- Physical Review Journals
All JournalsPhysics Magazine
Access Provided by Xinjiang University

Physical Review Fluids

Export citation

Export citation

Choose format for download:

Download Citation
  • Access by Xinjiang University

Generalized theoretical framework for spatial attenuation rates of gravity waves in a stepwise-stratified fluid system with multiple layers of arbitrary depths

L. Suswanth and Girish Kumar Rajan*

  • *Contact author: gkrpsu@gmail.com
PDF

Phys. Rev. Fluids 10, 054802 – Published 19 May, 2025

DOI: https://doi.org/10.1103/PhysRevFluids.10.054802

Abstract

This study presents a theoretical model for gravity wave attenuation in a multilayered fluid system comprising fluidized mud, water, an interfacial fluid layer (typically enriched with surface-active compounds including oil), and air, which closely mimics the atmosphere-ocean configuration. A rigorous mathematical analysis initiated from the boundary value problem yields the dispersion relation, which is then solved numerically to obtain the complex-valued wave number, whose imaginary part is the spatial attenuation rate. Parametric studies are conducted to investigate how the attenuation rate responds to changes in wave frequencies, fluid and interfacial properties, and layer depths. Results obtained are compared with published experimental data in select cases, showing good agreement. In addition, the model is applied to reduced systems by setting the densities or depths of select layers to zero. For a water-mud system, maximum attenuation rates occur when the mud layer's depth is comparable to the wavelength and exceeds the boundary layer thickness by 100%–200%. Similarly, in an oil-water-mud system, maximum attenuation rates are obtained when the mud layer's depth exceeds the boundary layer thickness by 10%. Further analysis highlights the significance of air dynamics through the ratio, α, defined as the attenuation rate in an air-oil-water-mud system relative to that in a corresponding vacuum-oil-water-mud system. Results reveal that α≫1 for low-frequency waves in deep water systems, with values reaching up to 103 in specific cases. Such high values underscore the critical role of air dynamics in predicting attenuation rates accurately. Moreover, in an air-oil-water-mud system, the maximum value of α and the corresponding frequency at which it occurs exhibit power-law dependencies on water layer depth (d2), scaling as d22/3 and d2−1/2, respectively. It is inferred that maximum attenuation rates are linked to strong velocity gradients in the fluid layers; they may also be understood by analyzing variations in the group velocity and the penetration depth, which is related to the rotational part of the velocity field.

View figure in article

Physics Subject Headings (PhySH)

Article Text

References (155)

