Some recently obtained results in classical vibration and acoustics: Coexistence of traveling and standing waves in a one-dimensional non-dispersive continuum

The problem of free vibration of a linear uniform axial bar, fixed at one end and connected to ground at the other end through a linear viscous damper has been carefully studied by several researchers. It's known that, for a fixed set of bar parameters and the special case of the damping coefficient λ equal to EA / c (c being the speed of sound in the continuum), no eigenvalues exist. Thus, energy imparted to the bar via harmonic motion of the fixed support will propagate through the bar and be fully dissipated in the damper, in effect making the bar appear to be semi-infinite. I will show some recent results by the present authors in which this phenomenon has been exploited in several other nondispersive media, the taut string and the circular acoustic duct, incorporating viscoelastic supports or absorbers to produce responses to harmonic motion at one or both boundaries that exhibit complete separation of traveling and standing waves, in effect localizing the vibration over a portion of the domain.