This paper considers closed loop control of systems that are represented by the one dimensional wave equation with distributed damping. The model and the control algorithm are based on the traveling wave approach, which yields irrational and fractional order transfer functions in the frequency domain. In previously studied cases, with damping acting only at the boundaries and a conservative flexible media, it was shown that the wave form did not change between reflections. In the current case of distributed damping it is shown that the waves are distorted during motion. The closed loop algorithm is designed to eliminate the wave reflections at the control end. It extends the absolute vibration suppression (AVS) method to the irrational realm. The resulting controller is of fractional order and is implemented by appropriate approximations. The effectiveness of the control algorithm is demonstrated through rejection of external disturbances.