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

 Let 



y x = a sin — (vt— x), 



2lT 



y 2 = 6sin — (yt + x), 



Y^yt+yt, 



dY 2ttv 

 dt 



dY 

 dx 



= -r— ( 6 cos— (^ + ^) + acos — (yt — x)u 



277/,. 2tT. -. 27T. ■ s \ 



= — ( 6cos — {yt + x)— acos t— v^~" x ) )' 



The total energy in any disk dx, estimated as kinetic + poten- 

 tial, is (Phil. Mag. [4] vol. xlv. p. 174) 



pdx( SdY\* 2 /dY\ 2 \ 



The coefficient of 2ah disappears from this expression, leaving 

 only the terms involving a 2 and b 2 , which give on integration 

 the same value of the energy per second as the sum of the 

 values for the component streams. This, of course, includes 

 the result of prop. L 



Prop. III. — If two equal and similar pendulum wave-sys- 

 tems, in the same phase and travelling in the same direction, 

 join each other in air, they cannot be superposed without 

 alteration. 



Let each of the wave-systems have an amplitude a ; then, if 

 they are simply superposed, the combined stream has an am- 

 plitude 2a, and the energy per second carried by the combined 

 stream is four times that of each of the single streams, or 

 twice the sum of the energy of the two streams together. 

 Hence the energy required for the superposition is twice the 

 total energy supplied, and simple superposition cannot take 

 place. 



Cor. — Reflexion will generally take place at the point of 

 junction. 



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

 same phase and travelling in the same direction, join each 

 other in air ; to determine the transmission and reflexion. 

 Let A be the amplitude of each of the original streams, 

 a that of the reflected stream, 

 b that of the transmitted stream. 

 At the common surface, 



2A + a=b; 



