184 Lord Rayleigh on Porous Bodies 
which is the solution of the problem proposed. It is under- 
stood that waves which have once entered the wall do not 
return. When dissipative forces act, this condition may 
always be satisfied by supposing the channels long enough. 
The necessary length of channel, or thickness of wall, will 
depend upon the properties of the gas and upon the size and 
shape of the channels. 
Even in the absence of dissipative forces there must be 
reflection, except in the extreme case # = 0. Putting k=k 
in (8), we have 
B =4? <») 
If 0=1 (that is, if half the wall be cut away), B = ^, B 2 = £, 
so that the reflection is but small. If the channels be cir- 
cular, and arranged in square order as close as possible to each 
other, <7 = (4— 7r)/7r, whence B = -121, B 2 = '015, nearly all the 
motion being transmitted. 
It remains to consider the value of k. The problem of the 
propagation of sound in a circular tube, having regard to the 
influence of viscosity and heat-conduction, has been solved 
analytically by KirchhofP, on the suppositions that the tan- 
gential velocity and the temperature-variation vanish at the 
walls. In discussing the solution, Kirchhoff takes the case in 
which the dimensions of the tube are such that the immediate 
effects of the dissipative forces are confined to a relatively thin 
stratum in the neighbourhood of the walls. In the present 
application interest attaches rather to the opposite extreme, 
viz. when the diameter is so small that the frictional layer 
pretty well fills the tube. Nothing practically is lost by an- 
other simplification which it is convenient to make (following 
Kirchhoff) — that the velocity of propagation of viscous and 
thermal effects is negligible in comparison with that of sound. 
One result of the investigation may be foreseen. When 
the diameter of the tube is very small, the conduction of heat 
from the centre to the circumference of the column of air 
becomes more and more free. In the limit the temperature of 
the solid walls controls that of the included gas, and the expan- 
sions and rarefactions take place isothermally. Under these 
circumstances there is no dissipation due to conduction, and 
everything is the same as if no heat were developed at all. 
Consequently the coefficient of heat-conduction will not appear 
in the result, which will involve, moreover, the Newtonian 
value of the velocity of sound (b) and not that of Laplace (a). 
Starting from Kirchhoff's formula?, we find as the value of 
* Pogg. Ann. cxxxiv. 1868. 
