186 On Porous Bodies in relation to Sound. 
By (12), (15), . xn.,y W _ 
If we suppose r= yo'oo centim., and g = 1, we shall have a wall 
of pretty close texture. In this case, by (18), p / =47*4, and 
I/ = '0412. A four-per-cent. loss may not appear to be much; 
but we must remember that in prolonged resonance we are 
concerned with the accumulated effects of a large number of 
reflections, so that rather a small loss in a single reflection 
may well be material. The thickness of the porous layer 
necessary to produce this effect is less than one centimetre. 
Again, suppose r=Yoo centim., g = l. We findj/ = 4*74, 
I / = -342, and the necessary thickness would be less than 10 
centimetres. 
If r be much greater than T ^ 5 centim., the exchange of heat 
between the air and the walls of the channels is no longer 
sufficiently free for the expansions to be treated as isothermal. 
When r is so great that the thermal and viscous effects extend 
only through a small fraction of it, we have the case discussed 
by Kirchhoff. If we suppose for simplicity # = (a state of 
things, it is true, not strictly consistent with channels of cir- 
cular section*), we have 
I=/-2, (19) 
in which , , x 
' =v »' + Ce-^ w 
The incident sound is absorbed more and more completely as 
the diameter of the channels increases ; but at the same time 
a greater thickness becomes necessary in order to prevent a 
return from the further side. If g = Q, there is no theoretical 
limit to the absorption ; and, as we have seen, a moderate 
value of g does not by itself entail more than a comparatively 
small reflection. A loosely compacted hay- or straw-stack 
would seem to be as effective an absorbent of sound as any- 
thing likely to be met with. 
In large spaces bounded by non-porous walls, roof, and floor, 
and with few windows, a prolonged resonance seems inevitable. 
The mitigating influence of thick carpets in such cases is well 
known. The application of similar material to the walls, or 
to the roof, appears to offer the best chance of further im- 
provement. 
* The problem in two dimensions is somewhat simpler than that treated 
by Kirchhoff. Although it would allow us without violence to suppose 
g=0, it seems scarcely worth while to enter upon it here, as the results 
are of precisely the same character. The principal difference is that the 
hyperbolic functions cosh &c. replace that of Bessel. 
