62 DYNAMICAL THEORY OF SOUND 



the most general free motion of the string may be regarded as 

 made up of two such wave-systems superposed. 



The form of the expression \/(P/p) for the wave-velocity is 

 to be noticed. As in all analogous cases the wave- velocity 

 appears as the square root of the ratio of two quantities, one 

 of which represents (in a -general sense) the elasticity, and the 

 other the inertia, of the medium concerned. 



A simple proof of the formula for the wave-velocity has 

 been given by Prof. Tait*. Imagine a string to be drawn with 

 constant velocity v through a smooth curved tube, the portions 

 outside the tube being straight and in the same line. Since 

 there is no tangential acceleration the tension P is uniform. 

 Also the resultant of the tensions on the ends of an element 

 Ss, at any point of the tube, will be a force PSs/R in the 

 direction of the normal, where R is the radius of curvature. 

 This will balance the "centrifugal force" p$s.v 2 /R (fv z =P/p. 

 Under this condition the tube may be abolished, since it exerts 

 no pressure, and we have a standing wave on a moving string. 

 If we now impress on everything a velocity v in the opposite 

 direction to the former, we have a wave progressing without 

 change of form, on a string which is otherwise at rest, with the 

 velocity \/(P/p). It will be noticed that this investigation does 

 not require the displacements to be small. 



The motion of an unlimited string consequent on arbitrary 

 initial conditions 



y=4>(x), y = + (x\ [* = 0], ............ (4) 



may be deduced from (1), but it will be sufficient to write down 

 the result, viz. 



rx+ct 



+(z)dz. (5) 



J x-ct 



This may be immediately verified. 



If the initial disturbance be restricted to a finite extent 

 of the string, the motion finally resolves itself into two 

 distinct waves travelling without change in opposite directions. 

 In these separate waves we have 



* Encyc. Brit. 9th ed. Art. " Mechanics." 



