Detailed review of the problem of wave removal of energy through the border of finite area of a foundation is given in the paper. Variation statement, methods and algorithms of solution of different dynamic problems for viscous-elastic system with account of non-reflective conditions on the border of finite foundation are given. The possibility to use non-reflective conditions is shown in tests problems. Dynamic behavior of heterogeneous viscous-elastic system ”dam – foundation” is studied under short-time intensive load subjected to the foundation. Weak dependence of energy dissipation in a system with hereditary viscous-elastic properties of material on the frequency of natural oscillations, as well as dependence of wave removal of energy on principal natural frequencies of system oscillations are revealed. In investigation of steady forced oscillations, the possibility of application of finite region of foundation is shown when non-reflective conditions are absent on the border and in exclusive cases only, when the frequency of external effect differs from artificial natural oscillations of discussed finite region; whereas the presence of non-reflective conditions on the border of finite region permits to avoid artificial resonance, unavailable in reality.