Finite-Element Modeling of Acoustic Modes in Solid $^4$He

Using a finite-element method we have calculated the low-frequency eigenmodes of various torsional oscillators (TO) in the literature. To elucidate the relationship between the elasticity [1] and apparent nonclassical rotatation inertia (NCRI) of solid $^4$He, we have investigated the dependence of the torsion mode on the shear modulus of $^4$He. If we exclude supersolidity, we find the inferred increase in the shear modulus that is necessary to account for typical frequency shifts in TO studies is significantly larger than that reported in Ref. [1] and nearly unphysical. Experiments are in progress to understand the connection between NCRI and the increased shear modulus. [1] J. Day and J. R. Beamish, arXiv:0709.4666v1 (2007).