BARS 133 



a permanent tension P is merely to add a term P?/' to the 

 equation (13) of 45, so that 





where c z -P/pa> (2) 



This equation has been employed to estimate the effect of 

 stiffness of a piano-wire on the sequence of proper tones, but 

 the matter is complicated by the uncertainty as to the nature 

 of the terminal conditions. A wire, where it passes over a 

 bridge, cannot be quite accurately regarded either as merely 

 "supported" or as "clamped." The question will perhaps be 

 sufficiently illustrated if we consider a wave-system 



7] = Ccosk(ct-x) (3) 



on an unlimited wire. We find, on substitution in (1) 



c 2 = c 2 +cr (4) 



E 



where c^ = .k z /c', (5) 



i.e. Cj is the velocity of transverse waves of length 2-7T/& on a bar 

 free from tension. We have seen that in the case of a piano- 

 string E/p is large compared with c 2 ; on the other hand K is 

 usually an exceedingly minute fraction of the wave-length. In 

 the graver modes of a piano-string this second influence pre- 

 dominates, and (Cj/Co) 2 is small ; the wave-velocity is practically 

 unaffected by stiffness, and the harmonic sequence is not 

 disturbed. It is only in the case of the modes of very high 

 order, where the length is divided into a large number of 

 vibrating segments, that a sensible effect could be looked for. 

 It has already been stated that in the pianoforte such modes are, 

 so far as may be, discouraged on independent grounds. In any 

 case it appears from (4) that the effect of stiffness is relatively 

 less important, the greater the value of c , i.e. the tighter the 

 wires are strung. 



51. Vibrations of a Ring. Flexural and Extensional 

 Modes. 



The theory of the vibrations of a circular ring is important 

 as throwing light on some later questions which can only be 



