European Journal of Applied Mathematics


Free surface flow past topography: A beyond-all-orders approach


a1 Mathematical Institute, University of Oxford, 24–29 St Giles, Oxford OX1 3LB, UK

a2 School of Mathematical Sciences, Queensland University of Technology, Brisbane QLD 4101, Australia email:

a3 School of Mathematical Sciences, University of Adelaide, Adelaide SA 5005, Australia


The problem of steady subcritical free surface flow past a submerged inclined step is considered. The asymptotic limit of small Froude number is treated, with particular emphasis on the effect that changing the angle of the step face has on the surface waves. As demonstrated by Chapman & Vanden-Broeck, (2006) Exponential asymptotics and gravity waves. J. Fluid Mech. 567, 299–326, the divergence of a power series expansion in powers of the square of the Froude number is caused by singularities in the analytic continuation of the free surface; for an inclined step, these singularities may correspond to either the corners or stagnation points of the step, or both, depending on the angle of inclination. Stokes lines emanate from these singularities, and exponentially small waves are switched on at the point the Stokes lines intersect with the free surface. Our results suggest that for a certain range of step angles, two wavetrains are switched on, but the exponentially subdominant one is switched on first, leading to an intermediate wavetrain not previously noted. We extend these ideas to the problem of flow over a submerged bump or trench, again with inclined sides. This time there may be two, three or four active Stokes lines, depending on the inclination angles. We demonstrate how to construct a base topography such that wave contributions from separate Stokes lines are of equal magnitude but opposite phase, thus cancelling out. Our asymptotic results are complemented by numerical solutions to the fully nonlinear equations.

(Received June 03 2011)

(Revised January 05 2012)

(Accepted January 09 2012)

(Online publication February 09 2012)

Key words:

  • exponential asymptotics;
  • free surface flows;
  • small-Froude number limit;
  • intermediate waves;
  • trapped-wave solutions