The problem considered is that of absorbing two-dimensional water waves in a channel by means of a moving termination at the end of the channel. The problem is formulated for a semi-infinite channel and solutions are determined according to a linearized theory. The motion of the termination that is needed for absorption is determined in the form of a linear operation on the measured surface elevation at a fixed point in the channel so a self-actuating wave-absorbing system can be devised. A theoretical method of studying the stability of such a system is presented. A system of this type was built and experiments with it are described. Wave absorption is demonstrated both for monochromatic waves and for wave pulses. The absorption of a wave pulse is compared with the absorption of the same pulse by a fixed beach making a ten degree angle with the horizontal direction.