310 On the Propagation of Sound between Parallel Walls. 



layer is very small. If in c.G.s. measure we take ya' = *16, 

 w = 27rx256, we find s/(n/'2[x f ) = 61. Thus i£ we suppose 

 the thickness of the Motional layer in (41) to be defined by 



we get 



y ] —y='lo millim. 



If the point under consideration be a few multiples of this 

 (say 1 millim.) from the walls, the ratio Q(y)"^Q(?/i) may be 

 neglected. 



The thickness of the layer through which Q 2 {y) -r- Q 2 (yi) 

 is sensible is of the same order of magnitude. 



Let us next consider what value of (]J\—y) makes (rf — y) 

 in (54) equal to unity. By (52), (55) 



v'2.a 2 



If we take /// = *16, v — -256, we find from (33) y'=*6 ; and 

 .,2 = 33200 ; so that for a frequency of 256 we get 



i/2.33200 2 .,._.. 

 y i — y=- 7i — r- — - — =40000. 



For air and for a sound of this pitch the falling off becomes 

 important at a distance of about 400 metres from the walls. 



As has already been suggested, this solution fails to answer 

 the practical question for the sake of which it was originally 

 attempted. It was desired to know whether in the propa- 

 gation of sound for long distances over smooth water, there 

 was any important shadow formed near the surface under the 

 influence of viscosity and heat conduction. It would appa- 

 rently be a matter of some difficulty to formulate and solve a 

 definite problem in which this question is involved. But, as 

 Lord Kelvin has pointed out to me, a sufficient answer to 

 the practical question may be arrived at by very simple 

 reasoning on the basis of a solution originally given by Stokes 

 (see 'Theory of Sound,' § 347). If U be the tangential 

 velocity of a plane vibrating rigidly in an atmosphere of 

 viscous fluid with a frequency n/27r, the work required to 

 maintain the motion is, for unit of area, 



jV(i/»^) . Wdt, 



or 



Ui. t . s/(kpn(M), 



where U m denotes the maximum value of U during the period. 

 The same expression may be applied to find the work lost 



