BARS 117 



or places of zero stress, by sin (STTX/I) = 0. In the gravest mode 

 (s 1) we have a node at the centre*. 



On the principles explained in 16, 32 the most general 

 free motion of the bar, under the present conditions, may be 

 expressed by a series 



STTCt D . S7TCt\ S7TX 



cos j- + B 8 sin =- \ cos -y-, ...... (15) 



where 5 = 0, 1 , 2, 3, .... Thus if the bar be started from rest 

 in the state of strain defined by 



= /<*) [ = 0], .................. (16) 



we have B 8 = Q; and we infer that it must be possible to 

 determine the coefficient A a so that 



^ / A 



= 2,(A a 



(17) 



for values of x ranging from to /. This is the result referred 

 to by anticipation in 33. 



The longitudinal vibrations of bars or wires have hardly 

 any practical application of importance, except in some primitive 

 forms of telephone. As regards bars, the pitch is very high 

 compared with that of the transverse vibrations, which it is 

 difficult to avoid exciting simultaneously. Again, if we compare 

 the frequencies of longitudinal vibration of a tense wire with 

 those of the corresponding transverse modes, the ratio will be 

 that of the wave-velocities, i.e. of *J(E/p) to ^(P/pco), where P 

 denotes the permanent tension. If e be the extension due 

 to P, we have P = Ee .(o, and the ratio is l/\/e , which is 

 usually very great f. Longitudinal vibrations may be elicited 

 on the monochord by rubbing the wire lengthwise with a piece 

 of leather sprinkled with resin ; the resulting note is very 

 shrill. 



It is assumed in the preceding theory that the extension 

 and the accompanying stress are at any instant uniform over 

 the cross-section ; in other words, we have assumed that the 



* The case * = needs, in strictness, separate examination. It leads to 

 = AQ (l + at), which may be interpreted as an oscillation of infinitely long 

 period. If the mass-centre be at rest we have A =0. 



t This comparison is due to Poisson (1828). 



