in relation to Sound. 183 
we may here neglect the effects of friction and heat- conduction. 
If a be the velocity of sound in the open, and K Q —n/a, we 
may write 
s= ( g*o* + Ber***)^**, .... (2) 
u=a(—e iK o x + Be- iK ^)e int ; .... (3) 
so that the incident wave is 
S==e i(nt+K x) • f£\ 
or, on throwing away the imaginary part, 
s — cos (nt + k ss) (5) 
These expressions are applicable when x exceeds a moderate 
multiple of the distance between the channels. Close up to 
the face the motion will be more complicated; but we have no 
need to investigate it in detail. The ratio of u and s at a 
place near the wall is given with sufficient accuracy by put- 
ting a?=0 in (2) and (3), 
u _ q(-l + B) , fi . 
s~ 1 + B W 
We now assume that a region about x=0, on one side of 
which (6) is applicable and on the other side of which (1) is 
applicable, may be taken so small relatively to the wave-length 
that the mean pressures are sensibly the same at the two boun- 
daries, and that the flow into the region at the one boundary 
is sensibly equal to the flow out of the region at the other 
boundary. The equality of flow does not imply an equality 
of mean velocities, since the areas concerned are different. 
The mean velocities will be inversely proportional to the cor- 
responding areas — that is, in the ratio cr : <t + (/, if a' denote 
the area of the unperforated part of the wall corresponding to 
each channel. By (1) and (6) the connexion between the 
inside and outside motion is expressed by 
n (B — Y)a . ,. 
We will denote the ratio of the imperforated to the perforated 
parts of the wall by g, so that g—a'ja. Thus, 
1-B _ kq m 
l + B~K(l+g) {<) 
If <7=0, k=k , there is no reflection; if there are no perfora- 
tions, <7 = co , and then B = l, signifying a complete reflection. 
In place of (7) we may write 
*(l+ff)-* ' 
*- <1+9) + Ko > ^ 
