FROM A VIBRATING- BODY TO A SURROUNDING- GAS. 
457 
of the string varies sensibly on proceeding even a moderate distance along it, and that 
the propagation of the sound-wave produced by no means takes place in two dimensions 
only. But the question how far a sound-wave is produced at all, and how far the dis- 
placement of the gas by the cylinder merely produces a local motion to and fro, is de- 
cided by what takes place in the immediate neighbourhood of the string, such as 
within a distance of a few diameters ; and though the sound-wave, when once produced, 
in its subsequent progress diverges in three dimensions, the same takes place with the 
hypothetical sound-wave which would be produced if lateral motion were prevented, so 
that the comparison which it is the object of the investigation to institute is not affected 
thereby. 
Assuming, then, the motion to be in two dimensions, and referring the fluid to polar 
coordinates, r, 0, r being measured from the axis of the undisturbed cylinder, we shall 
have for the fundamental equation derived from (1) 
rm 
dt* — a \dr 2 + 7- dr + r* d8*J ’ 
and if u\ v' be the components of the velocity along and perpendicular to the radius 
vector, 
1 d<f 
If c be the radius of the cylinder, and V the normal component of the velocity of the 
surface of the cylinder, we must have 
^=V, when r=c. 
As before, I will suppose the motion of the cylinder, and consequently of the fluid, to be 
regularly periodic, but instead of using circular functions directly I will employ the ima- 
ginary exponential e imat , i denoting as before \J — 1, and will put accordingly ~V=e imat \J, 
U being a given function of 6, and <p—-he imat . For a given value of r, 4 1 may by a known 
theorem be developed in a series of sines and cosines of Q and its multiples, and therefore 
for general values of r can be so developed, the coefficients being functions of r. If \f/„ 
be the coefficient of cos nQ or sin nQ, we find 
d^n .ldtyn W 2 
dr 2 ' r dr r 2 
0 . 
(15) 
If we suppose the normal velocity of the surface of the cylinder to vary in a given 
manner from one generating line to another, so that U is a given function of Q, we may 
expand U in a series of the form 
U=U 0 -J-Ui cos Q -{-U 2 cos 2S -(- . . . 
+U 7 ! sin 0+U' 2 sin 2Q-f-. . . 
On applying now the equation of condition which has to be satisfied at the surface of 
the cylinder, we see that a term U n cos nQ or sin nQ of the wth order in the expression 
3 s 2 
