Department of Mathematical Sciences Master's Thesis Defense: Cameron Norton
1:00 p.m. to 2:00 p.m.
Department of Mathematical Sciences
Master's Thesis Defense
Friday, December 12th, 2025
1:00PM-2:00PM
Stratton Hall 207
Speaker: Cameron Norton
Title: Long water waves on graphs
Abstract: The physical problem studied accounts for surface water waves in a channel network. The mathematical model refers to partial differential equations (PDEs) on a graph. The goal is to study wave reflection due to compatibility conditions at the vertex of the graph. The PDE system studied is the shallow water (or long wave) model, as considered by Jacovkis (SIAM Appl. Math., 1991). Jacovkis considered converging and diverging Y-shaped junctions, constrained by the Neumann-Kirchhoff vertex conditions. Jacovkis proved a solvability condition for the linearized hyperbolic system. This dissertation extends his framework to the reflection-transmission of waves at junctions, in different channel configurations, considering balanced and non-balanced graphs. These different regimes are expressed through scattering matrices at the vertices. New numerical simulations, not present in Jacovkis, illustrate the differences between these junction configurations and reveal when a balanced junction is reflectionless, leading to a transparent graph as called in the literature. Examples include novel configurations such as symmetric and asymmetric islands.