STRINGS 65 



or "incident" wave represented by the first term, and a "re- 

 flected" wave represented by the second. The amplitude of 

 the reflected wave is equal, at corresponding points, to that 

 of the incident wave, so that there is no alteration in the 

 energy, but the sign of y is reversed. It is otherwise obvious 

 that if on an unlimited string we start two waves which are 

 antisymmetrical with respect to 0, in opposite directions, the 



| Fig. 25. 



point of the string which is at will remain at rest, even if 

 it be free. Hence by the crossing of the waves the circum- 

 stances of reflection at a fixed point are exactly represented. 

 It will be noticed that a lateral force is exerted on the fixed 

 point during the process of reflection. 



In the case of a finite string whose ends are (say) at the 

 points x 0, x = I, we have the further condition that 



f(ct-l)-f(ct + l) = (2) 



for all values of t. If we write z for ct I, this becomes 



/(*)-/(* + SJ) (3) 



so that f(z) is a periodic function, its values recurring when- 

 ever z increases by 2/. It follows that the motion of the string 

 is periodic with respect to t, the period 2l/c being the time 

 which a wave would take to travel twice the length. It is 

 otherwise evident that a disturbance starting from any point 

 P of the string, in either direction, will after two successive 

 reflections at the ends pass P again, in the same direction as 

 at first, with its original amplitude and sign. 



L. 5 



