### Abstract

The steady problem of free surface flow due to a submerged line source is revisited for the case in which the fluid depth is finite and there is a stagnation point on the free surface directly above the source. Both the strength of the source and the speed of the fluid in the far field are measured by a dimensionless parameter - the Froude number. By applying techniques in exponential asymptotics, it is shown that there is a train of periodic waves on the surface of the fluid with an amplitude which is exponentially small in the limit that the Froude number vanishes. This study clarifies that periodic waves do form for flows due to a source, contrary to a suggestion by Chapman & Vanden-Broeck (2006, Exponential asymptotics and gravity waves. J. Fluid Mech., 567, 299-326). The exponentially small nature of the waves means that they appear beyond all orders of the original power-series expansion; this result explains why attempts at describing these flows using a finite number of terms in an algebraic power series incorrectly predict a flat free surface in the far field.

Original language | English |
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Pages (from-to) | 697-713 |

Number of pages | 17 |

Journal | IMA Journal of Applied Mathematics (Institute of Mathematics and Its Applications) |

Volume | 78 |

Issue number | 4 |

DOIs | |

Publication status | Published - Aug 2013 |

Externally published | Yes |

### Keywords

- exponential asymptotics
- free surface flow
- line source
- periodic waves
- Stokes lines

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## Cite this

*IMA Journal of Applied Mathematics (Institute of Mathematics and Its Applications)*,

*78*(4), 697-713. https://doi.org/10.1093/imamat/hxt016