1917]
COLLAPSE OF A SPHERICAL CAVITY
507
The maximum value of^> occurs when
— 4
.(14)
and then
4>r
.(15)
So long as z, which always exceeds 1, is less than 4, the greatest value of p, viz. P, occurs at infinity; but when 2 exceeds 4, the maximum p occurs at a finite distance given by (14) and is greater than P. As the cavity fills up, z becomes great, and (15) approximates to
...........................(16)
P~~ ** ' A3"
corresponding to r = 4<$R = T587.R............................(17)
It appears from (16) that before complete collapse the pressure near the boundary becomes very great. For example, if R = -^R^, p — 1260P.
This pressure occurs at a relatively moderate distance outside the boundary. At the boundary itself the pressure is zero, so long as the motion is free. Mr Cook considers the pressure here developed when the fluid strikes an absolutely rigid sphere of radius R. If the supposition of incompressibility is still maintained, an infinite pressure momentarily results; but if at this stage we admit compressibility, the instantaneous pressure P' is finite, and is given by the equation
/3' being the coefficient of compressibility. P, P', ft1 may all be expressed in atmospheres. Taking (as for water) /3' = 20,000, P — 1, and R = -^R0) Cook
finds
P' — 10,300 atmospheres = 68 tons per sq. inch,
and it would seem that this conclusion is not greatly affected by the. neglect of compressibility before impact.
The subsequent course of events might be traced as in Theory of Sound, § 279, but it would seem that for a satisfactory theory compressibility would have to be taken into account at an earlier stage.ero 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)