The time evolution of an oscillator coupled to an infinite string with a discontinuous mass density is investigated. It is shown that the equation of motion of the oscillator leads to a nonlinear characteristic equation due to the frequency-dependent nature of the point impedance of the string. A graphical method is applied to discuss the physical and unphysical roots associated with the characteristic equation. Then, the relevant eigen-frequencies (physical roots) are evaluated numerically to explain the motion of oscillator. It is found that the oscillator executes a damped oscillatory motion. However, due to the loss of homogeneity the real and imaginary parts of the eigen-frequencies simultaneously depend on the distance of oscillator from the discontinuity point and the difference between the densities of the semi-finite strings.
