Bragg resonance of waves in a two-layer fluid propagating over bottom ripples. Part I. Perturbation analysis
We investigate, via perturbation analyses, the mechanisms of nonlinear resonant interaction of surface-interfacial waves with a rippled bottom in a two-layer density-stratified fluid. As in a one-layer fluid, three classes of Bragg resonances are found to exist if nonlinear interactions up to the th...
| Main Authors: | , , |
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| Other Authors: | |
| Format: | Article |
| Language: | en_US |
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Cambridge University Press
2013
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| Online Access: | http://hdl.handle.net/1721.1/80267 https://orcid.org/0000-0003-1273-9964 |
| _version_ | 1826192529253466112 |
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| author | Alam, Mohammad-Reza Liu, Yuming Yue, Dick K. P. |
| author2 | Massachusetts Institute of Technology. Center for Ocean Engineering |
| author_facet | Massachusetts Institute of Technology. Center for Ocean Engineering Alam, Mohammad-Reza Liu, Yuming Yue, Dick K. P. |
| author_sort | Alam, Mohammad-Reza |
| collection | MIT |
| description | We investigate, via perturbation analyses, the mechanisms of nonlinear resonant interaction of surface-interfacial waves with a rippled bottom in a two-layer density-stratified fluid. As in a one-layer fluid, three classes of Bragg resonances are found to exist if nonlinear interactions up to the third order in the wave/ripple steepness are considered. As expected, the wave system associated with the resonances is more complicated than that in a one-layer fluid. Depending on the specifics of the resonance condition, the resonance-generated wave may be a surface or internal mode and may be transmitted or reflected. At the second order, class I Bragg resonance occurs involving two surface and/or internal waves and one bottom-ripple component. The interaction of an incident surface/internal wave with the bottom ripple generates a new surface or internal wave that may propagate in the same or the opposite direction as the incident wave. At the third order, class II and III Bragg resonances occur involving resonant interactions of four wave/ripple components: two surface and/or internal waves and two bottom-ripple components for class II resonance; three surface and/or internal waves and one bottom-ripple components for class III resonance. As in class I resonance, the resonance-generated wave in class II resonance has the same frequency as that of the incident wave. For class III resonance, the frequency of the resonant wave is equal to the sum or difference of the two incident wave frequencies. We enumerate and represent, using Feynman-like diagrams, the possible cases and combinations for Bragg resonance up to the third order (in two dimensions). Analytical regular perturbation results are obtained and discussed for all three classes of Bragg resonances. These are valid for limited bottom patch lengths and initial/finite growth of the resonant waves. For long bottom patches, a uniformly valid solution using multiple scales is derived for class I resonance. A number of applications underscoring the importance and implication of these nonlinear resonances on the evolution of ocean waves are presented and discussed. For example, it is shown that three internal/surface waves co-propagating over bottom topography are resonant under a broad range of Bragg conditions. The present study provides the theoretical basis and understanding for the companion paper (Alam, Liu & Yue 2008), where a direct numerical solution for the general nonlinear problem is pursued. |
| first_indexed | 2024-09-23T09:19:12Z |
| format | Article |
| id | mit-1721.1/80267 |
| institution | Massachusetts Institute of Technology |
| language | en_US |
| last_indexed | 2024-09-23T09:19:12Z |
| publishDate | 2013 |
| publisher | Cambridge University Press |
| record_format | dspace |
