Head-on collision between two hydroelastic solitary waves in shallow water. (English) Zbl 1394.35346
A comprehensive theoretical study on the head-on collision between two solitary waves in a thin elastic plate floating on an inviscid fluid of finite depth is investigated analytically by means of a singular perturbation method. The effects of plate compression are also taken into account. It is found that after collision, both the hydroelastic solitary waves preserves their original shape and positions. However a collision does have imprints on colliding waves with non-uniform phase shift up to the third order which creates tilting in the wave profile. Maximum run-up amplitude, wave speed, phase shift and distortion profile have also been calculated and plotted for two colliding solitary waves.
Reviewer: Dongbing Zha (Shanghai)
MSC:
| 35Q35 | PDEs in connection with fluid mechanics |
| 35Q74 | PDEs in connection with mechanics of deformable solids |
| 35R35 | Free boundary problems for PDEs |
| 76B25 | Solitary waves for incompressible inviscid fluids |
| 74K20 | Plates |
| 74F10 | Fluid-solid interactions (including aero- and hydro-elasticity, porosity, etc.) |
| 74B20 | Nonlinear elasticity |
| 35B25 | Singular perturbations in context of PDEs |
Keywords:
head-on collision; solitary waves; thin elastic plate; perturbation solutions; compressive forceReferences:
| [1] | Forbes, LK, Surface waves of large amplitude beneath an elastic sheet. part 1. high-order series solution, J. Fluid Mech., 169, 409-428, (1986) · Zbl 0607.76015 · doi:10.1017/S0022112086000708 |
| [2] | Forbes, LK, Surface waves of large amplitude beneath an elastic sheet. part 2. Galerkin solution, J. Fluid Mech., 188, 491-508, (1988) · Zbl 0643.76013 · doi:10.1017/S0022112088000813 |
| [3] | Părău, EI; Vanden-Broeck, JM, Three-dimensional waves beneath an ice sheet due to a steadily moving pressure, Philos. Trans. R. Soc. A Math. Phys. Eng. Sci., 369, 2973-2988, (2011) · Zbl 1228.86008 · doi:10.1098/rsta.2011.0115 |
| [4] | Deike, L; Bacri, JC; Falcon, E, Nonlinear waves on the surface of a fluid covered by an elastic sheet, J. Fluid Mech., 733, 394-413, (2013) · Zbl 1294.76009 · doi:10.1017/jfm.2013.379 |
| [5] | Chen, XJ; Wu, YS; Cui, WC; Tang, XF, Nonlinear hydroelastic analysis of a moored floating body, Ocean Eng., 30, 965-1003, (2003) · doi:10.1016/S0029-8018(02)00078-1 |
| [6] | Plotnikov, PI; Toland, JF, Modelling nonlinear hydroelastic waves, Philos. Trans. R. Soc. A Math. Phys. Eng. Sci., 369, 2942-2956, (2011) · Zbl 1228.76027 · doi:10.1098/rsta.2011.0104 |
| [7] | Blyth, MG; Pru, EI; Vanden-Broeck, JM, Hydroelastic waves on fluid sheets, J. Fluid Mech., 689, 541-551, (2011) · Zbl 1241.76083 · doi:10.1017/jfm.2011.451 |
| [8] | Vanden-Broeck, JM; Părău, EI, Two-dimensional generalized solitary waves and periodic waves under an ice sheet, Philos. Trans. R. Soc. A Math. Phys. Eng. Sci., 369, 2957-2972, (2011) · Zbl 1228.86002 · doi:10.1098/rsta.2011.0108 |
| [9] | Milewski, PA; Vanden-Broeck, JM; Wang, Z, Hydroelastic solitary waves in deep water, J. Fluid Mech., 679, 628-640, (2011) · Zbl 1241.76102 · doi:10.1017/jfm.2011.163 |
| [10] | Guyenne, P; Părău, EI, Forced and unforced flexural-gravity solitary waves, Procedia IUTAM, 11, 44-57, (2014) · doi:10.1016/j.piutam.2014.01.047 |
| [11] | Alam, MR, Dromions of flexural-gravity waves, J. Fluid Mech., 719, 1-13, (2013) · Zbl 1284.76062 · doi:10.1017/jfm.2012.590 |
| [12] | Sorensen, MP; Christiansen, PL; Lomdahl, P, Solitary waves on nonlinear elastic rods. I, J. Acoust. Soc. Am., 76, 871-879, (1984) · Zbl 0564.73035 · doi:10.1121/1.391312 |
| [13] | Dai, SQ, The interaction of two pairs of solitary waves in a two-fluid system, Sci. China Math., 81, 1718-1722, (1984) |
