Say I push down on a very large drum head that lies in the z = 0 plane at equili

Say I push down on a very large drum head that lies in the z = 0 plane at equilibrium. Suppose there is no friction between my finger and the drum head. Suppose my finger tip pushes the drum head down some distance d. Suppose I accelerate my finger tip to some constant velocity v in the x direction while always keeping my finger tip in the plane z = -d. Are waves produced during the acceleration of my finger? Once my finger is in motion does it tend to stay in constant motion that is does the drum head resist acceleration or deceleration of my finger tip?