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Transformation of Narrowband Wavetrains of Surface GravityWaves Passing over a Bottom Step

Published online by Cambridge University Press:  17 July 2014

A. R. Giniyatullin ,
A. A. Kurkin ,
S. V. Semin  and
Y. A. Stepanyants
A. R. Giniyatullin
Affiliation:
Nizhny Novgorod State Technical University n.a. R.E. Alekseev, Nizhny Novgorod, Russia
A. A. Kurkin
Affiliation:
Nizhny Novgorod State Technical University n.a. R.E. Alekseev, Nizhny Novgorod, Russia
S. V. Semin
Affiliation:
Nizhny Novgorod State Technical University n.a. R.E. Alekseev, Nizhny Novgorod, Russia University of Southern Queensland, Toowoomba, Australia
Y. A. Stepanyants*
Affiliation:
Nizhny Novgorod State Technical University n.a. R.E. Alekseev, Nizhny Novgorod, Russia University of Southern Queensland, Toowoomba, Australia
*
⋆ The authors adhere the principle of alphabetical order of the names.Corresponding author. E-mail: [email protected]
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Abstract

The problem of transformation of quasimonochromatic wavetrains of surface gravity waveswith narrow spatial-temporal spectra on the bottom shelf is considered in the linearapproximation. By means of numerical modeling, the transmission and reflectioncoefficients are determined as functions of the depth ratio and wave number (frequency) ofan incident wave. The approximation formulae are proposed for the coefficients of wavetransformation. The characteristic features of these formulae are analyzed. It is shownthat the numerical results agree quite satisfactorily with the proposed approximationformulae.

Keywords

wavetrainwave transformationnumerical modelingquasimonochromatic wavereflected and transmitted wavesbottom step
Type
Research Article
Information
Mathematical Modelling of Natural Phenomena , Volume 9 , Issue 5: Spectral problems , 2014 , pp. 73 - 82
Copyright
© EDP Sciences, 2014

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References

J. Adcroft et al. MITgcm User Manual. MIT Department of EAPS, Cambridge, MA, 2008.
Bartholomeusz, L.F.. The reflection of long waves at a step. Proc. Camb. Philos. Soc., 54 (1958), 106–118. CrossRefGoogle Scholar
G.P. Germain. Coefficients de réflexion et de transmission en eau peu profonde. Instytut Budownictwa Wodnego, Gdansk, Rozprawy Hydrotechniczne, Rep. no. 46, 5–13 (in French).
Gorshkov, K.A., Ostrovsky, L.A., Papko, V.V., Pelinovsky, E.N.. Electromodeling of finite amplitude water waves. Bull. Roy. Soc. New Zealand, 15 (1976), 123–131. Google Scholar
Grimshaw, R., Pelinovsky, E., Talipova, T.. Fission of a weakly nonlinear interfacial solitary wave at a step. Geophys. Astrophys. Fluid Dyn., 102 (2008), no. 2, 179–194. CrossRefGoogle Scholar
H. Lamb. Hydrodynamics. 6-th ed., Cambridge Univ. Press, Cambridge, 1932.
Losada, M.A., Vidal, C., Medina, R.. Experimental study of the evolution of a solitary wave at the abrupt junction. J. Geophys. Res., 94 (1989), no. C10, 14, 557–14, 566. CrossRefGoogle Scholar
Maderich, V., Talipova, T., Grimshaw, R., Pelinovsky, E., Choi, B. H., Brovchenko, I., Terletska, K., Kim, D.C.. The transformation of an interfacial solitary wave of elevation at a bottom step. Nonlin. Processes Geophys., 16 (2009), 33–42. CrossRefGoogle Scholar
Maderich, V., Talipova, T., Grimshaw, R., Terletska, K., Brovchenko, I., Pelinovsky, E., Choi, B.H.. Interaction of a large amplitude interfacial solitary wave of depression with a bottom step. Phys. Fluids, 22 (2010), 076602. CrossRefGoogle Scholar
V.A. Makarov, A.B. Menzin. Electrical Analog modeling in Oceanology, Leningrad, Gidrometeoizdat, 1976 (in Russian).
Marshal, J., Hill, C., Perelman, L., Adcroft, A.. Hydrostatic, quasi-hydrostatic, and nonhydrostatic ocean modeling. J. Geophys. Res., 102 no. C3, (1997), 5,733–5,752. CrossRefGoogle Scholar
Marshal, J., Adcroft, A., Hill, C., Perelman, L., Heisey, C.. A finite-volume, incompressible Navier–Stokes model for studies of the ocean on parallel computers. J. Geophys. Res., 102 (1997), no. C3, 5,753–5,766. CrossRefGoogle Scholar
Marshal, J.S., Naghdi, P.M.. Wave reflection and transmission by steps and rectangular obstacles in channels of finite depth. Theoret. Comput. Fluid Dynamics, 1 (1990), 287–301. CrossRefGoogle Scholar
Massel, S.R.. Harmonic generation by waves propagating over a submerged step. Coastal Eng., 7 (1983), 357–380. CrossRefGoogle Scholar
S.R. Massel. Hydrodynamics of the coastal zone, Elsevier, Amsterdam, 1989.
Miles, J.W.. Surface-wave scattering matrix for a shelf. J. Fluid Mech., 28 (1967), pt. 4, 755–767. CrossRefGoogle Scholar
Mirchina, N., Pelinovsky, E.. Nonlinear transformation of long waves at a bottom step. J. Korean Soc. Coastal Ocean Eng., 4 (1992), no. 3, 161–167. Google Scholar
Newman, J.N.. Propagation of water waves over an infinite step. J. Fluid Mech. 23 (1965), pt. 2, 339–415. CrossRefGoogle Scholar
E.N. Pelinovsky. On the solitary wave transformation on a shelf with the horizontal bottom. In: Theoretical and Experimental Studies of Tsunami, Eds. S.L. Solov’yov, A.I. Ivashchenko, and V.M. Kaistrenko, Moscow, Nauka, (1977), 61–63 (in Russian).
E.N. Pelinovsky. Hydrodynamics of Tsunami Waves, Nizhny Novogrod, IAP RAS, 1996 (in Russian).
Pelinovsky, E., Choi, B.H., Talipova, T., Wood, S.B., Kim, D.Ch.. Solitary wave transformation on the underwater step: Asymptotic theory and numerical experiments. Appl. Math. and Comp., 217 (2010), no. 1, 704–1,718. CrossRefGoogle Scholar
Seabra-Santos, F.J., Renouard, D.P., Temperville, A.M.. Numerical and experimental study of the transformation of a solitary wave over a shelf or isolated obstacle. J. Fluid Mech., 176 (1987), 17–134. CrossRefGoogle Scholar
L.N. Sretensky. The Theory of Wave Motions of a Liquid, Moscow, Nauka, 1977 (in Russian).
Stepanyants, Y.A.. On soliton propagation in the inhomogeneous long line. Radiotekhnika i Elektronika, 22, (1977), no. 5, 995–1,002 (in Russian).Google Scholar