Gain and efficiency equations are derived for a traveling wave heat engine, a device in which acoustic traveling waves force gas within a differentially heated regenerator to undergo a Stirling thermodynamic cycle and transform energy between thermal and acoustic forms. This derivation assumes nonturbulent flow conditions, a linear drag coefficient, a constant heat exchange coefficient, and neglects regenerator end effects. The complex characteristic impedance, gain, and efficiency are calculated for a thin slice of the regenerator in terms of dimensionless variables. With a Prandlt number of 0.7, the equations predict an efficiency of 70% that of an ideal Carnot cycle, and gain of 85% of that of theoretical maximum gain when fN ≡ωτ=0.003 and T′N≡ (dT/dx)T−1CIτ=0.4, where ω is the acoustic angular frequency, τ is the thermal time constant for the heat exchange process, dT/dx is the regenerator temperature gradient, and CI is the isothermal velocity of sound. In general, the equations predict that efficient high gain regenerators are quite short, have a large temperature gradient, and operate at low acoustic frequencies. Traveling wave heat pumps are also discussed and are very similar.