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Volumn 127, Issue , 2016, Pages 80-93

Derivation of dissipative Boussinesq equations using the Dirichlet-to-Neumann operator approach

Author keywords

Boundary layer; Boussinesq equations; Dispersive waves; Dissipation; Viscosity

Indexed keywords

BOUNDARY LAYERS; COASTAL ENGINEERING; ENERGY DISSIPATION; VISCOSITY;

EID: 84899042580     PISSN: 03784754     EISSN: None     Source Type: Journal    
DOI: 10.1016/j.matcom.2013.12.008     Document Type: Article
Times cited : (13)

References (67)
  • 1
    • 0001865962 scopus 로고
    • Model equations for long waves in nonlinear dispersive systems
    • [1] Benjamin, T.B., Bona, J.L., Mahony, J.J., Model equations for long waves in nonlinear dispersive systems. Phil. Trans. R. Soc. Lond. A 272 (1972), 47–78.
    • (1972) Phil. Trans. R. Soc. Lond. A , vol.272 , pp. 47-78
    • Benjamin, T.B.1    Bona, J.L.2    Mahony, J.J.3
  • 2
    • 0011913755 scopus 로고    scopus 로고
    • Boussinesq equations and other systems for small-amplitude long waves in nonlinear dispersive media. I: Derivation and linear theory
    • [2] Bona, J.L., Chen, M., Saut, J.-C., Boussinesq equations and other systems for small-amplitude long waves in nonlinear dispersive media. I: Derivation and linear theory. J. Nonlinear Sci. 12 (2002), 283–318.
    • (2002) J. Nonlinear Sci. , vol.12 , pp. 283-318
    • Bona, J.L.1    Chen, M.2    Saut, J.-C.3
  • 4
    • 0016883840 scopus 로고
    • A model for the two-way propagation of water waves in a channel
    • [4] Bona, J.L., Smith, R., A model for the two-way propagation of water waves in a channel. Math. Proc. Camb. Phil. Soc. 79 (1976), 167–182.
    • (1976) Math. Proc. Camb. Phil. Soc. , vol.79 , pp. 167-182
    • Bona, J.L.1    Smith, R.2
  • 5
    • 84873315670 scopus 로고    scopus 로고
    • Numerical study of a family of dissipative KdV equations
    • [5] Chehab, J.-P., Sadaka, G., Numerical study of a family of dissipative KdV equations. Commun. Pur. Appl. Anal. 12:1 (2013), 519–546.
    • (2013) Commun. Pur. Appl. Anal. , vol.12 , Issue.1 , pp. 519-546
    • Chehab, J.-P.1    Sadaka, G.2
  • 6
    • 77954344192 scopus 로고    scopus 로고
    • Decay of solutions to a water wave model with a nonlocal viscous dispersive term
    • [6] Chen, M., Dumont, S., Dupaigne, L., Goubet, O., Decay of solutions to a water wave model with a nonlocal viscous dispersive term. Discrete Contin. Dyn. Syst. 27:4 (2010), 1473–1492.
    • (2010) Discrete Contin. Dyn. Syst. , vol.27 , Issue.4 , pp. 1473-1492
    • Chen, M.1    Dumont, S.2    Dupaigne, L.3    Goubet, O.4
  • 7
    • 34247261590 scopus 로고    scopus 로고
    • Long-time asymptotic behaviour of dissipative Boussinesq systems
    • [7] Chen, M., Goubet, O., Long-time asymptotic behaviour of dissipative Boussinesq systems. Discrete Contin. Dyn. Syst. 17 (2007), 61–80.
    • (2007) Discrete Contin. Dyn. Syst. , vol.17 , pp. 61-80
    • Chen, M.1    Goubet, O.2
  • 8
    • 81355160297 scopus 로고    scopus 로고
    • Practical use of variational principles for modeling water waves
    • [8] Clamond, D., Dutykh, D., Practical use of variational principles for modeling water waves. Phys. D Nonlinear Phenom. 241:1 (2012), 25–36.
    • (2012) Phys. D Nonlinear Phenom. , vol.241 , Issue.1 , pp. 25-36
    • Clamond, D.1    Dutykh, D.2
  • 9
    • 0002592298 scopus 로고
    • Nonlinear harmonic analysis and analytic dependence
    • [9] Coifman, R.R., Meyer, Y., Nonlinear harmonic analysis and analytic dependence. Pseudodiffer. Operat. Appl. 43 (1985), 71–78.
