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Obliquely interacting solitary waves and wave wakes in free-surface flows

Published online by Cambridge University Press:  14 May 2025

Lei Hu ,
Xudan Luo Open the ORCID record for Xudan Luo [Opens in a new window]  and
Zhan Wang Open the ORCID record for Zhan Wang [Opens in a new window]
Lei Hu
Affiliation:
Key Laboratory for Mechanics in Fluid Solid Coupling Systems, Institute of Mechanics, Chinese Academy of Sciences, Beijing 100190, PR China
Xudan Luo
Affiliation:
State Key Laboratory of Mathematical Sciences, Academy of Mathematics and Systems Science, Chinese Academy of Sciences, Beijing 100190, PR China
Zhan Wang*
Affiliation:
Key Laboratory for Mechanics in Fluid Solid Coupling Systems, Institute of Mechanics, Chinese Academy of Sciences, Beijing 100190, PR China State Key Laboratory of High Temperature Gas Dynamics, Institute of Mechanics, Chinese Academy of Sciences, Beijing 100190, PR China
*
Corresponding author: Zhan Wang, zwang@imech.ac.cn
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Abstract

This paper investigates the weakly nonlinear isotropic bidirectional Benney–Luke (BL) equation, which is used to describe oceanic surface and internal waves in shallow water, with a particular focus on soliton dynamics. Using the Whitham modulation theory, we derive the modulation equations associated with the BL equation that describe the evolution of soliton amplitude and slope. By analysing rarefaction waves and shock waves within these modulation equations, we derive the Riemann invariants and modified Rankine–Hugoniot conditions. These expressions help characterise the Mach expansion and Mach reflection phenomena of bent and reverse bent solitons. We also derive analytical formulae for the critical angle and the Mach stem amplitude, showing that as the soliton speed is in the vicinity of unity, the results from the BL equation align closely with those of the Kadomtsev–Petviashvili (KP) equation. Corresponding numerical results are obtained and show excellent agreement with theoretical predictions. Furthermore, as a far-field approximation for the forced BL equation – which models wave and flow interactions with local topography – the modulation equations yield a slowly varying similarity solution. This solution indicates that the precursor wavefronts created by topography moving at subcritical or critical speeds take the shape of a circular arc, in contrast to the parabolic wavefronts observed in the forced KP equation.

JFM classification

Waves/Free-surface Flows: Solitary waves Waves/Free-surface Flows: Surface gravity waves

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Type
JFM Papers
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Journal of Fluid Mechanics , Volume 1011 , 25 May 2025 , A8
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© The Author(s), 2025. Published by Cambridge University Press

