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Effects of the sound speed vertical profile on the evolution of hydroacoustic waves

Published online by Cambridge University Press:  26 November 2019

S. Michele Open the ORCID record for S. Michele [Opens in a new window]  and
E. Renzi
S. Michele*
Affiliation:
Department of Mathematical Sciences, Loughborough University, LoughboroughLE11 3TU, UK
E. Renzi
Affiliation:
Department of Mathematical Sciences, Loughborough University, LoughboroughLE11 3TU, UK
*
†Email address for correspondence: [email protected]
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Abstract

We present a novel analytical model for the evolution of hydroacoustic waves in weakly compressible fluids characterised by depth variations of the sound speed profile. Using a perturbation expansion in terms of the small vertical variation of the sound speed, we derive a novel expression for the second-order velocity potential and show that this solution does not exist in the case of homogeneous sound speed. At the third order, we derive a linear Schrödinger equation governing the evolution of the wave envelope for large length and time scales, which features new terms depending on the sound speed distribution. We show that for generalised sound speed vertical profiles the frequency of the hydroacoustic signal can increase or decrease with respect to the constant sound speed case, depending on the profile. This has substantial implications on the speed of the wavetrain envelope. Our findings suggest the need to extend existing models that neglect the sound speed vertical variation, especially in view of applications to tsunami early warning.

JFM classification

Waves/Free-surface Flows: Waves/Free-surface Flows Compressible Flows: Compressible Flows
Type
JFM Papers
Information
Journal of Fluid Mechanics , Volume 883 , 25 January 2020 , A28
Copyright
© 2019 Cambridge University Press

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References

Abdolali, A., Cecioni, C., Bellotti, G. & Kirby, J. T. 2015a Hydro-acoustic and tsunami waves generated by the 2012 Haida Gwaii earthquake: modeling and in situ measurements. J. Geophys. Res. Oceans 120, 958–971.CrossRefGoogle Scholar
Abdolali, A., Kadri, U., Parsons, W. & Kirby, J. T. 2018 On the propagation of acoustic-gravity waves under elastic ice sheets. J. Fluid Mech. 837, 640–656.CrossRefGoogle Scholar
Abdolali, A., Kirby, J. T. & Bellotti, G. 2015b Depth-integrated equation for hydro-acoustic waves with bottom damping. J. Fluid Mech. 766, R1.CrossRefGoogle Scholar
Bender, C. M. & Orszag, S. A. 1999 Advanced Mathematical Methods for Scientists and Engineers. Asymptotic Methods and Perturbation Theory. Springer.Google Scholar
Brekhovskikh, L. M. & Godin, O. A. 1992 Acoustics of Layered Media II. Springer.CrossRefGoogle Scholar
Cecioni, C., Abdolali, A., Bellotti, G. & Sammarco, P. 2015 Large-scale numerical modeling of hydro-acoustic waves generated by tsunamigenic earthquakes. Nat. Hazards Earth Syst. Sci. 15, 627–636.CrossRefGoogle Scholar
Debnath, L. 2005 Nonlinear Partial Differential Equations for Scientists and Engineers. Birkhäuser.CrossRefGoogle Scholar
Ewing, W. M. & Worzel, J. L. 1948 Long-range sound transmission. Geol. Soc. Am. Mem. 27, 1–32.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
Hildebrand, F. B. 1962 Advanced Calculus for Applications. Prentice-Hall Inc.Google Scholar
Jensen, F., 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. 2016 Triad resonance between a surface-gravity wave and two high frequency hydro-acoustic waves. Eur. J. Mech. (B/Fluids) 55, 157–161.CrossRefGoogle Scholar
Kadri, U. & Stiassnie, M. 2012 Acoustic-gravity waves interacting with the shelf break. J. Geophys. Res. 117, C03035.CrossRefGoogle Scholar
Levin, B. & Nosov, M. 2009 Physics of Tsunamis. Springer.Google Scholar
Liao, S., Xu, D. & Stiassnie, M. 2016 On the steady-state nearly resonant waves. J. Fluid Mech. 794, 175–199.CrossRefGoogle Scholar
Longuet-Higgins, M. S. 1950 A theory of the origin of microseisms. Phil. Trans. R. Soc. Lond. A 243, 1–35.CrossRefGoogle Scholar
Mei, C. C. & Kadri, U. 2018 Sound signals of tsunamis from a slender fault. J. Fluid Mech. 836, 352–373.CrossRefGoogle Scholar
Mei, C. C., Stiassnie, M. & Yue, D. K.-P. 2005 Theory and Application of Ocean Surface Waves. World Scientific.Google Scholar
Michele, S. & Renzi, E. 2019 A second-order theory for an array of curved wave energy converters in open sea. J. Fluids Struct. 88, 315–330.CrossRefGoogle Scholar
Michele, S., Renzi, E. & Sammarco, P. 2019 Weakly nonlinear theory for a gate-type curved array in waves. J. Fluid Mech. 869, 238–263.CrossRefGoogle Scholar
Michele, S., Sammarco, P. & d’Errico, M. 2018 Weakly nonlinear theory for oscillating wave surge converters in a channel. J. Fluid Mech. 834, 55–91.CrossRefGoogle Scholar
Munk, W. H. 1974 Sound channel in an exponentially stratified ocean, with application to SOFAR. J. Acoust. Soc. Am. 55, 220–226.CrossRefGoogle Scholar
Renzi, E. 2017 Hydro-acoustic frequencies of the weakly compressible mild-slope equation. J. Fluid Mech. 812, 5–25.CrossRefGoogle Scholar
Renzi, E. & Dias, F. 2014 Hydro-acoustic precursors of gravity waves generated by surface pressure disturbances localised in space and time. J. Fluid Mech. 754, 250–262.CrossRefGoogle 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.CrossRefGoogle Scholar
Sells, C. C. L. 1965 The effect of a sudden change of shape of the bottom of a slightly compressible ocean. Phil. Trans. R. Soc. Lond. Ser. A 258, 495–528.CrossRefGoogle Scholar
Stiassnie, M. 2010 Tsunamis and acoustic-gravity waves from underwater earthquakes. J. Engng Maths 67, 23–32.CrossRefGoogle Scholar
Wilson, J. D. & Makris, N. C. 2008 Quantifying hurricane destructive power, wind speed, and air–sea material exchange with natural undersea sound. Geophys. Res. Lett. 35, L10603.CrossRefGoogle Scholar
Xu, D., Lin, Z., Liao, S. & Stiassnie, M. 2012 On the steady-state fully resonant progressive waves in water of finite depth. J. Fluid Mech. 710, 379–418.CrossRefGoogle Scholar
Yamamoto, T. 1982 Gravity waves and acoustic waves generated by submarine earthquakes. Soil Dyn. Earthq. Engng 1, 75–82.Google Scholar