220 Forms of the Vibrations of twitched and stroked Strings. 



appear in Neumann's drawings agrees with the formulae given 

 in no. 16*. 



For other points of the string both the descending and the 

 ascending portion of the velocity-curve are rippledf. For 



— L < a: < L for example, the velocity of the ascending 



motion fluctuates, according to no. 17, when B= —A, between 



the values 



AT m— 2i — 1 , AT m — 2i 



AL and AL ■, 



m m 



that of the descending motion between 



AT 2i , AT 2^ + 1 



— AL — and — AL . 



m m 



If the vibration-form considered be resolved into two elemen- 

 tary vibrations 97 and A77 (conf. no. 16), and the amplitude of the 

 central point of the string for the motion rj be denoted by P, 

 while the amplitude of the central p oint of any one part of the 

 string for the component At; is denoted by P', we shall have, 

 according to (28), 



. _ 8P p 8m 2 P' 

 A -LT' -^TT"' 



and therefore, when B= — A, 



m z 



The form of the string at a given time will be approximately 

 the same as in the case described in no. 13, except that for the 

 straight lines there occurring slightly rippled lines are to be 

 substituted. 



19. We may remark that Helmholtz gives the following 

 two equations in place of the three equations (27): — . 



8P rT 



y=g(L-*)*for 0<t<*±; 



8P rT 



They only agree with the first two of the equations (27) if 



* Disregarding isolated cases in which large)- secondary ripples appear, 

 and thus the motion is still more complicated. 



t These ripplings are recognized also in the following manner : — If a 

 bright violin-string vibrates over a dark ground, the field of vibration 

 appears to the naked eye uniformly bright ; but if observed by means of 

 a lens, there appear in it a number of brighter lines running parallel with 

 the string. 



