Recent observations of polar plumes in the southern solar coronal hole by the Extreme-Ultraviolet Imaging Telescope (EIT) on board the SOHO spacecraft show signatures of quasi-periodic compressional waves with periods of 10-15 minutes. The relative wave amplitude was found to increase with height in the plumes up to about 1.2 R. Using a one-dimensional linear wave equation for the magnetosonic wave, we show that the waves are propagating and that their amplitude increases with height. The observed propagation velocity agrees well with the expected sound velocity inside the plumes. We present the results of the first nonlinear, two-dimensional, magnetohydrodynamic (MHD) simulation of the magnetosonic waves in plumes for typical coronal conditions consistent with observations and gravitationally stratified solar corona. We find numerically that outward-propagating slow magnetosonic waves are trapped, and nonlinearly steepen in the polar plumes. The nonlinear steepening of the magnetosonic waves may contribute significantly to the heating of the lower corona by compressive dissipation.