Mr. R. H. M. Bosanquet on the Theory of Sound. 421 



and equating values of energy per second, 



2A 2 = a 2 + 6 2 ; 

 eliminating A, we find 



a + b = 0. 



The reflected and transmitted streams are equal ; and each of 

 them is equal to either of the original streams. 



This proposition finds an application in some cases of inter- 

 ference. The circumstances supposed here cannot, however, 

 be realized in an accurate manner physically ; for the two ori- 

 ginal streams can only be kept apart before their junction by 

 travelling in different channels. In this case there are two 

 reflected streams, and the solution is different. 



Prop. V. — Two equal and similar wave-systems, in the 

 same phase and travelling in the same direction by two sepa- 

 rate channels, join each other by the two channels uniting into 

 one, of the same size as either ; to determine the transmission 

 and reflexion, 



At the common surface at the entry to the channel of union, 



2(A + a) = b, 



2(A 2 -a 2 ) = 6 2 = 4(A + a) 2 , 



whence 



or 



A 2 + 4Aa + 3a 2 =0, 

 (A + a)(A + 3a) = 0. 



A + a = involves 5 = and is inadmissible ; 



and of the total energy incident per second, § is transmitted 

 in the combined vibration, the remaining ^ being reflected back 

 along the paths of the incident wave-systems. 



These circumstances may be realized in the case where the 

 wave-length is great compared with all the dimensions, so that 

 any difference of direction between the original and united 

 channels becomes immaterial. 



Prop. VI. — In the general case, where two systems of the 

 same wave-length join in air, to determine the reflexion and 

 transmission, considering one channel, i. e. under the circum- 

 stances of prop. IV. 



Let A, B be the amplitudes of the incident wave-systems, 

 7 the difference of phase, 

 a the amplitude of the reflected system, 

 b the amplitude of the transmitted system. 



