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On the Mach reflection of a solitary wave: revisited

Published online by Cambridge University Press:  11 February 2011

WENWEN LI ,
HARRY YEH  and
YUJI KODAMA
WENWEN LI
Affiliation:
School of Civil and Construction Engineering, Oregon State University, Corvallis, OR 97331-3212, USA
HARRY YEH*
Affiliation:
School of Civil and Construction Engineering, Oregon State University, Corvallis, OR 97331-3212, USA
YUJI KODAMA
Affiliation:
Department of Mathematics, Ohio State University, Columbus, OH 43210-1174, USA
*
†Email address for correspondence: [email protected]
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Abstract

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Reflection of an obliquely incident solitary wave at a vertical wall is studied experimentally in the laboratory wave tank. Precision measurements of water-surface variations are achieved with the aid of laser-induced fluorescent (LIF) technique and detailed features of the Mach reflection are captured. During the development stage of the reflection process, the stem wave is not in the form of a Korteweg–de Vries (KdV) soliton but a forced wave, trailing by a continuously broadening depression. Evolution of stem-wave amplification is in good agreement with the Kadomtsev–Petviashvili (KP) theory. The asymptotic characteristics and behaviours are also in agreement with the theory of Miles (J. Fluid Mech., vol. 79, 1977b, p. 171) except those in the neighbourhood of the transition between the Mach reflection and the regular reflection. The predicted maximum fourfold amplification of the stem wave is not realized in the laboratory environment. On the other hand, the laboratory observations are in excellent agreement with the previous numerical results of the higher-order model of Tanaka (J. Fluid Mech., vol. 248, 1993, p. 637). The present laboratory study is the first to sensibly analyse validation of the theory; note that substantial discrepancies exist from previous (both numerical and laboratory) experimental studies. Agreement between experiments and theory can be partially attributed to the large-distance measurements that the precision laboratory apparatus is capable of. More important, to compare the laboratory results with theory, the corrected interaction parameter is derived from proper interpretation of the theory in consideration of the finite incident wave angle. Our laboratory data indicate that the maximum stem wave can reach higher than the maximum solitary wave height. The wave breaking near the wall results in the substantial increase in wave height and slope away from the wall.

JFM classification

Nonlinear Dynamical Systems: Pattern formation Waves/Free-surface Flows: Solitary waves Waves/Free-surface Flows: Waves/Free-surface Flows
Type
Papers
Information
Journal of Fluid Mechanics , Volume 672 , 10 April 2011 , pp. 326 - 357
Copyright
Copyright © Cambridge University Press 2011. The online version of this article is published within an Open Access environment subject to the conditions of the Creative Commons Attribution-NonCommercial-ShareAlike licence <http://creativecommons.org/licenses/by-nc-sa/2.5/>. The written permission of Cambridge University Press must be obtained for commercial re-use.

References

REFERENCES

Barakhnin, V. B. & Khakimzyanov, G. S. 1999 Numerical simulation of an obliquely incident solitary wave. J. Appl. Mech. Tech. Phys. 40, 1008–1015.CrossRefGoogle Scholar
Chakravarty, C. & Kodama, Y. 2009 Soliton solutions of the KP equation and application to shallow water waves. Stud. Appl. Math. 123, 83–151.CrossRefGoogle Scholar
Courant, R. & Friedrichs, K. O. 1948 Supersonic Flow and Shock Waves. Intersciences.Google Scholar
Diorio, J. D., Liu, X. & Duncan, J. H. 2009 An experimental investigation of incipient spilling breakers. J. Fluid Mech. 633, 271–283.CrossRefGoogle Scholar
Duncan, J. H., Philomin, V., Behres, M. & Kimmel, J. 1994 The formation of spilling breaking water waves. Phys. Fluids 6 (8), 2558–2560.CrossRefGoogle Scholar
Duncan, J. H., Qiao, H., Philomin, V. & Wenz, A. 1999. Gentle spilling breakers: crest profile evolution. J. Fluid Mech. 379, 191–222.CrossRefGoogle Scholar
Funakoshi, M. 1980. Reflection of obliquely incident solitary waves. J. Phys. Soc. Japan 49, 2371–2379.CrossRefGoogle Scholar
Gardarsson, S. M. & Yeh, H. 2007 Hysteresis in shallow water sloshing. J. Engng Mech. ASCE 133, 1093–1100.CrossRefGoogle Scholar
Goring, D. G. 1979 Tsunami: the propagation of long waves onto a shelf. PhD thesis, California Institute of Technology.Google Scholar
Guizien, K. & Barthélemy, R. 2002 Accuracy of solitary wave generation by a piston wave maker. J. Hydraul. Res. 40, 321–331.Google Scholar
Kato, S., Takagi, T. & Kawahara, M. 1998 A finite element analysis of Mach reflection by using the Boussinesq equation. Intl J. Numer. Meth. Fluids 28, 617–631.3.0.CO;2-L>CrossRefGoogle Scholar
Kodama, Y., Oikawa, M. & Tsuji, H. 2009 Soliton solutions of the KP equation with V-shape initial waves. J. Phys. A: Math. Theor. 42, 1–9.CrossRefGoogle Scholar
LeVeque, R. J. 2002 Finite Volume Methods for Hyperbolic Problems. Cambridge University Press.CrossRefGoogle Scholar
Liu, X. & Duncan, J. H. 2006 An experimental study of surfactant effects on spilling breakers. J. Fluid Mech. 567, 433–455.CrossRefGoogle 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
Melville, W. K. 1980 On the Mach reflexion of a solitary wave. J. Fluid Mech. 98, 285–297.CrossRefGoogle Scholar
Miles, J. W. 1977 a Obliquely interacting solitary waves. J. Fluid Mech. 79, 157–169.CrossRefGoogle Scholar
Miles, J. W. 1977 b Resonantly interacting solitary waves. J. Fluid Mech. 79, 171–179.CrossRefGoogle Scholar
von Neumann, J. 1943 Oblique reflection of shocks. Explosives Research Rep. 12. Navy Department, Bureau of Ordnance, Washington, DC. (Also in John von Neumann Collected Works (ed. Taub, A. H.), vol. 6, pp. 238–299, MacMillan, 1963.)Google Scholar
Perroud, P. H. 1957 The solitary wave reflection along a straight vertical wall at oblique incidence. Tech. Rep. 99/3. University of California, Berkeley.Google Scholar
Ramsden, J. D. 1993 Tsunamis: forces on a vertical wall caused by long waves, bores, and surges on a dry bed. PhD thesis, California Institute of Technology.Google Scholar
Tanaka, M. 1993 Mach reflection of a large-amplitude solitary wave. J. Fluid Mech. 248, 637–661.CrossRefGoogle Scholar
Strang, G. 1968 On the construction and comparison of difference schemes. SIAM J. Numer. Anal. 5 (3), 506–517.CrossRefGoogle Scholar
Whitham, G. B. 1974 Linear and Nonlinear Waves. Wiley-Interscience.Google Scholar
Yeh, H. & Ghazali, A. 1986 A bore on a uniformly sloping beach. In Proc. 20th Intl Conf. on Coastal Engineering, pp. 877–888.Google Scholar
Yeh, H., Li, W. & Kodama, Y. 2010 Mach reflection and KP solitons in shallow water. Eur. Phys. J. Special Topics 185, 97–111.CrossRefGoogle Scholar