The processes of dissipation due to the crystal interaction of lattice with medium are taken into account using the friction forces. The methods of discrete lattice dynamics shows that the dissipative forces induce surface waves of new type which penetrate deeper to crystal and damp weaker than the waves existing without dissipation. The possibilities to discover and use the obtained surface vibrations are considered. Besides it is shown, that the accounted dissipation leads to essential modification of known surface waves: their frequency and damping is renormed, penetration to crystal decreases, at intensive enough dissipation the surface waves become aperiodic surface motions of different types. The estimations of surface vibration damping are given for contacts of crystal with gas and liquid.