SIMPLE-HAKMONIC WAVES. DIFFRACTION 225 



It is evident that p contributes nothing to the average effect, 

 since the mean value of d<f>/dr at any point is zero. If we 

 substitute from (13) we find that the average of the remaining 

 part is 



This quantity W is independent of r, as was to be anticipated, 

 since the mean energy in the space included between two 

 concentric spheres is constant. It measures the emission of 

 energy (per unit time) by the source. The formula may also 

 be inferred from the consideration that at a great distance the 

 waves may be regarded as plane. If in (8) we replace A by 

 the A/4>7rr of (13), and multiply by 4nrr*, we obtain the result 

 (15). 



It must be remembered that this calculation of the energy 

 emitted applies only to an isolated source in free space. A 

 source placed in an enclosure with rigid walls does no work on 

 the whole, since the energy of the gas is constant. Even in an 

 open space the emission of energy may be greatly modified by 

 the neighbourhood of an obstacle. Thus in the case of a source 

 P close to a rigid plane boundary the amplitude of vibration at 

 any point is doubled by the reflection as from the image P' 

 ( 73); the intensity is quadrupled, and the emission (on one side) 

 is therefore twice that of an equal source in free space. 



The equation -j^ ****>, .................. (16) 



7? / v\ 



or, in real form, </>= ^ - cos n U + -), 



^rirT \ c/ 



may likewise be interpreted as representing a " sink " of sound, 

 i.e. a point where energy is absorbed, under similar conditions, 

 at the rate ri 2 pB*l&Trc. This conception is however of no great 

 assistance in acoustics. 



The notion of a simple source, valuable as it is for theoretical 

 purposes, is seldom realized even approximately in practice. 

 A vibrating body such as a membrane, or either prong of a 

 tuning fork, is tending at any instant to produce a condensation 

 of the air in contact with it on the one side and a rarefaction 



L. 15 



