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The materials industry is under increasing pressure to enhance performance while lowering costs and environmental impact. Three layers of functionally graded materials (FGMs) were proposed as a solution to this problem. The general goal of this research is to develop a semi-analytical method for solving the free vibration problem of FGMs in an infinite-length composite cylinder made of stainless steel (SS) and silicon nitride (SN). The formulation is based on Legendre polynomial approach and the use of specific rectangle window functions. This approach will result in an eigenvalue/eigenvector issue in the computation of normalized frequencies and displacement profiles. The effective material properties of FGM cylinders vary following the radial direction in conformity with the power-law distribution of the volume fractions. The numerical iteration approach was generated to create data utilizing MATLAB apps to investigate the free vibration analysis from specified boundary conditions. The results are obtained for longitudinal, torsional, and flexural modes. Then, the propagation of guided waves in three-layers (SN/SS/SN) of functionally graded cylinder with diverse gradient for each mode is studied. Accordingly, the results obtained demonstrate that the volumetric fraction of the FGM cylinder is significantly impacted by the graded-index. In addition, the graded-index has a major influence on the variation of the material's properties in the radial direction. Furthermore, the dispersion curves of the normalized frequencies and phase velocities are significantly influenced by the graded-index. These results demonstrate that the phase velocities of the same mode increase as the exponents of the power-law increase. The displacement distributions are also explored to better understand the characteristics of the guiding waves inside the materials. Furthermore, the boundary conditions have a more significant influence on the normal stresses. High accuracy is found between our results and those published in the literature (error = 0.58%). This paper provides an important insight into the propagation of ultrasonic-guided waves in functionally graded multilayer cylindrical constructs for non-destructive testing. [ABSTRACT FROM AUTHOR]