- Physical Review Journals
All JournalsPhysics Magazine
Access Provided by Xinjiang University

Physical Review Fluids

Export citation

Export citation

Choose format for download:

Download Citation
  • Access by Xinjiang University

Ship waves on an elastic floating ice plate

Sergei Badulin*,†

Vladimir Gnevyshev‡

Yury Stepanyants¶,§

  • *Also at P.P. Shirshov Institute of Oceanology of the Russian Academy of Sciences, 36 Nakhimovsky Avenue, 117997 Moscow, Russia.
  • †Contact author: badulin.si@ocean.ru
  • ‡Contact author: avi9783608@gmail.com
  • Contact author: yury.stepanyants@unisq.edu.au
  • §Also at University of Southern Queensland, West Street, Toowoomba, Queensland 4350, Australia.
PDF

Phys. Rev. Fluids 10, 034801 – Published 4 March, 2025

DOI: https://doi.org/10.1103/PhysRevFluids.10.034801

Abstract

We study a wave wake produced by a finite-size source uniformly moving on an ice plate overlying deep water. We describe the kinematic characteristics of source-generated flexural-gravity waves in terms of isophase patterns scaled by the minimum phase speed and show that the wave picture is determined by ad hoc defined Mach and Bond numbers. Then, we show that several bifurcations occur in the wave wake when the source speed increases. In particular, cusps appear in the wake patterns at certain speeds due to the merging at specific points of two or three wave characteristics (wave rays) which correspond to longitudinal (divergent), transverse, and fan waves. To describe the distribution of wave amplitudes in the wake accounting for the source size and shape, we use the reference solution approach. This approach agrees with the stationary phase method in the far-field zone and reproduces also specific wave dynamics at short and intermediate distances from the source. The effect of a source shape on the wave wake pattern is graphically illustrated for the elliptic source as the dependence on the dimensionless Bond number and the source length-to-beam ratio. The illustrations of the source speed and shape effects are given for low-speed regimes when wave patterns are conformed by single-ray solutions. Ways of the generalization of the suggested approach are outlined.

View figure in article

Physics Subject Headings (PhySH)

Article Text

References (47)

