66 DYNAMICAL THEORY OF SOUND 



When the initial data are of displacement only, i.e. with 

 zero initial velocity, the successive forms assumed by the string 

 in the course of a period can be obtained by a graphical con- 

 struction. We suppose the initial form y </> (x), where <f> (as) is 

 originally defined only for values of x ranging from to I, to be 

 continued indefinitely both ways, subject to the conditions 

 4>(_#) = -</>(tf), (Z + aj) = -(Z-aj) (4) 



If we imagine curves of the type thus obtained to travel 

 both ways with velocity c, and if we take at each instant the 

 arithmetic mean of the ordinates, in accordance with 23 (7), it 

 is evident that the varying form thus obtained will represent 



Fig. 26. 



a possible motion on an unlimited string, in which the points 

 x = 0, x = I, x = + 21, . . . remain at rest. The portion between 

 x = and x = I will therefore satisfy all the conditions of the 

 question. The process is illustrated in the annexed Fig. 26 ; 

 the initial form here consists of two straight pieces meeting at 

 an angle, and the result after an interval l/8c is ascertained. 



In this way we might trace (after Young) the successive 

 forms assumed by a string excited by " plucking," one point of 

 the string being pressed aside out of its equilibrium position, 

 and then released from rest, but the actual construction can in 

 such a case be greatly simplified. It is easily seen that the 

 form of the string at any instant consists in general of three 

 portions; the outer portions have the same gradients as the 

 two pieces into which the string was initially divided, whilst 

 the gradient of the middle portion is the arithmetic mean of 