  1. E. Abolfazli, J.-H. Liang, Y. Fan, Q. J. Chen, N. D. Walker, and J. Liu, Surface gravity waves and their role in ocean-atmosphere coupling in the Gulf of Mexico, J. Geophys. Res.: Oceans 125, e2018JC014820 (2020).
  2. J. H. P. Studholme, M. Y. Markina, and S. K. Gulev, Role of surface gravity waves in aquaplanet ocean climates, J. Adv. Model. Earth Syst. 13, e2020MS002202 (2021).
  3. S. E. Brumer, V. Garnier, J.-L. Redelsperger, M.-N. Bouin, F. Ardhuin, and M. Accensi, Impacts of surface gravity waves on a tidal front: A coupled model perspective, Ocean Modell. 154, 101677 (2020).
  4. L. Wu, E. Sahlée, E. Nilsson, and A. Rutgersson, A review of surface swell waves and their role in air–sea interactions, Ocean Modell. 190, 102397 (2024).
  5. J. E. Stopa, F. Ardhuin, R. Husson, H. Jiang, B. Chapron, and F. Collard, Swell dissipation from 10 years of Envisat advanced synthetic aperture radar in wave mode, Geophys. Res. Lett. 43, 3423 (2016).
  6. D. M. Henderson and H. Segur, The role of dissipation in the evolution of ocean swell, J. Geophys. Res.: Oceans 118, 5074 (2013).
  7. G. K. Rajan, A three-fluid model for the dissipation of interfacial capillary-gravity waves, Phys. Fluids 32, 122121 (2020).
  8. C. R. Zaug and J. D. Carter, Dissipative models of swell propagation across the Pacific, Stud. Appl. Math. 147, 1519 (2021).
  9. W. G. Van Dorn, Boundary dissipation of oscillatory waves, J. Fluid Mech. 24, 769 (1966).
  10. D. Benielli and J. Sommeria, Excitation and breaking of internal gravity waves by parametric instability, J. Fluid Mech. 374, 117 (1998).
  11. D. Henderson, G. K. Rajan, and H. Segur, Dissipation of narrow-banded surface water waves, in Hamiltonian Partial Differential Equations and Applications, edited by P. Guyenne, D. Nicholls, and C. Sulem, Fields Institute Communications, Vol. 75 (Springer, New York, 2015), pp. 163–183.
  12. I. Sergievskaya, S. Ermakov, T. Lazareva, and J. Guo, Damping of surface waves due to crude oil/oil emulsion films on water, Mar. Pollut. Bull. 146, 206 (2019).
  13. R. A. Dalrymple and P. L. Liu, Waves over soft muds: A two-layer fluid model, J. Phys. Oceanogr. 8, 1121 (1978).
  14. H. G. Gade, Effects of a nonrigid, impermeable bottom on plane surface waves in shallow water, J. Mar. Res. 16, 61 (1958).
  15. J. P.-Y. Maa and A. J. Mehta, Soft mud response to water waves, J. Waterway Port Coastal Ocean Eng. 116, 634 (1990).
  16. C.-O. Ng, Water waves over a muddy bed: A two-layer Stokes' boundary layer model, Coastal Eng. 40, 221 (2000).
  17. P. L.-F. Liu and I.-C. Chan, A note on the effects of a thin visco-elastic mud layer on small amplitude water-wave propagation, Coastal Eng. 54, 233 (2007).
  18. S. J. Williams and D.-S. Jeng, Viscous attenuation of interfacial waves over a porous seabed, J. Coastal Res. SI 50, 338 (2007).
  19. S. Elgar and B. Raubenheimer, Wave dissipation by muddy seafloors, Geophys. Res. Lett. 35, 2008GL033245 (2008).
  20. M. Jain and A. J. Mehta, Role of basic rheological models in determination of wave attenuation over muddy seabeds, Cont. Shelf Res. 29, 642 (2009).
  21. M. A. Aleebrahim and M. Jamali, Experimental investigation of instability of fluid mud layer under surface wave motion, Phys. Fluids 34, 036602 (2022).
  22. H. Chou, J. Hunt, and M. Foda, Fluidization of marine mud by waves, Mar. Pollut. Bull. 23, 75 (1991).
  23. Y. Li and A. Mehta, Fluid mud in the wave-dominated environment revisited, in Coastal and Estuarine Fine Sediment Processes, edited by W. H. McAnally and A. J. Mehta, Proceedings in Marine Science, Vol. 3 (Elsevier, The Netherlands, 2000), pp. 79–93.
