I 9 S.J PLANE MOTION. IC >5 



the component waves and then adding their ordinates, or ana- 

 lytically by forming the equation of the resultant motion as in 

 Art. 189. 



196. Exercise. 



(i) Trace the wave produced by the superposition of two simple 

 harmonic motions in the same line of equal amplitudes, the periods 

 being as 2:1, (a) when they do not differ in phase, (b) when their 

 epochs differ by 7/16 of the period. 



197. The idea of wave motion implies that the displacement 

 y should be a periodic function of x and t such as to fulfil 

 the following conditions : y must assume the same value (a) 

 when x is changed into n\, (&) when t is changed into /+ T, 

 (c) when -both changes are made simultaneously; the constants 

 \ and T being connected by the relation \= VT. 



The condition (c) requires y to be of the form yf(Vtx] ; 

 for Vt x remains unchanged when x is replaced by x-\- VT 

 and at the same time /by /+ T. 



A particular case of such a function is y=a smc(Vtx). As 

 y should remain unchanged when / is replaced by /+ T t we 

 must have c=2Tr/VT=2Tr/\. Thus the function 



7=^ sin (Vt-x) 

 X 



fulfils the three conditions (a), (b), (c). Putting as before (Art. 

 193) 2irx/\ = e, we can write it 



198. The importance of this particular solution of our problem lies 

 in the fact that, according to Fourier's theorem, any single-valued 

 periodic function of period T can be expanded, between definite limits 

 of the variable, into a series of the form 



(21) 



