Inverse non-linear problem of the long wave run-up on coast
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arXiv
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| Main Authors: | , , , , , |
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| Format: | Preprint |
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2023
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| _version_ | 1866909397819588608 |
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| author | Rybkin, Alexei Pelinovsky, Efim Bobrovnikov, Oleksandr Palmer, Noah Pniushkova, Ekaterina Abramowicz, Daniel |
| author_facet | Rybkin, Alexei Pelinovsky, Efim Bobrovnikov, Oleksandr Palmer, Noah Pniushkova, Ekaterina Abramowicz, Daniel |
| contents | The study of the process of catastrophic tsunami-type waves on the coast makes it possible to determine the destructive force of waves on the coast. In hydrodynamics, the one-dimensional theory of the run-up of non-linear waves on a flat slope has gained great popularity, within which rigorous analytical results have been obtained in the class of non-breaking waves. In general, the result depends on the characteristics of the wave approaching (or generated on) the slope, which is usually not known in the measurements. Here we describe a rigorous method for recovering the initial displacement in a source localised in an inclined power-shaped channel from the characteristics of a moving shoreline. The method uses the generalised Carrier-Greenspan transformation, which allows one-dimensional non-linear shallow-water equations to be reduced to linear ones. The solution is found in terms of Erdélyi-Kober integral operator. Numerical verification of our results is presented for the cases of a parabolic bay and an infinite plane beach. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2309_14553 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Inverse non-linear problem of the long wave run-up on coast Rybkin, Alexei Pelinovsky, Efim Bobrovnikov, Oleksandr Palmer, Noah Pniushkova, Ekaterina Abramowicz, Daniel Fluid Dynamics Mathematical Physics Geophysics The study of the process of catastrophic tsunami-type waves on the coast makes it possible to determine the destructive force of waves on the coast. In hydrodynamics, the one-dimensional theory of the run-up of non-linear waves on a flat slope has gained great popularity, within which rigorous analytical results have been obtained in the class of non-breaking waves. In general, the result depends on the characteristics of the wave approaching (or generated on) the slope, which is usually not known in the measurements. Here we describe a rigorous method for recovering the initial displacement in a source localised in an inclined power-shaped channel from the characteristics of a moving shoreline. The method uses the generalised Carrier-Greenspan transformation, which allows one-dimensional non-linear shallow-water equations to be reduced to linear ones. The solution is found in terms of Erdélyi-Kober integral operator. Numerical verification of our results is presented for the cases of a parabolic bay and an infinite plane beach. |
| title | Inverse non-linear problem of the long wave run-up on coast |
| topic | Fluid Dynamics Mathematical Physics Geophysics |
| url | https://arxiv.org/abs/2309.14553 |