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Wave jumps and caustics in the propagation of finite-amplitude water waves

Published online by Cambridge University Press:  20 April 2006

D. H. Peregrine
D. H. Peregrine
Affiliation:
School of Mathematics, University of Bristol, Bristol BS8 1TW
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Abstract

Nonlinear effects on the refraction of water waves are discussed. The existence of conjugate solutions for wave fields, one set of which corresponds to the ‘anomalous’ refraction solutions of Peregrine & Ryrie (1983) provides the stimulus for consideration of jumps in wave-field properties. Just such a jump is described by Yue & Mei (1980) for a case where near-linear waves are reflected with small deflection by a rigid wall.

Wave jumps between conjugate solutions appear to be possible for finite-amplitude wavetrains. These are examined and the structure of wave jumps in the near-linear, small-deflection case is elucidated. They have a structure directly analogous to that of an undular bore on shallow water with surface tension (‘hydraulic analogy’). Numerical results of Yue & Mei (1980) provide valuable guidance and confirmation.

Linear waves are reflected at a caustic, and can be described with Airy functions. Although equivalent weakly nonlinear solutions exist, the results from reflection by a wall and from use of the hydraulic analogy show that, unlike the caustics of linear theory, nonlinear caustics should not be considered in isolation. Caustic cusps, or wave focusing, must be considered, unless bed topography has discontinuities. A qualitative discussion of focusing based on the behaviour of unsteady waves in the hydraulic analogy shows that wave jumps can be expected. The relationship to linear theory is also put in perspective. Nonlinearity causes the linear rays to split into two sets of characteristics. The splitting of a ray focus leads to two wave jumps.

Consideration of the case of a semi-infinite beach shows that anomalous refraction is most unlikely to occur because there is an offshore influence of the beach on the wave field which changes the incident wave conditions to prevent anomalous refraction.

Type
Research Article
Information
Journal of Fluid Mechanics , Volume 136 , November 1983 , pp. 435 - 452
Copyright
© 1983 Cambridge University Press

