We explain the reduction of the thermal conductance below the predicted universal value observed by Schwab et al. in terms of the scattering of thermal phonons off surface roughness using a scalar model for the elastic waves. Our analysis shows that the thermal conductance depends on two roughness parameters: the roughness amplitude $δ$ and the correlation length $a$. At sufficiently low temperatures the conductance decrease from the universal value quadratically with temperature at a rate proportional to $δ^{2}a$. Values of $δ$ equal to 0.22 and $a$ equal to about 0.75 of the width of the conduction pathway give a good fit to the data.

10 pages, 5 figures. Ref. added, typo corrected