Mr. R. H. M. Bosaiiquet on the Theori/ of Sound. 33 



rarefaction it would be reflected into a convergent pressure 

 by a process exactly the reverse of the above ; this would in 

 turn reenter the pipe^ travel back to the first source Sq, and 

 begin the cycle afresh. This would be a simple case of a 

 stream-line in an indefinitely extended frictionless fluid, the 

 flow through which would be maintained, by circulation of 

 energy, when once started. Such spherical divergence, how- 

 ever, cannot be actually set up at the end of a tube with any 

 approach to accuracy. 



In considering the energy of the motion, we saw that the 

 amount of it which lies beyond a distance r from the centre is 

 negligible when r is great in comparison with r^. This 

 remark becomes important subsequently, when it will illustrate 

 the extremely small influence on numerical results of great 

 diflcrences in the ultimate forms of motion at great distances. 

 But we shall employ the remark now to illustrate the passage 

 from the case of uniform flow of fluid to that of a periodic flow, 

 represented b}^ a circular function, according to solutions (i) 

 or (ii) of the differential equation. 



Instead of the velocity V being uniform, suppose it to be a 



f) 



function of the time, such as V sin kvt, where k (— ^ ) is small 



with respect to distances at which the energy is sensible. We 

 cannot suppose this as it stands to satisfy the conditions for a 

 flovf of energy. But if we notice that, kr and kc being 

 small, cos k (r — c) differs indefinitely little from 1, we can 

 put this function =Y ^\n kvt aos k(7' — c) , where c denotes an 

 arbitrary origin ; we see that we now have a motion analo- 

 gous to that in the neighbourhood of a loop surface with plane- 

 waves, which may be referred to the two streams of sound- 

 energy 



V 



an( 



^mk{vt—r + c) 



ij 



V 



-^sin A;(i5^ + r — c) 



(Prop. I. of note 4). The energy per second of each stream 

 will be , and that of the two together twice this amount, 



or half that in the case of continuous motion.^ 



Although I am going to prove this rigorously presently, yet 

 the derivation by the reasoning just employed is important, 

 because we shall want it later in an analogous case ; so I will 

 say a few words more on the details. 



The deduction of the distribution of energy required only 

 PUL Mag. S. 5. Vol. 4. No. 22. July 1877. D 