    • (1985) Pseudodiffer. Operat. Appl. , vol.43 , pp. 71-78
    • Coifman, R.R.1    Meyer, Y.2
  • 10
    • 0028443419 scopus 로고
    • Hamiltonian long-wave approximations to the water-wave problem
    • [10] Craig, W., Groves, M.D., Hamiltonian long-wave approximations to the water-wave problem. Wave Motion 19 (1994), 367–389.
    • (1994) Wave Motion , vol.19 , pp. 367-389
    • Craig, W.1    Groves, M.D.2
  • 11
    • 17144396016 scopus 로고    scopus 로고
    • Hamiltonian long-wave expansions for water waves over a rough bottom
    • [11] Craig, W., Guyenne, P., Nicholls, D., Sulem, C., Hamiltonian long-wave expansions for water waves over a rough bottom. Proc. R. Soc. A 461 (2005), 839–873.
    • (2005) Proc. R. Soc. A , vol.461 , pp. 839-873
    • Craig, W.1    Guyenne, P.2    Nicholls, D.3    Sulem, C.4
  • 12
    • 0002686768 scopus 로고
    • Numerical simulation of gravity waves
    • [12] Craig, W., Sulem, C., Numerical simulation of gravity waves. J. Comput. Phys. 108 (1993), 73–83.
    • (1993) J. Comput. Phys. , vol.108 , pp. 73-83
    • Craig, W.1    Sulem, C.2
  • 13
    • 0000784514 scopus 로고
    • Nonlinear modulation of gravity waves: a rigorous approach
    • [13] Craig, W., Sulem, C., Sulem, P.-L., Nonlinear modulation of gravity waves: a rigorous approach. Nonlinearity 5:2 (1992), 497–522.
    • (1992) Nonlinearity , vol.5 , Issue.2 , pp. 497-522
    • Craig, W.1    Sulem, C.2    Sulem, P.-L.3
  • 14
    • 1642309095 scopus 로고    scopus 로고
    • The origins of water wave theory
    • [14] Craik, A.D.D., The origins of water wave theory. Ann. Rev. Fluid Mech. 36 (2004), 1–28.
    • (2004) Ann. Rev. Fluid Mech. , vol.36 , pp. 1-28
    • Craik, A.D.D.1
  • 15
    • 33750379468 scopus 로고    scopus 로고
    • The numerical computation of freely propagating time-dependent irrotational water waves
    • [15] Dias, F., Bridges, T.J., The numerical computation of freely propagating time-dependent irrotational water waves. Fluid Dyn. Res. 38 (2006), 803–830.
    • (2006) Fluid Dyn. Res. , vol.38 , pp. 803-830
    • Dias, F.1    Bridges, T.J.2
  • 16
    • 34249100040 scopus 로고    scopus 로고
    • Dynamics of tsunami waves
    • A. Ibrahimbegovic I. Kozar Springer Netherlands
    • [16] Dias, F., Dutykh, D., Dynamics of tsunami waves. Ibrahimbegovic, A., Kozar, I., (eds.) Extreme Man-Made and Natural Hazards in Dynamics of Structures, 2007, Springer, Netherlands, 35–60, 10.1007/978-1-4020-5656-7_8.
    • (2007) Extreme Man-Made and Natural Hazards in Dynamics of Structures , pp. 35-60
    • Dias, F.1    Dutykh, D.2
  • 17
    • 38849134959 scopus 로고    scopus 로고
    • Theory of weakly damped free-surface flows: a new formulation based on potential flow solutions
    • [17] Dias, F., Dyachenko, A.I., Zakharov, V.E., Theory of weakly damped free-surface flows: a new formulation based on potential flow solutions. Phys. Lett. A 372 (2008), 1297–1302.
    • (2008) Phys. Lett. A , vol.372 , pp. 1297-1302
    • Dias, F.1    Dyachenko, A.I.2    Zakharov, V.E.3
  • 18
    • 0033489572 scopus 로고    scopus 로고
    • Nonlinear gravity and capillary-gravity waves
    • [18] Dias, F., Kharif, C., Nonlinear gravity and capillary-gravity waves. Ann. Rev. Fluid Mech. 31 (1999), 301–346.
