July  2012, 11(4): 1497-1522. doi: 10.3934/cpaa.2012.11.1497

A selection of nonlinear problems in water waves, analysed by perturbation-parameter techniques

1. 

School of Mathematics and Statistics, Newcastle University, Newcastle upon Tyne NE1 7RU, United Kingdom

Received  June 2011 Revised  September 2011 Published  January 2012

The methods of analysis based on asymptotic expansions, with a small parameter, are briefly outlined. These techniques are then applied to three examples in the theory of water waves, the aim being to demonstrate the effectiveness of this approach. Throughout, we relate this procedure to more rigorous methods.
The classical problem of water waves is formulated, and then various parameter choices are incorporated. The first problem examines the properties of small-amplitude, periodic waves over constant vorticity and, invoking the ideas of breakdown, scaling and matching, the possibility of stagnation is investigated. In the second case, a Camassa-Holm equation is derived; this is found to be relevant only for the horizontal velocity component in the flow at a specific depth. However, this special, integrable equation requires the retention of a number of terms of different asymptotic order--which is the least satisfactory way to use these methods. The final example, which shows how some new aspects of a familiar problem can be obtained by these methods, develops an asymptotic description of edge waves. This leads to a single equation that captures all aspects of the run-up pattern at a shoreline.
Citation: R. S. Johnson. A selection of nonlinear problems in water waves, analysed by perturbation-parameter techniques. Communications on Pure and Applied Analysis, 2012, 11 (4) : 1497-1522. doi: 10.3934/cpaa.2012.11.1497
References:
[1]

T. B. Benjamin, J. L. Bona and J. J. Mahoney, Model equations for long waves in nonlinear dispersive systems, Philos. Trans. R. Soc. Lond. A, 227 (1972), 47-78.

[2]

G. R. Burton and J. F. Toland, Surface waves on steady perfect-fluid flows with vorticity, Comm. Pure Appl. Math., 64 (2011), 975-1007.

[3]

A. W. Bush, "Perturbation Methods for Engineers and Scientists," CRC Press, Boca Raton, USA, 1992.

[4]

R. Camassa and D. Holm, An integrable shallow water equation with peaked solitons, Phys. Rev. Lett., 71 (1993), 1661-1664.

[5]

A. Constantin, On the scattering problem for the Camassa-Holm equation, Proc. R. Soc. Lond., A457 (2001), 953-970.

[6]

A. Constantin, Edge waves along a sloping beach, J. Phys. A Maths. Gen., 34 (2001), 9723-9731.

[7]

A. Constantin, M. Ehrnstrom and E. Wahlen, Symmetry of steady periodic waves with vorticity, Duke Math. J., 140 (2007), 591-603.

[8]

A. Constantin and J. Escher, Symmetry of steady deep-water waves with vorticity, Eur. J. Appl. Math., 15 (2004), 755-768.

[9]

A. Constantin and J. Escher, Analyticity of periodic traveling free surface water waves with vorticity, Ann. of Math., 173 (2011), 559-568.

[10]

A. Constantin, V. S. Gerdokov and R. I. Ivanov, Inverse scattering transform for the Camassa-Holm equation, Inverse Probl., 22 (2006), 2197-2207.

[11]

A. Constantin and D. Lannes, The hydrodynamical relevance of the Camassa-Holm and Degasperis-Procesi equations, Arch. Rat. Mech. Anal., 192 (2009), 165-186.

[12]

A. Constantin and W. Strauss, Exact periodic travelling water waves with vorticity, C. R. Math. Acad. Sci. Paris, 335 (2002), 797-800.

[13]

A. Constantin and W. Strauss, Exact steady periodic water waves with vorticity, Comm. Pure Appl. Math., 57 (2004), 481-527.

[14]

A. Constantin and W. Strauss, Rotational steady water waves near stagnation, Phil. Trans. Roy. Soc., A365 (2007), 2227-2239.

[15]

A. Constantin and W. Strauss, Periodic traveling gravity water waves with discontinuous vorticity, Arch. Rat. Mech. Anal., 202 (2011), 133-175.

[16]

A. Constantin and E. Varvaruca, Steady periodic water waves with constant vorticity: regularity and local bifurcation, Arch. Rat. Mech. Anal., 199 (2011), 33-67.

