Hamiltonian Formulation for Water Wave Equation

ABSTRACT

This paper concerns the development and application of the Hamiltonian function which is the sum of kinetic energy and potential energy of the system. Two dimensional water wave equations for irrotational, incompressible, inviscid fluid have been constructed in cartesian coordinates and also in cylindrical coordinates. Then Lagrangian function within a certain flow region is expanded under the assumption that the dispersion*μ* and the nonlinearity *ε* satisfied . Using Hamilton’s principle for water wave evolution Hamiltonian formulation is derived. It is obvious that the motion of the system is conservative. Then Hamilton’s canonical equation of motion is also derived.

This paper concerns the development and application of the Hamiltonian function which is the sum of kinetic energy and potential energy of the system. Two dimensional water wave equations for irrotational, incompressible, inviscid fluid have been constructed in cartesian coordinates and also in cylindrical coordinates. Then Lagrangian function within a certain flow region is expanded under the assumption that the dispersion

Cite this paper

S. Sultana and Z. Rahman, "Hamiltonian Formulation for Water Wave Equation,"*Open Journal of Fluid Dynamics*, Vol. 3 No. 2, 2013, pp. 75-81. doi: 10.4236/ojfd.2013.32010.

S. Sultana and Z. Rahman, "Hamiltonian Formulation for Water Wave Equation,"

References

[1] G. B. Whitham, “A General Approach to Linear and Non-Linear Dispersive Waves Using a Lagrangian,” Journal of Fluid Mechanics, Vol. 22, No. 2, 1965, pp. 273-283. doi:10.1017/S0022112065000745

[2] J. C. Luke, “A Variational Principle for a Fluid with a Free Surface,” Journal of Fluid Mechanics, Vol. 27, No. 2, 1967, pp. 395-397. doi:10.1017/S0022112067000412

[3] V. E. Zakharov, “Stability of Periodic Waves of Finite Amplitude on the Surface of a Deep Fluid,” Journal of Applied Mechanics and Technical Physics, Vol. 9, No. 2, 1968, pp. 190-194.

[4] A. C. Radder, “Hamiltonian Dynamics of Water Waves,” Advanced Series on Ocean Engineering, Vol. 4, 1999, pp. 21-59. doi:10.1142/9789812797551_0002

[5] R. Salmon, “Geophysical Fluid Dynamics,” Oxford University Press, Oxford, 1988.

[6] V. E. Zakharov and E. A. Kuznetsov, “Hamiltonian Formalism for Nonlinear Waves,” Physics Uspekhi, Vol. 40, No. 11, 1997, pp. 1087-1116. doi:10.1070/PU1997v040n11ABEH000304

[7] P. A. Madsen, H. R. Bingham and H. A. Schoffer, “Boussinesq-Type Formulations for Fully Non-Linear and Extremely Dispersive Water Waves: Derivation and Analysis,” Proceedings of the Royal Society of London, Vol. 459, No. 2033, 2003, pp. 1075-1104. doi:10.1098/rspa.2002.1067

[8] O. S. Madsen, S. Pahuja, H. Zhang and E. S. Chan, “A Diffusive Transport Mechanism for Fine Sediments,” Proceedings of the 28th International Conference on Coastal Engineering, Cardiff, 2003, pp. 741-753.

[9] G. B. Whitham, “Variational Methods and Applications to Water Waves,” Proceedings of the Royal Society A, Vol. 299, No. 1, 1967, pp. 6-25.

[10] J. W. Miles, “On Hamilton’s Principle for Surface Waves,” Journal of Fluid Mechanics, Vol. 83, No. 1, 1977, pp. 153-158. doi:10.1017/S0022112077001104

[11] D. M. Milder, “A Note Regarding ‘On Hamilton’s Principle for Surface Waves’,” Journal of Fluid Mechanics, Vol. 83, No. 1, 1977, pp. 159-161. doi:10.1017/S0022112077001116

[12] A. C. Radder, “An Explicit Hamiltonian Formulation of Surface Waves in Water of Finite Depth,” Journal of Fluid Mechanics, Vol. 237, 1992, pp. 435-455. doi:10.1017/S0022112092003483

[13] T. Y. Hou and P. Zhang, “Convergence of a Boundary Integral Method for 3-D Water Waves,” Discrete and Continuous Dynamical Systems, Series B, Vol. 2, No. 1, 2002, pp. 1-34. doi:10.3934/dcdsb.2002.2.1

[14] D. Ambrosi, “Hamiltonian Formulation for Surface Waves in a Layered Fluid,” Wave Motion, Vol. 31, No. 1, 2000, pp. 71-76. doi:10.1016/S0165-2125(99)00024-4

[15] Y. Lvov and E. G. Tabak, “A Hamiltonian Formulation for Long Internal Waves,” Physica D, Vol. 195, 2004, pp. 106-122. doi:10.1016/j.physd.2004.03.010

[16] Y. Hongli, S. Jinbao and Y. Liangui, “Water Wave Solutions Obtained by Variational Method,” Chinese Journal of Oceanology and Limnology, Vol. 24, No. 1, 2006, pp. 87-91. doi:10.1007/BF02842780

[17] J. J. Stoker, “Water Waves,” 1957.

[18] L. Debnath, “A Variational Principle for Nonlinear Water Waves,” Acta Mechanica, Vol. 72, No. 1-2, 1988, pp. 155-160. doi:10.1007/BF01176549

[19] L. Debnath, “On Initial Development of Axisymmetrio Waves in Fluids of Finite Depth,” Proceedings of the National Institute of Sciences of India, Vol. 85, 1969, pp. 567-585.

