction becomes extremelysmall an d the pressure of bowing necessary is therefore very low indeed.Passing o n to the cases in which the motion at the bowed point is still assumedto be a two-step zig-zag, but the value of o as 3 or some larger integer as itsnum erator, we see a t once a rema rkable difference (figure 20). Th e rictional forcehas no longer a constant uniform value while slipping ta kes place. 7his is inconsistent


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Position oft he bow edpo i n tValue of wa t the bowedpoint

Figure 20. Frictional-force curves and motion at !he bowed point. [Incompatible cases.]

with the assumption that the v elocity ofslipping is constant, and WPare thus orced toconclude that the v elo city of the bowed point during the slipping sta ge is necessarilynon-uniform in a greater o r less deg ree , when the pressure ofb owi ng is such that thevalue of w is greater than 2/r, th e rZ h arm onic being the principal memb er of themissing series.We see, therefore, that for the particular set of dam ping coefficients assumed , astrictly uniform v elocity of slipping is only possible when the bow is applied w ithsuch pressure tha t the nume rator o fw is either 1 or 2, and this may not be possiblea t all if the deno minator r is too large. Fo r othe r laws of damping , we wou ld get aneven m ore un favourab le result, that is, strict uniformity is never possible, thoughpractical uniformity may be attained in a number of cases.


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M E C H A N I C A L T H E O R Y O F T H E V I B R A T l O N S O F B O W E D S T R I N G S 313Th e foregoing results may, with ad vanta ge, be discussed a little more in detail.Ta ke the case in which the bow is applied exactly at the point 113. Th e value ofw isan d the m otion at the bow ed p oint is practically a perfect two-step zig-zag. Ifnow th e bow be removed to a point a little on one side of the node, say to the point161149, we find on a reference t o f igure 14 that if we ex clude the c ases inwhich the m otion at the bowed point is a four-step o r six-step zig-zag, etc. the onlypossible values of w are &,3 nd3.he value is evidently much too small tobe easily elicited. The v alue a$ is evidently of importa nce as it corresponds to thefirst principal mode of vibration of a bowed string. A very small pressure ofbowing is obv iously sufficient to elicit this type. But from what has already beensaid, it is evident th at for this value of o we cannot have strictly uniform velocityof slipping of the bowed point in the backward motion, though in its forwardmo tion, the velocity may be exactly equal to th at o fth e bow. If the third, sixth andoth er harm onics of the missing series are actually restored in the mo tion in nearlytheir pr oper amplitudes (n either more no r less), then the velocity in the backw ardmotion may be practically uniform, otherwise not. Thus we see that this non-uniformity of slipping and the restoration of the missing harmonics are closelyconnected with one another. If the bow is removed f urther still from the po int oftrisection a nd is applied a t, say 51/16, the possible values of w for a two -step zig-

B o w e d at Remarks

&-12 2 nd type&---A/-

Figure 21. Frictional-force curves and motion at the bowed point. (Four and six-step zig-zags).


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314 C V R A M A N : ACOUSTICSzag motion a t the bowed point w ould be&, &an d A.Th e value&may be elicitedif the pressure is sufficient. If the pressure is sufficiently reduced we may get thefirst principal type of yibration for wh ich w =&, but this would be modified bythe non-uniform slipping which inevitably occurs in this case.Figure 21 show s the frictional-force curves and the motio n at the bowed pointfor six specially selected cases in which th e latte r is a four-step zig-zag, an d o necase in which it is a six-step zig-zag. In the four-step zig-zags, the velocity isassumed to be constant an d the same in both stages during which slipping occurs,an d the frictional-force curves are seen to be entirely comp atible with this, as thefriction is constant and has the same value during bo th the stages. It mu st beund erstoo d th at this result holds good only in the special cases considered an d insome others, but not generally. In the case of the six-step zig-zag shown, thefrictional-force curve is evidently incom patible with it, as the friction in on e stageis less than in the two others.Th e form of the frictional-force curves for other cases in which the m otion atthe bowed point is representable by a six-step or a n eight-step zig-zag, etc. may beinvestigated by the method described above. In the great majority, i f not all, of

such cases, it would no doubt be found that the form of the frictional force isinco mp atible with the perfect parallelism a nd straightness of the descending linesof the vibration-curve at the bowed point. In other words, the velocity of thebow ed point w ould no t be quit e constan t and uniform in all the stages in which itslips past the bow.

