572 VISCOSITY. [CHAP, xi 



In the case of air- waves we have c=3*32 x 10 4 , j/='132, C.G.S., whence 



i/0-/c 2 = 27ri//Xc = 2'50X~ 1 x 10' 5 , =9'56X 2 x 10 3 , 

 if X be expressed in centimetres. 



To find the decay of free waves of any prescribed wave-length 

 (27T/&), we assume 



and, substituting in (4), we obtain 



- 2 c 2 (14). 



If we neglect the square of vk/c, this gives 



a = -^vk s ikc (15). 



Hence, in real form, 



u = ae- tlr cosk(xci) (16), 



where r = 3/2i/& 2 (17)*. 



The estimates of the rate of damping of aerial vibrations, 

 which are given by calculations such as the preceding, though 

 doubtless of the right order of magnitude, must be actually under 

 the mark, since the thermal processes of conduction and radiation 

 will produce effects of the same kind, of comparable amount, and 

 ought therefore, for consistency, to be included in our calculations. 

 This was first pointed out distinctly by Kirchhoff, who has 

 investigated, in particular, the theoretical velocity of sound-waves 

 in a narrow tube-f-. This problem is important for its bearing on 

 the well-known experimental method of Kundt. Lord Rayleigh 

 has applied the same principles to explain the action of porous 

 bodies in absorption of sound J. 



311. It remains to call attention to the chief outstanding 

 difficulty of our subject. 



It has already been pointed out that the neglect of the terms 

 of the second order (adujdx, &c.) seriously limits the application 

 of many of the preceding results to fluids possessed of ordinary 



* For a calculation, on the same assumptions, of the effect of viscosity in 

 damping the vibrations of air contained within spherical and cylindrical envelopes 

 reference may be made to the paper ' ' On the Motion of a Viscous Fluid contained 

 in a Spherical Vessel," cited on p. 558. 



f "Ueber den Einfluss der Warmeleitung in einem Gase auf die Schallbewegung," 

 Pogg. Ann., t. cxxxiv. (1868) ; Ges. Abh., p. 540. 



J " On Porous Bodies in relation to Sound," Phil. Mag., Sept. 1883. 



