Mr. R. H. M. Bosanquet on the Theory of Sound. 423 



reflected a 2 



incident ~~ A 2 + B 2 



-ifl-.A 4A»BW 7 \ 

 2 V V 1_ (A» + B»)" /" 



The transmission is complete only when y = — , or the incident 



wave-systems differ in phase by quarter of a period. 



When the phase-difference is either zero or half a period, 

 the transmission is a minimum and the reflexion a maximum ; 

 the values are : — 



A 2 



Transmission = A9 , n -„; 

 A 2 + B 2 



B 2 

 Reflexion = -rs — rw> 

 A 2 + B 2 



where A > B. 



(For C — C' = — 2a, and C + C'=25; andC, Care essen- 

 tially positive quantities.) 



Prop. VII. In the general case where two systems of the 

 same wave-length, travelling in the same direction by two 

 separate channels, join each other by the two channels uniting 

 into one, of the same size as either ; to determine the trans- 

 mission and reflexion, i. e. under the circumstances of prop. V. 



Following the notation of the last proposition, we have for 

 the common amplitude at the entry of the channel of union, 



C + 2a = b, (i) 



whence 



A 2 + B 2 + 2 AB cos 7 = (b - 2a) 2 = C 2 , 



and, equating values of energy per second, 



A 2 + B 2 = 2a 2 + Z> 2 , (ii) 



whence 



A 2 + B 2 -2ABcos r =C /2 



= b 2 + ±ab. 

 From this and (i), 



3&2_ 2 6C-C /2 =0, 



h= C± s /O t + SC f3 m 

 3 



and from (i), 



a= — 7z — • 



