### Abstract

Many interesting free-surface flow problems involve a varying bottom. Examples of such flows include ocean waves propagating over topography, the breaking of waves on a beach, and the free surface of a uniform flow over a localized bump. We present here a formulation for such flows that is general and, from the outset, demonstrates the wave character of the free-surface evolution. The evolut on of the free surface is governed by a system of equations consisting of a nonlinear wave-like partial differential equation coupled to a time-independent linear integral equation. We assume that the free-surface deformation is weakly nonlinear, but make no a priori assumption about the scale or amplitude of the topography. We also extend the formulation to include the effect of mean flows and surface tension. We show how this formulation gives some of the well-known limits for such problems once assumptions about the amplitude and scale of the topography are made.

Original language | English |
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Pages (from-to) | 95-106 |

Number of pages | 12 |

Journal | Studies in Applied Mathematics |

Volume | 100 |

Issue number | 1 |

DOIs | |

Publication status | Published - 1 Jan 1998 |

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### ASJC Scopus subject areas

- Applied Mathematics

### Cite this

**A formulation for water waves over topography.** / Milewski, Paul A.

Research output: Contribution to journal › Article

*Studies in Applied Mathematics*, vol. 100, no. 1, pp. 95-106. https://doi.org/10.1111/1467-9590.00071

}

TY - JOUR

T1 - A formulation for water waves over topography

AU - Milewski, Paul A.

PY - 1998/1/1

Y1 - 1998/1/1

N2 - Many interesting free-surface flow problems involve a varying bottom. Examples of such flows include ocean waves propagating over topography, the breaking of waves on a beach, and the free surface of a uniform flow over a localized bump. We present here a formulation for such flows that is general and, from the outset, demonstrates the wave character of the free-surface evolution. The evolut on of the free surface is governed by a system of equations consisting of a nonlinear wave-like partial differential equation coupled to a time-independent linear integral equation. We assume that the free-surface deformation is weakly nonlinear, but make no a priori assumption about the scale or amplitude of the topography. We also extend the formulation to include the effect of mean flows and surface tension. We show how this formulation gives some of the well-known limits for such problems once assumptions about the amplitude and scale of the topography are made.

AB - Many interesting free-surface flow problems involve a varying bottom. Examples of such flows include ocean waves propagating over topography, the breaking of waves on a beach, and the free surface of a uniform flow over a localized bump. We present here a formulation for such flows that is general and, from the outset, demonstrates the wave character of the free-surface evolution. The evolut on of the free surface is governed by a system of equations consisting of a nonlinear wave-like partial differential equation coupled to a time-independent linear integral equation. We assume that the free-surface deformation is weakly nonlinear, but make no a priori assumption about the scale or amplitude of the topography. We also extend the formulation to include the effect of mean flows and surface tension. We show how this formulation gives some of the well-known limits for such problems once assumptions about the amplitude and scale of the topography are made.

UR - http://www.scopus.com/inward/record.url?scp=0039938834&partnerID=8YFLogxK

U2 - 10.1111/1467-9590.00071

DO - 10.1111/1467-9590.00071

M3 - Article

AN - SCOPUS:0039938834

VL - 100

SP - 95

EP - 106

JO - Studies in Applied Mathematics

JF - Studies in Applied Mathematics

SN - 0022-2526

IS - 1

ER -