Transformation of Nonlinear Surface Gravity Waves under Shallow-Water Conditions

ABSTRACT

This article describes transformation of nonlinear surface gravity waves under shallow-water conditions with the aid of the suggested semigraphical method. There are given profiles of surface gravity waves on the crests steepening stages, their leading edges steepening. There are discussed the spectral component influence on the transformation of surface wave profile.

This article describes transformation of nonlinear surface gravity waves under shallow-water conditions with the aid of the suggested semigraphical method. There are given profiles of surface gravity waves on the crests steepening stages, their leading edges steepening. There are discussed the spectral component influence on the transformation of surface wave profile.

KEYWORDS

Nonlinear Surface Gravity Waves, Shallow-Water, Semigraphical Method, Transformation of Surface Wave Profile

Nonlinear Surface Gravity Waves, Shallow-Water, Semigraphical Method, Transformation of Surface Wave Profile

Cite this paper

nullI. Abbasov, "Transformation of Nonlinear Surface Gravity Waves under Shallow-Water Conditions,"*Applied Mathematics*, Vol. 1 No. 4, 2010, pp. 260-264. doi: 10.4236/am.2010.14032.

nullI. Abbasov, "Transformation of Nonlinear Surface Gravity Waves under Shallow-Water Conditions,"

References

[1] D. H. Peregrine, “Long Waves on a Beach,” Journal of Fluid Mechanics, Vol. 27, No. 0404, 1967, pp. 815-827.

[2] N. Е. Voltsinger, К. А. Klevanniy and Е. Н. Pelinovskiy, “Long-Wave Dynamics of Coastal Zone,” Gidrometeoizdat, 1989, p. 272.

[3] P. I. Naumkin and I. А. Shishmarev, “On the Existence and Breaking the Waves Described by the Whitham Equation,” Soviet Physics-Doklady, Vol. 384, No. 31, 1986, pp. 90-95.

[4] V. G. Galenin and V. V. Kuznetsov, “Simulation of Wave Transformation in Coastal Zone,” Water Resources, Vol. 7, No. 1, 1980, pp. 156-165.

[5] U. Kanoglu, “Nonlinear Evolution and Run Up-Run Down of Long Waves over a Sloping Beach,” Journal of Fluid Mechanics, Vol. 513, 2004, pp. 363-372.

[6] I. I. Didenkulova, N. Zaibo, А. А. Kurkin and Е. N. Peli-novskiy, “Steepness and Spectrum of Nonlinear Deformed Wave under Water Conditions,” Izvestiya RAN Atmos-pheric and Oceanic Physics, Vol. 42, No. 6, 2006, pp. 839-842.

[7] N. А. Kudryashov, Yu. I. Sytsko and S. А. Chesnokov, “Mathematic Simulation of Gravity Waves in the Ocean in ‘Shallow Water’ Approximation,” Pisma v ZhETF, Vol. 77, No. 10, 2003, p. 649.

[8] А. А. Litvinenko and G. А. Khabakhpashev, “Computer Simulation of Nonlinear Considerably Long Two-dimensional Waves under Water Conditions in Basins with Sloping Floor,” Computer technologies, Vol. 4, No. 3, 1999, pp. 95-105.

[9] S. Yu. Kuznetsov and Ya. V. Saprykina, “Experimental Researches of Wave Group Evolution in the Coastal Sea Zone,” Oceanology, Vol. 42, No. 3, 2002, p. 356-363.

[10] I. B. Abbasov, “Study and Simulation of Nonlinear Surface Waves under Shallow-Water Conditions,” Izvestiya RAN. Atmospheric and Oceanic Physics, Vol. 39, No. 4, 2003, pp. 506-511.

[11] H. Lamb, “Hydrodynamics,” Dover, New York, 1930, p. 524.

[12] G. Whitham, “Linear and Nonlinear Waves,” Wiley, New York, 1974, p. 622.

[13] L. М. Brekhovskikh and V. V. Goncharov, “Introduction to the Mechanics of Continuous Media,” Nauka, 1982, p. 325.

[14] М. B. Vinogradova, О. V. Rudenko and А. P. Sukhorukov, “Wave Theory,” Nauka, 1979, p. 383.

[15] А. K. Monin and V. P. Krasitskiy, “Phenomena at the Ocean Surface,” L. Gidrometeoizdat, 1985, p. 375.

[16] I. О. Leontiev, “Coastal Dynamics: Waves, Streams, Burden Streams,” M.: GEOS, 2001, p. 272.

[17] R. E. Flick, R. T. Guza and D. L. Inman, “Elevation and Velocity Measurements of Laboratory Shoaling Waves,” Journal of Geophysical Research, Vol. 86, No. 5, 1981, pp. 4149-4160.

[18] M. S. Lonquet-Higgins, “Grest Instabilities of Gravity Waves, Part 1,” Journal of Fluid Mechanics, Vol. 258, No. 1, 1994, pp. 115-129.

[19] V. Zaharov, “Weakly Non-Linear Waves on the Surface of an Ideal Finite Depth Fluid,” American Mathematical Society Transactions, Series 2, Vol. 182, 1998, pp. 167- 197.

[20] S. А. Gabov, “Introduction to the Theory on Nonlinear Waves,” Moscow State Unversity, 1988, p.176.

[21] Y. Goda and K. Morinobu, “Breaking Wave Heights on Horizontal Bed Affected by Approach Slope,” Coastal Engineering Journal, Vol. 40, No. 4, 1998, pp. 307-326.