    • (1999) Ann. Rev. Fluid Mech. , vol.31 , pp. 301-346
    • Dias, F.1    Kharif, C.2
  • 19
    • 84883168105 scopus 로고    scopus 로고
    • Theory and numerical analysis of Boussinesq systems: a review
    • N.A. Kampanis V.A. Dougalis J.A. Ekaterinaris CRC Press Boca Raton, FL
    • [19] Dougalis, V.A., Mitsotakis, D.E., Theory and numerical analysis of Boussinesq systems: a review. Kampanis, N.A., Dougalis, V.A., Ekaterinaris, J.A., (eds.) Effective Computational Methods in Wave Propagation, 2008, CRC Press, Boca Raton, FL, 63–110.
    • (2008) Effective Computational Methods in Wave Propagation , pp. 63-110
    • Dougalis, V.A.1    Mitsotakis, D.E.2
  • 20
    • 62649115260 scopus 로고    scopus 로고
    • Mathematical Modelling of Tsunami Waves
    • École Normale Supérieure de Cachan December
    • [20] Dutykh, D., Mathematical Modelling of Tsunami Waves. December 2007, École Normale Supérieure de Cachan http://tel.archives-ouvertes.fr/tel-00194763/.
    • (2007)
    • Dutykh, D.1
  • 21
    • 68349104241 scopus 로고    scopus 로고
    • Group and phase velocities in the free-surface visco-potential flow: new kind of boundary layer induced instability
    • [21] Dutykh, D., Group and phase velocities in the free-surface visco-potential flow: new kind of boundary layer induced instability. Phys. Lett. A 373 (2009), 3212–3216.
    • (2009) Phys. Lett. A , vol.373 , pp. 3212-3216
    • Dutykh, D.1
  • 22
    • 62649165873 scopus 로고    scopus 로고
    • Visco-potential free-surface flows and long wave modelling
    • [22] Dutykh, D., Visco-potential free-surface flows and long wave modelling. Eur. J. Mech. B/Fluids 28 (2009), 430–443.
    • (2009) Eur. J. Mech. B/Fluids , vol.28 , pp. 430-443
    • Dutykh, D.1
  • 23
    • 84987643941 scopus 로고    scopus 로고
    • Mathematical Modeling in the Environment
    • December, Habilitation à diriger des recherches, Université de Savoie
    • [23] Dutykh, D., Mathematical Modeling in the Environment. December, 2010, Habilitation à diriger des recherches, Université de Savoie http://tel.archives-ouvertes.fr/tel-00542937/.
    • (2010)
    • Dutykh, D.1
  • 24
    • 35548942232 scopus 로고    scopus 로고
    • Dissipative Boussinesq equations
    • [24] Dutykh, D., Dias, F., Dissipative Boussinesq equations. C. R. Mecanique 335 (2007), 559–583.
    • (2007) C. R. Mecanique , vol.335 , pp. 559-583
    • Dutykh, D.1    Dias, F.2
  • 25
    • 34447630128 scopus 로고    scopus 로고
    • Viscous potential free-surface flows in a fluid layer of finite depth
    • [25] Dutykh, D., Dias, F., Viscous potential free-surface flows in a fluid layer of finite depth. C. R. Acad. Sci. Paris Ser. I 345 (2007), 113–118.
    • (2007) C. R. Acad. Sci. Paris Ser. I , vol.345 , pp. 113-118
    • Dutykh, D.1    Dias, F.2
  • 26
    • 84892359394 scopus 로고    scopus 로고
    • Water waves generated by a moving bottom
    • A. Kundu Springer Verlag (Geo Sc.) Berlin, Heidelberg
    • [26] Dutykh, D., Dias, F., Water waves generated by a moving bottom. Kundu, A., (eds.) Tsunami and Nonlinear Waves, 2007, Springer Verlag (Geo Sc.), Berlin, Heidelberg, 65–96.
    • (2007) Tsunami and Nonlinear Waves , pp. 65-96
    • Dutykh, D.1    Dias, F.2
  • 27
    • 33750062247 scopus 로고    scopus 로고
    • Linear theory of wave generation by a moving bottom
    • [27] Dutykh, D., Dias, F., Kervella, Y., Linear theory of wave generation by a moving bottom. C. R. Acad. Sci. Paris. Ser. I 343 (2006), 499–504.