  1. W. Thomson, XLII. On stationary waves in flowing water. Part I, Philos. Mag. Ser. 5 22, 353 (1886).
  2. H. Lamb, Hydrodynamics, 6th ed. (Cambridge University Press, Cambridge, England, 1978).
  3. J. Lighthill, Waves in Fluids (Cambridge University Press, Cambridge, UK, 1978), p. 504.
  4. L. N. Sretenskii, The Theory of Wave Motion of Fluid, 2nd ed. (in Russian) (Nauka, Moscow, 1977).
  5. G. B. Whitham, Linear and Nonlinear Waves (Wiley, New York, 1974), p. 636.
  6. M. Kashiwagi, Transient responses of a VLFS during landing and take-off an airplane, J. Mar. Sci. Technol. 9, 14 (2004).
  7. J. W. Davys, R. J. Hosking, and A. D. Sneyd, Waves due to a steadily moving source on a floating ice plate, J. Fluid Mech. 158, 269 (1985).
  8. E. Dinvay, and H. Kalisch, and E. I. Părău, Fully dispersive models for moving loads on ice sheets, J. Fluid Mech. 876, 122 (2019).
  9. K. Johnsen, H. Kalisch, and E. I. Părău, Ship wave patterns on floating ice sheets, Sci. Rep. 12, 18931 (2022).
  10. W. S. Nugroho, K. Wang, R. J. Hosking, and F. Milinazzo, Time-dependent response of a floating flexible plate to an impulsively started steadily moving load, J. Fluid Mech. 381, 337 (1999).
  11. A. V. Pogorelova, V. M. Kozin, and A. A. Matyushina, Stress-strain state of ice cover during aircraft takeoff and landing, J. Appl. Mech. Tech. Phys. 56, 920 (2015).
  12. V. A. Squire, R. J. Hosking, A. D. Kerr, and P. J. Langhorne, Moving Loads on Ice Plates, Solid Mechanics and Its Applications (Kluwer Academic Publishers, Dordrecht, 1996).
  13. I. V. Sturova, Unsteady three-dimensional sources in deep water with an elastic cover and their applications, J. Fluid Mech. 730, 392 (2013).
  14. I. V. Sturova, The effect of non-uniform compression of an elastic plate floating on the fluid surface on the development of unsteady flexural-gravity waves, Fluid Dyn. 56, 211 (2021).
  15. I. V. Sturova, Motion of a load over an ice sheet with non-uniform compression, Fluid Dyn. 56, 503 (2021).
  16. A. E. Bukatov, Waves in Sea with Floating Ice Cover (Marine Hydrophysical Institute of RAS, Sebastopol, Russia, 2017).
  17. M. J. Lighthill, Contributions to the theory of waves in nonlinear dispersive systems, J. Inst. Maths. Appl. 1, 269 (1965).
  18. V. G. Gnevyshev and S. I. Badulin, On the asymptotics of multidimensional linear wave packets: Reference solutions, Moscow Univ. Phys. Bull. 72, 415 (2017).
  19. V. G. Gnevyshev and S. I. Badulin, On reference solutions for ship waves, AIP Adv. 10, 025014 (2020).
  20. V. Gnevyshev and S. Badulin, Wave patterns of gravity-capillary waves from moving localized sources, Fluids 5, 219 (2020).
  21. V. Gnevyshev and S. Badulin, Deep water waves from oscillating elliptic source, Water Waves 5, 239 (2023).
  22. H. Liang and X. Chen, Asymptotic analysis of capillary-gravity waves generated by a moving disturbance, Eur. J. Mech. B Fluids 72, 624 (2018).
  23. F. Moisy and M. Rabaud, Mach-like capillary-gravity wakes, Phys. Rev. E 90, 023009 (2014).
  24. F. Noblesse, J. He, Y. Zhu, L. Hong, C. Zhang, R. Zhu, and C. Yang, Why can ship wakes appear narrower than Kelvin's angle? Eur. J. Mech. B/Fluids 46, 164 (2014).
  25. G. Rousseaux and H. Kellay, Classical hydrodynamics for analogue spacetimes: Open channel flows and thin films, Philos. Trans. R. Soc. A 378, 20190233 (2020).
  26. P. N. Svirkunov and M. V. Kalashnik, Phase patterns of dispersive waves from moving localized sources, Phys.-Usp. 57, 80 (2014).
  27. L. Rayleigh, The form of standing waves on the surface of running water, Proc. London Math. Soc. s1-15, 69 (1883).
  28. G. G. Stokes, in 1905 Mathematical and Physical Papers (Cambridge University Press, Cambridge, reprinted by Johnson Reprint Co., New York, 1966), Vol. 5, pp. 362–368.
  29. G. D. Crapper, Surface waves generated by a travelling pressure point, Proc. Roy. Soc. A 282, 547 (1964).
  30. Z. F. Li, G. X. Wu, and Y. Y. Shi, Wave motions due to a point source pulsating and advancing at forward speed parallel to a semi-infinite ice sheet, Phys. Rev. Fluids 9, 014801 (2024).
  31. Y. A. Stepanyants and I. V. Sturova, Waves on a compressed floating ice plate caused by motion of a dipole in water, J. Fluid Mech. 907, A7 (2021).
  32. F. Milinazzo, M. Shinbrot, and N. W. Evans, A mathematical analysis of the steady response of floating ice to the uniform motion of a rectangular load, J. Fluid Mech. 287, 173 (1995).
  33. W. F. Weeks and D. L. Anderson, An experimental study of strength of young sea ice, Trans. Amer. Geophys. Union 39, 641 (1958).
  34. A. G. Voronovich, Propagation of internal and surface gravity waves in the approximation of geometrical optics, Izv. Acad. Sc. USSR. Atm. Ocean. Phys. 12, 850 (1976).
  35. L. Kelvin, Deep sea ship waves, Proc. Roy. Soc. Edinburgh 25, 1060 (1906).
  36. M. Rabaud and F. Moisy, Ship wakes: Kelvin or Mach angle? Phys. Rev. Lett. 110, 214503 (2013).
  37. M. V. Fedoryuk, Saddle Point Method, Encyclopedia of Mathematics (EMS Press, Berlin, 2001).
  38. M. Benzaquen, A. Darmon, and E. Raphael, Wake pattern and wave resistance for anisotropic moving disturbances, Phys. Fluids 26, 092106 (2014).
  39. J. W. Schulkes, R. J. Hosking, and A. D. Sneyd, Waves due to a steadily moving source on a floating ice plate. Part 2, J. Fluid Mech. 180, 297 (1987).
  40. J. W. Schulkes and A. D. Sneyd, Time-dependent response of floating ice to a steadily moving load, J. Fluid Mech. 186, 25 (1988).
  41. V. V. Bulatov and I. Yu. Vladimirov, Far fields at interface between an infinitely deep ocean and ice excited by a localized source, Izv. Atmos. Ocean. Phys. 59, 296 (2023).
  42. V. V. Bulatov, I. Yu. Vladimirov, and E. M. Morozov, Perturbations far fields of the interface surface of the deep ocean and the ice cover from localized sources, Doklady Russ. Acad. Sci., Earth Sciences. 512, 302 (2023).
  43. Zh. V. Malenko and A. A. Yaroshenko, Three-dimensional bending-gravitational waves in a floating ice sheet from a moving source of disturbances, Prikl. Matemat. Mehan. (J. Appl. Math. Mech.) 87, 1037 (2023).
  44. Zh. V. Malenko and A. A. Yaroshenko, Flexural-gravity waves in an ice cover exited by periodically varying moving perturbations, Fluid Dynamics 59, 415 (2024).
  45. V. V. Bulatov and I. Yu. Vladimirov, The phase structure of wave disturbances excited by a pulsating source at the interface of a liquid flow of finite depth and an ice sheet, Prikl. Matemat. Mehan. (J. Appl. Math. Mech.) 88, 392 (2024).
  46. L. A. Tkacheva, Motion of a load on an ice cover in the presence of a current with velocity shear, Fluid Dyn. 58, 263 (2023).
  47. G. F. Clauss and J. Bergmann, Gaussian wave packets—A new approach to seakeeping tests of ocean structures, Appl. Ocean Res. 8, 190 (1986).

Outline

Information

Phys. Rev. Fluids 10, 034801– Published 4 March, 2025
  • Received 16 October 2024
  • Accepted 17 January 2025
Crossmark - Check for updates button

©2025 American Physical Society

Sign In to Your Journals Account

Filter

Filter

Article Lookup

Enter a citation