  Over the last five years several experimental groups have reported anomalies
in the temperature dependence of the period and amplitude of a torsional
oscillator containing solid $^4$He. We model these experiments by assuming that
$^4$He is a viscoelastic solid--a solid with frequency dependent internal
friction. We find that while our model can provide a quantitative account of
the dissipation observed in the torsional oscillator experiments, it only
accounts for about 10% of the observed period shift, leaving open the
possibility that the remaining period shift is due to the onset of
superfluidity in the sample.
