The linearized hydrodynamic equations governing the expansion of a frictionless compressible coolant from a cylindrical untapered reactor core admit an exact solution. The pressure so obtained depends only on the time required for an acoustic signal to travel from the core boundary to the core center and is quite independent of the amount of fluid external to the reactor. The pressure should behave similarly in the case of moderate tapering of the core. In the case of extreme tapering it is necessary to consider the external fluid as incompressible. In this case it is possible to obtain an approximate solution for the pressure, which does depend on the amount of external fluid.