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References

Akhamanov, S. A., Sukhorukov, A. P. & Khokhlov, R. V. 1966 Self-focussing and self-trapping of intense light beams in a nonlinear medium. Sov. Phys. JETP 23, 1025–810 [transl. of Zh. Eksp. Teor. Fiz. 50, 1537–810].Google Scholar
Berger, U. & Kohlhase, S. 1976 Mach-reflection as a diffraction problem. In Proc. 15th Coastal Engng Conf., pp. 796–814. ASCE.
Benjamin, T. B. & Lighthill, M. J. 1954 On cnoidal waves and bores. Proc. R. Soc. Lond A 224, 448–460.Google Scholar
Chen, B. & Saffmann, P. G. 1980 Numerical evidence for the existence of new types of gravity waves of permanent form on deep water Stud. Appl. Maths 62, 1–21.Google Scholar
Chin, W. C. 1979 Kinematic wave approach to hydraulic jumps with waves J. Hydronautics 13, 120–124.Google Scholar
Cokelet, E. D. 1977 Steep gravity waves in water of arbitrary uniform depth. Phil. Trans. R. Soc. Lond A 286, 183–230.Google Scholar
Cramer, M. S. & Seebass, A. R. 1978 Focusing of weak shock waves at an arête J. Fluid Mech. 88, 209–222.Google Scholar
Davey, A. & Stewartson, K. 1974 On three-dimensional packets of surface waves. Proc. R. Soc. Lond A 338, 101–110.Google Scholar
Fornberg, B. & Whitham, G. B. 1978 A numerical and theoretical study of certain nonlinear wave phenomena. Phil. Trans. R. Soc. Lond A 289, 373–404.Google Scholar
Fridman, V. E. 1982 Self-refraction of small amplitude shock waves Wave Motion 4, 151–161.Google Scholar
Funakoshi, M. 1980 Reflection of obliquely incident solitary waves J. Phys. Soc. Japan 49, 2371–2379.Google Scholar
Grimshaw, R. 1981 The modulation of a weakly nonlinear wave packet in an inhomogeneous medium SIAM J. Appl. Maths 40, 10–34.Google Scholar
Gurevich, A. V. & Pitaevskii, L. P. 1974 Nonstationary structure of a collisionless shock wave. Sov. Phys. JETP 38, 291–810 [transl. of Zh. Eksp. Teor. Fiz. 65, 590–810].Google Scholar
Hayes, W. D. 1973 Group velocity and nonlinear dispersive wave-action. Proc. R. Soc. Londo A 232, 199–221.Google Scholar
Howe, M. S. 1968 Phase jumps J. Fluid Mech. 32, 779–789.Google Scholar
Kirby, J. T. & Dalrymple, R. A. 1983 A parabolic equation for the combined refraction-diffraction of Stokes’ waves by mildly varying topography J. Fluid Mech. 136, 453–466.Google Scholar
Korteweg, D. J. & DE VRIES, G 1895 On the change of form of long waves advancing in a rectangular channel, and on a new type of long stationary waves. Phil. Mag. (5) 39, 422–810.
Kulikovskii, A. G. & Reutov, V. A. 1976 Movement of solitary and periodic waves with an amplitude close to the limiting in a liquid layer of slowly varying depth. Fluid Dyn. 11, 884–810 [transl. of Izv. Akad. Nauk SSSR, Mekh. Zhid. Gaza no. 6, 76–810].Google Scholar
Kulikovskii, A. G. & Reutov, V. A. 1980 Propagation of nonlinear waves above semi-infinite underwater troughs and ridges. Fluid Dyn. 15, 217–810 [transl. of Izv. Akad. Nauk SSSR, Mekh. Zhid. Gaza no. 2, 53–810].Google Scholar
Longuet-Higgins, M. S. 1975 Integral properties of periodic gravity waves of finite amplitude. Proc. R. Soc. Lond A 342, 157–174.Google Scholar
Longuet-Higgins, M. S. 1978a The instabilities of gravity waves of finite amplitude in deep water. I. Super-harmonics. Proc. R. Soc. Lond A 360, 471–488.Google Scholar
Longuet-Higgins, M. S. 1978b The instabilities of gravity waves of finite amplitude in deep water. II. Sub-harmonics. Proc. R. Soc. Lond A 360, 489–505.Google Scholar
Longuet-Higgins, M. S. & Fenton, J. D. 1974 On the mass, momentum, energy and circulation of a solitary wave. II. Proc. R. Soc. Lond A 340, 471–493.Google Scholar
Mcintyre, M. E. 1981 On the ‘wave-momentum’ myth. J. Fluid Mech. 106, 331–810.Google Scholar
Mclean, J. W. 1982 Instabilities of finite-amplitude waves on water of finite depth J. Fluid Mech. 114, 331–341.Google Scholar
Melville, W. K. 1980 On the Mach reflexion of a solitary wave J. Fluid Mech. 98, 285–297.Google Scholar
Melville, W. K. 1982 The instability and breaking of deep-water waves J. Fluid Mech. 115, 165–185.Google Scholar
