We report measurements of the thickness of unsaturated superfluid ${}_{}{}^4{}_{}{}^{}\mathrm{He}$ films in persistent flow as a function of persistent current velocity. Our results are in quantitative agreement with the predictions of Kontorovich, and thus disagree with the conclusion of Rudnick and coworkers that $\frac{{\rho }_s}{\rho }$ has an enhanced velocity dependence in these films.

• Received 9 August 1978

DOI:https://doi.org/10.1103/PhysRevB.19.2485