Notes on osculations and mode tracing in semi-analytical waveguide modeling

TL;DR

This paper investigates the mathematical and computational aspects of mode crossings and osculations in semi-analytical waveguide models, proposing algorithms for matrix decomposition and mode tracing to better understand dispersion curve behaviors.

Contribution

It introduces a simple matrix flow decomposition algorithm and a mode tracing method based on ODE approximation for analyzing dispersion curves in waveguides.

Findings

Osculations occur where eigenvalue curves approach closely without crossing.

Decomposable matrix flows allow eigenvalue crossings in waveguide models.

The proposed methods improve understanding of mode interactions in waveguides.

Abstract

The dispersion curves of (elastic) waveguides frequently exhibit crossings and osculations (also known as veering, repulsion, or avoided crossing). Osculations are regions in the dispersion diagram where curves approach each other arbitrarily closely without ever crossing before veering apart. In semi-analytical (undamped) waveguide models, dispersion curves are obtained as solutions to discretized parameterized Hermitian eigenvalue problems. In the mathematical literature, it is known that such eigencurves can exhibit crossing points only if the corresponding matrix flow (parameter-dependent matrix) is uniformly decomposable. We discuss the implications for the solution of the waveguide problem. In particular, we make use of a simple algorithm recently suggested in the literature for decomposing matrix flows. We also employ a method for mode tracing based on approximating the eigenvalue problem for individual modes by an ordinary differential equation that can be solved by standard procedures.