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



Then . . 2tt , . 



y 1 =Asm — \yt — x), 



}l\ 



y 2 = Bsin | Y^Mi+TfJ 

 + z/ 2 = Csin |-^(r* — a?) + D j, 



where 



C 2 =A 2 + B 2 + 2ABcosy. 



It is clear that this combined vibration cannot generally be 

 transmitted unaltered ; for the energy per second would be 

 altered by a term depending on 2AB cos 7. 



Proceeding as before, we have, equating the values of the 

 amplitude at the common surface, 



C + a = b, 

 whence 



A 2 + B 2 + 2 AB cos 7 = (6 -^) 2 , 



and, equating values of energy per second, 



A 2 + B 2 =.a 2 + 6 2 ; 



whence 



Let 



then 



A 2 + B 2 - 2 AB cos y= (a + hf, 



A 2 + B 2 -2ABcos7^C /2 , 



C =6-a, 

 C' = b + a, 



c-c 

 c+c 



2 



- »; 



whence, if we draw from a point two radii equal to A, B 

 respectively, enclosing the angle 7, and complete the paralle- 

 logram of which they form two sides, half the difference of the 

 diagonals is the reflected amplitude a, and half the sum of the 

 diagonals is the transmitted amplitude b. The diagonal 

 through is of course C, the amplitude derived by geome- 

 trical superposition. 



The ratios of energy in the above case are :- — 



transmitted _ b 2 

 incident ~"A 2 + B 2 



-^i+V/i- (A , +B8) , )> 



