  We develop a fully nonlinear approximation to the short-wavelength limit of
eccentric waves in astrophysical discs, based on the averaged Lagrangian method
of Whitham (1965). In this limit there is a separation of scales between the
rapidly varying eccentric wave and the background disc. Despite having small
eccentricities, such rapidly varying waves can be highly nonlinear, potentially
approaching orbital intersection, and this can result in strong pressure
gradients in the disc. We derive conditions for the steepening of nonlinearity
and eccentricity as the waves propagate in a radially structured disc in this
short-wavelength limit and show that the behaviour of the solution can be
bounded by the behaviour of the WKB solution to the linearised equations.