    • (2006) C. R. Acad. Sci. Paris. Ser. I , vol.343 , pp. 499-504
    • Dutykh, D.1    Dias, F.2    Kervella, Y.3
  • 28
    • 17144414943 scopus 로고    scopus 로고
    • An efficient model for threedimensional surface wave simulations. Part I: Free space problems
    • [28] Fructus, D., Clamond, D., Kristiansen, O., Grue, J., An efficient model for threedimensional surface wave simulations. Part I: Free space problems. J. Comput. Phys. 205 (2005), 665–685.
    • (2005) J. Comput. Phys. , vol.205 , pp. 665-685
    • Fructus, D.1    Clamond, D.2    Kristiansen, O.3    Grue, J.4
  • 29
    • 76149098183 scopus 로고    scopus 로고
    • Numerical simulation of droplets, bubbles and waves: state of the art
    • [29] Fuster, D., Agbaglah, G., Josserand, C., Popinet, S., Zaleski, S., Numerical simulation of droplets, bubbles and waves: state of the art. Fluid Dyn. Res., 41, 2009, 65001.
    • (2009) Fluid Dyn. Res. , vol.41 , pp. 65001
    • Fuster, D.1    Agbaglah, G.2    Josserand, C.3    Popinet, S.4    Zaleski, S.5
  • 30
    • 0003498504 scopus 로고    scopus 로고
    • Tables of Integrals, Series, and Products
    • 6th ed. Academic Press Orlando, FL
    • [30] Gradshteyn, I.S., Ryzhik, M., Tables of Integrals, Series, and Products. 6th ed., 2000, Academic Press, Orlando, FL.
    • (2000)
    • Gradshteyn, I.S.1    Ryzhik, M.2
  • 31
    • 0037377588 scopus 로고    scopus 로고
    • Damping of large-amplitude solitary waves
    • [31] Grimshaw, R., Pelinovsky, E., Talipova, T., Damping of large-amplitude solitary waves. Wave Motion 37 (2003), 351–364.
    • (2003) Wave Motion , vol.37 , pp. 351-364
    • Grimshaw, R.1    Pelinovsky, E.2    Talipova, T.3
  • 33
    • 33244463286 scopus 로고    scopus 로고
    • Potential flow of viscous fluids: historical notes
    • [33] Joseph, D.D., Potential flow of viscous fluids: historical notes. Int. J. Multiphase Flow 32 (2006), 285–310.
    • (2006) Int. J. Multiphase Flow , vol.32 , pp. 285-310
    • Joseph, D.D.1
  • 34
    • 0016532331 scopus 로고
    • Effect of viscosity on long gravity waves
    • [34] Kakutani, T., Matsuuchi, K., Effect of viscosity on long gravity waves. J. Phys. Soc. Japan 39 (1975), 237–246.
    • (1975) J. Phys. Soc. Japan , vol.39 , pp. 237-246
    • Kakutani, T.1    Matsuuchi, K.2
  • 35
    • 51049102576 scopus 로고    scopus 로고
    • Dynamics of long spatial nonlinear waves in an ocean with a density jump and a gently sloping bottom
    • [35] Khabakhpashev, G., Dynamics of long spatial nonlinear waves in an ocean with a density jump and a gently sloping bottom. Oceanology 4 (2008), 457–465.
    • (2008) Oceanology , vol.4 , pp. 457-465
    • Khabakhpashev, G.1
  • 36
    • 0023124791 scopus 로고
    • Effect of bottom friction on the dynamics of gravity perturbations
    • [36] Khabakhpashev, G.A., Effect of bottom friction on the dynamics of gravity perturbations. Fluid Dyn. 22:3 (1987), 430–437.
    • (1987) Fluid Dyn. , vol.22 , Issue.3 , pp. 430-437
    • Khabakhpashev, G.A.1
  • 37
    • 62649138882 scopus 로고    scopus 로고
    • Nonlinear evolution equation for sufficiently long two-dimensional waves on the free surface of a viscous liquid
    • [37] Khabakhpashev, G.A., Nonlinear evolution equation for sufficiently long two-dimensional waves on the free surface of a viscous liquid. Comput. Technol. 2 (1997), 94–101.
