Abstract: In this paper, the cut-off frequencies of longitudinal wave propagating in functionally graded piezoelectric-piezomagnetic material (FGPPM) cylinder shells is investigated analytically. The mechanical, the electrical and the magnetical coupling waves equations of the longitudinal wave propagating in FGPPM cylinder shells in the cylindrical coordinate system expressed by the displacement function, the electric potential function and the magnetic potential function are established and the homogeneous boundary conditions are characterized. Considering that the wave number approaches zero, the Wentzel-Kramers-Brillouin (WKB) method is employed for solving these equations and derive an explicit approximate analytical solution for the cutoff frequency of longitudinal guided waves in an FGPPM shell. By comparing WKB solution with the Bessel function solution for the cut-off frequency of longitudinal guided waves in homogeneous piezoelectric and piezomagnetic cylinder shells and with the cut-off frequencies obtained from the dispersion curves of functionally graded piezoelectric and piezomagnetic cylinder shells, respectively, the high precision of the WKB solution has been verified. Further analysis and simplification shows that the cut-off frequency of longitudinal waves is a combination of two approximate arithmetic progressions with a common difference inversely proportional to the integral of reciprocal wave velocity along the radial direction. Numerical examples demonstrate that representing the cut-off frequency using the common differences of these two progressions yields sufficient accuracy. The effects of varying elastic parameters, density, electrical parameters, magnetic parameters, and shell thickness on the cut-off frequencies are investigated and compared. It is found that variations in elastic parameters and density have a significant impact on the cut-off frequency, while variations in electrical and magnetic parameters have a relatively smaller effect. Increasing the shell thickness leads to a decrease in the cut-off frequency. The analytical methods and conclusions presented in this paper are applicable to the cut-off frequencies of longitudinal guided waves in functionally graded elastic and piezoelectric material cylinder shells. These findings provide a theoretical basis for ultrasonic non-destructive testing of heterogeneous material cylinder shells based on cut-off frequencies.