  The specific heat of ultra-thin free-standing membranes is calculated using
the elastic continuum model. We first obtain the dispersion relations of the
discrete set of acoustic modes in the system. The specific heat is then
calculated by summing over the discrete out-of-plane wavevector component and
integrating over the continuous in-plane wavevector of these waves. In the
low-temperature regime (T < 4 K), the flexural polarization is seen to have the
highest contribution to the total specific heat. This leads to a linear
dependence with temperature, resulting in a larger specific heat for the
membrane compared to that of the bulk counterpart
