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Finite-amplitude acoustic-gravity waves: exact solutions

Published online by Cambridge University Press:  12 February 2015

Oleg A. Godin
Oleg A. Godin*
Affiliation:
Cooperative Institute for Research in Environmental Sciences, University of Colorado at Boulder, Boulder, CO 80309-0216, USA NOAA/Earth System Research Laboratory, Physical Sciences Division, Boulder, CO 80305-3328, USA
*
†Email address for correspondence: [email protected]
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Abstract

We consider strongly nonlinear waves in fluids in a uniform gravity field, and demonstrate that an incompressible wave motion, in which pressure remains constant in each fluid parcel, is supported by compressible fluids with free and rigid boundaries. We present exact analytic solutions of nonlinear hydrodynamics equations which describe the incompressible wave motion. The solutions provide an extension of the Gerstner wave in an incompressible fluid with a free boundary to waves in compressible three-dimensionally inhomogeneous moving fluids such as oceans and planetary atmospheres.

JFM classification

Compressible Flows: Compressible Flows Mathematical Foundations: General fluid mechanics Waves/Free-surface Flows: Waves/Free-surface Flows
Type
Papers
Information
Journal of Fluid Mechanics , Volume 767 , 25 March 2015 , pp. 52 - 64
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Copyright
© 2015 Cambridge University Press 

