  The most attractive application of fluid-based thermoacoustic (TA) energy
conversion involves traveling wave devices due to their low onset temperature
ratios and high growth rates. Recently, theoretical and numerical studies have
shown that thermoacoustic effects can exist also in solids. However, these
initial studies only focus on standing waves. This paper presents a numerical
study investigating the existence of self-sustained thermoelastic oscillations
associated with traveling wave modes in a looped solid rod under the effect of
a localized thermal gradient. Configurations having different ratios of the rod
radius $R$ to the thermal penetration depth $\delta_k$ were explored and the
traveling wave component (TWC) was found to become dominant as $R$ approaches
$\delta_k$. The growth-rate-to-frequency ratio of the traveling TA wave is
found to be significantly larger than that of the standing wave counterpart for
the same wavelength. The perturbation energy budgets are analytically
formulated and closed, shedding light onto the energy conversion processes of
solid-state thermoacoustic (SSTA) engines and highlighting differences with
fluids. Efficiency is also quantified based on the thermoacoustic production
and dissipation rates evaluated from the energy budgets.
