as the cross-sectional area is reduced. Each mode can be assigned to one of four 2-dimensional displacement symmetries: compressional, in-plane and out-of-plane flexural, and torsional. The lowest order mode in each group is a principal mode with no cut-off. Propagation of these four principal modes at all frequencies imposes a fundamental lower limit on the power transmitted ballistically along a straight bar, even as its dimensions are reduced such that the higher order modes carry negligible power. This is often called the ‘quantised limit’.

cd ![\[fig:Modes\_Plot\] (a) Frequency, $\nu$, against wavenumber, $q$, of elastic modes with cut-off frequencies up to 25 GHz for a SiN$_{\rm x}$ bar having cross section 500 $\times$ 200nm. Longitudinal, in-plane transverse, out-of-plane transverse and torsional symmetries are shown in blue, green, red and magenta respectively, with the principal modes dashed. (b) Total net power spectral density (PSD) for a bar with termination temperatures of 68mK and 135mK, with the ordinate matching that of (a). (c) Mode number $i$, against cut off frequency $\nu_{i}$, showing quadratic dependence.](Figure1.pdf "fig:"){width="3.37in"}

In the context of TESs, each leg consists of a SiN$_{\rm x}$ bar terminating at the central island with a temperature taken to be the superconducting transition temperature 