Exact expressions for the amplitudes and damped normal frequencies of a taut vibrating linear string of coupled masses

Abstract

We determine exact expressions for the amplitudes and vibrational eigenmodes of a taut linear string of identical masses that are subjected to a viscous damping that is proportional to their momentum. We first examine the damped displacements of the string in the more familiar limit of a continuous distribution of mass. We then derive a damped version of the linear, one-dimensional wave equation and describe the dispersion curve. Next, we find exact expressions for the damped normal eigenmodes for a finite system. We show that a fraction of these modes have imaginary frequencies that correspond to overdamped oscillations. Next, we show that the remaining modes exhibit underdamped oscillatory behaviour and represent that portion of the dispersion curve graphically. We use methods that are at an intermediate level of presentation.