  24. I. Safak and C. Sahin, Estimation of surface wave induced fluid mud thickness on the seafloor, Estuarine Coastal Shelf Sci. 267, 107735 (2022).
  25. J. T. Wells, Dynamics of coastal fluid muds in low-, moderate-, and high-tide-range environments, Can. J. Fish. Aquat. Sci. 40, s130 (1983).
  26. H. Macpherson, The attenuation of water waves over a non-rigid bed, J. Fluid Mech. 97, 721 (1980).
  27. N. M. Frew and R. K. Nelson, Spatial mapping of sea surface microlayer surfactant concentration and composition, in IEEE 1999 International Geoscience and Remote Sensing Symposium (IEEE, Hamburg, Germany, 1999), Vol. 3, pp. 1472–1474.
  28. I. R. Jenkinson, L. Seuront, H. Ding, and F. Elias, Biological modification of mechanical properties of the sea surface microlayer, influencing waves, ripples, foam and air-sea fluxes, Elem. Sci. Anth. 6, 26 (2018).
  29. B. Mithun Sundhar and G. K. Rajan, Characterizing ocean surface contamination: Composition, film thickness, and rheology, Mar. Pollut. Bull. 186, 114287 (2023).
  30. S. Ermakov, O. Danilicheva, I. Kapustin, O. Shomina, I. Sergievskazya, A. Kupaev, and A. Molkov, Film slicks on the sea surface: Their dynamics and remote sensing, in IGARSS 2020-2020 IEEE International Geoscience and Remote Sensing Symposium (IEEE, Waikoloa, HI, USA, 2020), pp. 3545–3548.
  31. M. Gade, H. Hühnerfuss, and G. M. Korenowski, Marine Surface Films: Chemical Characteristics, Influence on Air-Sea Interactions and Remote Sensing (Springer, Berlin, 2006).
  32. A. D. Jenkins and S. J. Jacobs, Wave damping by a thin layer of viscous fluid, Phys. Fluids 9, 1256 (1997).
  33. J. T. Davies and R. W. Vose, On the damping of capillary waves by surface films, Proc. R. Soc. London A 286, 218 (1965).
  34. J. Aitken, On the effect of oil on a stormy sea, Proc. R. Soc. Edinburgh 12, 56 (1884).
  35. B. Franklin, W. Brownrigg, and Farish, XLIV. Of the stilling of waves by means of oil. Extracted from sundry letters between Benjamin Franklin, LL. D. F. R. S. William Brownrigg, M. D. F. R. S. and the Reverend Mr. Farish, Philos. Trans. R. Soc. 64, 445 (1774).
  36. D. Lohse, Surfactants on troubled waters, J. Fluid Mech. 976, F1 (2023).
  37. B. D. Dore, Some effects of the air-water interface on gravity waves, Geophys. Astrophys. Fluid Dyn. 10, 215 (1978).
  38. W. Alpers and H. Hühnerfuss, The damping of ocean waves by surface films: A new look at an old problem, J. Geophys. Res.: Oceans 94, 6251 (1989).
  39. V. Žutić, B. Ćosović, E. Marčenko, N. Bihari, and F. Kršinić, Surfactant production by marine phytoplankton, Mar. Chem. 10, 505 (1981).
  40. E. J. Bock and N. M. Frew, Static and dynamic response of natural multicomponent oceanic surface films to compression and dilation: Laboratory and field observations, J. Geophys. Res.: Oceans 98, 14599 (1993).
  41. J. T. Mass, Dynamic properties of seawater surfactants, M.S. Thesis, Massachusetts Institute of Technology, 1996.
  42. R. Dorrestein, General linearized theory of the effect of surface films on water ripples, Proc. K. Ned. Akad. Wet. B 54, 260 (1951).
  43. J. Lucassen, Longitudinal capillary waves. Part 1. Theory, Trans. Faraday Soc. 64, 2221 (1968).
  44. E. H. Lucassen-Reynders and J. Lucassen, Properties of capillary waves, Adv. Colloid Interface Sci. 2, 347 (1970).
  45. J. Lucassen, Effect of surface-active material on the damping of gravity waves: A reappraisal, J. Colloid Interface Sci. 85, 52 (1982).
