Mr. R. II. M. Bosanquet on the Theory of S rund. 317 

 ;l-a> . 2A 



rt = A;— — , = 



or, replacing x by — > 



v 



J-v h _ 2Ar' 



?/ + v ? i/ + v 



The ratio of energy transmitted : original is 

 1/ i? 4v /2 4ot' 



A2 ^( u + ^-(^ +v y 

 The ratio of energy reflected : original is 



,2 ,„,/ ,,,v 2 



a 2 _ fv' — v\ 2 

 A 2 ~ \7+v) ' 



Since v, v' are symmetrically involved, the division takes place 

 in the same way whether we pass from the denser to the rarer, 

 or from the rarer to the denser medium. 

 Let 



4vv' /" v/ ~ v \ 2 _/d 



(v / + vf~ U > \V+i) ~"' 



Suppose the sound passes across a layer of gas or heated air 

 (two surfaces) ; then the following represents the distributions 

 of the sound-energy after reflexions and transmissions at the 

 two surfaces of the layer : — 



Back. Within the layer. Through. 



Ren. /3 Trans, a a 2 , 



Trans. a 2 /3 Refl. back a/3, 



Ren. forward a/3 2 ... a 2 /3 2 , 



Trans. a 2 /3 3 Refl. back a/3 3 , 



Refl. forward a/3 4 .., a 2 ^, 



and so on. 



The total transmission is 



1-/3 2 



Total reflexion back is ( ■= — -^ + 1 J /3 



Putting for a, /3 their values, 



a 2 (Aw') 2 2vv 



trans. 



VI— /3 2 J \v + v' 2 /\v + v'/ v 2 + v /2 



