Eliciting Harmonics on Strings

One may produce the qth harmonic of a string of length π by applying the ‘correct touch’ at the node π/q during a simultaneous pluck or bow. This notion was made precise by a model of Bamberger, Rauch and Taylor. Their ‘touch’ is a damper of magnitude b concentrated at π/q. The ‘correct touch’ is that b for which the modes, that do not vanish at π/q, are maximally damped. We here examine the associated spectral problem. We find the spectrum to be periodic and determined by a polynomial of degree q − 1. We establish lower and upper bounds on the spectral abscissa and show that the set of associated root vectors constitutes a Riesz basis and so identify ‘correct touch’ with the b that minimizes the spectral abscissa. Mathematics Subject Classification. 35P10, 35P15, 74K05, 74P10. Received September 5, 2006. Published online January 18, 2008.