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Propagation of Waves in Networks of Thin Fibers

In: Integral Methods in Science and Engineering, Volume 1

Author

Listed:
  • S. Molchanov

    (University of North Carolina)

  • B. Vainberg

    (University of North Carolina)

Abstract

This chapter 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 (ODEs) on the limiting graph. The vertices of the graph correspond to junctions of the wave guides. In order to determine the solutions of the ODEs 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.

Suggested Citation

  • S. Molchanov & B. Vainberg, 2010. "Propagation of Waves in Networks of Thin Fibers," Springer Books, in: Christian Constanda & M.E. Pérez (ed.), Integral Methods in Science and Engineering, Volume 1, chapter 25, pages 255-278, Springer.
    • Handle: RePEc:spr:sprchp:978-0-8176-4899-2_25
    DOI: 10.1007/978-0-8176-4899-2_25
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