transferred from the cold plate to the gas at a constant pressure, returning the parcel's temperature to its original value. Travelling wave devices can be described using the Stirling cycle. Temperature gradient Engines and heat pumps both typically use stacks and heat exchangers. The boundary between a prime mover and heat pump is given by the temperature gradient operator, which is the mean temperature gradient divided by the critical temperature gradient. The mean temperature gradient is the temperature difference across the stack divided by the length of the stack. The critical temperature gradient is a value that depends on characteristics of the device such as frequency, cross-sectional area and gas properties. If the temperature gradient operator exceeds one, the mean temperature gradient is larger than the critical temperature gradient and the stack operates as a prime mover. If the temperature gradient operator is less than one, the mean temperature gradient is smaller than the critical gradient and the stack operates as a heat pump. Theoretical efficiency In thermodynamics the highest achievable efficiency is the Carnot efficiency. The efficiency of thermoacoustic engines can be compared to Carnot efficiency using the temperature gradient operator. The efficiency of a thermoacoustic engine is given by The coefficient of performance of a thermoacoustic heat pump is given by Practical efficiency The most efficient thermoacoustic devices have an efficiency approaching 40% of the Carnot limit, or about 20% to 30% overall (depending on the heat engine temperatures). Higher hot-end temperatures may be possible with thermoacoustic devices because they have no moving parts, thus allowing the Carnot efficiency to be higher. This may partially offset their lower efficiency, compared to conventional heat engines, as a percentage of Carnot. The ideal Stirling cycle, approximated by traveling wave devices, is inherently more efficient than the ideal Brayton cycle, approximated by standing wave devices. However, the narrower pores required to give good thermal contact in a travelling wave device, as compared to a standing wave stack which requires deliberately imperfect thermal contact, also gives rise to greater frictional losses, reducing practical efficiency. The toroidal geometry often used in traveling wave devices, but not required for standing wave devices, can also boost losses due to Gedeon streaming around the loop. See also Sound amplification by stimulated emission of radiation (SASER) References Further reading Semipopular introduction to thermoacoustic effects and devices. Frank Wighard "Double Acting Pulse Tube Electroacoustic System" US Patent 5,813,234