GENERAL THEORY OF SOUND WAVES 209 



present case. So long as ct < r - a, the second part of (21) will 

 also vanish, but when ct lies between r a and r we shall have 



(23) 



When ct > r, the "second formula (22) applies, and we find 

 that, so long as ct < r + a, the result (23) will still hold. 

 Finally, when ct > r + a we have again 5 = 0. The results 

 are shewn graphically in the following figure which exhibits 

 the variation of s with t at a particular point, and the space- 



cb-a, 



ct+a 



Fig. 65. 



distribution of s at a particular instant, respectively. It 

 appears that after the lapse of a certain time (2a/c) we have a 

 diverging wave in the form of a spherical shell of thickness 2a, 

 and that s is positive through the outer half, and negative 

 through the inner half of the thickness. The changes in the 

 velocity may be inferred by means of the formula q = d<j>/dr. 

 For values of t between (r a)/c and (r + a)/c, i.e. during the 

 time of transit of the wave across the point considered, we find 



......... (24) 



whilst for other values of t we have <f> = 0. Hence within the 

 aforesaid limits of time we have 



(25) 



When r is large compared with a this changes sign for t = r/c, 

 approximately, the velocity being directed outwards in the 

 outer half, and inwards in the inner half of the shell. At the 

 boundaries of the disturbed region, where r = ct a, we have 

 q = + cas /2r. As the diverging wave reaches any point the 

 velocity suddenly rises from zero to the former of these values, 

 and as it leaves it the velocity falls suddenly from the latter 

 (negative) value to 0, The origin of the discontinuities in this 

 L. 14 



