### Abstract

A narrow-banded spectrum of gravity waves propagating in one horizontal direction at the surface of water with finite depth may be governed by a nonlinear Schrödinger equation (NLSE). If the depth is nonuniform, then the coefficients in the NLSE are variable and an additional linear term due to conservation of wave action flux is present, which may act as a dissipative term that causes the wave envelope to broaden and decrease in amplitude or which may act as a growth term that causes the wave envelope to focus and increase in amplitude. The addition of linear, viscous dissipation either enhances the dissipative process or competes with the growth effect. Here we consider such a variable-coefficient, dissipative NLSE. We find a solution that has uniform amplitude in the appropriate reference frame and examine its linear stability. We find that for waves propagating into shallower water, both bathymetry and viscous dissipation act to stabilize the modulational instability. For waves propagating into deeper water, bathymetry acts to enhance instability, but viscous dissipation eventually causes stabilization for small enough initial perturbation amplitudes.

Original language | English (US) |
---|---|

Pages (from-to) | 2030-2041 |

Number of pages | 12 |

Journal | SIAM Journal on Applied Mathematics |

Volume | 76 |

Issue number | 5 |

DOIs | |

State | Published - Jan 1 2016 |

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### All Science Journal Classification (ASJC) codes

- Applied Mathematics

### Cite this

*SIAM Journal on Applied Mathematics*,

*76*(5), 2030-2041. https://doi.org/10.1137/16M1055700

}

*SIAM Journal on Applied Mathematics*, vol. 76, no. 5, pp. 2030-2041. https://doi.org/10.1137/16M1055700

**The linear stability of a wavetrain propagating on water of variable depth.** / Rajan, Girish K.; Henderson, Diane M.

Research output: Contribution to journal › Article

TY - JOUR

T1 - The linear stability of a wavetrain propagating on water of variable depth

AU - Rajan, Girish K.

AU - Henderson, Diane M.

PY - 2016/1/1

Y1 - 2016/1/1

N2 - A narrow-banded spectrum of gravity waves propagating in one horizontal direction at the surface of water with finite depth may be governed by a nonlinear Schrödinger equation (NLSE). If the depth is nonuniform, then the coefficients in the NLSE are variable and an additional linear term due to conservation of wave action flux is present, which may act as a dissipative term that causes the wave envelope to broaden and decrease in amplitude or which may act as a growth term that causes the wave envelope to focus and increase in amplitude. The addition of linear, viscous dissipation either enhances the dissipative process or competes with the growth effect. Here we consider such a variable-coefficient, dissipative NLSE. We find a solution that has uniform amplitude in the appropriate reference frame and examine its linear stability. We find that for waves propagating into shallower water, both bathymetry and viscous dissipation act to stabilize the modulational instability. For waves propagating into deeper water, bathymetry acts to enhance instability, but viscous dissipation eventually causes stabilization for small enough initial perturbation amplitudes.

AB - A narrow-banded spectrum of gravity waves propagating in one horizontal direction at the surface of water with finite depth may be governed by a nonlinear Schrödinger equation (NLSE). If the depth is nonuniform, then the coefficients in the NLSE are variable and an additional linear term due to conservation of wave action flux is present, which may act as a dissipative term that causes the wave envelope to broaden and decrease in amplitude or which may act as a growth term that causes the wave envelope to focus and increase in amplitude. The addition of linear, viscous dissipation either enhances the dissipative process or competes with the growth effect. Here we consider such a variable-coefficient, dissipative NLSE. We find a solution that has uniform amplitude in the appropriate reference frame and examine its linear stability. We find that for waves propagating into shallower water, both bathymetry and viscous dissipation act to stabilize the modulational instability. For waves propagating into deeper water, bathymetry acts to enhance instability, but viscous dissipation eventually causes stabilization for small enough initial perturbation amplitudes.

UR - http://www.scopus.com/inward/record.url?scp=84992655479&partnerID=8YFLogxK

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U2 - 10.1137/16M1055700

DO - 10.1137/16M1055700

M3 - Article

AN - SCOPUS:84992655479

VL - 76

SP - 2030

EP - 2041

JO - SIAM Journal on Applied Mathematics

JF - SIAM Journal on Applied Mathematics

SN - 0036-1399

IS - 5

ER -