Mr. R. IT. M. Bosaiiquet on the TJieory of Sound. 37 



the total velocity of tlie reflected streiiin at Sy is expressed in 

 the form 



N' Q^\nk(yt + r — 2r). 



And the reflected stream is determined in phase in a manner 

 which is represented by supposing it to originate wholly at a 

 distance r from the centre, where r is defined by 







The energy per second of the reflected stream at Sq is 



2 '* 



If we suppose this quantity to be varied by the abstraction of 

 the energy-element per second reflected at surface S, 



The energy per second of the divergent stream through sur- 

 face S is 



pt,V2Sf_K. 



and the element reflected at distance r from the centre is 



and 2Y\ = Y,; 





The reflected stream is then determined in phase by the equa- 

 tions 





*y J-o */ '•o 



and if S = 47rr'^, 



that is to say J the reflected stream is the same in phase as if 

 it came from a single reflexion, at a distance beyond S^ equal 

 to its radius r^. 



We can employ this at once to get an approximation to 

 Helmholtz's result for hemispherical divergence from the end 



* Prop. I., cor., of Note 4. The total energy per second in periodic How 

 = that in constant flow, if the -periodic Jiow he that of an ordinary stream 

 ivith pressures and velocities. In these loop-surface motions with velocities 

 only, the total energy per second in terms of the velocity is half that in 

 the case of constant flow or in the case of an ordinary transmitted sound- 

 wave. 



