We present the results of three-dimensional (3D) ideal magnetohydrodynamics (MHD) simulations on the dynamics of a perpendicularly inhomogeneous plasma disturbed by propagating Alfvénic waves. Simpler versions of this scenario have been extensively studied as the phenomenon of phase mixing. We show that, by generalizing the textbook version of phase mixing, interesting phenomena are obtained, such as turbulence-like behavior and complex current-sheet structure, a novelty in longitudinally homogeneous plasma excited by unidirectionally propagating waves. This study is in the setting of a coronal hole. However, it constitutes an important finding for turbulence-related phenomena in astrophysics in general, relaxing the conditions that have to be fulfilled in order to generate turbulent behavior.