  46. H. Segur, D. Henderson, J. Carter, J. Hammack, C.-M. Li, D. Pheiff, and K. Socha, Stabilizing the Benjamin-Feir instability, J. Fluid Mech. 539, 229 (2005).
  47. S. J. Brown, M. S. Triantafyllou, and D. K. P. Yue, Complex analysis of resonance conditions for coupled capillary and dilational waves, Proc. R. Soc. London, Ser. A 458, 1167 (2002).
  48. S. A. Ermakov, Resonance damping of gravity-capillary waves on the water surface covered with a surface active film, Izv. Atmos. Oceanic Phys. 39, 624 (2003).
  49. S. Ermakov, I. Sergievskaya, and L. Gushchin, Damping of gravity-capillary waves in the presence of oil slicks according to data from laboratory and numerical experiments, Izv. Atmos. Oceanic Phys. 48, 565 (2012).
  50. G. K. Rajan, Dissipation of interfacial Marangoni waves and their resonance with capillary-gravity waves, Int. J. Eng. Sci. 154, 103340 (2020).
  51. S. A. Ermakov and G. E. Khazanov, Resonance damping of gravity–capillary waves on water covered with a visco-elastic film of finite thickness: A reappraisal, Phys. Fluids 34, 092107 (2022).
  52. H. Manikantan and T. M. Squires, Surfactant dynamics: Hidden variables controlling fluid flows, J. Fluid Mech. 892, P1 (2020).
  53. F. Ardhuin, B. Chapron, and F. Collard, Observation of swell dissipation across oceans, Geophys. Res. Lett. 36, 2008GL037030 (2009).
  54. F. Collard, F. Ardhuin, and B. Chapron, Monitoring and analysis of ocean swell fields from space: New methods for routine observations, J. Geophys. Res.: Oceans 114, 2008JC005215 (2009).
  55. H. Jiang, J. E. Stopa, H. Wang, R. Husson, A. Mouche, B. Chapron, and G. Chen, Tracking the attenuation and nonbreaking dissipation of swells using altimeters, J. Geophys. Res.: Oceans 121, 1446 (2016).
  56. F. E. Snodgrass, G. W. Groves, K. F. Hasselmann, G. R. Miller, W. H. Munk, and W. H. Powers, Propagation of ocean swell across the Pacific, Philos. Trans. R. Soc. London, Ser. A 259, 431 (1966).
  57. N. E. Canney and J. D. Carter, Stability of plane waves on deep water with dissipation, Math. Comput. Simul. 74, 159 (2007) .
  58. J. D. Carter and C. C. Contreras, Stability of plane-wave solutions of a dissipative generalization of the nonlinear Schrödinger equation, Physica D 237, 3292 (2008).
  59. A. D. Craik, George Gabriel Stokes on water wave theory, Annu. Rev. Fluid Mech. 37, 23 (2005).
  60. G. G. Stokes, On the theory of oscillatory waves, in Mathematical and Physical Papers (Cambridge University Press, Cambridge, 2009), Vol. 1, pp. 197–229.
  61. H. Lamb, Hydrodynamics, 6th ed. (Cambridge University Press, Cambridge, 1932).
  62. W. J. Harrison, The influence of viscosity on the oscillations of superposed fluids, Proc. London Math. Soc. s2-6, 396 (1908).
  63. V. G. Levich, Physicochemical Hydrodynamics: (by) Veniamin G. Levich, Transl. by Scripta Technica, Inc., Prentice-Hall International Series in the Physical and Chemical Engineering Sciences (Prentice-Hall, New York, 1962).
  64. L. M. Hocking, The damping of capillary-gravity waves at a rigid boundary, J. Fluid Mech. 179, 253 (1987).
  65. J. Kamphuis, Attenuation of gravity waves by bottom friction, Coastal Eng. 2, 111 (1978).
  66. J. Hunt, The viscous damping of gravity waves in shallow water, Houille Blanche 50, 685 (1964).
  67. C. C. Mei and L. F. Liu, The damping of surface gravity waves in a bounded liquid, J. Fluid Mech. 59, 239 (1973).
  68. K. M. Case and W. C. Parkinson, Damping of surface waves in an incompressible liquid, J. Fluid Mech. 2, 172 (1957).
