Nachbin lectures on conformal mapping and complex topography

toposAndre Nachbin lectured at the summer school on conformal mapping and complex topography.  The motivation is to understand waves over rapidly-varying topographies, inspired by theory for acoustic waves in heterogeneous random media.  In the theory, the principal tool in conformal mapping is the Schwarz-Christoffel transformation and the Schwarz-Christoffel Toolbox.  The talk started with a review of previous work of Hamilton (1977) and Nachbin (2003).  Then a derivation of the Boussinesq model in shallow water in the mapping coordinates was derived, based on three small parameters: nonlinearity, dispersion, and a topography parameter.  A key feature of the reduced system is the metric M which appears as a coefficient in the reduced Boussinesq equation.  It is an approximation to the Jacobian along the mapped surface. The second part of the talk discussed implementation details of the theory, implications for water waves, and extension to 3D.  Background reading includes Fokas & Nachbin (2012). Slides from the lectures can be downloaded here.  Lecture notes will be forthcoming.

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