STRINGS 79 



The point of discontinuity Q (of the gradient) must therefore 

 travel right or left with the velocity c. 



Let us suppose that Q starts from A at the instant t = 0, 

 and that & is at first positive. The observations of Helmholtz 

 shew that the velocity at a point x, viz. 



is during an interval x/c constant, whence 



= (7a(Z-a), ..................... (6) 



no additive constant being admissible, since ft must vanish with 

 a. This is the equation of a parabolic arc passing through A, B. 

 The conditions of the problem are therefore all fulfilled if we 

 imagine Q to travel backwards and forwards along two such 

 arcs, with velocity c, in the manner indicated in Fig. 32. In 

 terms of the maximum displacement j3 we have G = 4/8 /Z 2 , 

 and the equations of the two portions of the string at any 

 instant are therefore 



y,-^(i-), -^<l-*> ...... (7) 



It only remains to resolve this motion into its simple- 

 harmonic constituents. The details of the calculation are 

 given in 37. The result is 



STTCt 



where the summation embraces all integral values of s. Com- 

 paring with 25 (7) we have 



A = 0, B.-3| ................... (9) 



These results, and indeed the whole investigation, take no 

 account of the position of the point to which the bow is applied. 

 It is plain, however, that the position of the bow must have 

 some influence on the character of the vibration; and it is 

 found in fact that those normal modes are absent which have a 

 node at the point in question. It is for this reason that the 

 somewhat idealized vibration-form which is adopted as a basis 

 of calculation is only obtained in its purity at corresponding 

 nodes. 