[17]

E. T. Copson, "Asymptotic Expansions," Cambridge University Press, Cambridge, 1967.

[18]

M.-L. Dubreil-Jacotin, Sur la détermination rigoureuse des ondes permanentes périodiques d'ampleur finie, J. Math. Pures Appl., 13 (1934), 217-291.

[19]

M. Ehrnstrom and G. Villari, Linear water waves with vorticity: rotational features and particle paths, J. Differential Equations, 244 (2008), 1888-1909.

[20]

A. S. Fokas and B. Fuchssteiner, Symplectic structures, their B?cklund transformation and hereditary symmetries, Physica D, 4 (1981), 821-831.

[21]

W. B. Ford, "Divergent Series, Summability and Asymptotics," Chelsea, New York, 1960.

[22]

N. C. Freeman and R. S. Johnson, Shallow water waves on shear flows, J. Fluid Mech., 42 (1970), 401-409.

[23]

F. Gerstner, Theorie der Wellen. Abh. d. k. böhm. Ges. d. Wiss, (1802) [Also see Ann. Phys. Lpz. 2, 412 (1809) and Lamb (1932). Art. 251.]

[24]

G. H. Hardy, "Divergent Series," Clarendon, Oxford, 1949.

[25]

E. J. Hinch, "Perturbation Methods," Cambridge University Press, Cambridge, 1991.

[26]

M. H. Holmes, "Introduction to Perturbation Methods," Springer-Verlag, New York, 1995.

[27]

R. S. Johnson, On the development of a solitary wave over an uneven bottom, Proc. Camb. Phil. Soc., 73 (1973), 183-203.

[28]

R. S. Johnson, "A Modern Introduction to the Mathematical Theory of Water Waves," CUP, Cambridge, 1997.

[29]

R. S. Johnson, Camassa-Holm, Korteweg-de Vries and related models for water waves, J. Fluid Mech., 457 (2002), 63-82.

[30]

R. S. Johnson, The Camassa-Holm equation for water waves moving over a shear flow, Fluid Dyn. Res., 33 (2003), 97-111.

[31]

R. S. Johnson, "Singular Perturbation Theory," Springer, New York, 2004.

[32]

R. S. Johnson, Some contributions to the theory of edge waves, J. Fluid Mech., 524 (2005), 81-97.

[33]

R. S. Johnson, Edge waves: theories past and present, Phil. Trans. R. Soc. Lond., A365 (2007), 2359-2376.

[34]

R. S. Johnson, Water waves near a shoreline in a flow with vorticity: two classical examples, J. Nonlinear Math. Phys., 15 (2008), 133-156.

[35]

R. S. Johnson, Periodic waves over constant vorticity: some asymptotic results generated by parameter expansions, Wave Motion, 46 (2009), 339-349.

[36]

J. Kevorkian and J. D. Cole, "Perturbation Methods in Applied Mathematics," Springer-Verlag, New York, 1981.

[37]

J. Kevorkian and J.D. Cole, "Multiple Scale and Singular Perturbation Methods," Springer-Verlag, New York, 1996.

[38]

J. Ko and W. Strauss, Large-amplitude steady rotational water waves, Eur. J. Mech. B Fluids, 27 (2008), 96-109.

[39]

J. Ko and W. Strauss, Effect of vorticity on steady water waves, J. Fluid Mech., 608 (2008), 197-215.

[40]

D. J. Korteweg and G. de Vries, On the change of form of long waves advancing in a rectangular channel and on anew type of long stationary waves, Philos. Mag., 39 (1895), 422-443.

[41]

H. Lamb, "Hydrodynamics," (6th edition), CUP, Cambridge, 1932.

[42]

E. Mollo-Christensen, Allowable discontinuities in a Gerstner wave, Phys. Fluids, 25 (1982), 586-587.

[43]

J. D. Murray, "Asymptotic Analysis," Clarendon, Oxford, 1974.

[44]

D. H. Peregrine, Calculations of the development of an undular bore, J. Fluid Mech., 25 (1966), 321-330.

[45]

W. J. M. Rankine, On the exact form of waves near the surface of deep water, Phil. Trans. R. Soc. Lond., 153 (1863), 127-138.

