The eigenfrequencies of a two-mass oscillator moving uniformly along a string on a visco-elastic foundation are analysed. It is shown that in the case of purely elastic foundation, the oscillator has either one or two real positive eigenfrequencies dependent on the system parameters. Taking into account the viscosity of the foundation, the complex eigenfrequencies of the oscillator are investigated. The study shows that eigenfrequencies, which are related to attenuating vibrations of the oscillator, are not uniquely determined. It is found that the existence of an eigenfrequency ω = ω0 + iδ with a small imaginary part δ «ω0 is not a sufficient condition for resonance under an external force P exp (iω0t).