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Nonlinear long waves over a muddy beach

  • Erell-Isis Garnier (a1), Zhenhua Huang (a2) and Chiang C. Mei (a3)


We analyse theoretically the interaction between water waves and a thin layer of fluid mud on a sloping seabed. Under the assumption of long waves in shallow water, weakly nonlinear and dispersive effects in water are considered. The fluid mud is modelled as a thin layer of viscoelastic continuum. Using the constitutive coefficients of mud samples from two field sites, we examine the interaction of nonlinear waves and the mud motion. The effects of attenuation on harmonic evolution of surface waves are compared for two types of mud with distinct rheological properties. In general mud dissipation is found to damp out surface waves before they reach the shore, as is known in past observations. Similar to the Eulerian current in an oscillatory boundary layer in a Newtonian fluid, a mean displacement in mud is predicted which may lead to local rise of the sea bottom.


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Bryant, P. 1976 Periodic waves in shallow water. J. Fluid Mech. 59, 625644.
Coussot, P. 1997 Mud Flow Rheology and Dynamics. Bakema.
Dalrymple, R. & Liu, P. L.-F. 1978 Waves over soft muds: a two-layer fluid model. J. Phys. Oceanogr. 11211131.
Dalrymple, R. A., Nouri, Y. & Sabouri-Shargh, Z. 2008 Water waves propagating over mud. In Proceedings of 31st International Conference of Coastal Engineering, vol. 1 (ed. Smith, J. M.), vol. 1, pp. 314323.
Elgar, S. & Raubenheimer, B. 2008 Wave dissipation by muddy seafloors. Geophys. Res. Lett. 35, L07611.
Grataloup, G. & Mei, C. C. 2003 Localization of harmonics in nonlinear shallow water waves. Phy. Rev. E 68, 02634.
Huang, Z. H., Huhe, A. & Zhang, Y. 1992 An experimental study of the properties of fluid mud in Xishu, Lianyungang (in Chinese). Tech. Rep. IMCAS STR-92019. Institute of Mechanics, Chinese Academy of Sciences.
Huang, Z. H. & Huhe, A. 2009 A laboratory study of rheological properties of mudflows in Hangzhou Bay, China. Intl J. Sedim. Res. 24 (4), 410424.
Huhe, A., Huang, Z. H. & Chi, G. P. 1996 Experiments on the properties of cohesive sediments in Zhongshan Harbor, Jiangsu Province, Part II. In Report in Chinese by Institute of Mechanics. Chinese Academy of Sciences.
Jiang, F. & Mehta, A. J. 1995 Mudbanks of the southwest coast of India IV: mud viscoelastic properties. J. Coast. Res. 11 (3), 918926.
Liu, P. L.-F. & Chan, I.-C. 2007a A note on the effects of thin visco-elastic mud layer on small amplitude water-wave propagation. Coast. Engng 54, 233247.
Liu, P. L.-F. & Chan, I.-C. 2007b On long-wave propagation over a fluid-mud seabed. J. Fluid Mech. 579, 467480.
Liu, K. F. & Mei, C. 1989 Effects of water-induced friction on a muddy seabed modelled as a Bingham-plastic Fluid. J. Coast. Res. 5, 4, 777789.
Longuet-Higgins, M. S. & Stewart, R. W. 1964 Radiation stresses in water waves, a physical discussion with applications. Deep-Sea Res. 11, 529562.
Maa, J. P-Y. & Mehta, A. J. 1988 Soft mud properties: Voigt Model. J. Waterways Port Coast. Ocean Engng 114 (6), 765770.
MacPherson, H. 1980 The attenuation of water waves over a non-rigid bed. J. Fluid Mech. 97, 721742.
Malvern, L. E. 1969 Introduction to the Mechanics of a Continuous Medium. Prentice Hall.
Mathew, J. & Baba, M. 1995 Mudbanks of the southwest coast of India, II: wave-mud interactions. J. Coast. Res. 11 (1), 179187.
Mehta, A. J. 1996 Interaction between fluid-mud and water waves. In Environmental Hydraulics, pp. 153187. Kluwer, Chapter 5.
Mei, C. C. 1989 The Applied Dynamics of Ocean Surface Waves. World Scientific.
Mei, C. C. & Liu, K.-F. 1987 A Bingham-plastic model for a muddy seabed under long waves. J. Geophys. Res. (Oceans) 92 (C13), 1458114594.
Mei, C. C., Krotov, M., Huang, Z. H. & Huhe, A. 2010 Short and long waves over a muddy seabed. J. Fluid Mech. 643, 3358.
Mei, C. C. & Ünlüata, U. 1972 Harmonic generation in shallow water waves. In Waves on Beaches (ed. Meyer, R. E.), pp. 181202. Academic.
Miles, J. W. 1983 Solitary wave evolution over a gradual slope with turbulent friction. J. Phys. Oceanogr. 13, 551553.
Ng, C. 2000 Water waves over a muddy bed: a two-layer Stokes’ boundary layer model. Coast. Engng 40, 221242.
Ng, C. & Zhang, X. 2007 Mass Transport in Water waves over a thin layer of soft viscoelastic mud. J. Fluid Mech. 573, 105130.
Piedra-Cueva, I. 1993 On the response of a muddy bottom to surface water waves. J. Hydraul. Res. 31, 681695.
Rogers, S. R. & Mei, C. C. 1978 Nonlinear resonant excitation of a long and narrow bay. J. Fluid Mech. 88, 161180.
Wan, Z. H. 1994 Hyperconcentrated Flows. Balkema.
Wells, J. T. 1978 Shallow water waves and fluid-mud dynamics. Coast of Surinam, South America. Tech Rep. 257. Coastal Studies Institute, Lousiana State University.
Wells, J. T. & Coleman, J. M. 1981 Physical processes and fine-grained sediment dynamics, coast of Surinam South America. J. Sedim. Petrol. 51 (4), 10531068.
Zhang, X. & Ng, C. 2006 On the oscillatory and mean motions due to waves in a thin viscoelastic layer. Wave Motion 43, 387405.
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