[46]

D. R. Smith, "Singular-perturbation Theory: An Introduction with Applications," Cambridge University Press, Cambridge, 1985.

[47]

G. G. Stokes, Report on recent researches in hydrodynamics, Rep. 16th Brit. Assoc. Adv. Sci. (1846) 1-20. [See also Papers, Vol.1, 157-187. Cambridge University Press, 1880.]

[48]

M. D. van Dyke, "Perturbation Methods in Fluid Mechanics," Academic Press, New York, 1964. (Also The Parabolic Press, Stanford, 1975).

[49]

E. Wahlén, Steady water waves with a critical layer, J. Differential Equations, 246 (2009), 2468-2483.

[50]

C.-S. Yih, Note on edge waves in a stratified fluid, J. Fluid Mech., 24 (1966), 765-767.

show all references

References:
[1]

T. B. Benjamin, J. L. Bona and J. J. Mahoney, Model equations for long waves in nonlinear dispersive systems, Philos. Trans. R. Soc. Lond. A, 227 (1972), 47-78.

[2]

G. R. Burton and J. F. Toland, Surface waves on steady perfect-fluid flows with vorticity, Comm. Pure Appl. Math., 64 (2011), 975-1007.

[3]

A. W. Bush, "Perturbation Methods for Engineers and Scientists," CRC Press, Boca Raton, USA, 1992.

[4]

R. Camassa and D. Holm, An integrable shallow water equation with peaked solitons, Phys. Rev. Lett., 71 (1993), 1661-1664.

[5]

A. Constantin, On the scattering problem for the Camassa-Holm equation, Proc. R. Soc. Lond., A457 (2001), 953-970.

[6]

A. Constantin, Edge waves along a sloping beach, J. Phys. A Maths. Gen., 34 (2001), 9723-9731.

[7]

A. Constantin, M. Ehrnstrom and E. Wahlen, Symmetry of steady periodic waves with vorticity, Duke Math. J., 140 (2007), 591-603.

[8]

A. Constantin and J. Escher, Symmetry of steady deep-water waves with vorticity, Eur. J. Appl. Math., 15 (2004), 755-768.

[9]

A. Constantin and J. Escher, Analyticity of periodic traveling free surface water waves with vorticity, Ann. of Math., 173 (2011), 559-568.

[10]

A. Constantin, V. S. Gerdokov and R. I. Ivanov, Inverse scattering transform for the Camassa-Holm equation, Inverse Probl., 22 (2006), 2197-2207.

[11]

A. Constantin and D. Lannes, The hydrodynamical relevance of the Camassa-Holm and Degasperis-Procesi equations, Arch. Rat. Mech. Anal., 192 (2009), 165-186.

[12]

A. Constantin and W. Strauss, Exact periodic travelling water waves with vorticity, C. R. Math. Acad. Sci. Paris, 335 (2002), 797-800.

[13]

A. Constantin and W. Strauss, Exact steady periodic water waves with vorticity, Comm. Pure Appl. Math., 57 (2004), 481-527.

[14]

A. Constantin and W. Strauss, Rotational steady water waves near stagnation, Phil. Trans. Roy. Soc., A365 (2007), 2227-2239.

[15]

A. Constantin and W. Strauss, Periodic traveling gravity water waves with discontinuous vorticity, Arch. Rat. Mech. Anal., 202 (2011), 133-175.

[16]

A. Constantin and E. Varvaruca, Steady periodic water waves with constant vorticity: regularity and local bifurcation, Arch. Rat. Mech. Anal., 199 (2011), 33-67.

[17]

E. T. Copson, "Asymptotic Expansions," Cambridge University Press, Cambridge, 1967.

[18]

M.-L. Dubreil-Jacotin, Sur la détermination rigoureuse des ondes permanentes périodiques d'ampleur finie, J. Math. Pures Appl., 13 (1934), 217-291.

[19]

M. Ehrnstrom and G. Villari, Linear water waves with vorticity: rotational features and particle paths, J. Differential Equations, 244 (2008), 1888-1909.

[20]

A. S. Fokas and B. Fuchssteiner, Symplectic structures, their B?cklund transformation and hereditary symmetries, Physica D, 4 (1981), 821-831.

