Article thumbnail
Location of Repository

Exponential asymptotics and gravity waves

By S. J. Chapman and J. Vanden-Broeck


The problem of irrotational inviscid incompressible free-surface flow is examined in the limit of small Froude number. Since this is a singular perturbation, singularities in the flow field (or its analytic continuation) such as stagnation points, or corners in submerged objects or on rough beds, lead to a divergent asymptotic expansion, with associated Stokes lines. Recent techniques in exponential asymptotics are employed to observe the switching on of exponentially small gravity waves across these Stokes lines. As a concrete example, the flow over a step is considered. It is found that there are three possible parameter regimes, depending on whether the dimensionless step height is small, of the same order, or large compared to the square of the Froude number. Asymptotic results are derived in each case, and compared with numerical simulations of the full nonlinear problem. The agreement is remarkably good, even at relatively large Froude number. This is in contrast to the alternative analytical theory of small step height, which is accurate only for very small steps

Topics: Fluid mechanics
Year: 2006
DOI identifier: 10.1017/S0022112006002394
OAI identifier:

Suggested articles


  1. 1886 On stationary waves in flowing water.
  2. (2002). A comparison of linear and nonlinear computations of waves made by slender submerged bodies.
  3. (1984). A cusp-like free-surface flow due to a submerged source or sink. doi
  4. (1989). A semi-inverse method for free-surface flow over a submerged body.
  5. (1997). An axisymmetric free surface with a 120 degree angle along a circle. doi
  6. (1988). Analytic theory of the Saffman–Taylor fingers.
  7. (1973). Asymptotic Expansions: their Derivation and Interpretation .
  8. (1991). Asymptotics beyond all orders in a model of crystal growth.
  9. (1991). Computation of transcendental effects in growth problems: linear solvability conditions and nonlinear methods.
  10. (1999). Cusp flows due to an extended sink in two dimensions.
  11. (2002). Exponential asymptotics and capillary waves.
  12. (1998). Exponential asymptotics and Stokes lines in nonlinear ordinary differential equations.
  13. (1990). Flow caused by a point sink in a fluid having a free surface.
  14. (1982). Free surface flow over a semi-circular obstruction.
  15. (1990). Free-surface flow of a stream obstructed by an arbitrary bed topography.
  16. (1987). Free-surface flow over a step.
  17. (1993). Fully nonlinear two-layer flow over arbitrary topography.
  18. (1973). Generation of waves of small amplitude by an obstacle placed on the bottom of a running stream.
  19. (1977). Nonlinear free surface effects – the dependence on Froude number.
  20. (1995). Nonlinear free-surface flow computations for submerged cylinders.
  21. (1995). On short-scale oscillatory tails of long-wave disturbances.
  22. (1913). On some cases of wave motion on deep water.
  23. (1991). On the existence of homoclinic and heteroclinic orbits for differential equations with a small parameter.
  24. (1987). On the motion of a small two-dimensional body submerged beneath surface waves. doi
  25. (1948). On the reflection of surface waves by a submerged circular cylinder.
  26. (1999). On the roˆle of Stokes lines in the selection of Saffman–Taylor fingers with small surface tension.
  27. (1998). Ploughing flows. doi
  28. (1988). Structural stability of the Korteweg–de Vries solitons under a singular perturbation.
  29. (1950). Surface waves on deep water in the presence of a submerged circular cylinder. doi
  30. (1927). The method of images in some problems of surface waves.
  31. (2003). The selection of Saffman–Taylor fingers by kinetic undercooling.
  32. (2004). Trapped waves between submerged obstacles.
  33. (1998). Two-dimensional steady free surface flow into a semi-infinite mat sink.
  34. (1989). Uniform asymptotic smoothing of Stokes discontinuities.
  35. (1995). Weakly nonlocal solitary waves in a singularly perturbed Korteweg– de Vries equation.

To submit an update or takedown request for this paper, please submit an Update/Correction/Removal Request.