210 DYNAMICAL THEORY OF SOUND 



solution is to be sought of course in the discontinuity of the 

 initial distribution of density. Any difficulty which may be 

 felt on such grounds may in general be removed by substituting 

 in imagination an initial distribution in which the discontinuity 

 is replaced by a very rapid but continuous transition. 



The solution of (6) in terms of the general initial con- 

 ditions (16) may be investigated in a similar manner, but it 

 must suffice to quote the results. It may easily be verified 

 that they satisfy all the conditions of the question. They are 



n/> = i (r + ct) fa(r + ct) + $(r- ct) fa (r - ct) 



1 fr+ct 



+ il %<*)<* -(26) 



&&J r-ct 



for ct < r, and 



r$ = | (ct + r) fa(ct + r)-(ct- r) fa(ct-r) 



1 rct+r 



+ 2-J -()*... (27) 



*C J ct-r 



for ct > r. 



Since the origin evidently occupies an exceptional position 

 in the theory of spherical waves it is desirable to calculate the 

 value of (f> there, more especially as the result will be of service 

 presently when we come to the solution of the general equation 

 70 (4) of sound waves. The result may be deduced from 

 (27), or more directly from (15). We find 



...(28) 



and therefore from (17) and (18) 



(29) 



For example, in the special problem above considered, where 

 fa (r) = 0, whilst % (r) c\ or according as r a, we find 

 $ = c\t or according as t a/c. The consequent value of 

 s at is s for t < a/c and zero for t > a/c, whilst at the instant 

 t = a/c it is negative infinite. To escape this result we must 

 slightly modify the data, replacing the original distribution 

 of density by a continuous one. The figure is an attempt to 



