Hostname: page-component-586b7cd67f-t7czq Total loading time: 0 Render date: 2024-11-23T18:08:25.702Z Has data issue: false hasContentIssue false
  • English
  • Français

On higher order Bragg resonance of water waves by bottom corrugations

Published online by Cambridge University Press:  12 July 2010

JIE YU  and
LOUIS N. HOWARD
JIE YU*
Affiliation:
Department of Civil, Construction and Environmental Engineering, North Carolina State University, Raleigh, NC 27695-7908, USA
LOUIS N. HOWARD
Affiliation:
Department of Mathematics, Massachusetts Institute of Technology, Cambridge, MA 02139, USA
*
†Email address for correspondence: [email protected]
Get access Rights & Permissions [Opens in a new window]

Abstract

The exact theory of linearized water waves in a channel of indefinite length with bottom corrugations of finite amplitude (Howard & Yu, J. Fluid Mech., vol. 593, 2007, pp. 209–234) is extended to study the higher order Bragg resonances of water waves occurring when the corrugation wavelength is close to an integer multiple of half a water wavelength. The resonance tongues (ranges of water-wave frequencies) are given for these higher order cases. Within a resonance tongue, the wave amplitude exhibits slow exponential modulation over the corrugations, and slow sinusoidal modulation occurs outside it. The spatial rate of wave amplitude modulation is analysed, showing its quantitative dependence on the corrugation height, water-wave frequency and water depth. The effects of these higher order Bragg resonances are illustrated using the normal modes of a rectangular tank.

JFM classification

Geophysical and Geological Flows: Coastal engineering Waves/Free-surface Flows: Surface gravity waves Waves/Free-surface Flows: Wave scattering
Type
Papers
Information
Journal of Fluid Mechanics , Volume 659 , 25 September 2010 , pp. 484 - 504
Copyright
Copyright © Cambridge University Press 2010

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

REFERENCES

Abramowitz, M. & Stegun, I. A. 1964 Handbook of Mathematical Functions. National Bureau of Standards, Applied Mathematics Series 55. US Government Printing Office.Google Scholar
Alam, M.-R., Liu, Y. & Yue, D. K. P. 2009 Bragg resonance of waves in a two-layer fluid propagating over bottom ripples. Part 1. Perturbation analysis. J. Fluid Mech. 624, 191–224.CrossRefGoogle Scholar
Bragg, W. H. & Bragg, W. L. 1913 The reflection of X-rays by crystals. Proc. R. Soc. Lond. A 88 (605), 428–438.Google Scholar
Dalrymple, R. A. & Kirby, J. T. 1986 Water waves over ripples. J. Waterw. Port Coast. Ocean Engng Div. ASCE 112, 309–319.CrossRefGoogle Scholar
Davies, A. G. 1982 The reflection of the wave energy by undulations of the seabed. Dyn. Atmos. Oceans 6, 207–232.CrossRefGoogle Scholar
Davies, A. G., Guazzelli, E. & Belzons, M. 1989 The propagation of long waves over an undulating bed. Phys. Fluids A 1, 1331–1340.CrossRefGoogle Scholar
Davies, A. G. & Heathershaw, A. D. 1984 Surface wave propagation over sinusoidally varying topography. J. Fluid Mech. 144, 419–443.CrossRefGoogle Scholar
Guazzelli, E., Rey, V. & Belzons, M. 1992 Higher-order Bragg reflection of gravity surface waves by periodic beds. J. Fluid Mech. 245, 301–317.CrossRefGoogle Scholar
Howard, L. N. & Yu, J. 2007 Normal modes of a rectangular tank with corrugated bottom. J. Fluid Mech. 593, 209–234.CrossRefGoogle Scholar
Kirby, J. T. 1986 A general wave equation for waves over rippled beds. J. Fluid Mech. 162, 171–186.CrossRefGoogle Scholar
Kirby, J. T. 1989 Propagation of surface waves over an undulating bed. Phys. Fluids A 1, 1898–1899.CrossRefGoogle Scholar
Liu, P. L.-F. 1987 Resonant reflection of water waves in a long channel with corrugated boundaries. J. Fluid Mech. 179, 371–381.CrossRefGoogle Scholar
Liu, Y. & Yue, D. K. P. 1998 On generalized Bragg scattering of surface waves by bottom ripples. J. Fluid Mech. 356, 297–326.Google Scholar
Mathieu, É. 1868 Mémoire sur le mouvement vibratoire d'une membrane de forme elliptique. J. Math. Pure Appl. 13, 137–203.Google Scholar
Mei, C. C. 1985 Resonant reflection of surface waves by periodic sandbars. J. Fluid Mech. 152, 315–337.CrossRefGoogle Scholar
Mei, C. C., Hara, T. & Naciri, M. 1988 Note on Bragg scattering of water waves by parallel bars on the seabed. J. Fluid Mech. 186, 147–162.CrossRefGoogle Scholar
Phillips, O. M. 1960 On the dynamics of unsteady gravity waves of finite amplitude. Part 1. J. Fluid Mech. 9, 193–217.CrossRefGoogle Scholar
Phillips, O. M. 1977 The Dynamics of the Upper Ocean. Cambridge University Press.Google Scholar
Rey, V., Guazzelli, E. & Mei, C. C. 1996 Resonant reflection of surface gravity waves by one-dimensional doubly sinusoidal beds. Phys. Fluids 8, 1525–1530.CrossRefGoogle Scholar
Yu, J. & Mei, C. C. 2000 a Do longshore bars shelter the shore? J. Fluid Mech. 404, 251–270.CrossRefGoogle Scholar
Yu, J. & Mei, C. C. 2000 b Formation of sand bars under surface waves. J. Fluid Mech. 416, 315–348.CrossRefGoogle Scholar