The plucked string: an example of non-normal dynamics

TL;DR

This paper analyzes the non-normal dynamics of a plucked string, highlighting its unique features like non-orthogonal eigenvectors and lack of normal modes, and provides an analytical solution relevant to musical instruments.

Contribution

It presents an analytical approach to understanding the transient decay of a plucked string, emphasizing its non-normal dynamical properties.

Findings

The plucked string exhibits non-normal dynamics with non-orthogonal eigenvectors.

An analytical solution approximates the string's motion during decay.

The model is relevant for understanding musical instrument vibrations.

Abstract

Motion of a single Fourier mode of the plucked string is an example of transient, free decay of coupled, damped oscillators. It shares the rarely discussed features of the generic case, e.g., possessing a complete set of non-orthogonal eigenvectors and no normal modes, but it can be analyzed and solved analytically by hand in an approximation that is appropriate to musical instruments' plucked strings.