78 DYNAMICAL THEOKY OF SOUND 



applied at a node of one of the harmonics, and the point 

 observed should be at another node of the same. 



Except at the two instants in each period when the velocity 

 suddenly changes, the acceleration of the point (P) examined is 

 zero. It follows from 22 (2) that the curvature of the string 

 in the neighbourhood of P vanishes, and that the form of the 

 string at any instant is accordingly made up of straight pieces. 



Fig. 32. 



It appears that all the conditions of the problem can be satisfied 

 if we assume that the form is always that of two such pieces 

 meeting at a variable point Q. In Fig. 32 let AB (= I) be the 

 undisturbed position of the string, and let a (= AN) and /3 

 (= NQ) be the coordinates of Q referred to A as origin and AB 

 as axis of abscissae. The equations of the two portions of the 

 string are 



yi = M> y* = P(i-x)l(i-*), ......... (i) 



and the difference of the velocities near Q on the two sides 

 is accordingly 



In the time St a length d8t of the string is traversed by the 

 point Q, so that a mass pd&t has its velocity increased by the 

 above amount. This is the effect of the transverse force 



where P is the tension, acting for the time St. Equating the 

 change of momentum to the impulse of the force we find 



* ...................... (4) 



