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Localization of gravity waves on a random floor: weak and strong disorder analysis

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Abstract

In the absence of any forcing and rotational effects, the 1D linearized Boussinesq’s equation for the evolution of a surface gravity wave propagating in a random bottom has been studied. The time-dependent evolution of plane-wave-like modes of gravity waves in the presence of weak disorder and for Fourier number outside the localized gap are shown to be well approximated by monochromatic telegrapher’s waves. For strong disorder the one-to-one correspondence for any Fourier number has been revisited (Cáceres in AIP Adv 11:045218, 2021).

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Data Availability Statement

All data that support the findings of this study are included within the article (and any supplementary files).

Notes

  1. Space stationary in the sense that any \(n-\)moment of the random field \(\xi \left( x\right) \) fulfills translational invariance: \(\left\langle \xi \left( x_{1}-X\right) \xi \left( x_{2}-X\right) \cdots \xi \left( x_{n-1}-X\right) \xi \left( x_{n}-X\right) \right\rangle \) \( =\left\langle \xi \left( x_{1}\right) \xi \left( x_{2}\right) \cdots \xi \left( x_{n-1}\right) \xi \left( x_{n}\right) \right\rangle ,\forall X\). We note here that for the present symmetric binary disorder any \(2n-\)moment breaks in the form: \(\left\langle \xi \left( x_{1}\right) \xi \left( x_{2}\right) \right\rangle \cdots \left\langle \xi \left( x_{2n-1}\right) \xi \left( x_{2n}\right) \right\rangle \) if the sequence is ordered: \( x_{2}\ge \cdots \ge x_{2n-1}\ge x_{2n}\). This is the key ingredient to prove that Terwiel’s cumulants, for the present binary disorder, cancel for order higher than the second one, see [14].

References

  1. J.K. Cochran, H.J. Bokuniewicz, P.L. Yager, Encyclopedia of Ocean Sciences (Academic Press, New York, 2019)

    Google Scholar 

  2. J. Thomas, R. Yamada, An amplitude equation for surface gravity wave-topography interactions. Phys. Rev. Fluids 3, 124802 (2018)

    Article  ADS  Google Scholar 

  3. B. Cho, N.C. Makris, Predicting the effects of random ocean dynamic processes on underwater acoustic sensing and communication. Sci. Rep. 10, 4525 (2020). https://doi.org/10.1038/s41598-020-61043-w

    Article  ADS  Google Scholar 

  4. Mohammad-Reza. Alam, Chiang C. Mei, Attenuation of long interfacial waves over a randomly rough seabed. J. Fluid Mech. 587, 73–96 (2007)

    Article  ADS  MathSciNet  Google Scholar 

  5. F. Ardhuin, Large-scale forces under surface GravityWaves at a wavy bottom: a mechanism for the generation of primary microseisms. Geophys. Res. Lett. 45, 8173–8181 (2018). https://doi.org/10.1029/2018GL078855

    Article  ADS  Google Scholar 

  6. P. Devillard, F. Dunlop and B. Souillard, Localization of gravity waves on a channel with a random bottom, J. Fluid. Mech. 186, 521–538 (1988). https://doi.org/10.1017/S0022112088000254

  7. M. Belzons, E. Guazzelli, B. Souillard, Localization of surface gravity waves on a random bottom, Waves Random Media pp. 541–562 (1990), https://doi.org/10.1142/9789814340687_0010

  8. W. Craig, P. Guyenne and C. Sulem, Water waves over a random bottom, J. Fluid. Mech., 640, 79–107 (2009). https://doi.org/10.1017/S0022112009991248

  9. C. Lawrence, O. Gramstad, K. Trulsen, Variational Boussinesq model for kinematics calculation of surface gravity waves over bathymetry, J. Wave Motion 100, 102665 (2021). https://doi.org/10.1016/j.wavemoti.2020.102665