  69. G. H. Keulegan, Energy dissipation in standing waves in rectangular basins, J. Fluid Mech. 6, 33 (1959).
  70. T. B. Benjamin and F. Ursell, The stability of the plane free surface of a liquid in vertical periodic motion, Proc. R. Soc. London A 225, 505 (1954).
  71. J. Lucassen and E. Lucassen-Reynders, Wave damping and gibbs elasticity for nonideal surface behavior, J. Colloid Interface Sci. 25, 496 (1967).
  72. K. Dysthe and Y. Rabin, Damping of short waves by insoluble surface films, Technical report C-11-86, U. S. Office of Naval Research, London (1986).
  73. V. G. Levich, Damping of waves by surface-active substances. I, Acta Physicochim. URSS 14, 307 (1941).
  74. P. P. Lombardini, F. Piazzese, and R. Cini, The marangoni wave in ripples on an air-water interface covered by a spreading film, Nuovo Cimento C 5, 256 (1982).
  75. H. Hühnerfuss, W. Alpers, W. D. Garrett, P. A. Lange, and S. Stolte, Attenuation of capillary and gravity waves at sea by monomolecular organic surface films, J. Geophys. Res. 88, 9809 (1983).
  76. P. L. Ghia and P. Trivero, On the vibration modes of the air-water interface in the presence of surface films, Nuovo Cimento C 11, 450 (1988).
  77. G. K. Batchelor, An Introduction to Fluid Dynamics (Cambridge University Press, 1967).
  78. B. Fiscella, P. P. Lombardini, P. Trivero, and R. Cini, Ripple damping on water surface covered by a spreading film: Theory and experiment, Nuovo Cimento C 8, 491 (1985).
  79. M. P. Tulin, On the transport of energy in water waves, J. Eng. Math. 58, 339 (2007).
  80. B. D. Dore, Oscillatory response of a monomolecular slick to progressive waves, Arch. Meteorol. Geophys. Bioklimatol., Ser. A 31, 385 (1982).
  81. A. Shmidt, Damping of capillary-gravitational waves in a viscous fluid of finite depth by surfactants, J. Appl. Math. Mech. 30, 901 (1966).
  82. M. Van Den Tempel and R. P. Van De Riet, Damping of waves by surface-active materials, J. Chem. Phys. 42, 2769 (1965).
  83. R. Cini, P. P. Lombardini, and H. Hühnerfuss, Remote sensing of marine slicks utilizing their influence on wave spectra, Int. J. Remote Sens. 4, 101 (1983).
  84. J. Lucassen and M. V. D. Tempel, Longitudinal waves on visco-elastic surfaces, J. Colloid Interface Sci. 41, 491 (1972).
  85. J. W. Miles, Surface-wave damping in closed basins, Proc. R. Soc. London A 297, 459 (1967).
  86. F. C. Goodrich, The mathematical theory of capillarity. II, Proc. R. Soc. London A 260, 490 (1961).
  87. D. Dimitrov, I. Panaiotov, P. Richmond, and L. Ter-Minassian-Saraga, Dynamics of insoluble monolayers: I. dilatational or elastic modulus, friction coefficient, and Marangoni effect for dipalmitoyl lecithin monolayers, J. Colloid Interface Sci. 65, 483 (1978).
  88. I. Sergievskaya and S. Ermakov, Damping of gravity–capillary waves on water surface covered with a visco-elastic film of finite thickness, Izv. Atmos. Oceanic Phys. 53, 650 (2017).
  89. I. Sergievskaya and S. Ermakov, A phenomenological model of wave damping due to oil films, in Remote Sensing of the Ocean, Sea Ice, Coastal Waters, and Large Water Regions 2019, edited by C. R. Bostater, Jr., X. Neyt, and F. Viallefont-Robinet (SPIE, International Society for Optics and Photonics, Philadelphia, 2019), Vol. 11150, pp. 153–158.
  90. D. M. Henderson and J. W. Miles, Surface-wave damping in a circular cylinder with a fixed contact line, J. Fluid Mech. 275, 285 (1994).
