In some inertial confinement fusion target designs a spherical shell collapses on a void or compresses a small amount of gaseous material. There can be a period during which both the outside (driver) pressure and the inside pressure have a negligible effect on the implosion dynamics, and the motion is essentially ballistic. The changes in the aspect ratio occur mainly because of geometrical convergence. For reasonable parameters the inner surface does not begin to decelerate until shortly before convergence is complete. An approximate description of this ‘‘coasting’’ phase has been developed and applied to study the evolution of perturbations on the inner and outer surfaces of the shell in the limit where the fluid is incompressible. The two surfaces are strongly coupled as long as the shell remains thin. When the shell becomes thick compared to the inner radius, the inner and outer surface perturbations decouple. Under these conditions the surface wave action is a good adiabatic invariant, which can be used to estimate the change in the amplitude of a perturbation as a function of the shell inner radiusR1. Detailed analysis confirms the adiabatic invariance argument and extends the results. The adiabatic invariant may also be applicable in the case of compressible fluids.