THEORY OF VIBRATIONS 15 



Since (6) is of the same type as 6 (1), with 



n* = c/a, (7) 



the variation of q is simple-harmonic, say 



q = C cos (nt + e), (8) 



with the frequency 



AT 



Moreover, since the displacement of any particle of the 

 system along its path, from its equilibrium position, is pro- 

 portional to q (being equal to aq in the above notation), we see 

 that each particle will execute a simple-harmonic vibration of 

 the above frequency, and that the different particles will keep 

 step with one another, passing through their mean positions 

 simultaneously. The amplitudes of the respective particles are 

 moreover in fixed ratios to one another, the absolute amplitude, 

 and the phase, being alone arbitrary, i.e. dependent on the 

 particular initial conditions. 



The kinetic and potential energies are respectively 



T = i of = \ n*aC* sin 2 (nt + e), ) 

 V= I c(? = IcC 2 cos 2 (nt + e), j 



the sum being 



T+ V=\n\iV* = \cG\ ............... (11) 



in virtue of (7). Since the mean values of sin 2 (w + e) and 

 cos 2 (nt -he) are obviously equal, and therefore each =J, the 

 energy is on the average half kinetic and half potential. 



The application of the theory to particular cases requires 

 only the calculation of the coefficients a and c, the latter being 

 (in mechanical problems) usually the more troublesome. In 

 the case of a body attached to a vertical wire, and making 

 torsional oscillations about the axis of the wire, a is the moment 

 of inertia about this axis, and c is the modulus of torsion, 

 i.e. cq is the torsional couple when the body is turned through 

 an angle q. 



Again in the case of a mass suspended by a coiled spring 

 (Fig. 4), if we assume that the vertical displacement of any 

 point of the spring is proportional to its depth z below the 



