Abstract: The frequency equation of the axisymmetric coupled vibration of a ultrasonic tubular resonator with both ends free is derived by an exact solution method,based on Fliigge classical thin shell theory.The dependence of the dimensionless frequency Ω on the ratio of length to radius l/a,the ratio of radius to thickness a/h and Poisson's ratio v is investigated,and the axisymmetric coupled vibration of a ultrasonic tubular resonator with both ends shear diaphragm is compared with that with both ends free.The accuracy of two-dimensional and three-dimensional apparent elasticity method is also assessed by the comparison with the exact solution method.Finally,the mode shape of the ultrasonic tubular resonator is calculated by the exact solution method,and its transform efficient is analyzed.The results show that as for two boundary conditions of both ends free and shear diaphragm,the difference of the effect of the boundary condition on the dimensionless frequency of the axisymmetric coupled vibration of a ultrasonic tubular resonator becomes smaller as l/a increases,and the larger a/h,the smaller the difference.Moreover,for free-free boundary conditions,the change of a/h has nearly no effect on the dimensionless frequency of the thin tubular resonator usually considered to mean a/h>10.Furthermore,the dimensionless frequency Ω decreases as v increases for a given l/a,and the impact of v onΩis different for different l/a,and the maximum impact will appear when l/a=π.Finally,the study also shows the three-dimensional apparent elasticity method has very high calculation accuracy.