Mode veering and symmetry-protected crossings in conservative elastic waveguides: unified perturbation-theoretic interpretation and adaptive tracking

Accurate mode tracking is essential for elastic waveguide dispersion analysis in ultrasonic nondestructive evaluation and structural health monitoring. Its reliability, however, deteriorates near mode veering and closely spaced eigenvalues, where rapid eigenvector exchange causes mode misidentification. Although widely observed, the quantitative relationship between veering, eigenvector evolution, and tracking robustness has not been systematically established. By specializing classical perturbation theory to the single-parametric Hermitian SAFE eigenproblem--exemplified by conservative elastic waveguides--we obtain explicit expressions for eigenvector derivatives and modal coupling strength. This yields a unified, quantitative interpretation of mode veering, symmetry-protected crossings, and degeneracies, and clarifies their distinct tracking implications: eigenvector sensitivity scales inversely with the eigengap, explaining modal repulsion and the degradation of correlation-based tracking near avoided crossings, whereas symmetry-protected crossings remain benign because symmetry-induced decoupling preserves smooth eigenvector evolution, and symmetry-protected degeneracies require rotation-invariant subspace tracking. A numerical consistency condition and an existence result for a critical step size are then derived, motivating a two-level adaptive strategy with an a posteriori error indicator that separates numerical tracking consistency from symmetry-based physical correctness. Numerical examples validate the theoretical predictions and demonstrate improved robustness in regions of strong modal interaction, providing practical guidance for reliable dispersion calculations and ultrasonic inspection.