The Error in Rayleigh's Approximative Period

TL;DR

This paper provides rigorous bounds and explicit formulas for the error in Rayleigh's approximation of the period of a stretched string, revealing how initial conditions affect accuracy.

Contribution

It introduces precise bounds and a new formula for the relative error in Rayleigh's period approximation, improving understanding of its accuracy.

Findings

Rayleigh's approximative period overestimates the true period

The relative error is proportional to initial fractional displacement

The relative error is inversely proportional to initial stretch

Abstract

We obtain rigorous a priori upper and lower bounds to the exact period of the celebrated Rayleigh stretched string differential equation. We use them to show that Rayleigh's approximative period overestimates the true period and that the relative error is, to a first approximation, directly proportional to the initial fractional displacement and inversely proportional to the initial stretch. Thus, for a given length and stretch, one can determine the initial displacement so as to guarantee a prescribed accuracy in Rayleigh's period while for a given displacement one can see why the relative error blows up of the initial stretch is tiny. We have replaced the big-O terms with explicit inequalities and a new elegant formula for the relative error.