The article presents the research results of acoustic oscillations excitation (occurrence of thermoacoustic instability) in the traveling-wave thermoacoustic engine, in which the temperature of the heat exchanger-heater is maintained at 300 K, and the temperature of the heat exchanger-cooler is at the cryogenic temperature level. The mathematical model is described that allows to determine the boundaries of occurrence of thermoacoustic instability in the thermoacoustic engine, taking into account its geometric and thermophysical characteristics, as well as the working fluid used. The results obtained can be used to create low-temperature thermoacoustic engines for converting heat into acoustic oscillations, which can be used in various practical applications, for example, for generating electrical energy and reducing the cost of regasification of cryogenic products.