A terminal–boundary value problem that describes the process of damping the vibrations of a rod consisting of two segments with different densities and elasticity coefficients but with identical wave travel times

In this paper, in terms of a finite-energy weak solution, we study a terminal–boundary value problem describing the complete damping, at a terminal time $T$, of the longitudinal vibrations of a rod consisting of two segments with different densities and elasticity coefficients under the condition that the lengths of the segments are such that the wave travel times along these segments are equal. We find an explicit analytic expression for a solution to this problem and prove its uniqueness. This problem is important for the design of acoustic systems in which one can completely damp an acoustic signal by a terminal time instant by applying boundary controls at the ends of a vibrating rod.