Forerunning mode transition in a continuous waveguide

TL;DR

This paper introduces a new forerunning mode transition in a uniform waveguide, where the separation front propagates ahead of the wave, with detailed analytical and numerical analysis of the transition and its dependence on wave amplitude.

Contribution

The study reveals a novel forerunning mode transition in waveguides, with analytical solutions and numerical simulations showing how steady-state transitions evolve into forerunning regimes.

Findings

Steady-state separation mode exists only within a bounded amplitude range.

Beyond critical amplitude, porous-like local separations emerge ahead of the main front.

The separation front speed approaches the wave group velocity as amplitude increases.

Abstract

We have discovered a new, forerunning mode transition as the periodic transition wave propagating in a uniform continuous waveguide. The latter is represented by an elastic beam separating from the elastic foundation under the action of sinusoidal waves. The critical displacement is the separation criterion. We show that the steady-state separation mode, where the separation front speed is independent of the wave amplitude, exists only in a bounded speed-dependent range of the wave amplitude. As the latter exceeds the critical value the steady-state mode is replaced by a more complicated regime with porous-like local separations emerging at a distance ahead of the main transition front. The crucial feature of this simple model is that the wave group speed is greater than the phase speed. The analytical solution allows us to show in detail how the steady-state mode transforms into the forerunning one. The established forerunning mode studied numerically manifests itself as a periodic process. As the incident wave amplitude grows the period decreases, while the separation front speed averaged over the period increases to the group velocity of the wave. In addition, the complete set of relations is presented for the waves excited by the oscillating force moving along the free beam, including the resonant case corresponding to a certain relation between the load's speed and frequency.