Vibrations of twitched and stroked Strings. 211 



The constant A is determined by the amplitude of any point 



AT 



of the string which is equal to ^-(L— x)x. Therefore, if the 



amplitude of the central point of the string is denoted by P, 

 we have P= — n— , and consequently 



A=fg. ...... (28) 



According to Neumann's experiments (conf. no. 8) P de- 

 pends on the place at which the string is stroked, and on the 

 velocity with which the bow is drawn across ; that is to say, 

 the velocity of the stroked point is, in ascending, equal to that 

 of the bow. Let V be this velocity, and a=X the stroking- 

 place ; then, according to (27) and (28), 



VLT 



This, however, holds good only when the stroking-place lies 

 pretty near one end of the string (conf. the end of no. 1). 



13. In detail, the motion of the string takes place in the fol- 

 lowing manner : — 



At the time £ = it is in the position of equilibrium. 



T 



If0<£<^, there is one point of the string for which 



T— st" would be < t; hence it consists of the two straight lines 

 To their point of intersection corresponds the abscissa 



It moves upon the parabola 



AT 

 2L 



AT 

 y=f±(L-aO* (29) 



T 

 from #=0 to #=L while t increases from to -~. 



For t— r% only the middle equation of (27) holds good ; 



we have therefore y = ; that is, at the expiration of the half 

 vibration-period all the points of the string pass through the 

 position of equilibrium. 



T 



For t > s the two equations 



