The nonlinear coupling between exothermic chemical reactions and a nanowire or nanotube with large axial heat conduction results in a self-propagating thermal wave guided along the nanoconduit. The resulting reaction wave induces a concomitant thermopower wave of high power density (>7 kW/kg), resulting in an electrical current along the same direction. We develop the theory of such waves and analyze them experimentally, showing that for certain values of the chemical reaction kinetics and thermal parameters, oscillating wavefront velocities are possible. We demonstrate such oscillations experimentally using a cyclotrimethylene-trinitramine/multiwalled carbon nanotube system, which produces frequencies in the range of 400 to 5000 Hz. The propagation velocity oscillations and the frequency dispersion are well-described by Fourier's law with an Arrhenius source term accounting for reaction and a linear heat exchange with the nanotube scaffold. The frequencies are in agreement with oscillations in the voltage generated by the reaction. These thermopower oscillations may enable new types of nanoscale power and signal processing sources.