FROM A VIBRATING- BODY TO A SURROUNDING GAS. 
449 
ting sphere and a long vibrating cylinder, the motion of the fluid in the latter case being 
supposed to be in two dimensions. The sphere is chosen as the best representative of a 
bell, among the few geometrical forms of body for which the problem can be solved. 
The cylinder is chosen as the representative of a vibrating string. In the case of the 
sphere the problem is identical with that solved by Poisson in his memoir “ Sur les 
movements simultanes d’un pendule et de l’air environnant”*, but the solution is dis- 
cussed with a totally different object in view, and is obtained from the beginning, to 
avoid the needless complexity introduced by taking account of the initial circumstances, 
instead of supposing the motion already going on. 
A. Solution of the Problem in the case of a Vibrating Sphere. 
Suppose an elastic solid, spherical externally in its undisturbed position, to vibrate 
in the manner of a bell, the amplitude of vibration being very small. Suppose it 
surrounded by a homogeneous gas, which is at rest except in so far as it is set in mo- 
tion by the sphere ; and let it be required to determine the motion of the gas in terms of 
that of the sphere supposed given. We may evidently for the purposes of the present 
problem suppose the gas not to be subject to the action of external forces. 
Let the gas be referred to the rectangular axes of x, y , z, and let u, v, w be the com- 
ponents of the velocity. Since the gas is at rest except as to the disturbance commu- 
nicated to it from the sphere, u, v, w are by a well-known theorem the partial differen- 
tial coefficients with respect to x , y, z of a function <p of the coordinates ; and if (f be 
the constant expressing the ratio of the small variations of pressure to the corresponding 
small variations of density, we must have 
df~ a 
\dx^dy^ dz *) ’ 
( 1 ) 
and if s be the small condensation, 
a 2 dt 
As we have to deal with a sphere, it will be convenient to refer the gas to polar coor- 
dinates r, 6, co, the origin being in the centre. In terms of these coordinates, equation 
(1) becomes 
,2 dtp , 1 d 
dr' r 1 sin $ d6 
+1 
d 2 <p 
'■6 do j 2 
( 2 ) 
and if u', v 1 , w' be the components of the velocity along the radius vector and in two 
directions perpendicular to the radius vector, the first in and the second perpendicular 
to the plane in which 6 is measured, 
i_ dtp 
( 3 ) 
dr t dQ r sin 0 dw ’ 
Let c be the radius of the sphere, and V the velocity of any point of its surface 
* Memoires de l’Aeademie des Sciences, t. xi. p. 521. 
3 r 2 
