196 DYNAMICAL THEORY OF SOUND 



The case of an elliptic section can be solved in a similar 

 manner. The result, first given by Boussinesq (1868), is 

 E = V(a 2 + 6 2 )/a 2 & 2 , (35) 



where a, b are the semi-axes. If we put a = oo we get the 

 case of a narrow crevice, bounded by parallel planes, the 

 breadth being 26, viz. 



E = V/6 2 (36) 



This can of course be obtained more easily by an independent 

 process. 



The formula (30), when combined with (34) or (36), agrees 

 with the result of the more complete investigation given by 

 Lord Rayleigh (1883). It appears that u goes through its 

 cycle of phases in a distance 27r/r, but that within this space 

 the amplitude is diminished in the ratio e~ 2n = 1/535. In the 

 case of circular section we have 



OT 2 = ifi'n/pda*, (37) 



by (28) and (34). Hence when the circumstances are such that 

 the ratio v/na? is large, the distance in question is small com- 

 pared with the wave-length (X = ^TTC/U) in the open ; for we 

 have 



(Xs7/27r) 2 - *7 2 c 2 /tt 2 = 4>v/na 2 (38) 



Hence in a sufficiently narrow tube the waves are rapidly 

 stifled, the mechanical energy lost being of course converted 

 into heat. 



The investigation has been employed by Lord Rayleigh to 

 illustrate the absorption of sound by porous bodies. When 

 a sound-wave impinges on a slab which is permeated by a large 

 number of very minute channels, part of the energy is lost, so 

 far as sound is concerned, by dissipation within these channels, 

 in the way just explained. The interstices in hangings and 

 carpets act in a similar manner, and it is to this cause that the 

 effect of such appliances in deadening echoes in a room is to be 

 ascribed, a certain proportion of the energy being lost at each 

 reflection. It is to be observed that it is only through the 

 action of true dissipative forces, such as viscosity and thermal 

 conduction, that sound can die out in an enclosed space, no mere 

 modification of the waves by irregularities being of any avail. 



