Variable Tension, Large Deflection Ideal String Model For Transverse Motions

TL;DR

This paper develops a new, more general model for transverse vibrations of an ideal string that relaxes common assumptions, yet still results in the classical linear wave equation, and extends to variable density and area.

Contribution

It introduces a novel approach removing traditional assumptions, demonstrating the linearity of the wave equation under more general conditions, and connects classical, finite, and variational methods.

Findings

The classical wave equation remains valid without constant tension or small deformation assumptions.

The force distribution in static cases is proportional to the second derivative of displacement.

An extended equation accounts for variable initial density and cross-sectional area.

Abstract

In this study a new approach to the problem of transverse vibrations of an ideal string is presented. Unlike previous studies, assumptions such as constant tension, inextensibility, constant crosssectional area, small deformations and slopes are all removed. The main result is that, despite such relaxations in the model, not only does the final equation remain linear, but, it is exactly the same equation obtained in classical treatments. First, an "infinitesimals" based analysis, similar to historical methods, is given. However, an alternative and much stronger approach, solely based on finite quantities, is also presented. Furthermore, it is shown that the same result can also be obtained by Lagrangian mechanics, which indicates the compatibility of the original method with those based on energy and variational principles. Another interesting result is the relation between the force distribution and string displacement in static cases, which states that the force distribution per length is proportional to the second spatial derivative of the displacement. Finally, an equation of motion pertaining to variable initial density and area is presented.