Mathematical Sciences Department Applied Math Seminar - Binan Gu, WPI
1:00 p.m. to 2:00 p.m.
Applied Math Seminar
Binan Gu, WPI
Thursday, September 3
1-2pm
SH 313
Title: Shallow-Water Wave Scattering in Channel Networks: From Two-Dimensional Junctions to Quantum Graphs
Abstract: Consider shallow-water waves propagating through a two-dimensional network of channels. At a junction, an incoming wave divides into reflected and transmitted components. We ask how this division depends on channel widths and branching angles. An idealized model-reduction replaces each channel by its centreline, producing a 1D metric graph; a wave differential operator together with vertex conditions define what is known as a quantum graph. We study the linearized shallow-water equations, where (quantum-graphs-like) Neumann--Kirchhoff vertex conditions lead to a scattering matrix. This matrix maps incoming waves onto outgoing waves, revealing the roles of geometric channel parameters.
The 1D graph reduction is ultimately a modelling hypothesis whose regime of validity must be determined, in particular regarding the role of the 2D branching angles. For wave propagation, the relevant small parameter is the channel width relative to wavelength, κ. We examine the limit κ → 0, corresponding to long waves. We explore different scenarios, varying the channel-width ratios, the branching angles, and the junction symmetry in order to determine when the two-dimensional results are close to the reduced/simplified 1D scattering predictions.