Modelling Water Waves on Graphs

Waves on graphs are a current subject of research interest. As opposed to flows on graphs, the reflection–transmission of waves at the graph’s vertex is a problem that needs to be further modelled mathematically. The literature on the reflection and transmission of waves at a vertex is scarce. Some articles are reviewed and discussed. Water waves are a good topic for comparing different mathematical models, from hyperbolic conservation laws to weakly nonlinear, weakly dispersive systems of partial differential equations on a two-dimensional fattened (thick) graph and the respective one-dimensional graph-model reduction. In this study, we present a particular water wave model in which junction angles are systematically included in the mathematical model. Comparing the solutions with the fattened-graph model gave rise to a more general compatibility condition at the vertex. Current research topics of interest are outlined at the end.