The proposal of a novel thermoacoustic regenerator using multi-temperature heat sources (MTHS) makes it possible to utilize lower-grade energy and keep relatively high efficiency in a thermoacoustic engine (TE) simultaneously. Based on thermodynamic laws combined with linear thermoacoustic theory, the time-averaged total power, enthalpy flux, acoustic power, entropy flux, and exergy flux in each component are derived and calculated to further understand the mechanism of a TE with the regenerator using two-temperature heat sources (TTHS). The comparison of the energy flows between the traditional TEs and those utilizing TTHS shows that the improvement of the temperature gradient in the regenerator by adding a mid-heater with appropriate heating power can increase the acoustic power and efficiency of a TE.