  91. L. A. Gushchin and S. A. Ermakov, Laboratory study of surfactant redistribution in the flow field induced by surface waves, Izv. Atmos. Oceanic Phys. 40, 244 (2003).
  92. S. A. Ermakov and S. V. Kijashko, Laboratory study of the damping of parametric ripples due to surfactant films, in Marine Surface Films, edited by M. Gade, H. Hühnerfuss, and G. M. Korenowski (Springer, Berlin, Heidelberg, 2006), pp. 113–128.
  93. R. I. Slavchov, B. Peychev, and A. S. Ismail, Characterization of capillary waves: A review and a new optical method, Phys. Fluids 33, 101303 (2021).
  94. D. M. Henderson, Effects of surfactants on Faraday-wave dynamics, J. Fluid Mech. 365, 89 (1998).
  95. H. Hühnerfuss, P. Lange, and W. Walter, Wave damping by monomolecular surface films and their chemical structure. Part I: Variation of the hydrophobic part of carboxylic acid esters, J. Mar. Res. 40, 209 (1982).
  96. H. Hühnerfuss, P. A. Lange, and W. Walter, Wave damping by monomolecular surface films and their chemical structure. Part II: Variation of the hydrophilic part of the film molecules including natural substances, J. Mar. Res. 42, 737 (1984).
  97. H. Hühnerfuss, P. A. Lange, and W. Walter, Relaxation effects in monolayers and their contribution to water wave damping: I. Wave-induced phase shifts, J. Colloid Interface Sci. 108, 430 (1985).
  98. H. Hühnerfuss, P. A. Lange, and W. Walter, Relaxation effects in monolayers and their contribution to water wave damping: II. The Marangoni phenomenon and gravity wave attenuation, J. Colloid Interface Sci. 108, 442 (1985).
  99. G. Sutherland, T. Halsne, J. Rabault, and A. Jensen, The attenuation of monochromatic surface waves due to the presence of an inextensible cover, Wave Motion 68, 88 (2017).
  100. A. Nägeli and F. Schanz, The influence of extracellular algal products on the surface tension of water, Int. Rev. Gesamten Hydrobiol. Hydrogr. 76, 89 (1991).
  101. Q. Wang, A. Feder, and E. Mazur, Capillary wave damping in heterogeneous monolayers, J. Phys. Chem. 98, 12720 (1994).
  102. J. Lucassen and R. S. Hansen, Damping of waves on monolayer-covered surfaces: I. Systems with negligible surface dilational viscosity, J. Colloid Interface Sci. 22, 32 (1966).
  103. R. Cini, P. P. Lombardini, C. Manfredi, and E. Cini, Ripples damping due to monomolecular films, J. Colloid Interface Sci. 119, 74 (1987).
  104. J. Lucassen and M. V. D. Tempel, Dynamic measurements of dilational properties of a liquid interface, Chem. Eng. Sci. 27, 1283 (1972).
  105. J. T. Mass and J. H. Milgram, Dynamic behavior of natural sea surfactant films, J. Geophys. Res.: Oceans 103, 15695 (1998).
  106. A. Z. Mazurek and S. J. Pogorzelski, Elastic properties of natural sea surface films incorporated with solid dust particles: Model Baltic sea studies, Int. J. Oceanogr. 2012, 638240 (2012).
  107. B. Noskov and T. Zubkova, Dilational surface properties of insoluble monolayers, J. Colloid Interface Sci. 170, 1 (1995).
  108. F. De Voeght and P. Joos, Damping of a disturbance on a liquid surface, J. Colloid Interface Sci. 98, 20 (1984).
  109. R. S. Hansen and J. Ahmad, Waves at interfaces, in Progress in Surface and Membrane Science, edited by J. F. Danielli, M. D. Rosenberg, and D. A. Cadenhead (Academic Press, USA, 1971), Vol. 4, pp. 1–56.
  110. G. K. Rajan and D. M. Henderson, Linear waves at a surfactant-contaminated interface separating two fluids: Dispersion and dissipation of capillary-gravity waves, Phys. Fluids 30, 072104 (2018).
