  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.