| [14] | Sorensen, MP; Christiansen, PL; Lomdahl, P; Skovgaard, O, Solitary waves on nonlinear elastic rods. II, J. Acoust. Soc. Am., 81, 1718-1722, (1987) · doi:10.1121/1.394786 |
| [15] | Huang, G; Lou, S; Xu, Z, Head-on collision between two solitary waves in a Rayleigh-benard convecting fluid, Phys. Rev. E, 47, r3830, (1993) · doi:10.1103/PhysRevE.47.R3830 |
| [16] | Huang, G; Velarde, MG, Head-on collision of two concentric cylindrical ion acoustic solitary waves, Phys. Rev. E, 53, 2988, (1996) · doi:10.1103/PhysRevE.53.2988 |
| [17] | Dai, HH; Dai, SQ; Huo, Y, Head-on collision between two solitary waves in a compressible mooneyrivlin elastic rod, Wave Motion, 32, 93-111, (2000) · Zbl 1074.74586 · doi:10.1016/S0165-2125(00)00029-9 |
| [18] | Părău, EI; Dias, F, Nonlinear effects in the response of a floating ice plate to a moving load, J. Fluid Mech., 460, 281-305, (2002) · Zbl 1045.76006 |
| [19] | Demiray, H, Head-on-collision of nonlinear waves in a fluid of variable viscosity contained in an elastic tube, Chaos Solitons Fract., 41, 1578-1586, (2009) · Zbl 1198.76164 · doi:10.1016/j.chaos.2008.06.022 |
| [20] | Ozden, AE; Demiray, H, Re-visiting the head-on collision problem between two solitary waves in shallow water, Int. J. Non-Linear Mech., 69, 66-70, (2015) · Zbl 1348.76040 · doi:10.1016/j.ijnonlinmec.2014.11.022 |
| [21] | Laitone, EV, The second approximation to cnoidal and solitary waves, J. Fluid Mech., 9, 430-444, (1960) · Zbl 0095.22302 · doi:10.1017/S0022112060001201 |
| [22] | Grimshaw, R, The solitary wave in water of variable depth. part 2, J. Fluid Mech., 46, 611-622, (1971) · Zbl 0222.76017 · doi:10.1017/S0022112071000739 |
| [23] | Gardner, CS; Greene, JM; Kruskal, MD; Miura, RM, Method for solving the Korteweg-devries equation, Phys. Rev. Lett., 19, 1095, (1967) · Zbl 1061.35520 · doi:10.1103/PhysRevLett.19.1095 |
| [24] | Lighthill, M.J.: A technique for rendering approximate solutions to physical problems uniformly valid. Philosophical Magaz. Ser. 7. 40(5), 1179-1120 (1949) · Zbl 0035.20504 |
| [25] | Lin, CC, On a perturbation theory based on the method of characteristics, J. Math. Phys., 33, 117-134, (1954) · Zbl 0056.19904 · doi:10.1002/sapm1954331117 |
| [26] | Su, CH; Mirie, RM, On head-on collisions between two solitary waves, J. Fluid Mech., 98, 509-525, (1980) · Zbl 0434.76021 · doi:10.1017/S0022112080000262 |
| [27] | Kuo, YH, On the flow of an incompressible viscous fluid past a flat plate at moderate Reynolds number, J. Math. Phys., 32, 81-101, (1953) · Zbl 0051.17301 · doi:10.1002/sapm195332183 |
| [28] | Dai, SQ, Poincare-Lighthill-kuo method and symbolic computation, Appl. Math. Mech. Engl. Ed., 22, 261-269, (2001) · Zbl 1116.70006 · doi:10.1023/A:1015565502306 |
| [29] | Van Dyke, M.: Perturbation methods in fluid mechanics/annotated edition. NASA STI/Recon Technical Report A 75, 46926 (1975) · Zbl 0329.76002 |
| [30] | Zhu, Y; Dai, SQ, On head-on collision between two GKDV solitary waves in a stratified fluid, Acta Mech. Sin., 7, 300-308, (1991) · Zbl 0753.76031 · doi:10.1007/BF02486737 |
| [31] | Zhu, Y, Head-on collision between two mkdv solitary waves in a two-layer fluid system, Appl. Math. Mech. Engl. Ed., 13, 407-417, (1992) · Zbl 0758.76073 · doi:10.1007/BF02450731 |
| [32] | Xia, X; Shen, HT, Nonlinear interaction of ice cover with shallow water waves in channels, J. Fluid Mech. Engl. Ed., 467, 259-268, (2002) · Zbl 1034.74020 |
| [33] | Bhatti, M.M., Lu, D.Q.: Effect of compression on the head-on collision between two hydroelastic solitary waves. In: Proceedings of the 12th International Conference on Hydrodynamics. Egmond aan Zee, The Netherlands (2016) · Zbl 0051.17301 |
| [34] | Guyenne, P; Părău, EI, Finite-depth effects on solitary waves in a floating ice sheet, J. Fluids Struct., 49, 242-262, (2014) · doi:10.1016/j.jfluidstructs.2014.04.015 |
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
