Communications in Mathematical Sciences

Volume 15 (2017)

Number 8

Transport of power in random waveguides with turning points

Pages: 2327 – 2371



Liliana Borcea (Department of Mathematics, University of Michigan, Ann Arbor, Mich., U.S.A.)

Josselin Garnier (Centre de Mathématiques Appliquées, Ecole Polytechnique, Palaiseau, France)

Derek Wood (Department of Mathematics, University of Michigan, Ann Arbor, Mich., U.S.A.)


We present a mathematical theory of time-harmonic wave propagation and reflection in a two-dimensional random acoustic waveguide with sound soft boundary and turning points. The boundary has small fluctuations on the scale of the wavelength, modeled as random. The waveguide supports multiple propagating modes. The number of these modes changes due to slow variations of the waveguide cross-section. The changes occur at turning points, where waves transition from propagating to evanescent or the other way around. We consider a regime where scattering at the random boundary has significant effect on the wave traveling from one turning point to another. This effect is described by the coupling of its components, the modes. We derive the mode coupling theory from first principles, and quantify the randomization of the wave and the transport and reflection of power in the waveguide. We show in particular that scattering at the random boundary may increase or decrease the net power transmitted through the waveguide depending on the source.


mode coupling, turning waves, scattering, random waveguide

2010 Mathematics Subject Classification

35Q99, 60F05, 78M35

Received 13 February 2017

Accepted 23 August 2017

Published 20 December 2017