Tsunami waves induced by submarine volcanic eruption have multi-physical mechanisms, such as underwater explosions, pyroclastic flows, flank and lava bench failures, column and caldera collapses, and atmospheric pressure disturbances. Based on the cylindrical Boussinesq model proposed by Chwang & Wu (1977), a numerical model for tsunami waves driven by axisymmetric moving atmospheric disturbances is developed to understand how moving pressure fields lead to tsunamis and explore how pressure parameters affect water wave patterns. The model is verified against numerical results of cylindrical wave propagation on different terrains. The computed wave elevations induced by the atmospheric disturbance of the 2022 Tonga volcanic tsunami event agree well with the DART buoy data in the Pacific Ocean. The effects of the radial velocity, spatial scale, and strength of pressure disturbances are discussed. It turns out that the water wave pattern is highly related to the pressure radial velocity, and the Proudman resonance is triggered when the pressure velocity approaches the shallow water wave speed. However, there is less influence of the spatial scale and strength of the pressure disturbance on the amplitude amplification factor.