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References

Bertaina, G., Pitaevskii, L. & Stringari, S. 2010 First and second sound in cylindrically trapped gases. Phys. Rev. Lett. 105, 150402.CrossRefGoogle ScholarPubMed
Bruun, G. M. & Clark, C. W. 1999 Hydrodynamic excitations of trapped Fermi gases. Phys. Rev. Lett. 83, 5415–5418.Google Scholar
Chernyak, A. V., Karelsky, K. V. & Petrosyan, A. S. 2013 Simple waves and the Riemann problem in compressible shallow water flows. Phys. Scr. T 155, 014041.Google Scholar
Christensen-Dalsgaard, J., Duvall, T. L. Jr., Gough, D. O., Harvey, J. W. & Rhodes, E. J. Jr. 1985 Speed of sound in the solar interior. Nature 315, 378–382.Google Scholar
Constantin, A. 2001a On the deep water wave motion. J. Phys. A: Math. Gen. 34, 1405–1417.Google Scholar
Constantin, A. 2001b Edge waves along a sloping beach. J. Phys. A: Math. Gen. 34, 9723–9731.Google Scholar
Constantin, A. 2012 An exact solution for equatorially trapped waves. J. Geophys. Res. 117, C05029.Google Scholar
Constantin, A. 2013 Some three-dimensional nonlinear equatorial flows. J. Phys. Oceanogr. 43, 165–175.Google Scholar
Constantin, A. 2014 Some nonlinear, equatorially trapped, nonhydrostatic internal geophysical waves. J. Phys. Oceanogr. 44, 781–789.CrossRefGoogle Scholar
Constantin, A. & Escher, J. 2004 Symmetry of steady periodic surface water waves with vorticity. J. Fluid Mech. 498, 171–181.Google Scholar
Constantin, A., Sattinger, D. & Strauss, W. 2006 Variational formulations for steady water waves with vorticity. J. Fluid Mech. 548, 151–163.Google Scholar
Csordas, A. & Graham, R. 2001 Finite-temperature hydrodynamic modes of trapped quantum gases. Phys. Rev. A 64, 013619.CrossRefGoogle Scholar
Dubreil-Jacotin, M. L. 1932 Sur les ondes de type permanent dans les liquides hétérogenes. Atti Accad. Naz. Lincei Rend (6) 15, 814–819.Google Scholar
Duvall, T. L., Jefferies, S. M., Harvey, J. W. & Pomerantz, M. A. 1993 Time–distance helioseismology. Nature 362, 430–432.CrossRefGoogle Scholar
Fouchet, T., Guerlet, S., Strobel, D. F., Simon-Miller, A. A., Bézard, B. & Flasar, F. M. 2008 An equatorial oscillation in Saturn’s middle atmosphere. Nature 453, 200–202.Google Scholar
Fritts, D. C. & Alexander, M. J. 2003 Gravity wave dynamics and effects in the middle atmosphere. Rev. Geophys. 41, Art. 1003, 3-1–3-64.CrossRefGoogle Scholar
Froude, W. 1862 On the rolling of ships. Trans. Inst. Naval Arch. 3, 45–62.Google Scholar
Gerstner, F. 1809 Theorie der Wellen samt einer daraus abgeleiteten Theorie der Deichprofile. Ann. Phys. 2, 412–445.Google Scholar
Gill, A. E. 1982 Atmosphere–Ocean Dynamics. Academic.Google Scholar
Godin, O. A. 2012 Incompressible wave motion of compressible fluids. Phys. Rev. Lett. 108, 194501.Google Scholar
Godin, O. A. 2014 Shear waves in inhomogeneous, compressible fluids in a gravity field. J. Acoust. Soc. Am. 135, 1071–1082.Google Scholar
Godin, O. A. & Fuks, I. M. 2012 Transmission of acoustic-gravity waves through gas–liquid interfaces. J. Fluid Mech. 709, 313–340.Google Scholar
Gossard, E. & Hooke, W. 1975 Waves in the Atmosphere. Elsevier.Google Scholar
Hargreaves, J. K. & Gadsden, M. 1992 The Solar-Terrestrial Environment. Cambridge University Press.CrossRefGoogle Scholar
Hecht, J. H., Walterscheid, R. L., Hickey, M. P., Rudy, R. J. & Liu, A. Z. 2002 An observation of a fast external atmospheric acoustic-gravity wave. J. Geophys. Res. 107, Art. 4444, 12-1–12-12.Google Scholar
Henry, D. 2008 On Gerstner’s water wave. J. Nonlinear Math. Phys. 15, 87–95.Google Scholar
Henry, D. 2013 An exact solution for equatorial geophysical water waves with an underlying current. Eur. J. Mech. (B/Fluids) 38, 18–21.Google Scholar
Johnson, R. S. 2005 Some contributions to the theory of edge waves. J. Fluid Mech. 524, 81–97.Google Scholar
Kinnersley, W. 1976 Exact large amplitude capillary waves on sheets of fluid. J. Fluid Mech. 77, 229–241.Google Scholar
Lamb, H. 1932 Hydrodynamics, 6th edn. Cambridge University Press.Google Scholar
Landau, L. D. & Lifshitz, E. M. 2004 Course of theoretical physics. In Fluid Mechanics, 2nd edn, vol. 6. Elsevier.Google Scholar
Leblanc, S. 2004 Local stability of Gerstner’s waves. J. Fluid Mech. 506, 245–254.CrossRefGoogle Scholar
Matioc, A.-V. 2012 An exact solution for geophysical equatorial edge waves over a sloping beach. J. Phys. A: Math. Theor. 45, 365501.Google Scholar
Matioc, A.-V. 2013 Exact geophysical waves in stratified fluids. Appl. Anal. 92, 2254–2261.Google Scholar
Mollo-Christensen, E. 1978 Gravitational and geostrophic billows: some exact solutions. J. Atmos. Sci. 35, 1395–1398.Google Scholar
Mollo-Christensen, E. 1979 Edge waves in a rotating stratified fluid, an exact solution. J. Phys. Oceanogr. 9, 226–229.Google Scholar
Mollo-Christensen, E. 1982 Allowable discontinuities in a Gerstner wave field. Phys. Fluids 26, 586–587.CrossRefGoogle Scholar
Monismith, S. G., Cowen, E. A., Nepf, H. M., Magnaudet, J. & Thais, L. 2007 Laboratory observations of mean flows under surface gravity waves. J. Fluid Mech. 573, 131–147.Google Scholar
Pekeris, C. L. 1948 The propagation of a pulse in the atmosphere. Part II. Phys. Rev. 73, 145–154.Google Scholar
Podesta, J. J. 2005 Compressible fluid model for the seismic waves generated by a sunquake. Solar Phys. 232, 1–23.Google Scholar
Pollard, R. T. 1970 Surface waves with rotation: an exact solution. J. Geophys. Res. 75, 5895–5898.Google Scholar
Price, G. H. 1992 Application of the phase-integral method to the trapping of acoustic waves in a gravitating fluid. I. Planar polytrope and turning-point behavior. Astrophys. J. 391, 845–853.Google Scholar
Rankine, W. J. M. 1863 On the exact form of waves near the surface of deep water. Phil. Trans. R. Soc. Lond. 153, 127–138.Google Scholar
Reech, F. 1869 Sur la théorie des ondes liquides périodiques. C. R. Acad. Sci. Paris 68, 1099–1101.Google Scholar
Riemann, G. F. B. 1858 Ueber die Fortpflanzung ebener Luftwellen von endlicher Schwingungsweite. Abhandl. Ges. Wiss. Göttingen, Math.-Physik Kl. 8, 43–65.Google Scholar
Sidorenkov, L. A., Tey, M. K., Grimm, R., Hou, Y.-H., Pitaevskii, L. & Stringari, S. 2013 Second sound and the superfluid fraction in a Fermi gas with resonant interactions. Nature 498, 78–81.CrossRefGoogle Scholar
Stuhlmeier, R. 2011 On edge waves in stratified water along a sloping beach. J. Nonlinear Math. Phys. 18, 127–137.Google Scholar
Weber, J. E. H. 2012 A note on trapped Gerstner waves. J. Geophys. Res. 117, C03048.Google Scholar
Yih, C.-S. 1966 Note on edge waves in a stratified fluid. J. Fluid Mech. 24, 765–767.Google Scholar