    • (1997) Comput. Technol. , vol.2 , pp. 94-101
    • Khabakhpashev, G.A.1
  • 38
    • 77955896733 scopus 로고    scopus 로고
    • Under which conditions the Benjamin-Feir instability may spawn an extreme wave event: a fully nonlinear approach
    • [38] Kharif, C., Touboul, J., Under which conditions the Benjamin-Feir instability may spawn an extreme wave event: a fully nonlinear approach. Eur. Phys. J. Special Topics 185 (2010), 159–168.
    • (2010) Eur. Phys. J. Special Topics , vol.185 , pp. 159-168
    • Kharif, C.1    Touboul, J.2
  • 39
    • 0004168443 scopus 로고
    • Hydrodynamics
    • Cambridge University Press Cambridge
    • [39] Lamb, H., Hydrodynamics. 1932, Cambridge University Press, Cambridge.
    • (1932)
    • Lamb, H.1
  • 40
    • 22544482964 scopus 로고    scopus 로고
    • Well-posedness of the water-waves equations
    • [40] Lannes, D., Well-posedness of the water-waves equations. J. Am. Math. Soc. 18:3 (2005), 605–654.
    • (2005) J. Am. Math. Soc. , vol.18 , Issue.3 , pp. 605-654
    • Lannes, D.1
  • 41
    • 59649093678 scopus 로고    scopus 로고
    • Derivation of asymptotic two-dimensional time-dependent equations for surface water wave propagation
    • [41] Lannes, D., Bonneton, P., Derivation of asymptotic two-dimensional time-dependent equations for surface water wave propagation. Phys. Fluids, 21, 2009, 16601.
    • (2009) Phys. Fluids , vol.21 , pp. 16601
    • Lannes, D.1    Bonneton, P.2
  • 42
    • 12144283394 scopus 로고    scopus 로고
    • Viscous effects on transient long-wave propagation
    • [42] Liu, P.-F., Orfila, A., Viscous effects on transient long-wave propagation. J. Fluid Mech. 520 (2004), 83–92.
    • (2004) J. Fluid Mech. , vol.520 , pp. 83-92
    • Liu, P.-F.1    Orfila, A.2
  • 43
    • 33847083590 scopus 로고    scopus 로고
    • Boundary layer flow and bed shear stress under a solitary wave
    • [43] Liu, P.-F., Park, Y.S., Cowen, E.A., Boundary layer flow and bed shear stress under a solitary wave. J. Fluid Mech. 574 (2007), 449–463.
    • (2007) J. Fluid Mech. , vol.574 , pp. 449-463
    • Liu, P.-F.1    Park, Y.S.2    Cowen, E.A.3
  • 44
    • 84974513241 scopus 로고
    • A variational principle for a fluid with a free surface
    • [44] Luke, J.C., A variational principle for a fluid with a free surface. J. Fluid Mech. 27 (1967), 375–397.
    • (1967) J. Fluid Mech. , vol.27 , pp. 375-397
    • Luke, J.C.1
  • 45
    • 0008466732 scopus 로고
    • A free surface vortex method with weak viscous effects
    • SIAM
    • [45] Lundgren, T.S., A free surface vortex method with weak viscous effects. Mathematical Aspects of Vortex Dynamics, 1989, SIAM, 68–79.
    • (1989) Mathematical Aspects of Vortex Dynamics , pp. 68-79
    • Lundgren, T.S.1
  • 46
    • 0026191546 scopus 로고
    • A new form of the Boussinesq equations with improved linear dispersion characteristics
    • [46] Madsen, P.A., Murray, R., Sorensen, O.R., A new form of the Boussinesq equations with improved linear dispersion characteristics. Coast. Eng. 15 (1991), 371–388.
    • (1991) Coast. Eng. , vol.15 , pp. 371-388
    • Madsen, P.A.1    Murray, R.2    Sorensen, O.R.3
  • 47
    • 3242825992 scopus 로고    scopus 로고
    • Higher-order Boussinesq-type equations for surface gravity waves: derivation and analysis
    • [47] Madsen, P.A., Schaffer, H.A., Higher-order Boussinesq-type equations for surface gravity waves: derivation and analysis. Phil. Trans. R. Soc. Lond. A 356 (1998), 3123–3184.
