Density of states of the acoustical vibration field in a medium with piecewise constant parameters: Application to superfluid helium in a porous material

We calculate the effect of separation and boundary surfaces of any kind on the density of the acoustical vibration field in a system consisting of small solid and liquid domains. The results obtained are used for calculating thermodynamic characteristics of the phonon gas (such as the free energy, the entropy, the internal energy, and the heat capacity) in a medium with piecewise constant parameters at low temperatures. Liquid helium (${}_{}{}^4{}_{}{}^{}\mathrm{He}$) in a porous material (or a fine-grained solid powder immersed in liquid helium) is considered as a particular case and an important example of such media.