100 DYNAMICAL THEORY OF SOUND 



distributed continuously in -space and in time, the differential 

 equation being 



as in 28. Suppose, in the first place, that 



y * /*\ rjrx /\ - ^Tne .c *+\ m7nr . . 



- =/j (t) sm -j- +/, (0 sin -j- + . . . +f m (t) sin -y- , (2) 



the coefficients being known functions of t. The equation (1) 

 is then satisfied by 



7TO) . ?? mirx 



-- + 77 2 sm-- + ...+7; m sin - , ...(3) 



provided ***/. ................... (4) 



The solution of this equation has been given in 8. If we 

 assume that r) s = 0, 17, = for t = oo , and that/^ (t) is sensible 

 only for a finite range of t, the resulting value of rj 8 is 



57TC^ f 00 ,. ,.. . STTCt 



If as a particular case we put 



we have ^ = e-^^sin, ........... (7) 



STTC L 



by 8 (18). As a function of t, Y now follows the special law 

 indicated by the last factor in (6), at every point of the string, 

 but we have not yet made any special assumption as to the 

 distribution of the force over the length. Its time-integral is 

 given by 



1 T 00 



P J - 



TT7 , ~ . ~ . ^ . mirx 



Ydt= Ojsm-y- + O 2 sm- 7 - + ... + C m sm y- . (8) 



We may now seek to determine the values of the coefficients 

 so that this expression may be sensible only in the neighbour- 

 hood of the point x a. We assume, then, that 



PJ -oo 



