Eliciting Harmonics on Strings

Steven J. Cox (1) , Antoine Henrot (2)

1 Computational & Applied Mathematics
2 CORIDA - Robust control of infinite dimensional systems and applications

One may produce the $q$th harmonic of a string of length $\pi$ by applying the 'correct touch' at the node $\pi/q$ 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 $\pi/q$. The 'correct touch' is that $b$ for which the modes, that do not vanish at $\pi/q$, 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.

Mots clés: Point-wise damping; Spectral abscissa; Riesz basis

Domaines: Equations aux dérivées partielles [math.AP]

Steven J. Cox, Antoine Henrot. Eliciting Harmonics on Strings. ESAIM: Control, Optimisation and Calculus of Variations, 2008, 14 (4), pp.657-677. ⟨10.1051/cocv:2008004⟩. ⟨hal-00015299v2⟩