[22] Ya. V. Saprykina, S. Yu. Kuznetsov, Zh. Tcherneva and N. Andreeva, “Space-Time Unsteadiness of the Amplitude-Phase Structure of Storm Waves in the Coastal Sea Zone,” Oceanology, Vol. 49, No. 2, 2009, pp. 198-208.

[23] K. Kawasaki, “Numerical Simulation of Breaking and Post-Breaking Wave Deformation Process around a Submerged Breakwater,” Coastal Engineering Journal, Vol. 41, No. 3-4, 1999, pp. 201-223.

[1] D. H. Peregrine, “Long Waves on a Beach,” Journal of Fluid Mechanics, Vol. 27, No. 0404, 1967, pp. 815-827.

[2] N. Е. Voltsinger, К. А. Klevanniy and Е. Н. Pelinovskiy, “Long-Wave Dynamics of Coastal Zone,” Gidrometeoizdat, 1989, p. 272.

[3] P. I. Naumkin and I. А. Shishmarev, “On the Existence and Breaking the Waves Described by the Whitham Equation,” Soviet Physics-Doklady, Vol. 384, No. 31, 1986, pp. 90-95.

[4] V. G. Galenin and V. V. Kuznetsov, “Simulation of Wave Transformation in Coastal Zone,” Water Resources, Vol. 7, No. 1, 1980, pp. 156-165.

[5] U. Kanoglu, “Nonlinear Evolution and Run Up-Run Down of Long Waves over a Sloping Beach,” Journal of Fluid Mechanics, Vol. 513, 2004, pp. 363-372.

[6] I. I. Didenkulova, N. Zaibo, А. А. Kurkin and Е. N. Peli-novskiy, “Steepness and Spectrum of Nonlinear Deformed Wave under Water Conditions,” Izvestiya RAN Atmos-pheric and Oceanic Physics, Vol. 42, No. 6, 2006, pp. 839-842.

[7] N. А. Kudryashov, Yu. I. Sytsko and S. А. Chesnokov, “Mathematic Simulation of Gravity Waves in the Ocean in ‘Shallow Water’ Approximation,” Pisma v ZhETF, Vol. 77, No. 10, 2003, p. 649.

[8] А. А. Litvinenko and G. А. Khabakhpashev, “Computer Simulation of Nonlinear Considerably Long Two-dimensional Waves under Water Conditions in Basins with Sloping Floor,” Computer technologies, Vol. 4, No. 3, 1999, pp. 95-105.

[9] S. Yu. Kuznetsov and Ya. V. Saprykina, “Experimental Researches of Wave Group Evolution in the Coastal Sea Zone,” Oceanology, Vol. 42, No. 3, 2002, p. 356-363.

[10] I. B. Abbasov, “Study and Simulation of Nonlinear Surface Waves under Shallow-Water Conditions,” Izvestiya RAN. Atmospheric and Oceanic Physics, Vol. 39, No. 4, 2003, pp. 506-511.

[11] H. Lamb, “Hydrodynamics,” Dover, New York, 1930, p. 524.

[12] G. Whitham, “Linear and Nonlinear Waves,” Wiley, New York, 1974, p. 622.

[13] L. М. Brekhovskikh and V. V. Goncharov, “Introduction to the Mechanics of Continuous Media,” Nauka, 1982, p. 325.

[14] М. B. Vinogradova, О. V. Rudenko and А. P. Sukhorukov, “Wave Theory,” Nauka, 1979, p. 383.

[15] А. K. Monin and V. P. Krasitskiy, “Phenomena at the Ocean Surface,” L. Gidrometeoizdat, 1985, p. 375.

[16] I. О. Leontiev, “Coastal Dynamics: Waves, Streams, Burden Streams,” M.: GEOS, 2001, p. 272.

[17] R. E. Flick, R. T. Guza and D. L. Inman, “Elevation and Velocity Measurements of Laboratory Shoaling Waves,” Journal of Geophysical Research, Vol. 86, No. 5, 1981, pp. 4149-4160.

[18] M. S. Lonquet-Higgins, “Grest Instabilities of Gravity Waves, Part 1,” Journal of Fluid Mechanics, Vol. 258, No. 1, 1994, pp. 115-129.

[19] V. Zaharov, “Weakly Non-Linear Waves on the Surface of an Ideal Finite Depth Fluid,” American Mathematical Society Transactions, Series 2, Vol. 182, 1998, pp. 167- 197.

[20] S. А. Gabov, “Introduction to the Theory on Nonlinear Waves,” Moscow State Unversity, 1988, p.176.

[21] Y. Goda and K. Morinobu, “Breaking Wave Heights on Horizontal Bed Affected by Approach Slope,” Coastal Engineering Journal, Vol. 40, No. 4, 1998, pp. 307-326.

[22] Ya. V. Saprykina, S. Yu. Kuznetsov, Zh. Tcherneva and N. Andreeva, “Space-Time Unsteadiness of the Amplitude-Phase Structure of Storm Waves in the Coastal Sea Zone,” Oceanology, Vol. 49, No. 2, 2009, pp. 198-208.

[23] K. Kawasaki, “Numerical Simulation of Breaking and Post-Breaking Wave Deformation Process around a Submerged Breakwater,” Coastal Engineering Journal, Vol. 41, No. 3-4, 1999, pp. 201-223.