THEORY OF UNDULATION —VIBRATION. 147 



PAI£T II, 



UNDULATORY THEORY. 



\S I. VIBRATION. 



In order to imderstancl the mode in wliicli undulations are propagated through 

 an elastic medium, it is necessary to attend first to the subject of vibration. If 

 a body be so held in equilibrium between opposing forces, as that, when dis- 

 placed from its position of repose, it is urged to return by a force always pro- 

 portional to its distance from that point, then the time occupied in returning, 

 supposing it to be left at liberty to obey this impulse, will be the same, what- 

 ever may be the magnitude of the displacement. Moreover, if the extent or 

 amplitude of the displacement be taken as a radius, and a circle described about 

 the point of repose as a centre, the velocity of the body in its returning move- 

 ment will be proportional, at every point of its path, to the per- 

 pendicular to the path drawn from that point and cut off by the 

 circular arc. Thus, if be the point of repose, and CA the 

 amplitude of displacement, the force urging the body to return 

 being proportional to CA, CF, CH, &c., when the body is at the 

 jDoints A, F, and II, then the velocities at these several points will 

 ^^S 27. |,g proportional to zero at A, to EF at F, and to Gil at H. 

 These elementary propositions in physics, which admit of easy demonstration,* 



*The demonstration refeiTed to is the following: Piit CA = «, CII = 3;. Put also t for the 

 time from the beginning of movement, v for the velocity, and jj- for the force drawing the body 

 toward C at the distance unity. Then if dx he the small sjiace described in the small time 

 dt, vdt= — dx; dx being negative because, when v is increasing, it diminishes x. 



Also, if dv be the small increase of velocity in the time dt, we shall have the force, at the 

 distance x, equal to p-x, and p-xdt^dv. Whence, 



tdp = —p-xdx ; and v" = — p-x^ -j- C. 



But when t' = 0, x = a: consequently, 



v"^p-(a- — Z-) ; or, v^=pVa- — x- = (in the figure) ^^(JA- — CH-=;j.GH. 



Therefore v is proportional to GH. Also, the time of vibration is constant, irrespective of 

 the value of a. For, substituting in the first ec^uation the value of v just found, 



pVa- — x-.dt = — d'x; or, dt ^= — . , 



pVa^ — x'^ 



1 -'a; 

 This gives t= — -sin - +C. 

 p a 



When < = 0, z^a: whence i = - 1 90° — sin _ ] =_ cos -. 



p^ a) p a 



Now, taking t' and t" for two particular values of <, one at the beginning and the other at 

 the end of a complete double vibration, l" — t' will be the duration of the vibration, and will 



be measured by the increase which the arc cos - undergoes during one complete series of 



a 



all the possible values of ~ in diminishing and in increasing order — that is, from z = 4-a to c 

 a 



^=-\- a again. Hence, putting T=t" — <'= the duration of a vibration, we have, 



1 2tt 



T^- (277 {m-{-\) — 27rm) = — , which is constant. 



P P 



The symbol a disappears from this expression, showing that the duration of the vibration is 

 independent of the amplitude. 

 We have here also a proposition essentially the same as that demonstrated bj a different 