[21]

W. B. Ford, "Divergent Series, Summability and Asymptotics," Chelsea, New York, 1960.

[22]

N. C. Freeman and R. S. Johnson, Shallow water waves on shear flows, J. Fluid Mech., 42 (1970), 401-409.

[23]

F. Gerstner, Theorie der Wellen. Abh. d. k. böhm. Ges. d. Wiss, (1802) [Also see Ann. Phys. Lpz. 2, 412 (1809) and Lamb (1932). Art. 251.]

[24]

G. H. Hardy, "Divergent Series," Clarendon, Oxford, 1949.

[25]

E. J. Hinch, "Perturbation Methods," Cambridge University Press, Cambridge, 1991.

[26]

M. H. Holmes, "Introduction to Perturbation Methods," Springer-Verlag, New York, 1995.

[27]

R. S. Johnson, On the development of a solitary wave over an uneven bottom, Proc. Camb. Phil. Soc., 73 (1973), 183-203.

[28]

R. S. Johnson, "A Modern Introduction to the Mathematical Theory of Water Waves," CUP, Cambridge, 1997.

[29]

R. S. Johnson, Camassa-Holm, Korteweg-de Vries and related models for water waves, J. Fluid Mech., 457 (2002), 63-82.

[30]

R. S. Johnson, The Camassa-Holm equation for water waves moving over a shear flow, Fluid Dyn. Res., 33 (2003), 97-111.

[31]

R. S. Johnson, "Singular Perturbation Theory," Springer, New York, 2004.

[32]

R. S. Johnson, Some contributions to the theory of edge waves, J. Fluid Mech., 524 (2005), 81-97.

[33]

R. S. Johnson, Edge waves: theories past and present, Phil. Trans. R. Soc. Lond., A365 (2007), 2359-2376.

[34]

R. S. Johnson, Water waves near a shoreline in a flow with vorticity: two classical examples, J. Nonlinear Math. Phys., 15 (2008), 133-156.

[35]

R. S. Johnson, Periodic waves over constant vorticity: some asymptotic results generated by parameter expansions, Wave Motion, 46 (2009), 339-349.

[36]

J. Kevorkian and J. D. Cole, "Perturbation Methods in Applied Mathematics," Springer-Verlag, New York, 1981.

[37]

J. Kevorkian and J.D. Cole, "Multiple Scale and Singular Perturbation Methods," Springer-Verlag, New York, 1996.

[38]

J. Ko and W. Strauss, Large-amplitude steady rotational water waves, Eur. J. Mech. B Fluids, 27 (2008), 96-109.

[39]

J. Ko and W. Strauss, Effect of vorticity on steady water waves, J. Fluid Mech., 608 (2008), 197-215.

[40]

D. J. Korteweg and G. de Vries, On the change of form of long waves advancing in a rectangular channel and on anew type of long stationary waves, Philos. Mag., 39 (1895), 422-443.

[41]

H. Lamb, "Hydrodynamics," (6th edition), CUP, Cambridge, 1932.

[42]

E. Mollo-Christensen, Allowable discontinuities in a Gerstner wave, Phys. Fluids, 25 (1982), 586-587.

[43]

J. D. Murray, "Asymptotic Analysis," Clarendon, Oxford, 1974.

[44]

D. H. Peregrine, Calculations of the development of an undular bore, J. Fluid Mech., 25 (1966), 321-330.

[45]

W. J. M. Rankine, On the exact form of waves near the surface of deep water, Phil. Trans. R. Soc. Lond., 153 (1863), 127-138.

[46]

D. R. Smith, "Singular-perturbation Theory: An Introduction with Applications," Cambridge University Press, Cambridge, 1985.

[47]

G. G. Stokes, Report on recent researches in hydrodynamics, Rep. 16th Brit. Assoc. Adv. Sci. (1846) 1-20. [See also Papers, Vol.1, 157-187. Cambridge University Press, 1880.]

[48]

M. D. van Dyke, "Perturbation Methods in Fluid Mechanics," Academic Press, New York, 1964. (Also The Parabolic Press, Stanford, 1975).

[49]

E. Wahlén, Steady water waves with a critical layer, J. Differential Equations, 246 (2009), 2468-2483.

[50]

C.-S. Yih, Note on edge waves in a stratified fluid, J. Fluid Mech., 24 (1966), 765-767.

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