    • (1998) Phil. Trans. R. Soc. Lond. A , vol.356 , pp. 3123-3184
    • Madsen, P.A.1    Schaffer, H.A.2
  • 48
    • 0003440256 scopus 로고
    • The Applied Dynamics of Ocean Surface Waves
    • World Scientific
    • [48] Mei, C.C., The Applied Dynamics of Ocean Surface Waves. 1994, World Scientific.
    • (1994)
    • Mei, C.C.1
  • 49
    • 0027789687 scopus 로고
    • Alternative form of Boussinesq equations for nearshore wave propagation
    • [49] Nwogu, O., Alternative form of Boussinesq equations for nearshore wave propagation. J. Waterway Port Coast. Ocean Eng. 119 (1993), 618–638.
    • (1993) J. Waterway Port Coast. Ocean Eng. , vol.119 , pp. 618-638
    • Nwogu, O.1
  • 50
    • 0040470906 scopus 로고
    • Damping of solitary waves
    • [50] Ott, E., Sudan, R.N., Damping of solitary waves. Phys. Fluids 13 (1970), 1432–1434.
    • (1970) Phys. Fluids , vol.13 , pp. 1432-1434
    • Ott, E.1    Sudan, R.N.2
  • 51
    • 84958426983 scopus 로고
    • Long waves on a beach
    • [51] Peregrine, D.H., Long waves on a beach. J. Fluid Mech. 27 (1967), 815–827.
    • (1967) J. Fluid Mech. , vol.27 , pp. 815-827
    • Peregrine, D.H.1
  • 52
    • 0011790417 scopus 로고
    • Variational statement of the problem of liquid motion in a container of finite dimensions
    • [52] Petrov, A.A., Variational statement of the problem of liquid motion in a container of finite dimensions. Prikl. Math. Mekh. 28:4 (1964), 917–922.
    • (1964) Prikl. Math. Mekh. , vol.28 , Issue.4 , pp. 917-922
    • Petrov, A.A.1
  • 53
    • 76349123163 scopus 로고    scopus 로고
    • Numerical simulation of viscous, nonlinear and progressive water waves
    • [53] Raval, A., Wen, X., Smith, M.H., Numerical simulation of viscous, nonlinear and progressive water waves. J. Fluid Mech. 637 (2009), 443–473.
    • (2009) J. Fluid Mech. , vol.637 , pp. 443-473
    • Raval, A.1    Wen, X.2    Smith, M.H.3
  • 54
    • 0026219343 scopus 로고
    • Numerical simulations of the quasi stationary stage of ripple excitation by steep-capillary waves
    • [54] Ruvinsky, K.D., Feldstein, F.I., Freidman, G.I., Numerical simulations of the quasi stationary stage of ripple excitation by steep-capillary waves. J. Fluid Mech. 230 (1991), 339–353.
    • (1991) J. Fluid Mech. , vol.230 , pp. 339-353
    • Ruvinsky, K.D.1    Feldstein, F.I.2    Freidman, G.I.3
  • 55
    • 0000691222 scopus 로고    scopus 로고
    • Nonlinear evolution of water surface waves: the frequency down-shift phenomenon
    • [55] Skandrani, C., Kharif, C., Poitevin, J., Nonlinear evolution of water surface waves: the frequency down-shift phenomenon. Contemp. Math. 200 (1996), 157–171.
    • (1996) Contemp. Math. , vol.200 , pp. 157-171
    • Skandrani, C.1    Kharif, C.2    Poitevin, J.3
  • 56
    • 0003585575 scopus 로고
    • Water Waves, The Mathematical Theory With Applications
    • Wiley New York
    • [56] Stoker, J.J., Water Waves, The Mathematical Theory With Applications. 1958, Wiley, New York.
    • (1958)
    • Stoker, J.J.1
  • 57
    • 0026135818 scopus 로고
    • Burgers equation with a fractional derivative; hereditary effects on nonlinear acoustic waves
    • [57] Sugimoto, N., Burgers equation with a fractional derivative; hereditary effects on nonlinear acoustic waves. J. Fluid Mech. 225 (1991), 631–653.