Transition from the rational to the irrational modes ofvibration and vice versaTh e preceding investigatio n of the form of the frictional-force curves has alreadyenabled us to form a general idea of the conditions which under the kinematicaltheory o utlined in sections V to XI1 breaks dow n to an appreciable extent. Wehave seen that this departure from the comparatively simple vibration-forms sofar investigated is due in the first instance to the velocity of slipping at th e bowedpoint becoming non-uniform, and is closely connected with the progressive (asdistinguished from the discontinuous) restoration of the missing harmonicswhich occurs when the bow is gradually moved away from some impo rtant nodalpoint. We are thus led to investigate the transitional forms of vibration, as theymay be called, which are intermediate between the irrational types discussed insections V to IX and their ration al modifications wo rked o ut in section X. Thesetransitional forms may be found, a priori, by a general method which will be bestunderstood by considering some specific cases.We may confine attention at first to the cases in which the motion a t the bowedpoint is a two-step zig-zag or a close approximation thereto. Let the bow beapplied at some point intermediate between, say, 115 and 116, with pressure and


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M E C H A N I C A L T H E OR Y O F T H E V I B R A T I O N S O F B O W E D S T R I N G S 315velocity approximately those necessary to elicit the first type of vibration (seefigure 13). When the bow is applied exactly at 1/5, the vibrational form of thestring is defined by on e large positive discontinuity in the velocity-diagram an dfour small negative discontinuities these are so situated th at the mo tion a t thebowed po int is a perfect two-stepzig-zag a nd in the resulting motio n, the 5th, 10thharmo nics etc. are abse nt. Similarly when th e bow is applied a t 116, we have o nelarge discontinuous change of velocity and five small ones: the 6th, 12thharmon ics etc. are absent. When th e bow is gradually moved from 1/5 to 1/15, he5th, 10th harmonics etc. gradually reappear, an d the 6th, 12th harmonics etc.become feebler and disappear. In these transitiona l stages, the forward mo tion ofthe bowed point m ust still take place with uniform velocity equ al to th at of thebow, but the velocity in the backwa rd m otion need no t be strictly con stant. Weare thu s naturally led to assume that the transitional modes should be capable ofbeing deduced from a velocity-diagram with one large discontinuity and fivesmall ones. O n proceeding to find the relation tha t the positions and magnitudesof these discontinuities must satisfy in order that the bowed point may have auniform velocity in the forward motion, it is found by trial that the velocity-diag ram m ust be similar in certain respects to tha t for the case of bowing at lj6.The five small discontinuities should all be equal to one another, and theirpositions o n the string with respect to the bowed poin t are perfectly determinate.If the m otio n of the string is assumed to be of the perfectly symm etrical type, th atis involving on ly sine compon ents, th e position of the sixth (large) discontinuity isalso unique ly determined . If, further, the inc lination of the lines in the velocity-diag ram an d the velocity of the bow a re assigned specific values, the magn itude ofthe discontinuities is known, and the vibration-form is completely fixed. Ingeneral, however, an asymm etrical vibration-form is possible, and it is found t ha twhen the velocity of the bow is given, the form of vibration has a possiblevaria tion determ ined by (a) the position assigned to the large discontinuity whichmust lie within certain limits and (b) the inclination of the lines of the velocity-diagram to the x-axis.Except when th e slope of the lines of the velocity-diagram is such tha t the fivesmall discontinuities all vanish (in which case the transitional form becomesidentical with the first irrational type of vibration), all the transitional formsreferred to in the preceding paragraph have the common feature that thevibration-curve at the bowed point instead of being a perfect two-step zig-zag,con sists of a zig-zag in which the ste ep descending line is not perfectly straig ht b utconsists of three (or in th e extrem e cases two) straight lines, all of which how everare m uch steeper than the ascending line which is perfectly straight. Ten cases oftransitional forms thus worked ou t, a priori, for bowed points lying between 114an d 1/6, are sho wn in figure 22, in which the corresponding vibration-curvescalculated for points near the end of the string are also shown.It will be seen that in all the ten curves, the vibration-curve at the bowed pointclosely app roach es the simple two-step zig-zag form an d in som e cases (e.g. the


