Excitation of Josephson currents by aerogel vibrations in superfluid $^3$He

The problem of mechanical vibrations of an aerogel attached to an elastic wire in superfluid $^3$He is solved for the case, where the aerogel is also in a superfluid state. The hydrodynamic boundary conditions at the aerogel surface formulated in this work allow one to explain the anomalously rapid increase in the frequency of mechanical vibrations of the system on cooling. The found relation between the phase jump at the aerogel boundary and the superfluid current flowing across the boundary suggests the Josephson character of such current.