Magnetosonic wave formation driven by an expanding cylindrical piston is numerically simulated to obtain better physical insight into the initiation and evolution of largescale coronal waves caused by coronal eruptions. Several very basic initial configurations are employed to analyze intrinsic characteristics of MHD wave formation that do not depend on specific properties of the environment. It turns out that these simple initial configurations result in piston/wave morphologies and kinematics that reproduce common characteristics of coronal waves. In the initial stage, the wave and the expanding source region cannot be clearly resolved; i.e. a certain time is needed before the wave detaches from the piston. Thereafter, it continues to travel as what is called a “simple wave.” During the acceleration stage of the source region inflation, the wave is driven by the piston expansion, so its amplitude and phase-speed increase, whereas the wavefront profile steepens. At a given point, a discontinuity forms in the wavefront profile; i.e. the leading edge of the wave becomes shocked. The time/distance required for the shock formation is shorter for a more impulsive source-region expansion. After the piston stops, the wave amplitude and phase speed start to decrease. During the expansion, most of the source region becomes strongly rarefied, which reproduces the coronal dimming left behind the eruption. However, the density increases at the source-region boundary, and stays enhanced even after the expansion stops, which might explain stationary brightenings that are sometimes observed at the edges of the erupted coronal structure. Also, in the rear of the wave a weak density depletion develops, trailing the wave, which is sometimes observed as weak transient coronal dimming. Finally, we find a well-defined relationship between the impulsiveness of the source-region expansion and the wave amplitude and phase speed. The results for the cylindrical piston are also compared with the outcome for a planar wave that is formed by a one-dimensional piston, to find out how different geometries affect the evolution of the wave. An in-depth analysis of numerical simulations is performed to obtain a better insight into the nature of various phenomena occurring in the solar atmosphere as a consequence of the eruption of unstable coronal structures. Although the simulations take into account only the most basic characteristics of a flux-rope eruption, the simulation analysis reveals important information on various eruption-related effects. It quantifies the relation v between the eruption dynamics and the evolution of the large-amplitude coronal magnetohydrodynamic wave and the associated chromospheric downward-propagating perturbation. We show that the downward propagation of the chromospheric Moreton-wave disturbance can be approximated by a constant-amplitude switch-on shock that moves through a medium of rapidly decreasing Alfvén velocity. The presented analysis reveals the nature of secondary effects that are observed as coronal upflows, secondary shocks, various forms of wave-trains, delayed large-amplitude slow disturbances, transient coronal depletions, etc. We also show that the eruption can cause an observable Moreton wave and a secondary coronal front only if it is powerful enough and is preferably characterized by significant lateral expansion. In weaker eruptions, only the coronal and transition region signatures of primary waves are expected to be observed. In powerful events, the primary wave moves at an Alfvén Mach number significantly larger than 1 and steepens into a shock due to the nonlinear evolution of the wavefront. After the eruption-driven phase, the perturbation evolves as a freely propagating simple wave, characterized by a significant deceleration, amplitude decrease, and wave-profile broadening. In weak events the coronal wave does not develop into a shock and propagates at a speed close to the ambient magnetosonic speed.