Eliciting harmonics on strings

Abstract

Abstract.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.

The damped string problem revisited

“…We note that the area of damped wave equations remains incredibly active up to this day. Since we cannot possibly describe the recent developments in detail in this paper, we refer, for instance, to [6], [7], [8], [9], [11], [13], [15], [16], [17], [19], [20], [21], [22], [23], [24], [31], [32], [34], [36], [37], [38], [39], [46], [53], [54], [55], [56], [57], [58], [59], [60], [66], [74], [75], [76], [77], [78], [79], [86], [87], [88], [90], [91], [92], [94], [101], [103], and the references therein, which lead the interested reader into a variety of directions.…”

Spectral Theory of Operator Pencils, Hermite-Biehler Functions, and their Applications

“…In [19] a model was proposed to explain the playing of 'harmonics' on stringed instruments. This has a long history, see [59].…”