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31 6 c v R A M A N : A C O U S T I C SBowed andobserved at Observed a t

3 - 1 2-3 13

3- 1617 1AA/ w J 7Vibrati o n-curve at Corresponding vibrationthe bowed point curve at a point nea r theend of the stringFigure 22. Transitional forms of vibration. (Mod ifications of the first irrational type).

curves for bowing at 31/16 and 31/17) is nearly indistinguishable from it. Thevibration-curve for the point near the end of the string, however, shows thedifferences clearly enough.Most of the curves on the right-hand side of figure 22 are seen to beasymmetrical; in other words, the appearance of the curves is not the same if th e


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page w ere held upside d own. T he three curves for the case of bowing a t 2/11 1 areparticularly instructive. T he first of these three curves (which is an extrem e case)shows a shar p peak at the top and five horizontal steps, below. The second (whichis the sym metrical case) show s six steps, of which the first and the last are s horte rthan the rest. T he third curve (which is also a n extreme case) shows five horizontalsteps above, and one sharp point below.As an exam ple of the method of draw ing these curves, we may take th e case ofbow ing at 31/16. Th e velocity-diagram for the case is draw n in a mann er sim ilar totha t for bowing at 116, the p ositions of the discon tinuities being as follows: Th efirst small discontinuity is initially a t the en d of the string (x = 0).The point ofbowing is at 31/16, and two other small discontinuities are therefore situate at61/16. Another pair must be at 121116. For, 61/16 + 31/16 = 12 /16 - 1/16. Theinclination of the lines of the velocity-diagram and the position of the largediscontinuity are arb itrary , though the latte r must initially lie between the limits141/16 and I. If the large discontinuity is initially at 141116 and belongs to thepositive wave, we have o ne extrem e asymmetrical case in which the descendingpa rt of the vibration-curve a t the bowed poin t consists of two straight lines only.[For, (I- 141/16)+ 1- 1/16)= (121116+ 31/16)]. If the large discontinuity isinitially at the end 1 of the string, we have the symmetrical case in which thedescending part o fth e vibration-curve at the bowed point consists of three straightlines. When the large discontinuity is initially at 141116 and belongs to thenegative wave, we have the other extreme asymmetrical case: [(141/16 + 31/16)= (I- 121116)+ 1- 1/16)]. Th e slope of the lines of the velocity-d iagram an d th evelocity of the bow determ ine the m agnitudes of the discontinuities. If the lineswere horizontal, the mag nitudes of the discontinuities would be the sam e as if thestring were bowed at 116. Whatever the slope might be, and wherever the largediscontinuity may lie initially within the limits referred to, the descending motio nat the bow ed p oint occup ies exactly 3/16 of the whole period of vibration. In oth erword s, th e trans itiona l form also obeys the kinematical law given in equa tion (17),i.e.w = (nx,/l), n being put e qua l to 1, as this is a modification of the first irratio naltype. The sam e kinem atical law is satisfied for all the ten cases shown in figure 22,and is, in fact, true for all the trans itional modifications of the first irratio nal type,

(n = 1) .In the preceding treatment, we have obviously neglected to take into accountthe existence of certain n oda l points of minor im porta nce lying between 115 and116 or between 114 and 115. For instance, between 115 and 116, we have the nodalpoin ts 31/16,21/11 and 31/17. Fo r a m ore com plete theory, we have also to ta keinto account the gradu al dropping out and reappearance of the 16th, the 1,lthandthe 17th harmonics and their trains of harmonics of high order, as the bow isgradually moved across from 115 to 116. As however these harmonics are of verysmall amplitudes, the correction may be neglected altogether if the bow be notapplied exactly at any one of these nodal points. If it is applied exactly at such

nod al poin t, the correction may be effected by the metho d described in section X