  111. Ø. Saetra, Effects of surface film on the linear stability of an air–sea interface, J. Fluid Mech. 357, 59 (1998).
  112. K. Mahshid, S. Ghader, S. H. Shamsnia, and S. A. Haghshenas, An analytical solution to wave dissipation induced by interacting waves and currents with muddy deposits, Ocean Dyn. 72, 715 (2022).
  113. S. H. Shamsnia, S. A. Haghshenas, S. Ghader, and K. Mahshid, An analytical model for mass transport calculations in a viscous muddy layer, Appl. Ocean Res. 115, 102816 (2021).
  114. S. H. Shamsnia and D. Dutykh, An analytical study on wave-current-mud interaction, Water 12, 2899 (2020).
  115. Y.-Z. Xia and K.-Q. Zhu, A study of wave attenuation over a Maxwell model of a muddy bottom, Wave Motion 47, 601 (2010).
  116. E.-I. Garnier, Z. Huang, and C. C. Mei, Nonlinear long waves over a muddy beach, J. Fluid Mech. 718, 371 (2013).
  117. I.-C. Chan, Theoretical model for nonlinear long waves over a thin viscoelastic muddy seabed, Mathematics 10, 2715 (2022).
  118. D. J. Robillard, A. J. Mehta, and I. Safak, Comments on wave-induced behavior of a coastal mud, Coastal Eng. 186, 104400 (2023).
  119. W.-Y. Hsu, H.-H. Hwung, R.-Y. Yang, and C.-M. Liu, Interfacial wave motion caused by wave-mud interaction, J. Visualization 15, 215 (2012).
  120. W. Hsu, H. Hwung, T. Hsu, A. Torres-Freyermuth, and R. Yang, An experimental and numerical investigation on wave-mud interactions, J. Geophys. Res.: Oceans 118, 1126 (2013).
  121. H. Mac Pherson and P. Kurup, Wave damping at the Kerala mudbanks, Indian J. Marine Sci. 10, 154 (1981).
  122. Y. Nouri, Experimental study of water wave interactions with a layer of mud, Ph.D. thesis, The Johns Hopkins University, 2013.
  123. N. Almashan and R. A. Dalrymple, Damping of waves propagating over a muddy bottom in deep water: Experiment and theory, Coastal Eng. 105, 36 (2015).
  124. S. A. Haghshenas and M. Soltanpour, An analysis of wave dissipation at the Hendijan mud coast, the Persian Gulf, Ocean Dyn. 61, 217 (2011).
  125. A. Quesada, E. Bautista, and F. Méndez, Damping coefficient by long waves—Viscoelastic mud–current interaction, Phys. Fluids 34, 093111 (2022).
  126. S. H. Shamsnia, M. Soltanpour, M. Bavandpour, and C. Gualtieri, A study of wave dissipation rate and particles velocity in muddy beds, Geosciences 9, 212 (2019).
  127. Y.-Z. Xia, The attenuation of shallow-water waves over seabed mud of a stratified viscoelastic model, Coastal Eng. J. 56, 1450021 (2014).
  128. L. Huang and Q. Chen, Spectral collocation model for solitary wave attenuation and mass transport over viscous mud, J. Eng. Mech. 135, 881 (2009).
  129. K. Newyear and S. Martin, A comparison of theory and laboratory measurements of wave propagation and attenuation in grease ice, J. Geophys. Res.: Oceans 102, 25091 (1997).
  130. R. Wang and H. H. Shen, Experimental study on surface wave propagating through a grease–pancake ice mixture, Cold Reg. Sci. Technol. 61, 90 (2010).
  131. D. K. Sree, A. W.-K. Law, and H. H. Shen, An experimental study on the interactions between surface waves and floating viscoelastic covers, Wave Motion 70, 195 (2017), Recent Advances on Wave Motion in Fluids and Solids.
  132. D. K. Sree, A. W.-K. Law, and H. H. Shen, An experimental study on gravity waves through a floating viscoelastic cover, Cold Reg. Sci. Technol. 155, 289 (2018).
  133. D. K. Sree, A. W.-K. Law, and H. H. Shen, An experimental study of gravity waves through segmented floating viscoelastic covers, Appl. Ocean Res. 101, 102233 (2020).
