Hostname: page-component-78c5997874-m6dg7 Total loading time: 0 Render date: 2024-11-14T02:56:10.896Z Has data issue: false hasContentIssue false
  • English
  • Français

Hydro-acoustic frequencies of the weakly compressible mild-slope equation

Published online by Cambridge University Press:  22 December 2016

Emiliano Renzi
Emiliano Renzi*
Affiliation:
Department of Mathematical Sciences, Loughborough University, Loughborough, Leics LE11 3TU, UK
*
†Email address for correspondence: E.Renzi@lboro.ac.uk
Get access Rights & Permissions [Opens in a new window]

Abstract

We present a novel analytical solution for hydro-acoustic waves in a weakly compressible fluid flow over a slowly varying bottom. Application of a multiple-scale perturbation technique and matched asymptotic analysis leads to a uniform analytical solution of the depth-averaged governing equations in three dimensions. We show that the slow depth variation has a leading-order effect on the evolution of the normal mode amplitude and direction. This dynamics is much richer than the two-dimensional limit analysed in previous studies. For tsunamigenic disturbances, we show that the hydro-acoustic wave field is made up by longshore trapped and offshore propagating components, which were not explicated in previous work. For a plane beach, we find an exact analytical solution of the model equation in terms of integrals of Bessel functions. Our model offers a physical insight into the evolution of hydro-acoustic waves of interest for the design of tsunami early warning systems.

JFM classification

Compressible Flows: Compressible Flows Waves/Free-surface Flows: Waves/Free-surface Flows
Type
Papers
Information
Journal of Fluid Mechanics , Volume 812 , 10 February 2017 , pp. 5 - 25
Check for updates
Copyright
© 2016 Cambridge University Press 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

Ballard, M. S. 2012 Modeling three-dimensional propagation in a continental shelf environment. J. Acoust. Soc. Am. 131 (3), 1969–1977.Google Scholar
Bender, C. M. & Orszag, S. A. 1999 Advanced Mathematical Methods for Scientists and Engineers. Springer.CrossRefGoogle Scholar
Cecioni, C., Abdolali, A., Bellotti, G. & Sammarco, P. 2015 Large-scale numerical modeling of hydro-acoustic waves generated by tsunamicenic earthquakes. Nat. Hazards Earth Syst. Sci. 15, 627–636.CrossRefGoogle Scholar
Cheng, H. 2007 Advanced Analytic Methods in Applied Mathematics, Science and Engineering. LuBan Press.Google Scholar
Dingemans, M. S. 1997 Water Wave Propagation over Uneven Bottoms. World Scientific.Google Scholar
Eyov, E., Klar, A., Kadri, U. & Stiassnie, M. 2013 Progressive waves in a compressible ocean with an elastic bottom. Wave Motion 50, 929–939.CrossRefGoogle Scholar
Hendin, G. & Stiassnie, M. 2013 Tsunami and acoustic-gravity waves in water of constant depth. Phys. Fluids 25, 086103.Google Scholar
Hildebrand, F. B. 1976 Advanced Calculus for Applications. Prentice-Hall.Google Scholar
Jensen, F. B., Kuperman, W. A., Porter, M. B. & Schmidt, H. 2011 Computational Ocean Acoustics. Springer.CrossRefGoogle Scholar
Kadri, U. 2015 Wave motion in a heavy compressible fluid: revisited. Eur. J. Mech. (B/Fluids) 49, 50–57.CrossRefGoogle Scholar
Kadri, U. & Stiassnie, M. 2013 A note on the shoaling of acoustic-gravity waves. WSEAS Trans. Fluid Mech. 8 (2), 43–49.Google Scholar
Longuet-Higgins, M. S. 1950 A theory on the origin of microseisms. Phil. Trans. R. Soc. Lond. A 243, 1–35.Google Scholar
Massel, S. R. 2013 Ocean Surface Waves. World Scientific.Google Scholar
Mei, C. C. 1997 Mathematical Analysis in Engineering. Cambridge University Press.Google Scholar
Mei, C. C., Stiassnie, M. & Yue, D. K.-P. 2005 Theory and Application of Ocean Surface Waves. World Scientific.Google Scholar
Olver, F. W. J., Lozier, D. W., Boisvert, R. F. & Clark, C. W. 2010 NIST Handbook of Mathematical Functions. Cambridge University Press.Google Scholar
Renzi, E. & Dias, F. 2014a Hydro-acoustic precursors of gravity waves generated by surface pressure disturbances localised in space and time. J. Fluid Mech. 754, 250–262.CrossRefGoogle Scholar
Renzi, E. & Dias, F. 2014b Time-domain hydroacoustic Green function for surface pressure disturbance. In Proceedings of the 29th Intl. Workshop on Water Waves and Floating Bodies, Osaka (Japan). IWWWFB.Google Scholar
Sammarco, P., Cecioni, C., Bellotti, G. & Abdolali, A. 2013 Depth-integrated equation for large-scale modelling of low-frequency hydroacoustic waves. J. Fluid Mech. 722, R6.Google Scholar
Sammarco, P. & Renzi, E. 2008 Landslide tsunamis propagating along a plane beach. J. Fluid Mech. 598, 107–119.CrossRefGoogle Scholar
Stiassnie, M. 2010 Tsunamis and acoustic-gravity waves from underwater earthquakes. J. Engng Maths 67, 23–32.Google Scholar
Yamamoto, T. 1982 Gravity waves and acoustic waves generated by submarine earthquakes. Soil Dyn. Earthq. Engng 1 (2), 75–82.Google Scholar