<p>This study investigates the dispersion characteristics of axisymmetric torsional-compression coupled waves in a functionally graded material (FGM) cylindrical tube with free boundary. The effects of various parameters on the dispersion characteristics of waves in FGM cylindrical tubes are analyzed. Firstly, the material properties of FGM are characterized by the mixed power law and the exponential models. Secondly, based on the elastic wave theory, the wave equation for an infinitely long functionally graded cylindrical tube is established. Then, the Wentzel–Kramers–Brillouin–Jeffreys (WKBJ) method is innovatively used to obtain the dispersion equation of the coupled wave in the FGM cylindrical tube. The FGM is degenerated into isotropic material, and the validity and correctness of the proposed method are verified by comparing with the existing literature results. Finally, the mathematical software is used to solve the dispersion equation, and the dispersion curves of different FGM cylindrical tube radius ratios and different material graded indexes under two graded models are obtained. The results show that the radius ratio, the graded model and the material graded index have a significant effect on the dispersion effect. This study reveals a torsional wave coupling weakening in FGM tubes, establishes the radius ratio's regulatory role, and pioneers a wave velocity difference method for quantitative characterization, advancing the relevant theoretical framework.</p>