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Surface waves are waves that propagate along an interface, with energy that is localized near the surface. A planar discontinuity in vorticity in two-dimensional inviscid, incompressible fluid flows supports surface waves. In this talk, we consider a scenario when an infinite vorticity front is described as a graph. We first discuss the derivation of contour dynamics equations for vorticity fronts. Then we prove that, using the modified energy method, the weakly nonlinear solutions to both the vorticity front equation and the Burgers-Hilbert equation satisfy the same cubically nonlinear, quasilinear, nonlocal asymptotic equation. Therefore, we conclude that the Burgers-Hilbert equation is an approximate equation for the dynamics of surface waves on the vorticity discontinuity over a cubic timescale. This is joint work with John K. Hunter, Ryan Christopher Moreno-Vasquez, and Qingtian Zhang.