Mode complexity in a harmonically forced string with a local spring–damper and transitions from vibrations to waves

We numerically and analytically investigate the dynamics of a linear, uniform, homogeneous, undamped string coupled to a linear spring–dashpot system at its center and fixed to ground at its ends. Both ends of the string are excited through identical and synchronous harmonic motion, and the steady-state dynamics of the system is analytically studied. The localized damping introduces mode complexity, which results in spatial shifting of the peak resonance amplitudes to different locations of the string and highly nontrivial phase variations confined to certain predictable boundary layers of the string. We find that there exists a unique combination of system parameters such that mode complexity reaches an absolute maximum: in such an optimal case, damping destroys all the normal modes of vibration, and instead, traveling waves are formed. This peak in complexity marks the transition between very weak and very strong damping effects and provides a direct mode complexity-induced transition from vibrations (standing waves) to traveling waves.