1917] PRESSURE DEVELOPED DURING COLLAPSE OF A SPHERICAL CAVITY 505
and if p be the density, the whole kinetic energy of the motion is
(2)
Again, if P be the pressure at infinity and R0 the initial value of R, the work done is
. j; ii J. , -if) « TJ<)\
—5— (_K0' — IV).
Q
When we equate (2) and (3) we get
™ 2P fRns ,
•(3)
.(4)
expressing the velocity of the boundary in terms of the radius. Also, since U=dR/dt, •
t==
2pr
if /3 = R/R0. The time of collapse to a given fraction of the original radius is thus proportional to R0p^P~^, a result which might have been anticipated by a consideration of "dimensions." The time T of complete collapse is obtained by making /3 == 0 in (5). An equivalent expression is given by Besant, who refers to Cambridge Senate House Problems of 1847.
Writing ft3 = z, we have
/* 1 /QU/2 W M r 1 -
I p~t \A}[>J ii _T/T \ —. * 7
Jo(l-/33)^~ hZ which may be expressed by means of F functions. Thus
.........(6)
According to (4) U increases without limit as R diminishes. This indefinite increase may be obviated if we introduce, instead of an internal pressure zero or constant, one which increases with sufficient rapidity. We may suppose such a pressure due to a permanent gas obedient to Boyle's law. Then, if the initial pressure be Q, the work of compression is 4>7rQRo8 log (R0/R), which is to be subtracted from (3). Hence
a,nd U= 0 when
(8)
2 denoting (as before) the ratio of volumes RS/R0S. Whatever be the (positive) yalue of Q, U comes again to zero before complete collapse, and if Q >P the first movement of the boundary is outwards. The boundary oscillates between bwo positions, of which one is the initial..............................(1)