On some Points in the received Theory of Sound. 53 



mitting such waves as would produce musical sounds to a 

 considerable distance without important alteration, even if the 

 waves were plane ; although sooner or later a wave of this 

 kind would be converted into what Professor Challis calls a 

 breaker^ but which I think is more nearly analogous to a bore. 

 An integral of the exact equation which applies to s])herical 

 waves has never been obtained ; but it is evident that the di- 

 vergence of such a wave must tend to counteract the formation 

 of a bore. In fact, if we suppose the velocity v represented, as 



it would be approximately, by- / (r — at)^ we have, neglecting 



the term involving— 2> 7-; =— /' [r— at)'i so that for a given 



phase of the wave, that is for a given value ofr — at, the tangent 

 to the velocity curve (vol. xxxiii. p. 350) would vary inversely 

 as r; and therefore, as far as depended only on divergence, 

 the inclination of the curve would become more and more 

 gentle in the anterior, as well as the posterior portion. I feel 

 however almost certain, in consequence of an investigation in 

 which the effect of divergence was very approximately taken 

 account of, that the formation of what I have called a bore, 

 although much retarded by divergence, is not ultimately pre- 

 vented. I speak of course with reference to the ideal fluid in 

 which potp. I see no reason for supposing that the develop- 

 ment of heat and cold by sudden condensation and rarefaction 

 would have any tendency to prevent the formation of a bore. 

 I have already alluded to one cause which would have such a 

 tendency (vol. xxxiii. p. 356), namely the internal friction of 

 the fluid. If, during the rapid condensation and rarefaction 

 of the fluid, there should be time for any sensible quantity of 

 heat to pass off" or be received by way of radiation, that would 

 apparently have much the same effect as internal friction. 

 The effect of distance upon the quality of sound, and the 

 causes why sounds are mellowed by distance in air, whereas 

 under water the sound of a distant bell is heard as a crash, 

 would form an interesting field of inquiry. 



I proceed now to notice the apparent contradiction at which 

 Professor Challis has arrived by considering spherical waves, 

 a contradiction which it is the chief object of this communi- 

 cation to consider. The only reason why I took no notice of 

 it in a former communication was, that it was expressed with 

 such brevity by Professor Challis (vol. xxxii. p. 497), that I 

 did not perceive how the conclusion that the condensation 

 varies inversely as the square of the distance was arrived at. 

 On mentioning this circumstance to Professor Challis, he 



