West proposed a looped Stirling engine with a liquid-column which the shape of is U. After this proposal, Martini worked the engine with 4 U-tubes, Li worked it with 3U-tubes. However, it's not clear that they undergo Stirling cycle and how the number of U-tubes affects the design. In this paper, we show that the engine of this type undergoes Stirling cycle, and the effect of the number of the U-tubes. By calculating the acoustic impedance in the middle of the gas-column theoretically, the possibility is showed that the engine undergoes Stirling cycle. And the number of the U-tubes can be changed by adjusting the length of gas-column or liquid-column to approach an acoustic impedance.