664 . .ON EESONANT REFLEXION OF SOUND [446
and we may take
<j> = A" cos k' (an + I) e*n*.........................(11)
From (11) when no is very small,
u = d<j>/dx=:-kfA"smk/l.eint,........................(12)
os = — orld$ldt = -ikA" cosŁ7. eint, ...............(13)
u k'
so that - = .~tan&7 ............................... (14)
as IK
Now, under the conditions supposed, where the transition from the state of things outside to that inside, at a distance from the mouth large compared with the diameter of a channel, occupies a space which is small compared with the wave-length, we may assume that s is the same in (6) and (14), and
that
(cr + cr') u in (6) = an in (14),
where cr represents the perforated area and a-' the unperforated. Accordingly, if we put A = 1, as we may do without loss of generality, the condition to determine B is
j5-l a- k't&nk'l
If there be no dissipation in the channels, h = 0, and k' = Jc. In this case R _ (°" + cr') cos 6 cos kl — icr sin kl ,- „,
(a- + cr') cos 6 cos kl + icr sin kl ...................
Here Mod B — I, or the reflexion is total, as of course it should be. If in (16) cr = 0, B = I, the wall being unperforated. On the other hand, if a' = 0, the partitions between the channels being infinitely thin,
R __ cos 6 cos kl — i sin kl
cos 6 cos kl + isinkl In the case of perpendicular incidence 6 = 0, and
B = e~*m, ......... . ....................... (18)
the wall being in effect transferred from x = 0 to x = - L
We have now to consider the form assumed when k' is complex. In (15) cos k'l = cos kil cos ik2l + sin k-J/ sin ik2l,
. ............(19)
sin k'l = sink^ cos ik2l — cos k^l sinik2l. j
Before proceeding further it may be worth while to deal with the case where h, and consequently &2, is very small, but k.2l so large that vibrations in the channels are sensibly extinguished before the stopped end is reached. In this case
cos ikzI = ^e^1, sin ikzl ~ ^ie**1,
so that in (19), tan k'l = — i. Also by (9), k'/k = 1, and (15) becomes
"" (0-+0-7)cos 8' ........................(20)— I,5, " On porous bodies in relation to Sound."] can involve the concentrations only as ratios; otherwise the element of mass would enter into the result uncompensated. In like manner the diffusibilities can be involved only as ratios, or the element of time would enter. And since these ratios are all pure numbers, dx must be proportional to x. In words, the linear period at any place is proportional, cceteris paribus, to the distance from the original boundary. In this argument the thickness of the film— another linear quantity—is omitted, as is probably for the most part legitimate. In imagination we may suppose the film to be infinitely thin or, if it be of finite thickness, that the diffusion takes place strictly in one dimension.