adopting a general basis for the displacement field, $$\label{eq:BasisExp}
u_i =  a_{ir}\psi_{ir},$$ where $a_{ir}$ is the $r$’th expansion coefficient of the $i$-directed displacement and $\psi_{ir}$ is the associated basis function. Equation (\[eq:BasisExp\]) may be substituted into Eq. (\[eq:ElasticWave\]), and the resulting algebraic equations solved numerically to give the dispersion curves of the propagating modes. Although a variety of basis functions, such as Gaussian-Hermite polynomials, could be used for this purpose, we have found power-series expansions to be particularly effective [@nishiguchi1997acoustic].

In the case of a homogeneous, isotropic, insulating dielectric such as SiN$_{\rm x}$, the stiffness tensor simplifies, and the modal calculation requires only the mass density, $\rho$, Young’s modulus, $E$, and Poisson’s ratio, $\nu_p$, of the material, in addition to the height, $H$, and width, $W$, of the bar. Averaging over any specific microstructure in favour of the bulk elastic properties is appropriate given the long dominant phonon wavelengths ($>$1$\mu$m) at low temperatures.

Figure \[fig:Modes\_Plot\](a) shows the dispersion profiles of the low-order modes of the experimentally considered geometry $H=$200 nm, $W=$ 500nm, with $\rho =$ 3.14gcm$^{-3}$, $E=$ 280GPa and $\nu_p =$0.28 [@vlassak1992new; @walmsley2007poisson; @MaterialsCRC]. With the exception of the four lowest-order modes, all modes have a cut-off frequency that increases 