Scientific Papers - Vi

1919] ON EESONANT REFLEXION OF SOUND FROM A PERFORATED WALL 663
distributed, and bounded by surfaces everywhere perpendicular to the face of the wall. If the channels be sufficiently numerous relatively to the wavelength of vibration, the transition, when sound impinges, from simple plane waves on the outside to the waves of simple form in the interior of the channels occupies a space which is small relatively to the wave-length, and then the connexion between the condition of things outside and inside admits of simple expression.
On the outside, where the dissipation is neglected, the velocity potential (<£) of the plane waves, incident and reflected in the plane of say, at angle 6, is subject to
dtyldtf^a^dtyldtf + dty/df), ..................... (1)
or if ^ oc eini, where n is real,
d*d>ldx2 + d2<j>/dyz + k*ct>=0, ........................ (2)
k being equal to n/a. The solution of (1) appropriate to our purpose is
the first term representing the incident wave travelling towards — x, and the second the reflected wave. From (3) we obtain for the velocity u parallel to x, and the condensation s, when as = 0,
5), ..................... (4)
doc .
i^ = --e<(n*+*tf8infl)(jA+m .................. (5)
a dt a
., , u /) B — A ,£N
so that — = 0080 jj- — -, ............................... (6)
as B + A /
For the motion inside a channel we introduce in (1) on the left a term hd<j>/dt, h being positive, to represent the dissipation. Thus, if <£ be still proportional to eint, we have in place of (2)
d*<j>/da? + d^dyz + dz<j>/dz*'+ k'*<j> = 0, .................. (7)
where k'z is now complex, being given by
V*=*&- ink/a* ............................... (8)
If we write k' = k^ — ikz, where klt k2 are real and positive, we have
^2 _ &22 = ^ ]Clk2 = %nh/a? ......................... (9)
At a very short distance from the mouth of the channel dtyfdy*, dtyfdz2 in (7) may be neglected, and thus
^ = e^(A'cos^ + J5/sin/c/«!} ...................... (10)
If the channel be closed at x ~ — 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.