A New Wave Technique for Free Vibration of a String With Time-Varying Length

We introduce a new technique to analyze free vibration of a string with time-varying length by dealing with traveling waves. When the string length is varied, the natural frequencies and vibration energy are not constant. Thus, free response is not represented by discrete standing waves but by traveling waves, and a given phase of oscillation travels along the string. String tension and nonzero instantaneous transverse velocity at the moving boundary results in energy variation. When the string undergoes retraction, free vibration energy increases exponentially with time, causing dynamic instability. The new wave technique gives the time-varying natural frequency and the exact amount of energy transferred into the vibrating string at the moving boundary.