36 Mr. R. H. M. Bosanquet on tlte Theory of Sound, 



wave, and the velocities become : — 



VS. 

 divergent = —^^ sin k[vt — r + r) , 



V s 



convergent = -^^ sin k{vt -\-r — r)j 



V s 



both of which become ■ ^^^ sin fa, if r = r. 



Now the transmission of energy takes place with the A^elo- 

 city of sound ; so that the consequence of the loop being at a 

 distance r from the centre is that the reflected system is later 

 in phase than it would have been if reflected all at once at Sq, 

 by the time sound takes to travel over 2(r— ^o). 



For convenience in estimating this retardation, let us alter 

 our origin of time in the above equations by the time sound 

 takes to travel over r ; the equations become : — 



V s 



divergent velocity = -~^iAnk{yt — r) ; 



VS.. 

 convergent velocity = ^J^ sin k{yt + r—2r). 



The first is now in a form independent of the reflexion ; and 

 the latter expresses the features of a wave reflected at a distance 

 r from the centre. If we regard this latter stream as made 

 up of elements of energy per second reflected from the succes- 

 sive surfaces traversed by the former, we have for the velocity- 

 element reflected from distance p (V^= maximum velocity of 



V 



reflected vibration = ^ ), 



\dN' ^\\\k{yt^T — 'lp). 



Expanding this expression, and remembering that kp is small, 

 we have 



<iV^(sin k{yt + v") — 2kp cos k{vt + r)). 



If these elements are all transmitted to Sq, as is indicated by 

 the original equations, so as to make up there the reflected 

 velocity across So, this becomes 



pdY'QCo^k{yt-\-r); 



and if we put 