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References

Ablowitz, M.J. & Baldwin, D.E. 2012 Nonlinear shallow ocean-wave soliton interactions on flat beaches. Phys. Rev. E 86 (3), 036305.CrossRefGoogle ScholarPubMed
Ablowitz, M.J., Biondini, G. & Wang, Q. 2017 Whitham modulation theory for the Kadomtsev–Petviashvili equation. Proc. R. Soc. A 473 (2204), 20160695.CrossRefGoogle ScholarPubMed
Ablowitz, M.J. & Clarkson, P.A. 1991 Solitons, nonlinear evolution equations and inverse scattering. In London Mathematical Society Lecture Note Series, vol. 149. Cambridge University Press.Google Scholar
Ablowitz, M.J. & Curtis, C.W. 2011 On the evolution of perturbations to solutions of the Kadomtsev–Petviashvilli equation using the Benney–Luke equation. J. Phys. A 44 (19), 195202.CrossRefGoogle Scholar
Ablowitz, M.J. & Curtis, C.W. 2013 Conservation laws and non-decaying solutions for the Benney–Luke equation. Proc. R. Soc. A 469 (2152), 20120690.CrossRefGoogle Scholar
Ablowitz, M.J., Fokas, A.S. & Musslimani, Z.H. 2006 On a new non-local formulation of water waves. J. Fluid Mech. 562, 313.CrossRefGoogle Scholar
Bazhenova, T.V., Gvozdeva, L.G., Komarov, V.S. & Sukhov, B.G. 1975 Diffraction of strong shock waves by convex corners. Fluid Dyn. 8 (4), 611–619.CrossRefGoogle Scholar
Benney, D.J. & Luke, J.C. 1964 On the interactions of permanent waves of finite amplitude. J. Math. Phys. 43 (1–4), 309–313.CrossRefGoogle Scholar
Biondini, G. 2007 Line soliton interactions of the Kadomtsev–Petviashvili equation. Phys. Rev. Lett. 99 (6), 064103.CrossRefGoogle ScholarPubMed
Biondini, G. & Chakravarty, S. 2006 Soliton solutions of the Kadomtsev–Petviashvili II equation. J. Math. Phys. 47 (3), 033514.CrossRefGoogle Scholar
Biondini, G., Hoefer, M.A. & Moro, A. 2020 Integrability, exact reductions and special solutions of the KP–Whitham equations. Nonlinearity 33 (8), 4114–4132.CrossRefGoogle Scholar
Biondini, G., Maruno, K.-I., Oikawa, M. & Tsuji, H. 2009 Soliton interactions of the Kadomtsev–Petviashvili equation and generation of large-amplitude water waves. Stud. Appl. Math. 122 (4), 377–394.CrossRefGoogle Scholar
Bokhove, O. & Kalogirou, A. 2016 Variational water wave modelling: from continuum to experiment. In Lectures On the Theory of Water Waves. 1st edn (ed. Bridges T.J., Groves M.D. & Nicholls D.P.), pp. 226–260. Cambridge University Press.CrossRefGoogle Scholar
Byrd, P.F. & Friedman, M.D. 2013 Handbook of Elliptic Integrals for Engineers and Scientists. 2nd edn. Springer.Google Scholar
Cornish, V. 1910 Waves of the Sea and Other Water Waves. T. Fisher Unwin.CrossRefGoogle Scholar
Curtis, C.W. & Shen, S.S.P. 2015 Three-dimensional surface water waves governed by the forced Benney–Luke equation. Stud. Appl. Math. 135 (4), 447–465.CrossRefGoogle Scholar
Funakoshi, M. 1980 Reflection of obliquely incident solitary waves. J. Phys. Soc. Japan 49 (6), 2371–2379.CrossRefGoogle Scholar
Gidel, F., Bokhove, O. & Kalogirou, A. 2017 Variational modelling of extreme waves through oblique interaction of solitary waves: application to Mach reflection. Nonlinear Proc. Geophys. 24 (1), 43–60.CrossRefGoogle Scholar
Gilmore, F.R., Plesset, M.S. & Crossley, H.E. 1950 The analogy between hydraulic jumps in liquids and shock waves in gases. J. Appl. Phys. 21 (3), 243–249.CrossRefGoogle Scholar
Grava, T., Klein, C. & Pitton, G. 2018 Numerical study of the Kadomtsev–Petviashvili equation and dispersive shock waves. Proc. Roy. Soc. A 474 (2210), 20170458.CrossRefGoogle Scholar
Kadomtsev, B.B. & Petviashvili, V.I. 1970 On the stability of solitary waves in weakly dispersing media. Sov. Phys. Dokl. 15, 539–541.Google Scholar
Kao, C.-Y. & Kodama, Y. 2012 Numerical study of the KP equation for non-periodic waves. Maths Comput. Simulat. 82 (7), 1185–1218.CrossRefGoogle Scholar
Katsis, C. & Akylas, T.R. 1987 On the excitation of long nonlinear water waves by a moving pressure distribution. Part 2. Three-dimensional effects. J. Fluid Mech. 177, 49–65.CrossRefGoogle Scholar
Kodama, Y. 2004 Young diagrams and N-soliton solutions of the KP equation. J. Phys. A 37 (46), 11169–11190.CrossRefGoogle Scholar
Kodama, Y. 2010 KP solitons in shallow water. J. Phys. A 43 (43), 434004.CrossRefGoogle Scholar
Kodama, Y. 2018 Solitons in Two-Dimensional Shallow Water. Society for Industrial and Applied Mathematics.CrossRefGoogle Scholar
Kodama, Y., Oikawa, M. & Tsuji, H. 2009 Soliton solutions of the KP equation with V-shape initial waves. J. Phys. A 42 (31), 312001.CrossRefGoogle Scholar
Kodama, Y. & Yeh, H. 2016 The KP theory and Mach reflection. J. Fluid Mech. 800, 766–786.CrossRefGoogle Scholar
