Eliciting harmonics on strings

¹ Computational and Applied Mathematics, Rice University, Houston, TX, USA; cox@caam.rice.edu
² Institut Élie Cartan, UMR 7502, Nancy Université - CNRS - INRIA, Nancy, France.

Received: 5 September 2006

Abstract

One may produce the qth harmonic of a string of length π by applying the 'correct touch' at the node 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 . The 'correct touch' is that b for which the modes, that do not vanish at , are maximally damped. We here examine the associated spectral problem. We find the spectrum to be periodic and determined by a polynomial of degree . 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.