[20] C. R. Mondal, “Uniform Asymptotic Analysis of Shallow-Water Waves Due to a Periodic Surface Pressure,” Quarterly of Applied Mathematics, 1986, pp. 133-140.

[21] N. C. Mahanti, “Small-Amplitude Internal Waves Due to an Oscillatory Pressure,” Quarterly of Applied Mathematics, Vol. 37, 1997, pp. 92-97.

[22] T. B. Benjamin, “Instability of Periodic Wavetrains in Nonlinear Dispersive Systems,” Proceedings of the Royal Society of London Series A, Vol. 299, No. 1456, 1967, pp. 59-76. doi:10.1098/rspa.1967.0123

[1] G. B. Whitham, “A General Approach to Linear and Non-Linear Dispersive Waves Using a Lagrangian,” Journal of Fluid Mechanics, Vol. 22, No. 2, 1965, pp. 273-283. doi:10.1017/S0022112065000745

[2] J. C. Luke, “A Variational Principle for a Fluid with a Free Surface,” Journal of Fluid Mechanics, Vol. 27, No. 2, 1967, pp. 395-397. doi:10.1017/S0022112067000412

[3] V. E. Zakharov, “Stability of Periodic Waves of Finite Amplitude on the Surface of a Deep Fluid,” Journal of Applied Mechanics and Technical Physics, Vol. 9, No. 2, 1968, pp. 190-194.

[4] A. C. Radder, “Hamiltonian Dynamics of Water Waves,” Advanced Series on Ocean Engineering, Vol. 4, 1999, pp. 21-59. doi:10.1142/9789812797551_0002

[5] R. Salmon, “Geophysical Fluid Dynamics,” Oxford University Press, Oxford, 1988.

[6] V. E. Zakharov and E. A. Kuznetsov, “Hamiltonian Formalism for Nonlinear Waves,” Physics Uspekhi, Vol. 40, No. 11, 1997, pp. 1087-1116. doi:10.1070/PU1997v040n11ABEH000304

[7] P. A. Madsen, H. R. Bingham and H. A. Schoffer, “Boussinesq-Type Formulations for Fully Non-Linear and Extremely Dispersive Water Waves: Derivation and Analysis,” Proceedings of the Royal Society of London, Vol. 459, No. 2033, 2003, pp. 1075-1104. doi:10.1098/rspa.2002.1067

[8] O. S. Madsen, S. Pahuja, H. Zhang and E. S. Chan, “A Diffusive Transport Mechanism for Fine Sediments,” Proceedings of the 28th International Conference on Coastal Engineering, Cardiff, 2003, pp. 741-753.

[9] G. B. Whitham, “Variational Methods and Applications to Water Waves,” Proceedings of the Royal Society A, Vol. 299, No. 1, 1967, pp. 6-25.

[10] J. W. Miles, “On Hamilton’s Principle for Surface Waves,” Journal of Fluid Mechanics, Vol. 83, No. 1, 1977, pp. 153-158. doi:10.1017/S0022112077001104

[11] D. M. Milder, “A Note Regarding ‘On Hamilton’s Principle for Surface Waves’,” Journal of Fluid Mechanics, Vol. 83, No. 1, 1977, pp. 159-161. doi:10.1017/S0022112077001116

[12] A. C. Radder, “An Explicit Hamiltonian Formulation of Surface Waves in Water of Finite Depth,” Journal of Fluid Mechanics, Vol. 237, 1992, pp. 435-455. doi:10.1017/S0022112092003483

[13] T. Y. Hou and P. Zhang, “Convergence of a Boundary Integral Method for 3-D Water Waves,” Discrete and Continuous Dynamical Systems, Series B, Vol. 2, No. 1, 2002, pp. 1-34. doi:10.3934/dcdsb.2002.2.1

[14] D. Ambrosi, “Hamiltonian Formulation for Surface Waves in a Layered Fluid,” Wave Motion, Vol. 31, No. 1, 2000, pp. 71-76. doi:10.1016/S0165-2125(99)00024-4

[15] Y. Lvov and E. G. Tabak, “A Hamiltonian Formulation for Long Internal Waves,” Physica D, Vol. 195, 2004, pp. 106-122. doi:10.1016/j.physd.2004.03.010

[16] Y. Hongli, S. Jinbao and Y. Liangui, “Water Wave Solutions Obtained by Variational Method,” Chinese Journal of Oceanology and Limnology, Vol. 24, No. 1, 2006, pp. 87-91. doi:10.1007/BF02842780

[17] J. J. Stoker, “Water Waves,” 1957.

[18] L. Debnath, “A Variational Principle for Nonlinear Water Waves,” Acta Mechanica, Vol. 72, No. 1-2, 1988, pp. 155-160. doi:10.1007/BF01176549

[19] L. Debnath, “On Initial Development of Axisymmetrio Waves in Fluids of Finite Depth,” Proceedings of the National Institute of Sciences of India, Vol. 85, 1969, pp. 567-585.

[20] C. R. Mondal, “Uniform Asymptotic Analysis of Shallow-Water Waves Due to a Periodic Surface Pressure,” Quarterly of Applied Mathematics, 1986, pp. 133-140.

[21] N. C. Mahanti, “Small-Amplitude Internal Waves Due to an Oscillatory Pressure,” Quarterly of Applied Mathematics, Vol. 37, 1997, pp. 92-97.

[22] T. B. Benjamin, “Instability of Periodic Wavetrains in Nonlinear Dispersive Systems,” Proceedings of the Royal Society of London Series A, Vol. 299, No. 1456, 1967, pp. 59-76. doi:10.1098/rspa.1967.0123