Krehl, P. & van der Geest, M. 1991 The discovery of the Mach reflection effect and its demonstration in an auditorium. Shock Waves 1 (1), 3–15.CrossRefGoogle Scholar
Lee, S.-J. & Grimshaw, R.H.J. 1990 Upstream-advancing waves generated by three-dimensional moving disturbances. Phys. Fluids 2 (2), 194–201.CrossRefGoogle Scholar
Lee, S.-J., Yates, G.T. & Wu, T.Y. 1989 Experiments and analyses of upstream-advancing solitary waves generated by moving disturbances. J. Fluid Mech. 199, 569–593.CrossRefGoogle Scholar
Li, W., Yeh, H. & Kodama, Y. 2011 On the Mach reflection of a solitary wave: revisited. J. Fluid Mech. 672, 326–357.CrossRefGoogle Scholar
Mach, E. & Wosyka, J. 1875 Ueber einige mechanische Wirkungen des elektrischen Funkens. Ann. Phys. - Berlin 232 (11), 407–416.CrossRefGoogle Scholar
Melville, W.K. 1980 On the Mach reflexion of a solitary wave. J. Fluid Mech. 98 (2), 285–297.CrossRefGoogle Scholar
Miles, J.W. 1977 a Obliquely interacting solitary waves. J. Fluid Mech. 79 (1), 157–169.CrossRefGoogle Scholar
Miles, J.W. 1977 b Resonantly interacting solitary waves. J. Fluid Mech. 79 (1), 171–179.CrossRefGoogle Scholar
Milewski, P.A. 1998 A formulation for water waves over topography. Stud. Appl. Maths 100 (1), 95–106.CrossRefGoogle Scholar
Milewski, P.A. & Tabak, E.G. 1999 A pseudospectral procedure for the solution of nonlinear wave equations with examples from free-surface flows. SIAM J. Sci. Comput. 21 (3), 1102–1114.CrossRefGoogle Scholar
Neu, J.C. 2015 Singular perturbation in the physical sciences. In Graduate Studies in Mathematics, vol. 167. American Mathematical Society.Google Scholar
von Neumann, J. 1943 Oblique reflection of shock. Explos. Res. Rep. 12.Navy Dept., Bureau of Ordinance.Google Scholar
von Neumann, J. 1945 Refraction, intersection and reflection of shock waves. NAVORD Rep. 203–245.Navy Dept., Bureau of Ordinance.Google Scholar
Perroud, P.H. 1957 The solitary wave reflection along a straight vertical wall at oblique incidence. PhD thesis, University of California.Google Scholar
Reichenbach, H. 1983 Contributions of Ernst Mach to fluid mechanics. Annu. Rev. Fluid Mech. 15 (1), 1–29.CrossRefGoogle Scholar
Ryskamp, S., Maiden, M.D., Biondini, G. & Hoefer, M.A. 2021 Evolution of truncated and bent gravity wave solitons: the Mach expansion problem. J. Fluid Mech. 909, A24.CrossRefGoogle Scholar
Ryskamp, S.J., Hoefer, M.A. & Biondini, G. 2022 Modulation theory for soliton resonance and Mach reflection. Proc. Roy. Soc. A 478 (2259), 20210823.CrossRefGoogle Scholar
Schlatter, P., Adams, N.A. & Kleiser, L. 2005 A windowing method for periodic inflow/outflow boundary treatment of non-periodic flows. J. Comput. Phys. 206 (2), 505–535.CrossRefGoogle Scholar
Shu, C.-W. 1998 Essentially non-oscillatory and weighted essentially non-oscillatory schemes for hyperbolic conservation laws. In Advanced Numerical Approximation of Nonlinear Hyperbolic Equations, (ed. Cockburn B., Shu C.-W., Johnson C., Tadmor E. & Quarteroni A.), pp. 325–432. Springer.CrossRefGoogle Scholar
Tanaka, M. 1993 Mach reflection of a large-amplitude solitary wave. J. Fluid Mech. 248, 637–661.CrossRefGoogle Scholar
Tsuji, H. & Oikawa, M. 2007 Oblique interaction of solitons in an extended Kadomtsev–Petviashvili equation. J. Phys. Soc. Japan 76 (8), 084401.CrossRefGoogle Scholar
Wang, C. & Pawlowicz, R. 2012 Oblique wave-wave interactions of nonlinear near-surface internal waves in the Strait of Georgia. J. Geophys. Res. Oceans 117 (C6), 2012JC008022.CrossRefGoogle Scholar
Whitham, G.B. 1957 A new approach to problems of shock dynamics Part I two-dimensional problems. J. Fluid Mech. 2 (2), 145–171.CrossRefGoogle Scholar
Whitham, G.B. 1959 A new approach to problems of shock dynamics Part 2. Three-dimensional problems. J. Fluid Mech. 5 (3), 369–386.CrossRefGoogle Scholar
Whitham, G.B. 1999 Linear and nonlinear waves. In Pure and Applied Mathematics, vol. 1237. Wiley.Google Scholar
Yuan, C., Grimshaw, R.H.J., Johnson, E. & Chen, X. 2018 The propagation of internal solitary waves over variable topography in a horizontally two-dimensional framework. J. Phys. Oceanogr. 48 (2), 283–300.CrossRefGoogle Scholar
Yuan, C. & Wang, Z. 2022 On diffraction and oblique interactions of horizontally two-dimensional internal solitary waves. J. Fluid Mech. 936, A20.CrossRefGoogle Scholar
Yuan, C., Wang, Z. & Chen, X. 2020 The derivation of an isotropic model for internal waves and its application to wave generation. Ocean Model. 153, 101663.CrossRefGoogle Scholar
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