XIX. On the Motion of a small Sphere acted upon by the Vibrations of an Elastic Medium. By the Rev. JAMES CHALLIS, M.A., Plumian Professor of Astronomy in the University of Cambridge.

[Read April 26, 1841.]

It is proposed in this Essay to give a mathematical investigation respecting the motion of a small solid sphere submitted to the dynamical action of the vibrations of a medium so constituted that the pressure (p) and density (p) are related to each other by the equation p a certain constant.

=

a2p, a2 being

1. For this purpose it will be convenient to obtain, first, the equations which apply to the motion of such a medium when directed to or from a centre, whether the centre be moving or stationary.

Conceive P to be a fixed point in space at which the motion of the fluid is directed to or from a moving centre C. Describe about C as a centre a spherical surface always passing through the point P, and concentric with this another passing through P', a point in CP produced. Let, at a given time t, CP = r, and CP' = r', or r + dr, dr being supposed very small. Conceive now a conical surface, with an indefinitely small vertical angle, to have its vertex at C, and its axis coinciding with CPP', and let it always include the same portion (m2) of the interior spherical surface. Then if the velocity of the centre C resolved in the direction of r, the radius CP at the time t +7, ( being very small) will become r±ar, and CP' will become r+drar, the interval dr being supposed not to vary with the time. Hence the portion of the exterior surface included by the VOL, VII. PART III.

α =

conical surface at the time t + is m2.

2

(r+or+ar)', or m2. (1+

dr r±

and this, neglecting terms of the order dr x aт, is equal to

m2 r'2
72

Again, let v and p be the velocity and density of the fluid which passed the area m2 at the time t, and v, p, the values of the same quantities at any time t + T. Now the quantity of fluid which in the small time St passes m2 is equal to fm2p,v,dr, the integral being taken from 70 to 7 St. And because

Also if v, p, be the velocity and density of the fluid which is passing the m2 r'2

area of the exterior surface at the time t+7, the quantity of fluid

2

which passes in the interval &t is

in the interval it is m2 v'p'd, taken from 70 to 7 =ôt.

And, because v and p' are what v and p become by very small changes of time and place,

Consequently, supposing the velocity positive when directed from the centre, the increment of matter in the space between the two areas in the time dt, is ultimately,

Now if any point be selected between P and P', the radius to which at the time t is r,, by what has been already shewn, the transverse section of the cone through this point at the time t + dt is with sufficient approximam2 r2 and is therefore independent of St. Hence at any instant jiz during the interval at the content of the conical frustum is mr dr,

tion

2

It is plain that since P has been assumed to be a fixed point of space, the differential coefficients here are partial. The above equation, with

are the three equations which determine the circumstances of the motion. As the velocity (a) of the centre C in no way enters into them, we may conclude that the same equations apply to motion tending to or from a moving centre as to motion tending to or from a fixed centre.

2. From the equations (1), (2), (3), others more immediately applicable to the question proposed to be discussed will now be deduced. The equation (1) is equivalent to

and by substituting for p in (3) from (2) there results,

If now we assume o' to be such a function of r and t that the partial differential coefficient is equal to v, and substitute this ex

do' dr

pression for in (5), the equation is integrable with respect to r. The result is,

a2 Nap. log. p +

do' do'

+ = f(t).

dt 2dr2

3. Before making use of this equation it will be necessary to consider the comparative values of its terms under the circumstances in which we propose to apply it. The circumstances are, that v is very small compared to a, and always exceedingly small compared to the breadths of the waves whose dynamical action is to be investigated.

First, it is plain that the terms having a in their denominators will be small compared to the others. Neglecting those terms, or, which is the same thing, considering a infinite, we have the case of an incompressible fluid, and the equation applicable to it is,

The known equation which gives the pressure (p) of an incompressible fluid is

As this equation contains two arbitrary functions, two conditions of the motion may be arbitrarily assumed. Let us assume for one condition, that the excess of the pressure (p) above the pressure II, which would exist in the undisturbed state of the fluid, is solely owing to a velocity arbitrarily impressed in the direction of r. Then v and ƒ'(t) being supposed to vanish when p= II, we must have,

As a second condition, let us suppose that the velocity is impressed at a given distance (r), and is given at any time t by the expression. m sin bt. Hence f(t)=mr sin bt, and f(t)=bmr cos bt. Consequently by substituting,