Vibrations of tivit cited and stroked Strings. 213 



can only be designated as approximately accordant. On this 

 account we will now give another treatment of the motion of 

 the violin-string, more closely agreeing with the facts, for the 

 case that the bow acts at a node-point of an upper tone. 



15. For our purpose it is expedient at first to let the sup- 

 position made in no. 8 stand, but now to take into account the 

 occurrence of node-points, which has hitherto been excluded. 



The occurrence of node-points is usually settled with the aid 

 of Fourrier's series. This, however, is not necessary ; for 

 d'Alembert's solution conducts to similar results. The latter 

 can be applied in our case also ; we can therefore assume as 

 known that the points 



2 T 3 T m-lj 

 x— — L, #=— L, .... #= L 



m m m . 



must be node-points as soon as the point x— — is a node, and 



m 



that every portion of the string vibrates separately as an inde- 

 pendent string, each two adjacent parts vibrating symmetri- 

 cally to one another. If, then, A is a constant and T the vibra- 

 tion-period of a division of the string, we have, according to 

 (27), 



for Q< „.<-■— 



y=A Gr-*) for 



y=kxQ f -t) 



o<4 



< mx 

 = 2l? 



mx < t 

 2L = T 



^ 2L— mx 

 = 2L 



2L— mx 

 2L 



ir= lj 



, (32) 



2L 

 where now —p?, = a, if a denotes the fundamental constant of 



mi ? 



the differential equation (2). 



Further, for — < x< — : — 

 m= = m 



for 0<™< 



y=- A (S- a )S-0" 



»--Ag-.)cM> -S<^i. 



T = 2L } 

 2Jj—mx < t < mx 

 ~ 2L =T=2L 



(33) 



