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In this paper, the method of radial basis function (RBF) is employed to construct the approximating function of the reproducing kernel particle method (RKPM), which can reduce the adverse effect of different kernel functions on computational accuracy and improve stability in the problem domain, and the radial basis reproducing kernel particle method (RRKPM) is presented. Compared with the RKPM, the RRKPM has higher computational accuracy and better stability. Then RRKPM is applied to wave propagation, and the discretized system equation can be obtained using the integral weak form. The penalty method is applied to imposing the essential boundary condition, and the two-point difference method is selected to discretize the time. The accuracy and stability of the RRKPM for wave propagation problem are illustrated by the numerical examples.