The string of variable density: perturbative and non-perturbative results

TL;DR

This paper develops systematic iterative approximations for the vibrational modes of a fixed string with variable density, comparing perturbative, WKB, and numerical methods to analyze their accuracy and asymptotic behavior.

Contribution

It introduces three theorems for approximating string modes regardless of inhomogeneity, validated through comparisons with WKB and numerical results.

Findings

Accurate mode approximations using theorems for variable density strings

Perturbation theory results agree with WKB asymptotics for energy behavior

Different density dependence in one-dimensional strings compared to higher dimensions

Abstract

We obtain systematic approximations for the modes of vibration of a string of variable density, which is held fixed at its ends. These approximations are obtained iteratively applying three theorems which are proved in the paper and which hold regardless of the inhomogeneity of the string. Working on specific examples we obtain very accurate approximations which are compared both with the results of WKB method and with the numerical results obtained with a collocation approach. Finally, we show that the asymptotic behaviour of the energies of the string obtained with perturbation theory, worked to second order in the inhomogeinities, agrees with that obtained with the WKB method and implies a different functional dependence on the density that in two and higher dimensions.