Propagation of Waves in Networks of Thin Fibers

TL;DR

This paper analyzes wave propagation in thin fiber networks, showing that as fibers shrink, wave behavior can be described by differential equations on a graph with specific vertex conditions, especially when wave propagation is possible.

Contribution

The authors provide a simplified, improved framework for describing wave propagation in shrinking fiber networks, including cases where wave propagation occurs in cylindrical parts.

Findings

Asymptotic description of waves via ODEs on a graph

Vertex conditions expressed through scattering matrices

Extension of results below and around the spectral threshold

Abstract

The paper contains a simplified and improved version of the results obtained by the authors earlier. Wave propagation is discussed in a network of branched thin wave guides when the thickness vanishes and the wave guides shrink to a one dimensional graph. It is shown that asymptotically one can describe the propagating waves, the spectrum and the resolvent in terms of solutions of ordinary differential equations on the limiting graph. The vertices of the graph correspond to junctions of the wave guides. In order to determine the solutions of the ODE on the graph uniquely, one needs to know the gluing conditions (GC) on the vertices of the graph. Unlike other publications on this topic, we consider the situation when the spectral parameter is greater than the threshold, i.e., the propagation of waves is possible in cylindrical parts of the network. We show that the GC in this case can be expressed in terms of the scattering matrices related to individual junctions. The results are extended to the values of the spectral parameter below the threshold and around it.