238 DYNAMICAL THEORY OF SOUND 



there is merely a local reciprocating motion of the gas from the 

 anterior to the posterior region, and back again in the opposite 

 phase of the body's motion, in which the region that had been 

 anterior becomes posterior. If the rate of alternation of the 

 body's motion be taken greater and greater, or, in other words, 

 the periodic time less and less, the condensation and rarefaction 

 of the gas, which in the first instance was utterly insensible, 

 presently becomes sensible, and sound waves (or waves of the 

 same nature in case the periodic time be beyond the limits of 

 audibility) are produced, and exist along with the reciprocating 

 flow. As the periodic time is diminished, more and more of the 

 encroachment of the vibrating body on the gas goes to produce 

 a true sound wave, less and less a mere local reciprocating flow. 

 For a given periodic time, and given size, form, and mode of 

 vibration of the vibrating body, the gas behaves so much the 

 more nearly like an incompressible fluid as the velocity of 

 propagation of sound in it is greater ; and on this account the 

 intensity of the sonorous vibrations excited in air as compared 

 with hydrogen may be vastly greater than corresponds merely 

 with the difference of density of the two gases." 



These remarks are exemplified in the results of 77 (13), 

 (14). If we fix our attention on a point at a distance from the 

 sphere, supposed vibrating with the velocity 



U=Acoant, (1) 



the motion there is given, when the period is sufficiently long, 

 by the formula 



A a 3 

 <f> Y7 cos Q . cos nt, (2) 



as if the fluid were incompressible. But when the frequency is 

 increased until the wave-length is small compared with the 

 distance r from the centre, the appropriate formula is 



ka?A / r 



?A A . f.r\ /ox 

 cos . sin n(t j , (3) 



and the amplitude is accordingly greater than in the former 

 case in the ratio kr, or 2irr/\. For the same frequency, the 

 amplitude, which depends on k/c or n/c 2 , will in different gases 



