582 



>i>UND 



regard* number i if holes ami rate of rotation. It U 

 evident that if the blast of air U broken up into 

 siu-ci- i\ e |Hirt inn- which have i-sm-d through, gay, 

 triangular instead <if tin- usual circular holes, the 

 JIII-IH of the pulses which build up the note will be 

 changed. And such a change U recognised at once 

 by the car. 



Mii-t instructive in this connection arc the laws 

 of vibration of stretched strings. If we fix one end 

 of a pretty long rope to a wall, and, with tin- other 

 ciul in the hand, keep it in a stretched condition 

 free of the floor, we may observe any slight disturb- 

 ance given to it running ana solitary wave along 

 \\ie rojie and l>ack again after reflection at the wall. 

 The tighter the rope is stretched the quicker will 

 this disturbance travel to and fro along it. It is 

 not difficult to show that the speed at which such 

 a disturliance or wave wilj travel along a stretched 

 cord depends on the tension ( T ) and on the mass 

 ( HI ) per unit-length of the cord, being given by the 

 simple formula c=v(T/)). Suppose we have 

 such a stretched cord of indefinite length, and that 

 a series of exactly equal waves are running along 

 it from left to right, as shown in tig. 2, <i, in which 

 the straight line indicates the undisturbed position 

 of the string. Now let there le propagated along 

 the string from right to left an exactly equal series 

 of waves, which, if existing alone, would throw the 

 string into some such form as shown in 6 (fig. 2). 





Fig. 2. 



The superposition of these two exactly similar 

 series of waves propagated in opposite directions 

 gives rise to a resultant motion indicated in c (fig. 

 2). Here the points ABCDEF, being once at rest, 

 are always at rest ; since, whatever l>e the displace- 

 ment due to the a waves, an exactly equal and 

 opposite displacement is produced by the o waves, 

 now and forever. Intermediate points, however, 

 will move up and down between the limits indi- 

 cated by the doited loops in c (fig. 2). The string, 

 in fact, will vibrate in Mgnmti whose ends are 

 fixed at the point* ABCDEK. The segments will 

 be, at any instant, alternately above and In-low 

 the undistiirlied position of the string. The motion- 

 less points are called nodt* ; and it is evident 

 that we nmv fix any two of them, and cut away all 

 the string lying lieyond these chosen nodes with- 

 out in any way affecting the motion of the parl 

 lying lietween them. If we fix two contiguous 

 nodes, for example A, B, we have a definite length 

 of siring vibrating as a whole. If A and C are 

 fixed we ;jet twice that first length of string vibrat- 

 ing in two segment*; if A and D, we have three 

 time the length vibrating in three segment*; if A 

 and E, four times the length vibrating in four seg- 

 ment* ; if A and F, five tunes the length vibrating 

 in five segments; and so on. Now all these are 

 (imply different ways of producing exactly the 

 name vibration, so that a note which is given by 

 one length of stretched string vibrating as a whole 

 may In- given by I. lengths of a similar string 

 similarly stretched, vibrating in L segments. But 

 we may have this string of length L itself vibrat- 

 ing as a whole. We nave merely to suppose the 

 oppositely directed series of waves to be L times 

 'inger than those shown in fig. 2 (a, 6). If n is the 



frequency of any vibration, and / the ware-length 

 or distance from crest to crest, it is easy to see that 

 a given ware will travel over a distance nl in one 

 second. That is, we have til = v = V(T/m), a 

 quantity depending only on the t. n-i,m and mass 

 of the string. Consequently, if we double the wave- 

 length we halve the frequency ; if we halve the 

 wave-length we double the frequency ; ami .-o on. 

 Generally then the frequency of vibration of a 

 string of given tension and density varies inveisely 

 as its length. This is the principle on which all 

 instruments of the violin and guitar types are 

 played. The player, by pressing the string down 

 with the finger at diiferent points, can shorten 

 the string in the required ratio, thereby producing 

 a correspondingly higher note. 



From what precedes we see that a stretched 

 string, which vibrates as a whole, say, 100 times 

 per second, can also vibrate 200 times per second 

 in two segments, :H) times per second in three 

 segments, and, in general, n hundred times in n 

 segments, if the segments are not too short. l!\ 

 lightly touching, without pressing, a violin-string 

 at the proper point, so that that point is made a 

 node, any of these higher notes (overtones) may be 

 obtained. This is a very common practice in play- 

 ing the violoncello. Kot only, however, may a 

 string be so made to utter any of the overtones, 

 but it is practically impossible to prevent some of 

 them sounding along with the fundamental note. 

 It is, in fact, upon the presence of these orertones 

 that the quality or character of the sound depends. 

 They give the form to the vibration. In fig. 3 we 

 see now the form of the wave is changed by super- 

 posing upon a given vibration the first and second 

 overtones, having frequencies twice and three times 

 the frequency of the fundamental tone. The musi- 

 cal relations of these orertones of stretched strings 

 are discussed under HARMONICS (q.r.). The har- 

 monics of a note are the simple harmonic vibrations 



1:3 



Fig. 3. 



into which, according to Fourier's analysis, any 

 steadily recurring periodic motion can be decom- 

 posed. The prime or fundamental tone is the first 

 harmonic ; and higher harmonics are all overtones. 

 But, as we shall see hereafter, all overtones are not 

 necessarily harmonies. 



Air-columns, such as we have in organ-pipes (see 

 ORGAN), vibrate according to laws very similar to 

 those which rule the vibrations of strings. The 

 frequency of the note is inversely as the distance 

 between two successive nodes. One essential 

 difference is that the ends of a stretched string 

 must , lie nodes, whereas in pipes one or lx>th ends 

 may lie loops, where the velocities experience their 

 maximum change and the pressure is invariable. 

 In the open organ-pipe Imth ends are loops, lietween 

 which one node at least must exist if a sound is 

 produced. This gives the fundamental vibration, 

 and may be diagrammatical!}- indicated, as in fig. 

 4, n. The wave-length is (approximately) double 

 the length of the tube. The second harmonic U 

 produced when two nodi-- intervene, as in b (fig. 4); 

 the third when three nodes intervene, as in c (fig. 

 4); and so on. The wave-lengths of these are 

 respectively the length of the tube and two-third* 



