FOURIER'S THEOREM 



Fig. 32 starts from A to describe the upper parabolic arc. At 

 this instant we have y = 0, everywhere, whilst 



(i) 



We therefore begin with the finite series 



. 7TX . TTCt . 



y A l sm-j- sin -j- + A^ sm j- sm -j I- ... 



. mirx . mirct /sn 



4- A in sm sm . ...(2) 



This satisfies the differential equation, and makes y = for t = 0. 

 It only remains so to determine the constants that the series 



. D . D . mirx /ON 



1 sm -j- + J5 2 sm j~ + . . . + .O TO sin y , ...(3) 



where B t = --A s , ..................... (4) 



may represent the initial distribution (1) of velocity, as nearly 

 as possible. The determination of B 8 has virtually been made 

 in 32 (15). With the necessary modifications of notation we 

 find 



as stated in 27. The graph of the initial velocity, and the 

 approximation attained by taking the first eight terms of the 

 series (3), are shewn in Fig. 36. 



It will be noticed that our approximation has even an 

 advantage over the result obtained by carrying the series to 

 infinity. In the latter case, the initial velocity, as represented 

 by (1), is discontinuous when x = 0, being zero for # = 0, but 

 equal to 4y9 c/Z when x differs ever so little from 0. The 

 idealized representation of the motion in 27 is in this 

 respect imperfect ; the parabola in Fig. 32 should be slightly 

 modified so as to touch the line AB at its extremities. 



38. String Excited by Impact. 



As a final example we take the case of a string started by 

 an impact, as in 26. We begin with the case of a force 



72 