    • (1991) J. Fluid Mech. , vol.225 , pp. 631-653
    • Sugimoto, N.1
  • 58
    • 77957116046 scopus 로고    scopus 로고
    • Energy dissipation in two-dimensional unsteady plunging breakers and an eddy viscosity model
    • [58] Tian, Z., Perlin, M., Choi, W., Energy dissipation in two-dimensional unsteady plunging breakers and an eddy viscosity model. J. Fluid Mech 655 (2010), 217–257.
    • (2010) J. Fluid Mech , vol.655 , pp. 217-257
    • Tian, Z.1    Perlin, M.2    Choi, W.3
  • 59
    • 33846272362 scopus 로고    scopus 로고
    • An efficient method for the numerical calculation of viscous effects on transient long-waves
    • [59] Torsvik, T., Liu, P.-F., An efficient method for the numerical calculation of viscous effects on transient long-waves. Coast. Eng. 54 (2007), 263–269.
    • (2007) Coast. Eng. , vol.54 , pp. 263-269
    • Torsvik, T.1    Liu, P.-F.2
  • 60
    • 35548945169 scopus 로고
    • The effect of a surface layer of viscous fluid on the wave resistance of a thin ship
    • [60] Tuck, E.O., The effect of a surface layer of viscous fluid on the wave resistance of a thin ship. J. Ship Res. 18 (1974), 265–271.
    • (1974) J. Ship Res. , vol.18 , pp. 265-271
    • Tuck, E.O.1
  • 61
    • 84987673529 scopus 로고    scopus 로고
    • Sea Wave Pattern Evaluation, Technical Report
    • The University of Adelaide
    • [61] Tuck, E.O., Scullen, D.C., Lazauskas, L., Sea Wave Pattern Evaluation, Technical Report. 2007, The University of Adelaide.
    • (2007)
    • Tuck, E.O.1    Scullen, D.C.2    Lazauskas, L.3
  • 62
    • 0001360283 scopus 로고
    • The long-wave paradox in the theory of gravity waves
    • [62] Ursell, F., The long-wave paradox in the theory of gravity waves. Proc. Camb. Phil. Soc. 49 (1953), 685–694.
    • (1953) Proc. Camb. Phil. Soc. , vol.49 , pp. 685-694
    • Ursell, F.1
  • 63
    • 0013528960 scopus 로고
    • Nonlinear, dispersive, shallow-water waves developed by a moving bed
    • [63] Villeneuve, M., Savage, S.B., Nonlinear, dispersive, shallow-water waves developed by a moving bed. J. Hydraulic Res. 31 (1993), 249–266.
    • (1993) J. Hydraulic Res. , vol.31 , pp. 249-266
    • Villeneuve, M.1    Savage, S.B.2
  • 64
    • 84962545187 scopus 로고    scopus 로고
    • Linear and Nonlinear waves
    • John Wiley & Sons Inc. New York
    • [64] Whitham, G.B., Linear and Nonlinear waves. 1999, John Wiley & Sons Inc., New York.
    • (1999)
    • Whitham, G.B.1
  • 65
    • 10644246428 scopus 로고
    • The Laplace Transform. Princeton Mathematical Series, 6
    • Princeton Univ. Press Princeton
    • [65] Widder, D.V., The Laplace Transform. Princeton Mathematical Series, 6. 1946, Princeton Univ. Press, Princeton.
    • (1946)
    • Widder, D.V.1
  • 66
    • 33744917573 scopus 로고    scopus 로고
    • A note on stabilizing the Benjamin-Feir instability
    • [66] Wu, G., Liu, Y., Yue, D., A note on stabilizing the Benjamin-Feir instability. J. Fluid Mech. 556 (2006), 45–54.
    • (2006) J. Fluid Mech. , vol.556 , pp. 45-54
    • Wu, G.1    Liu, Y.2    Yue, D.3
  • 67
    • 34250447917 scopus 로고
    • Stability of periodic waves of finite amplitude on the surface of a deep fluid
    • [67] Zakharov, V.E., Stability of periodic waves of finite amplitude on the surface of a deep fluid. J. Appl. Mech. Tech. Phys. 9 (1968), 190–194.
    • (1968) J. Appl. Mech. Tech. Phys. , vol.9 , pp. 190-194
    • Zakharov, V.E.1


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