Abstract and Applied Analysis

Nonlinear Hydroelastic Waves beneath a Floating Ice Sheet in a Fluid of Finite Depth

Ping Wang and Zunshui Cheng

Full-text: Open access


The nonlinear hydroelastic waves propagating beneath an infinite ice sheet floating on an inviscid fluid of finite depth are investigated analytically. The approximate series solutions for the velocity potential and the wave surface elevation are derived, respectively, by an analytic approximation technique named homotopy analysis method (HAM) and are presented for the second-order components. Also, homotopy squared residual technique is employed to guarantee the convergence of the series solutions. The present formulas, different from the perturbation solutions, are highly accurate and uniformly valid without assuming that these nonlinear partial differential equations (PDEs) have small parameters necessarily. It is noted that the effects of water depth, the ice sheet thickness, and Young’s modulus are analytically expressed in detail. We find that, in different water depths, the hydroelastic waves traveling beneath the thickest ice sheet always contain the largest wave energy. While with an increasing thickness of the sheet, the wave elevation tends to be smoothened at the crest and be sharpened at the trough. The larger Young’s modulus of the sheet also causes analogous effects. The results obtained show that the thickness and Young’s modulus of the floating ice sheet all greatly affect the wave energy and wave profile in different water depths.

Article information

Abstr. Appl. Anal., Volume 2013, Special Issue (2013), Article ID 108026, 13 pages.

First available in Project Euclid: 26 February 2014

Permanent link to this document

Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier


Wang, Ping; Cheng, Zunshui. Nonlinear Hydroelastic Waves beneath a Floating Ice Sheet in a Fluid of Finite Depth. Abstr. Appl. Anal. 2013, Special Issue (2013), Article ID 108026, 13 pages. doi:10.1155/2013/108026. https://projecteuclid.org/euclid.aaa/1393450001

Export citation


  • A. G. Greenhill, “Wave motion in hydrodynamics,” American Journal of Mathematics, vol. 9, no. 1, pp. 62–96, 1886.
  • V. A. Squire, J. P. Dugan, P. Wadhams, P. J. Rottier, and A. K. Liu, “Of ocean waves and sea ice,” Annual Review of Fluid Mechanics, vol. 27, pp. 115–168, 1995.
  • V. A. Squire, “Of ocean waves and sea-ice revisited,” Cold Regions Science and Technology, vol. 49, no. 2, pp. 110–133, 2007.
  • V. A. Squire, “Synergies between VLFS hydroelasticity and sea ice research,” International Journal of Offshore and Polar Engineering, vol. 18, no. 4, pp. 241–253, 2008.
  • T. Kakinuma, K. Yamashita, and K. Nakayama, “Surface and internal waves due to a moving load on a very large floating structure,” Journal of Applied Mathematics, vol. 2012, Article ID 830530, 14 pages, 2012.
  • F. Xu and D. Q. Lu, “Wave scattering by a thin elastic plate floating on a two-layer fluid,” International Journal of Engineering Science, vol. 48, no. 9, pp. 809–819, 2010.
  • L. K. Forbes, “Surface waves of large amplitude beneath an elastic sheet. Part 1. High-order series solution,” Journal of Fluid Mechanics, vol. 169, pp. 409–428, 1986.
  • L. K. Forbes, “Surface waves of large amplitude beneath an elastic sheet. Part 2. Galerkin solution,” Journal of Fluid Mechanics, vol. 188, pp. 491–508, 1988.
  • J.-M. Vanden-Broeck and E. I. Părău, “Two-dimensional generalized solitary waves and periodic waves under an ice sheet,” Philosophical Transactions of the Royal Society A, vol. 369, no. 1947, pp. 2957–2972, 2011.
  • S.-J. Liao, The proposed homotopy analysis technique for the solution of nonlinear problems [Ph. D. Dissertation], Shanghai Jiao Tong University, 1992.
  • S.-J. Liao, Beyond Perturbation: Introduction to the Homotopy Analysis Method, Modern Mechanics and Mathematics, Chapman and Hall/ CRC Press, 1st edition, 2003.
  • S.-J. Liao, Homotopy Analysis Method in Nonlinear Differential Equations, Springer & Higher Education Press, Heidelberg, Germany, 2003.
  • S.-J. Liao, “On the analytic solution of magnetohydrodynamic flows of non-Newtonian fluids over a stretching sheet,” Journal of Fluid Mechanics, vol. 488, pp. 189–212, 2003.
  • S.-J. Liao and K. F. Cheung, “Homotopy analysis of nonlinear progressive waves in deep water,” Journal of Engineering Mathematics, vol. 45, no. 2, pp. 105–116, 2003.
  • L. Tao, H. Song, and S. Chakrabarti, “Nonlinear progressive waves in water of finite depth–-an analytic approximation,” Coastal Engineering, vol. 54, no. 11, pp. 825–834, 2007.
  • S.-J. Liao, “On the homotopy multiple-variable method and its applications in the interactions of nonlinear gravity waves,” Communications in Nonlinear Science and Numerical Simulation, vol. 16, no. 3, pp. 1274–1303, 2011.
  • D. Xu, Z. Lin, S.-J. Liao, and M. Stiassnie, “On the steady-state fully resonant progressive waves in water of finite depth,” Journal of Fluid Mechanics, vol. 710, pp. 379–418, 2012.
  • J. Cheng and S. Q. Dai, “A uniformly valid series solution to the unsteady stagnation-point flow towards an impulsively stretching surface,” Science China, vol. 53, no. 3, pp. 521–526, 2010.
  • S. Abbasbandy, “The application of homotopy analysis method to nonlinear equations arising in heat transfer,” Physics Letters A, vol. 360, no. 1, pp. 109–113, 2006.
  • S.-J. Liao and A. Campo, “Analytic solutions of the temperature distribution in Blasius viscous flow problems,” Journal of Fluid Mechanics, vol. 453, pp. 411–425, 2002.
  • W. Wu and S.-J. Liao, “Solving solitary waves with discontinuity by means of the homotopy analysis method,” Chaos, Solitons and Fractals, vol. 26, no. 1, pp. 177–185, 2005.
  • E. Sweet and R. A. van Gorder, “Analytical solutions to a generalized Drinfel'd-Sokolov equation related to DSSH and KdV,” Applied Mathematics and Computation, vol. 216, pp. 2783–2791, 2010.
  • R. A. van Gorder, “Analytical method for the construction of solutions to the Föppl-von Kármán equations governing deflections of a thin flat plate,” International Journal of Non-Linear Mechanics, vol. 47, pp. 1–6, 2012.
  • S.-J. Liao, “An optimal homotopy-analysis approach for strongly nonlinear differential equations,” Communications in Nonlinear Science and Numerical Simulation, vol. 15, no. 8, pp. 2003–2016, 2010.
  • L. Zou, Z. Zong, Z. Wang, and L. He, “Solving the discrete KdV equation with homotopy analysis method,” Physics Letters A, vol. 370, no. 3-4, pp. 287–294, 2007.
  • J. Cheng, S.-P. Zhu, and S.-J. Liao, “An explicit series approximation to the optimal exercise boundary of American put options,” Communications in Nonlinear Science and Numerical Simulation, vol. 15, no. 5, pp. 1148–1158, 2010.
  • A. J. Roberts, “Highly nonlinear short-crested water waves,” Journal of Fluid Mechanics, vol. 135, pp. 301–321, 1983.