A three-dimensional compressible magnetohydrodynamic (MHD) simulation in the spherical coordinates is presented for the propagation of MHD waves in a structured solar atmosphere by using the fractional step method. An initial magnetic field profile that has the characteristics of the open configuration at the pole and closed configuration at the equator is used. The initial state of the solar atmosphere consists of the chromosphere and corona; thus the transition region from the photosphere is approximated. The purpose of this study is to understand the nature of propagation MHD waves in a two-layer solar atmosphere. Thus a pressure pulse is added at the bottom of the chromosphere (i.e., r = 1 Rs). We compare the response of this case with one in which we introduce the pulse at the bottom of the corona (i.e., r = 1.018 Rs). We find that (1) where the disturbance is initiated at the bottom of the chromosphere, a pair of fast mode waves and a slow mode MHD wave are generated. The pair of fast mode MHD waves are expansion waves and their propagation speeds increase with increasing height, (2) if the disturbance is initiated at the bottom of the corona, (r = 1.018 Rs), we observed a pair of fast and slow mode MHD waves propagating upward to the corona and another pair propagating downward toward the photosphere, respectively. We suggest that the model has the potential to be used to show the correspondence between observed flare initiation Moreton waves of the chromosphere and Extreme Ultraviolet Imaging Telescope waves in the low corona.