24 DYNAMICAL THEORY OF SOUND 



as a simple vibration whose period is 27T/-J- (% + w 2 ), and whose 

 amplitude oscillates between the limits and 2a, in the time 

 7T/-| (n^ n z ). This is illustrated graphically, with x as ordinate 

 and t as abscissa, in Fig. 10, for the case of n^ : n z = 41 : 39. 



11. Free Oscillations with Friction. 



The conception of a dynamical system as perfectly isolated 

 and free from dissipative forces, which was adapted provisionally 

 in 4 10, is of course an ideal one. In practice the energy of 

 free vibrations is gradually used up, or rather converted into 

 other forms, although in most cases of acoustical interest the 

 process is a comparatively slow one, in the sense that the 

 fraction of the energy which is dissipated in the course of a 

 single period is very minute. 



To represent the effects of dissipation, whether this be due 

 to causes internal to the system, or to the communication of 

 energy to a surrounding medium, we introduce forces of resist- 

 ance which are proportional to velocity. The forces in question 

 are by hypothesis functions of the velocity*, and when the 

 motion is small, the first power only need be regarded. 



The equation of free motion of a particle about a position of 

 equilibrium thus becomes 



,, d z x rr dx 



M M = - Kx - R di< .................. < J > 



where R is the coefficient of resistance. If we write 



k, .................. (2) 



weget 



The solution of this equation may be made to depend on 

 that of 4 (3) by the following artifice f. We put 



//IX 

 (4) 



* We shall see at a later stage (Chap. VIII) that the resistance of a medium 

 may introduce additional forces depending on the acceleration. These have 

 the effect of a slight apparent increase of inertia, and contribute nothing to 

 the dissipation. It is unnecessary to take explicit account of them at present. 



t Another method of solution is given in 20. 



