### Abstract

The Maslov integral representation of slowly varying surface gravity wavetrains is developed, allowing for smooth but otherwise arbitrary variations of both bathymetry and horizontal currents. Although we focus on the surface gravity wave problem, the results presented can be applied - and are presented in a form which facilitates their application - to any type of scalar small amplitude dispersive wave motion in a slowly varying environment, with or without background flow. The Maslov integral provides an asymptotically valid solution, on the wavelength scale, to the initial value problem under conditions in which exact solutions are unavailable owing to nonseparability of the equations of motion. Caustics of arbitrary complexity are properly treated. Away from caustics, stationary phase evaluation of the Maslov integral reduces it to a superposition of locally plane, slowly varying dispersive wavetrains that conserve wave action. An important step in the development of the Maslov integral is the geometric construction of exact solutions to the time-dependent wave action conservation equation. It is shown that in an unbounded homogeneous environment the Maslov integral reduces as a special case to the usual Fourier integral solution to the initial value problem.

Original language | English (US) |
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Pages (from-to) | 247-266 |

Number of pages | 20 |

Journal | Wave Motion |

Volume | 32 |

Issue number | 3 |

DOIs | |

State | Published - Sep 2000 |

### ASJC Scopus subject areas

- Modeling and Simulation
- Physics and Astronomy(all)
- Computational Mathematics
- Applied Mathematics