Edge waves with longshore currents

Hsu, Hung-Chu

doi:10.1090/qam/1399

Author: Hung-Chu Hsu
Journal: Quart. Appl. Math. 73 (2015), 593-598
MSC (2010): Primary 76B15, 35B36, 74G05
DOI: https://doi.org/10.1090/qam/1399
Published electronically: June 16, 2015
MathSciNet review: 3400761
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: We present an exact solution to the nonlinear governing equations for nonlinear edge waves with an underlying longshore current, propagating over a plane-sloping beach. By performing an analysis in Lagrangian variables we describe the flow characteristics in great detail.

References

P. H. Leblond and L. A. Mysak, Waves in the Ocean, Elsevier, 1978, 602 pp.

G. G. Stokes, Report on recent researches in hydrodynamics, Rep. 16th Brit. Assoc. Adv. Sci. Southampton, Murray, London, 1846, 1-20.

Horace Lamb, Hydrodynamics, 6th ed., Cambridge Mathematical Library, Cambridge University Press, Cambridge, 1993. With a foreword by R. A. Caflisch [Russel E. Caflisch]. MR 1317348

F. Ursell, Edge waves on a sloping beach, Proc. Roy. Soc. London Ser. A 214 (1952), 79–97. MR 50420, DOI https://doi.org/10.1098/rspa.1952.0152

Ulf Torsten Ehrenmark, Oblique wave incidence on a plane beach: the classical problem revisited, J. Fluid Mech. 368 (1998), 291–319. MR 1640061, DOI https://doi.org/10.1017/S0022112098001888

G. B. Whitham, Nonlinear effects in edge waves, J. Fluid Mech. 74 (1976), no. 2, 353–368. MR 421326, DOI https://doi.org/10.1017/S0022112076001833

H. Yeh, Nonlinear progressive edge waves: their instability and evolution, J. Fluid Mech. 152 (1985), 479-499.

K. M. Mok and H. Yeh, On the mass transport of progressive edge waves, Physics of Fluids 11(10) (1999), 2906-2924.

C. Yih, Note on edge waves in a stratified fluid, J. Fluid Mech. 24 (1966), 765-767.

Erik Mollo-Christensen, Allowable discontinuities in a Gerstner wave field, Phys. Fluids 25 (1982), no. 4, 586–587. MR 658268, DOI https://doi.org/10.1063/1.863802

F. Gerstner, Theorie der Wellen samt einer daraus abgeleiteten Theorie der Deichprofile (in German), Ann. Phys. 2 (1809), 412-445.

Adrian Constantin, Edge waves along a sloping beach, J. Phys. A 34 (2001), no. 45, 9723–9731. MR 1876166, DOI https://doi.org/10.1088/0305-4470/34/45/311

Mats Ehrnström, Joachim Escher, and Bogdan-Vasile Matioc, Two-dimensional steady edge waves. I. Periodic waves, Wave Motion 46 (2009), no. 6, 363–371. MR 2598633, DOI https://doi.org/10.1016/j.wavemoti.2009.06.002

David Henry and Octavian G. Mustafa, Existence of solutions for a class of edge wave equations, Discrete Contin. Dyn. Syst. Ser. B 6 (2006), no. 5, 1113–1119. MR 2224873, DOI https://doi.org/10.3934/dcdsb.2006.6.1113

R. S. Johnson, Edge waves: theories past and present, Philos. Trans. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci. 365 (2007), no. 1858, 2359–2376. MR 2329153, DOI https://doi.org/10.1098/rsta.2007.2013

Raphael Stuhlmeier, On edge waves in stratified water along a sloping beach, J. Nonlinear Math. Phys. 18 (2011), no. 1, 127–137. MR 2786939, DOI https://doi.org/10.1142/S1402925111001210

Anca-Voichita Matioc, An exact solution for geophysical equatorial edge waves over a sloping beach, J. Phys. A 45 (2012), no. 36, 365501, 10. MR 2967917, DOI https://doi.org/10.1088/1751-8113/45/36/365501

Adrian Constantin, Nonlinear water waves with applications to wave-current interactions and tsunamis, CBMS-NSF Regional Conference Series in Applied Mathematics, vol. 81, Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, 2011. MR 2867413

David Henry, On Gerstner’s water wave, J. Nonlinear Math. Phys. 15 (2008), no. suppl. 2, 87–95. MR 2434727, DOI https://doi.org/10.2991/jnmp.2008.15.s2.7

A. Constantin, An exact solution for equatorially trapped waves, J. Geophys. Res. 117 (2012), C05029.

A. Constantin and P. Germain, Instability of some equatorially trapped waves, J. Geophys. Res. 118 (2013), 1-9.

Hung-Chu Hsu, An exact solution for equatorial waves, Monatsh. Math. 176 (2015), no. 1, 143–152. MR 3296207, DOI https://doi.org/10.1007/s00605-014-0618-2

A. Constantin, Some three-dimensional nonlinear equatorial flows, J. Phys. Oceanogr. 43 (2013), 165-175.

A. Constantin, Some nonlinear, equatorial trapped, non-hydrostatic, internal geophysical waves, J. Phys. Oceanogr. (2013), DOI 10.1175/JPO-D-13-0174.1.

Hung-Chu Hsu, Some nonlinear internal equatorial flows, Nonlinear Anal. Real World Appl. 18 (2014), 69–74. MR 3176298, DOI https://doi.org/10.1016/j.nonrwa.2013.12.011

Hung-Chu Hsu, An exact solution for nonlinear internal equatorial waves in the $f$-plane approximation, J. Math. Fluid Mech. 16 (2014), no. 3, 463–471. MR 3247362, DOI https://doi.org/10.1007/s00021-014-0168-3

Adrian Constantin, The trajectories of particles in Stokes waves, Invent. Math. 166 (2006), no. 3, 523–535. MR 2257390, DOI https://doi.org/10.1007/s00222-006-0002-5

D. Henry, On the deep-water Stokes flow, Int. Math. Res. Not. 22 (2008), DOI 10.1093/imrn/rnn071.

Adrian Constantin and Walter Strauss, Pressure beneath a Stokes wave, Comm. Pure Appl. Math. 63 (2010), no. 4, 533–557. MR 2604871, DOI https://doi.org/10.1002/cpa.20299

P. A. Howd, A. J. Bowen, and R. A. Holman, Edge waves in the presence of strong longshore currents, J. Geophys. Res. 97 (1992), 11357-11371.

R. S. Johnson, A modern introduction to the mathematical theory of water waves, Cambridge Texts in Applied Mathematics, Cambridge University Press, Cambridge, 1997. MR 1629555