  10. L.A. Mysak, Wave propagation in random media, with oceanic applications. Rev. Geophys. Space Phys. 16, 233 (1978)

    Article  ADS  Google Scholar 

  11. M. Sahimi, Heterogeneous Materials, Vol. I, Morphology and Linear Transport and Optical Properties Interdisciplinary Applied Mathematics. (Springer, New York, 2000)

    Google Scholar 

  12. M.O. Cáceres, Non-equilibrium Statistical Physics with Application to Disordered Systems (Springer, Berlin, 2017). (ISBN 978-3-319-51552-6)

    Book  Google Scholar 

  13. M.O. Cáceres, Finite-velocity diffusion in random media. J. Stat. Phys. 179, 729–747 (2020)

    Article  ADS  MathSciNet  Google Scholar 

  14. M.O. Cáceres, Comments on wave-like propagation with binary disorder. J. Stat. Phys. 182, 36 (2021). https://doi.org/10.1007/s10955-021-02699-0

    Article  ADS  MathSciNet  MATH  Google Scholar 

  15. M.O. Cáceres, Surface gravity waves on randomly irregular floor and the telegrapher’s equation. AIP Adv. 11, 045218 (2021). https://doi.org/10.1063/5.0049572

    Article  ADS  Google Scholar 

  16. J.M. Pearson, A Theory of Waves (Allyn and Bacon, Inc., Boston, 1966)

    Google Scholar 

  17. J. Masoliver, G.H. Weiss, Finite-velocity diffusion. Eur. J. Phys. 17, 190 (1996)

    Article  Google Scholar 

  18. A. Compte, R. Metzlerz, The generalized Cattaneo equation for the description of anomalous transport processes. J. Phys. A Math. Gen. 30, 7277–7289 (1997)

    Article  ADS  MathSciNet  Google Scholar 

  19. J. Stoker, Water Waves (Interscience, New York, 1957)

    MATH  Google Scholar 

  20. M.S. Howe, On wave scattering by random inhomogeneities, with application to the theory of weak bores. J. Fluid Mech. 45, 785 (1971)

    Article  ADS  Google Scholar 

  21. M.O. Cáceres, Gravity waves on a random bottom: exact dispersion-relation. Wave Random Complex Media (2021). https://doi.org/10.1080/17455030.2021.1918795

    Article  Google Scholar 

  22. I. McHardy, M. Nizama, A.A. Budini, M.O. Cáceres, Intermittent waiting-time noises through embedding processes. J. Stat. Phys. 177, 608 (2019)

    Article  ADS  MathSciNet  Google Scholar 

  23. M.O. Cáceres, AIP Adv. 11, 079902 (2021). https://doi.org/10.1063/5.0059975

  24. M.O. Cáceres, H.S. Wio, Non-Markovian diffusion-like equation for transport processes with anisotropic scattering. Physica 142A, 563 (1987)

    Article  ADS  Google Scholar 

  25. M.O. Cáceres, Exact results on Poisson’s noise, Poisson’s flights and Poisson’s fluctuations. J. Math. Phys. 62, 063303 (2021). https://doi.org/10.1063/5.0040819

    Article  ADS  MathSciNet  MATH  Google Scholar 

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Acknowledgements

M.O.C. thanks to funding provided by CONICET (Grant no. PIP 112-201501-00216, CO), and Grant: Secretaría de Ciencia Técnica y Postgrado: 06/C565. U. N. Cuyo (2019).

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Authors and Affiliations

  1. Comision Nacional de Energía Atómica, Centro Atómico Bariloche and Instituto Balseiro, Universidad Nacional de Cuyo, Av. E. Bustillo 9500, CP 8400, Bariloche, Argentina

    Manuel O. Cáceres

  2. CONICET, Centro Atómico Bariloche, Av. E. Bustillo 9500, CP 8400, Bariloche, Argentina

    Manuel O. Cáceres

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  1. Manuel O. Cáceres
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Correspondence to Manuel O. Cáceres.

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Cáceres, M.O. Localization of gravity waves on a random floor: weak and strong disorder analysis. Eur. Phys. J. Spec. Top. 231, 513–519 (2022). https://doi.org/10.1140/epjs/s11734-021-00401-9

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