| spelling | mit-1721.1/802672022-09-26T11:12:54Z Bragg resonance of waves in a two-layer fluid propagating over bottom ripples. Part I. Perturbation analysis Alam, Mohammad-Reza Liu, Yuming Yue, Dick K. P. Massachusetts Institute of Technology. Center for Ocean Engineering Massachusetts Institute of Technology. Department of Mechanical Engineering Massachusetts Institute of Technology. School of Engineering Alam, Mohammad-Reza Liu, Yuming Yue, Dick K. P. We investigate, via perturbation analyses, the mechanisms of nonlinear resonant interaction of surface-interfacial waves with a rippled bottom in a two-layer density-stratified fluid. As in a one-layer fluid, three classes of Bragg resonances are found to exist if nonlinear interactions up to the third order in the wave/ripple steepness are considered. As expected, the wave system associated with the resonances is more complicated than that in a one-layer fluid. Depending on the specifics of the resonance condition, the resonance-generated wave may be a surface or internal mode and may be transmitted or reflected. At the second order, class I Bragg resonance occurs involving two surface and/or internal waves and one bottom-ripple component. The interaction of an incident surface/internal wave with the bottom ripple generates a new surface or internal wave that may propagate in the same or the opposite direction as the incident wave. At the third order, class II and III Bragg resonances occur involving resonant interactions of four wave/ripple components: two surface and/or internal waves and two bottom-ripple components for class II resonance; three surface and/or internal waves and one bottom-ripple components for class III resonance. As in class I resonance, the resonance-generated wave in class II resonance has the same frequency as that of the incident wave. For class III resonance, the frequency of the resonant wave is equal to the sum or difference of the two incident wave frequencies. We enumerate and represent, using Feynman-like diagrams, the possible cases and combinations for Bragg resonance up to the third order (in two dimensions). Analytical regular perturbation results are obtained and discussed for all three classes of Bragg resonances. These are valid for limited bottom patch lengths and initial/finite growth of the resonant waves. For long bottom patches, a uniformly valid solution using multiple scales is derived for class I resonance. A number of applications underscoring the importance and implication of these nonlinear resonances on the evolution of ocean waves are presented and discussed. For example, it is shown that three internal/surface waves co-propagating over bottom topography are resonant under a broad range of Bragg conditions. The present study provides the theoretical basis and understanding for the companion paper (Alam, Liu & Yue 2008), where a direct numerical solution for the general nonlinear problem is pursued. 2013-08-23T16:24:45Z 2013-08-23T16:24:45Z 2009-04 2008-12 Article http://purl.org/eprint/type/JournalArticle 0022-1120 1469-7645 http://hdl.handle.net/1721.1/80267 ALAM, MOHAMMAD-REZA, YUMING LIU, and DICK K. P. YUE. Bragg Resonance of Waves in a Two-layer Fluid Propagating over Bottom Ripples. Part I. Perturbation Analysis. Journal of Fluid Mechanics 624 (April 30, 2009): 191. © Cambridge University Press. https://orcid.org/0000-0003-1273-9964 en_US http://dx.doi.org/10.1017/S0022112008005478 Journal of Fluid Mechanics Article is made available in accordance with the publisher's policy and may be subject to US copyright law. Please refer to the publisher's site for terms of use. application/pdf Cambridge University Press Other University Web Domain |
| spellingShingle | Alam, Mohammad-Reza Liu, Yuming Yue, Dick K. P. Bragg resonance of waves in a two-layer fluid propagating over bottom ripples. Part I. Perturbation analysis |
| title | Bragg resonance of waves in a two-layer fluid propagating over bottom ripples. Part I. Perturbation analysis |
| title_full | Bragg resonance of waves in a two-layer fluid propagating over bottom ripples. Part I. Perturbation analysis |
| title_fullStr | Bragg resonance of waves in a two-layer fluid propagating over bottom ripples. Part I. Perturbation analysis |
| title_full_unstemmed | Bragg resonance of waves in a two-layer fluid propagating over bottom ripples. Part I. Perturbation analysis |
| title_short | Bragg resonance of waves in a two-layer fluid propagating over bottom ripples. Part I. Perturbation analysis |
| title_sort | bragg resonance of waves in a two layer fluid propagating over bottom ripples part i perturbation analysis |
| url | http://hdl.handle.net/1721.1/80267 https://orcid.org/0000-0003-1273-9964 |
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