  We calculate the frequency-dependent mesoscopic acoustic phonon transmission
probability through the abrupt junction between a semi-infinite,
one-dimensional cylindrical quantum wire and a three-dimensional bulk
insulator, using a perturbative technique that is valid at low frequency. The
system is described using elasticity theory, and traction-free boundary
conditions are applied to all free surfaces. In the low-frequency limit the
transmission probability vanishes as the frequency squared, the transport being
dominated by the longitudinal channel, which produces a monopole source of
elastic radiation at the surface of the bulk solid. The thermal conductance
between an equilibrated wire nonadiabatically coupled to a bulk insulator
should therefore vanish with temperature at temperature cubed.
