This paper is devoted to the simulation of acoustic wave propagation in complex media, applying the finite difference scheme and the orthonormal compactly supported wavelet transform approximation,to the time and the space dimension of the wave equation, respectively. A new method, which is a fast adaptive arithmetic, named Multiresolution Finite Difference (MRFD) is first proposed to solve the problem of the wave propagation in multi-layered medium with nonperiodic boundary condition. Thus the problem is solved in the wavelet domain rather than in the traditional Euclidian space. Due to adaptive and vanishing-moment property of the wavelet basis, the difference operator and solution vector have sparseness in the wavelet domain. MRFD is a promising method because of some advantages such as lighter computational burden, efficiency of convergence and robustness. Numerical results in geophysics exploration show the effectiveness and potential of the method.