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31 8 c v R AM A N : ACOU S T I C Sfor rational po ints of bowing, and in an y case, the correction so made would haveno effect if some other node of the missing harmonic is chosen as the point ofobservation. When, however, the intervening nodal point is of importance, amore accurate method of correction is required. For example, when the bow ismoved step by step from 1/2 to 2115, the gradual reappearance of the secondharmonic and the gradual dropping out of the fifth harmon ic have bo th to betaken i nto account. This may be don e by draw ing a velocity-diagram similar tothat for bowing at 115, tha t is, with on e large and four small discontinuities, andreadjusting the positions of these discontinuities so as to give a uniform velocityin the forward motion a t the bowed point. Figure 23 (first three g raphs in th e left-

Bowed at O bser ved a t-Velocity- diagrams vibratio n-curv e at the bowedpoint and a ta point nea r theend of the string

Figure 23. Transitional forms of vibration. (Modifications of the first irrational type).


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hand column) shows three velocity-diagrams of this type, and against each ofthese diagrams is shown the corresponding motion at the bowed point and thevibration-curve at a point near the end of the string. The gradual transition fromthe type corresponding to a string bowed at 112 to one bowed at 2115 is clearlyseen.

The graphs in figure 23 for the case of bowing at 31/8 show the dropping out ofthe 3rd harmonic. One of the cases is of the symmetrical type and the other of theasymmetrical type. In the five vibration-curves at the bowed point shown on theright-hand side of figure 23, it will be seen that the ascent with the bow is madewith a uniform velocity, but that the velocity of descent is not constant.

Similarly when the bow is applied at some point intermediate between 113 and21/7, the velocity-diagram of the transitional mode may be drawn in a mannersimilar to that for bowing at 117, that is, with one large and six smalldiscontinuities, the positions of these being readjusted in the manner requisite togive a uniform velocity in the forward motion at the bowed point. Thesevibration-forms would show the gradual dropping out of the seventh harmonic asthe bow approaches the point 2117. There is however an alternative set oftransition-forms in which the velocity-diagram is similar to that for bowing at 1/4,i.e. has one large and three small discontinuities, and these forms securepredominance as the bow is removed farther and farther from 113 towards 1/4.Since both types of transition-forms give a forward motion with uniform velocityat the bowed point for identically the same fraction of the period, they may besuperposed in any desired proportion so as to secure the specified velocity at thispoint. In this way, the various modifications of the first irrational type ofvibration obtained by bowing at points between 1/3 and 114 may be accuratelydrawn. Results approximating to the truth may however be obtained withoututilizing the principle of superposition of transitional forms here suggested, bymerely choosing the form with seven discontinuities for bowed points lyingbetween 1/3 and say 31/10, and the form with four discontinuitiesfor bowed pointslying between 31/10 and 114. The curves thus drawn are shown in figure 24.

Of the five cases shown in figure 24, the second, third and fifth are of theasymmetrical type and the others are symmetrical forms.In all these transitional forms, the descending motion at the bowed point is not

executed with a uniform velocity but consists of two or three stages in which thevelocities are different. The relative duration of these stages and the velocity of thebowed point during these intervals are capable of adjustment within aconsiderable range, and any such alteration involves important changes in theamplitudes and, for the asymmetrical types, also in the phases of the harmonicswhich have a node near the bowed point. There is thus, prima facie, reason toconsider that these transitional forms should be capable of adjusting themselvesin such manner as to secure the maintenance of the motion, and also its stability,over a considerable range of bowing pressures and velocities.A further mode ofadjustment is provided by the principle of superposition of transitional forms


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320 c v R A M A N : A C O U S T I C SBowed at Observed at

910

Velocity-diagmms Vibration-curves at the bowedpoint and at a point near theend of the string

Figure 24. Transitional forms of vibration. (Modifications of the first irrational type).

referred to previously. The theory may be further elaborated by working ou t theform of the frictional-force curve in a representa tive set of transitiona l m odes ofvibration by the general method described in the preceding sub-section, andproving its compatibility with the mo tion a t the bowed point corresponding toeac h mod e. This detailed inv estigation m ust be reser

1918 bull indian assoc cultiv sci v15 p1-158 - 2

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