This thesis describes the characterization of the performance of a Two-Stage Travelling Wave Thermoacoustic Engine, called the DEMO 2 Engine. Preliminary experiments on the DEMO 2 Engine showed that its performance level was far below its design expectations. Thus, an analytical approach was developed to investigate the sources of losses that were affecting the performance of the DEMO 2 Engine. The approach involves studying the heat transfer and acoustic performance of the DEMO 2 Engine as well as constructing a numerical model of the DEMO 2 Engine. Consequently, a heat flow chart was constructed to investigate the heat flow across the individual components in the DEMO 2 Engine. Also, the Four Microphone Impedance Tube Method was modified to enable the determination of the Transfer Matrix for the components in the DEMO 2 Engine. The effects of heat input on the Transfer Matrix of the DEMO 2 Engine's Thermoacoustic Core was also investigated. Apart from that, a numerical model was constructed and validated with experimental observations. Subsequently, the model was able to carry out further investigations on the DEMO 2 Engine where experimental investigations are unable to do so. As a result, the losses affecting the DEMO 2 Engine was successfully identified. The losses were also characterized quantitatively such that each of its effect could be compared with each other.