[LANG] HIGH FREQUENCY VIBRATIONS, ETC. 165 



The solution of which is 



(nx nx \ 



A cos — j^ B sin ^- J cos (nt-{-e), 



where A and B are constants depending upon boundary conditions. 

 Thus if the origin is taken at one end of the bar then 



— =0 for x — o and x = I, 

 dx 



where / is the length of the bar. So that B = o and 



nl nl 



sin — = o or — = sir 

 c c 



where 5 has all integral values including zero. 



Ti SC 



The frequency N = -— so that we have N= — , or the frequency 



2'K 21 



varies inversely as the length of the bar if the velocity c is constant, 

 and the frequencies of the various modes may be calculated by 

 giving .y its integial values. 



The nodes (3' = o) are given by thos'e values of x which make 



cos I — ^ ) =^- ^^^ graves't node {s = 1) gives, therefore, a node at 



the centre and we see that a bar clamped at its central point is cap- 

 able of free vibrations. The only mode dealt with in this work is 

 that corresponding to 5 = 1 but for bars longer than one meter evi- 

 dence of vibrations corresponding to the nodes {s = 2) was also 

 found. 



Turning now to a discussion of the stationary waves in the air 

 in the glass tube we see that for free vibrations of the air column 

 exactly the same mathematical relation would apply, provided that 

 both ends of the tube were open. There would be a node of displace- 

 ment at the centre of the tube. 



If both ends are closed the boundary conditions lead to the 

 formation of a node at each end only, for the gravest mode. 



However, in the arrangement under discussion, while the vibra- 

 tion of the bar is free, at least after the first few oscillations, the air 

 column vibration is never free, but is due to an impressed force of 

 the form y = A cos {nt-{-e) situated at .r = 0. If the tube be closed 

 at x = l the motion of the gas will be given by 



A n{l — x) 



y= , sin cos (n/ + €,) 



. nl c 



stn — 

 c 



where the symbols are as before. 



