Resolution of the 300-Year-Old Vibrating String Controversy

TL;DR

This paper challenges the classical 1D vibrating string model, proposing a new 2D non-linear PDE model that better captures realistic string behaviors, supported by theoretical proofs, numerical simulations, and experimental validation.

Contribution

A novel 2D non-linear PDE model for vibrating strings is developed, overcoming limitations of the classical model and enabling analysis of complex initial conditions and motions.

Findings

Classical model cannot represent all realistic initial conditions.

Pure transverse motion is only possible in rare cases.

Experimental results support the new 2D model.

Abstract

The dispute about the well-known 1D vibrating string model and its solutions, known as The Vibrating String Controversy, spanned the whole of 1700s and involved a group of the most eminent scientists of the time. After that, the model stood undisputed for over two centuries. In this study, it is shown that not only this 300-year-old model cannot correspond to reality, but it is theoretically not quite plausible, either. A new 2D model is developed removing all the assumptions of the classical model. The result is a pair of non-linear partial differential equations modeling 2D motions of a finite 1D string. A theorem that can be used to determine the initial displacement functions from the initial shape of the string is proven. The new model is capable of representing initial conditions that cannot be handled in the classical model. It also allows initially non-taut/non-slack strings and self-intersecting shapes. The classical model and the non-taut strings emerge as special limit cases. It is proven that pure transverse motions of a 1D string are possible only in very rare cases. A theorem that sets the conditions for pure transverse motions is also presented. Numerical studies of interesting cases are presented in support of the new model. High-speed camera experiments are also conducted, the results of which also support the new theory.