Wave-based control (WBC) is a technique for motion control of under-actuated flexible systems. It envisages actuator motion as launching a motion wave into the system, while simultaneously absorbing any wave returning from the system. For rest-to-rest motion the net launch displacement is set at half the target displacement. In absorbing the returning wave and vibrations, WBC moves the system the remaining distance to the target, with zero steady-state error. The focus of this paper is on very small residual vibrations around the target position which can endure for a long time after arrival at target. This issue was discovered through a recent development within WBC context on controlling complex two-dimensional, mass-spring, beam-like arrays. To date their existence has been unidentified. This paper investigates and interprets the nature of these vibrations, explains and identifies them based on wave ideas, and finally offers a new wave-based approach to mitigate or suppress them. It also discusses their implication, not just for WBC but for the general problem of control of flexible systems.