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.

研究了耦合到具有不连续质量密度的无限弦的振荡器的时间演化。结果表明，由于弦的点阻抗的频率相关性，振荡器的运动方程导致非线性特征方程。应用图形方法来讨论与特征方程相关的物理和非物理根。然后，对相关的本征频率（物理根）进行数值评估以解释振荡器的运动。发现振荡器执行阻尼的振荡运动。但是，由于均匀性的损失，本征频率的实部和虚部同时取决于振荡器到不连续点的距离以及半有限弦的密度之间的差异。