Wave propagation in spatially modulated tubes

TL;DR

This paper studies how wave propagation in symmetric tubes with periodic geometric modulations is affected, deriving equations for wave velocity and identifying conditions for propagation failure and effects of bottlenecks.

Contribution

It introduces a combined asymptotic and projection method to model wave dynamics in modulated tubes, revealing velocity dependence and failure conditions.

Findings

Wave velocity depends nonlinearly on geometric modulation ratio.

Finite intervals of wave propagation failure caused by tube modulation.

Bottlenecks induce period changes in pulse trains by integer fractions.

Abstract

We investigate wave propagation in rotationally symmetric tubes with a periodic spatial modulation of cross section. Using an asymptotic perturbation analysis, the governing quasi two-dimensional reaction-diffusion equation can be reduced into a one-dimensional reaction-diffusion-advection equation. Assuming a weak perturbation by the advection term and using projection method, in a second step, an equation of motion for traveling waves within such tubes can be derived. Both methods predict properly the nonlinear dependence of the propagation velocity on the ratio of the modulation period of the geometry to the intrinsic width of the front, or pulse. As a main feature, we can observe finite intervals of propagation failure of waves induced by the tube's modulation. In addition, using the Fick-Jacobs approach for the highly diffusive limit we show that wave velocities within tubes are governed by an effective diffusion coefficient. Furthermore, we discuss the effects of a single bottleneck on the period of pulse trains within tubes. We observe period changes by integer fractions dependent on the bottleneck width and the period of the entering pulse train.