28 Mr. R. H. M. Bosanquet on the Theory of Sound. 



And the reflected and transmitted energy per second together 

 = incident flow ; 



From these equations we find, rejecting the solution b = 0, 



7 a \ ^^0 A ^^X *"^ ^^n 



which determine the reflected and transmitted streams. 



It is easy to treat similarly the more general case, where 

 two streams flowing in opposite directions meet at the change 

 of section ; but the form of analysis is not convenient for pro- 

 ceeding much further, so I pass on. 



We may deduce from the above the general principle, that 

 if (under the conditions stated) a stream of sound-energy 

 diverge, occupying at successive instants a variable surface S, 

 portions of the energy will be reflected back at every instant; 

 and since the total amount is constant (conservation of energy), 

 the total amount on surface S diminishes as the surface in- 

 creases. If we suppose it uniformly distributed over the 

 surface, we may then express the total energy on S in the 

 form* 



where Eq, E^, E2 are coeflicients depending on the circum- 

 stances. There can be no positive index terms, since the energy 

 supplied cannot augment itself. 



If S change to S + (iS, E changes to E— r/E; and since dE 

 is not sent forward, it is reflected back. Hence the general 

 expression for the reflected element of energy is 



c?E = El -^2 + 2E2 -^2- + • • • • 



In the cases I shall consider, the first term alone is important; 

 it would be possible to found subsequent developments very 

 simply on the first term of the above expansion. But it is 

 desirable to discuss the case of spherical divergence more fully. 



* In the case where the surface S diminishes as the stream flows on, 

 the expression is different : since the energy per second through succes- 

 sive surfaces S cannot increase, negative powers are excluded ; and the 

 expressions in the text are replaced by 



where dS is the element of diminution of surface. 