Miles, J. W. 1977a Obliquely interacting solitary waves J. Fluid Mech. 79, 157–169.Google Scholar
Miles, J. W. 1977b Resonantly interacting solitary waves J. Fluid Mech. 79, 171–179.Google Scholar
Miles, J. W. 1977c Diffraction of solitary waves Z. angew. Math. Phys. 28, 889–902.Google Scholar
Miles, J. W. 1980 The second Painlevé transcendent: a nonlinear Airy function. In Mechanics Today (ed. S. Nemat-Nasser), vol. 5, 297–810. Pergamon.
Ostrovskii, L. A. 1968 Envelope shock waves. Sov. Phys. JETP 27, 660–810 [transl. of Zh. Eksp. Teor. Fiz. 54, 1235–810].Google Scholar
Ostrovskii, L. A. & Shrira, V. I. 1976 Instability and self-refraction of solitons. Sov. Phys. JETP 44, 738–810 [transl. of Zh. Eksp. Teor. Fiz. 71, 1412–810].Google Scholar
Peregrine, D. H. 1966 Calculations of the development of an undular bore J. Fluid Mech. 25, 321–330.Google Scholar
Peregrine, D. H. 1981 Refraction of finite-amplitude water waves: deepwater waves approaching circular caustics. J. Fluid Mech. 109, 63–810.Google Scholar
Peregrine, D. H. 1983a Breaking waves on beaches Ann. Rev. Fluid Mech. 15, 149–178.Google Scholar
Peregrine, D. H. 1983b Water waves, nonlinear Schrödinger equations and their solutions. J. Austral. Math. Soc. B 25, 16–810.Google Scholar
Peregrine, D. H. & Ryrie, S. 1983 Anomalous refraction and conjugate solutions of finiteamplitude water waves J. Fluid Mech. 134, 91–101.Google Scholar
Peregrine, D. H. & Smith, R. 1979 Nonlinear effects upon waves near caustics. Phil. Trans. R. Soc. Lond A 292, 341–370.Google Scholar
Peregrine, D. H. & Thomas, G. P. 1979 Finite-amplitude deep-water waves on currents. Phil. Trans. R. Soc. Lond A 292, 371–390.Google Scholar
Reutov, V. A. 1976 Behaviour of perturbations of solitary and periodic waves on the surface of a heavy liquid. Fluid Dyn. 11, 778–810 [transl. of Izv. Akad. Nauk SSSR, Mekh. Zhid. Gaza no. 5, 156–810].Google Scholar
Rosales, R. 1978 The similarity solution for the Korteweg—de Vries equation and the related Painlevé transcendent. Proc. R. Soc. Lond A 361, 265–275.Google Scholar
Ryrie, S. & Peregrine, D. H. 1982 Refraction of finite-amplitude water waves obliquely incident to a uniform beach J. Fluid Mech. 115, 91–104.Google Scholar
Schwartz, L. W. 1974 Computer extension and analytic continuation of Stokes’ expansion for gravity waves J. Fluid Mech. 62, 553–578.Google Scholar
Schwartz, L. W. & Fenton, J. D. 1982 Strongly nonlinear waves Ann. Rev. Fluid Mech. 14, 39–60.Google Scholar
Stamnes, J. J., Løvhaugen, O., Spjelkavik, B., Mei, C. C., Lo, E. & Yue, D. K. P. 1983 Nonlinear lens focussing of surface waves—theory and experiment J. Fluid Mech. 135, 71–94.Google Scholar
Stiassnie, M. & Peregrine, D. H. 1979 On averaged equations for finite-amplitude water waves J. Fluid Mech. 94, 401–407.Google Scholar
Stiassnie, M. & Peregrine, D. H. 1980 Shoaling of finite-amplitude surface waves on water of slowly-varying depth J. Fluid Mech. 97, 783–805.Google Scholar
Sturtevant, B. & Kulkarny, V. A. 1976 The focusing of weak shock waves J. Fluid Mech. 73, 651–671.Google Scholar
Su, M.-Y., Bergin, M., Marler, P. & Myrick, R. 1982a Experiments on nonlinear instabilities and evolution of steep gravity-wave trains J. Fluid Mech. 124, 45–72.Google Scholar
Su, M.-Y., Bergin, M., Myrick, R. & Roberts, J. 1982b Experiments on shallow-water wave grouping and breaking. In Proc. 1st Intl Conf. on Meteorology and Air—Sea Interaction in the Coastal Zone. Koninklijk Nederlands Meteorologisch Instituut, The Hague.
Walker, J. R. 1976 Refraction of finite-height and breaking waves. In Proc. 15th Conf. on Coastal Engng, pp. 507–524 ASCE.
Whalin, R. W. 1972 Wave refraction theory in a convergence zone. Proc. 13th Conf. on Coastal Engng, pp. 451–470. ASCE.
Whitham, G. B. 1965 Non-linear dispersive waves. Proc. R. Soc. Lond A 283, 238–261.Google Scholar
Whitham, G. B. 1974 Linear and Non-linear Waves. Wiley-Interscience.
Yue, D. K. P. & Mei, C. C. 1980 Forward diffraction of Stokes waves by a thin wedge J. Fluid Mech. 99, 33–52.Google Scholar