  134. G. K. Rajan, Damping rate measurements and predictions for gravity waves in an air–oil–water system, Phys. Fluids 34, 022113 (2022).
  135. J. B. Keller, Gravity waves on ice-covered water, J. Geophys. Res.: Oceans 103, 7663 (1998).
  136. G. De Carolis and D. Desiderio, Dispersion and attenuation of gravity waves in ice: A two-layer viscous fluid model with experimental data validation, Phys. Lett. A 305, 399 (2002).
  137. R. Wang and H. H. Shen, Gravity waves propagating into an ice-covered ocean: A viscoelastic model, J. Geophys. Res.: Oceans 115, C06024 (2010).
  138. H. Chen, R. P. Gilbert, and P. Guyenne, Dispersion and attenuation in a porous viscoelastic model for gravity waves on an ice-covered ocean, Eur. J. Mech. B. Fluids 78, 88 (2019).
  139. X. Zhao and H. H. Shen, Three-layer viscoelastic model with eddy viscosity effect for flexural-gravity wave propagation through ice cover, Ocean Modell. 131, 15 (2018).
  140. M. G. Velarde and X.-L. Chu, The Harmonic Oscillator Description of Longitudinal (Marangoni-Lucassen) Waves at Liquid Interfaces (Springer US, Boston, 1988), pp. 183–187.
  141. M. G. Velarde and X.-L. Chu, The harmonic oscillator approach to sustained gravity-capillary (laplace) waves at liquid interfaces, Phys. Lett. A 131, 430 (1988).
  142. X.-L. Chu and M. G. Velarde, Transverse and longitudinal waves induced and sustained by surfactant gradients at liquid-liquid interfaces, J. Colloid Interface Sci. 131, 471 (1989).
  143. J. D. Barter, Surface strain modulation of insoluble surface film properties, Phys. Fluids 6, 2606 (1994).
  144. S. Berg, Marangoni-driven spreading along liquid-liquid interfaces, Phys. Fluids 21, 032105 (2009).
  145. Y. Hu, X. Guo, X. Lu, Y. Liu, R. A. Dalrymple, and L. Shen, Idealized numerical simulation of breaking water wave propagating over a viscous mud layer, Phys. Fluids 24, 112104 (2012).
  146. B.-Q. Deng, Y. Hu, X. Guo, R. A. Dalrymple, and L. Shen, Numerical study on the dissipation of water waves over a viscous fluid-mud layer, Comput. Fluids 158, 107 (2017).
  147. C. D. Troy and J. R. Koseff, The viscous decay of progressive interfacial waves, Phys. Fluids 18, 026602 (2006).
  148. N. J. Laxague, C. J. Zappa, S. Soumya, and O. Wurl, The suppression of ocean waves by biogenic slicks, J. R. Soc. Interface 21, 20240385 (2024).
  149. G. K. Rajan, Solutions of a comprehensive dispersion relation for waves at the elastic interface of two viscous fluids, Eur. J. Mech. B Fluids 89, 241 (2021).
  150. L. Debnath, Nonlinear Water Waves (Academic Press, Boston, 1994).
  151. B. R. Sutherland, Internal Gravity Waves (Cambridge University Press, Cambridge, UK, 2010).
  152. L. Shay, Upper ocean structure: Responses to strong atmospheric forcing events, in Encyclopedia of Ocean Sciences, 2nd ed., edited by J. H. Steele (Academic Press, Oxford, 2009), pp. 192–210.
  153. H. G. Gade, Effects of a nonrigid, impermeable bottom on plane surface waves in shallow water, Ph.D. thesis, Texas A&M University, 1957.
  154. F. Jiang, Bottom mud transport due to water waves, Ph.D. thesis, University of Florida, 1993.
  155. B. Xu and P. Guyenne, Assessment of a porous viscoelastic model for wave attenuation in ice-covered seas, Appl. Ocean Res. 122, 103122 (2022).

Outline

Information

Phys. Rev. Fluids 10, 054802– Published 19 May, 2025
  • Received 2 January 2025
  • Accepted 31 March 2025
Crossmark - Check for updates button

©2025 American Physical Society

Sign In to Your Journals Account

Filter

Filter

Article Lookup

Enter a citation