2026-07-072026-07-07http://salesiana.dossiersoluciones.com/handle/123456789/80632Sound propagation is considered in range-independent environments and environments consisting of a range-independent background on which a weak range-dependent perturbation is superimposed. Recent work on propagation of both types of environment, involving both ray- and mode-based wavefield descriptions, have focused on the importance of $α$, a ray-based ``stability parameter,'' and $β,$ a mode-based ``waveguide invariant.'' It is shown that, when $β$ is evaluated using asymptotic mode theory, $% β=α.$ Using both ray and mode concepts, known results relating to the manner by which $α$ (or $β$) controls both the unperturbed wavefield structure and the stability of the perturbed wavefield are briefly reviewed.Submitted to J. Acoust. Soc. AmChaotic Dynamicstext