290 Lord Rayleigh on the Production 



not, as might for a moment be supposed, imply an infinite 

 emission of energy. If the pressure be divided into two 

 parts, one of which has the same phase as the velocity, and 

 the other the same phase as the acceleration, it will be found 

 that the former part, on which the w^ork depends, is finite. The 

 infinite part of the pressure does no work on the whole, but 

 merely keeps up the vibration of the air immediately round 

 the source, whose effective inertia is indefinitely great. 



" We will now investigate the energy emitted from a simple 

 source of given magnitude, supposing for the sake of greater 

 generality that the source is situated at the vertex of a rigid 

 cone of solid angle (o. If the rate of introduction of fluid at 

 the source be A cos Icat, we have 



a)i^d(f>ldr = X cos Jcat 



ultimately, corresponding to 



6=^^Q0il:(at-r) (2) 



whence 

 and 



d(i> JcaA . y . V ,„. 



(or'^d(l>ldr = A{cos k (cit—r)—kr sin k [at — r)}. . (4) 



Thus if dy^ be the work transmitted in time dt, we get, since 

 Bp=—p d(f>/dt, 



-^r- = ~ — ■ sill ^ [O't — '^0 cos k(at — r) 



dt (or \ y \ J 



-to siBrk{at — 7n.. 



' CO ^ ^ 



Of the right-hand member the first term is entirely periodic,, 

 and in the second the mean value of sin^ k(at—r) is J. Thus 

 in the lono- run 



W=^-^% (5) 



'' It will be remarked that when the source is given, the 

 amplitude varies inversely as o), and therefore the intensity 

 inversely as o)^. For an acute cone the intensity is greater, 

 not only on account of the diminution in the solid angle 

 through which the sound is distributed, but also because the 

 total energy emitted from the source is itself increased. 



"When the source is in the open, we have only to mit 

 o) = 47r, and when it is close to a rigid plane, a> = 27r. 



" These results find an interesting application in the theory 



