Quantization of the hydrodynamic modes in superfluid ${}_{}{}^4{}_{}{}^{}\mathrm{He}$ with a free surface is carried out. A complete and orthonormal set of eigenfunctions for both ripplons and surface-reflected phonons is constructed. Applications of the formalism to the calculation of the thermal energy density near the surface, the temperature-dependent part of the surface tension, and the rate at which power is absorbed by the surface from an incident phonon beam are given.

• Received 2 December 1974

DOI:https://doi.org/10.1103/PhysRevB.12.163