104 KINEMATICS. [193. 



193. If the motion of A be simply harmonic, sayjy = # 



the motions of the successive points of the cord will differ from 

 the motion of A only in phase, and the displacements of all 

 these points at any time t can be represented by 



y = a sin(W e), (19) 



where e varies from o to 2 TT as we pass from A to K. 



As the time T in which the motion spreads from A to K is 

 equal to the period of a vibration of A (or of any other point of 

 the cord), we have co = 27r/T, or, by (18), o) = 27rF/X. And if x 

 be the distance of the point of the cord under consideration from 

 A, we must have x : \ = e : 2?r ; that is, e = 27rx/\. Substituting- 

 these values of co and e in (19), the equation of the wave motion 

 can be written in the form 



y=a*\TL(Vt-x). (20) 



X 



194. This equation can be looked upon from two different 

 points of view according as we regard / or x as variable. 



Let / be constant ; i.e. let us consider the displacements of 

 all points of the cord at a given instant. If for x in (20) we 

 substitute x+n\, where n is any positive or negative integer, 

 the angle (Vt x} 2ir/\ is changed by 2irn, so that the value 

 of y remains unchanged. The displacements of all particles 

 whose distances from A differ by whole wave lengths are there- 

 fore the same; in other words, the state of motion at any 

 instant is represented by a series of equal waves. 



Now let x be constant, and t variable. Substituting for t in 

 (20) the value t+nT=t+n\/V, the angle (Vtx) 2?r/X is again 

 changed by 27r, and y remains the same. This shows the 

 periodicity in the motion of any given particle. 



195. If the point A (Fig. 49) be subjected simultaneously to 

 more than one simple harmonic motion, the displacements 

 resulting from each must be added algebraically, thus forming- 

 a compound wave which can readily be traced by first tracing 



