Discontinuous Wave-Motion. 49 



reproduces that of a bowed string in a very perfect manner, 

 the discontinuous changes of velocity being clearly shown. 

 Though the motion is a free oscillation, it remains practically 

 unaltered in form for a considerable number of periods, and 

 the experiment c;m thus be readily projected on the screen, 

 and is suitable for lecture demonstration. 



We shall now briefly consider the theory of the experiment 

 described above from two distinct points of view. First, 

 by application of the Fourier Analysis : Taking the originally 

 fixed end as the origin (a? = 0) and the position of the string 

 at time t = Q as the axis of x, we obtain the expansion 



. irx . 2irt . 2irx . knt 



y = ai sm -=- sm -~j — \-a 2 sin —r- sm -pp- + &c, 



in which the cosine terms are entirely absent, y being equal 

 to zero when t = 0. The values of the coefficients a v a 2 , &c. 



have to be found from the initial condition I -—- ) = cox. 



10n \dt ) t=0 



By expanding cox in a series of sines, differentiating the 

 expression for y, and putting £ = 0, we obtain 





(dy 

 \dt 



\ 2tt ( . 1TX 



+ 



2 



2a 2 sin - 



7r.r 



&c 



) 







2lco / . irx 

 = sin -j-- 



•K \ I 



_1 



2 



. 2t™ 

 sin L 



+ &C 



•)• 





The values 



of the coefficients may 



now be 



writte 



n down : 







Zo>T 



«1=— 2~, a 2- 





1Z»T 



"4 7I 2 ' 









and 



so on. 



Finally we have 















y 



n==o 7 HP 



»=i n 2 7r 2 



sin 



T sm 



2-7771^ 



T 







To find the character of the motion expressed by the series, 

 we have to effect its summation. This is best done by 

 differentiating the series and writing it in the form 



PKil. Mag. S. 6. Vol. 31. No. 181. Jan. 1916. E 



