In order to analyze the far-field propagation characteristics of the new non-diffractive Airy beams deeply in theory on the condition of finite energy, firstly, the one-dimensional paraxial wave equation which decides the propagation characteristics of light waves in free space was analyzed by taking advantage of the Fourier analysis method. Moreover, the special characteristics of Airy function and the initial condition of finite energy with exponential decay were also taken into consideration. As a result, the analytical solution of the wave equation can describe the propagation characteristics of one-dimensional Airy beams accurately in free space under the condition of finite energy. Then, the propagation characteristics of one-dimensional and two-dimensional Airy beams were studied by utilizing the acquired analytical solution. What′s more, the influence of various parameters on non-diffractive propagation and transverse acceleration characteristics of Airy beams were studied in detail. Study results indicate the non-diffractive propagation distances of two-dimensional Airy beams are 1014, 624, 455, 338, 193 mm separately when the arbitrary transverse scale is 100 μm and the decay factors are 0.03, 0.05, 0.07, 0.1, 0.2, separately. The non-diffractive propagation distances will become longer if the decay factor becomes smaller when the arbitrary transverse scale keeps invariant. Moreover, the transverse acceleration of Airy beams will become larger with the decrease of arbitrary transverse scale when the decay factor keeps invariant. The adopted study method is also fit for analysis of Airy beams′ propagation characteristics in the media with refractive index.