Reappraisal of Whitham's 1967 theory for wave–meanflow interaction in shallow water

Thomas J. Bridges; Daniel J. Ratliff

doi:10.1016/j.wavemoti.2022.103050

Reappraisal of Whitham's 1967 theory for wave–meanflow interaction in shallow water

Thomas J. Bridges, Daniel J. Ratliff^*

^*Corresponding author for this work

Mathematics, Physics and Electrical Engineering

Research output: Contribution to journal › Article › peer-review

2 Citations (Scopus)

13 Downloads (Pure)

Abstract

The modulation equations for Stokes waves in shallow water coupled to wave-generated meanflow, derived in Whitham (1967), based on an averaged Lagrangian are re-visited. Firstly, it is shown that they can be recast into two coupled classical shallow water equations, with modified gravity having the sign of the Whitham index: sign(ω₀^′′ω₂). Secondly, it is shown that the amplitude of the meanflow and amplitude of the wave are, in general, independent. Thirdly, the implications of the coalescing characteristics, whose unfolding is associated with the Benjamin–Feir instability, are studied.

Original language	English
Article number	103050
Journal	Wave Motion
Volume	115
Early online date	27 Aug 2022
DOIs	https://doi.org/10.1016/j.wavemoti.2022.103050
Publication status	Published - 1 Nov 2022

Keywords

Benjamin–Feir instability
Coalescing characteristics
Stokes waves
Wave–mean flow interaction
Whitham modulation theory

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10.1016/j.wavemoti.2022.103050

AAMAccepted author manuscript, 371 KBLicence: CC BY-NC-ND

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title = "Reappraisal of Whitham's 1967 theory for wave–meanflow interaction in shallow water",

abstract = "The modulation equations for Stokes waves in shallow water coupled to wave-generated meanflow, derived in Whitham (1967), based on an averaged Lagrangian are re-visited. Firstly, it is shown that they can be recast into two coupled classical shallow water equations, with modified gravity having the sign of the Whitham index: sign(ω0′′ω2). Secondly, it is shown that the amplitude of the meanflow and amplitude of the wave are, in general, independent. Thirdly, the implications of the coalescing characteristics, whose unfolding is associated with the Benjamin–Feir instability, are studied.",

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AU - Bridges, Thomas J.

AU - Ratliff, Daniel J.

PY - 2022/11/1

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N2 - The modulation equations for Stokes waves in shallow water coupled to wave-generated meanflow, derived in Whitham (1967), based on an averaged Lagrangian are re-visited. Firstly, it is shown that they can be recast into two coupled classical shallow water equations, with modified gravity having the sign of the Whitham index: sign(ω0′′ω2). Secondly, it is shown that the amplitude of the meanflow and amplitude of the wave are, in general, independent. Thirdly, the implications of the coalescing characteristics, whose unfolding is associated with the Benjamin–Feir instability, are studied.

AB - The modulation equations for Stokes waves in shallow water coupled to wave-generated meanflow, derived in Whitham (1967), based on an averaged Lagrangian are re-visited. Firstly, it is shown that they can be recast into two coupled classical shallow water equations, with modified gravity having the sign of the Whitham index: sign(ω0′′ω2). Secondly, it is shown that the amplitude of the meanflow and amplitude of the wave are, in general, independent. Thirdly, the implications of the coalescing characteristics, whose unfolding is associated with the Benjamin–Feir instability, are studied.

KW - Benjamin–Feir instability

KW - Coalescing characteristics

KW - Stokes waves

KW - Wave–mean flow interaction

KW - Whitham modulation theory

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DO - 10.1016/j.wavemoti.2022.103050

M3 - Article

AN - SCOPUS:85138199805

SN - 0165-2125

VL - 115

JO - Wave Motion

JF - Wave Motion

M1 - 103050

ER -

Reappraisal of Whitham's 1967 theory for wave–meanflow interaction in shallow water

Abstract

Keywords

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