Hostname: page-component-cd9895bd7-jn8rn Total loading time: 0 Render date: 2024-12-21T15:08:44.913Z Has data issue: false hasContentIssue false
  • English
  • Français

On the coupled time-harmonic motion of deep water and a freely floating body: trapped modes and uniqueness theorems

Published online by Cambridge University Press:  14 June 2012

Nikolay Kuznetsov  and
Oleg Motygin
Nikolay Kuznetsov
Affiliation:
Laboratory for Mathematical Modelling of Wave Phenomena, Institute for Problems in Mechanical Engineering, Russian Academy of Sciences, V.O., Bol’shoy pr. 61, St Petersburg 199178, Russian Federation
Oleg Motygin*
Affiliation:
Laboratory for Mathematical Modelling of Wave Phenomena, Institute for Problems in Mechanical Engineering, Russian Academy of Sciences, V.O., Bol’shoy pr. 61, St Petersburg 199178, Russian Federation
*
†Email address for correspondence: [email protected]
Get access Rights & Permissions [Opens in a new window]

Abstract

We investigate the time-harmonic small-amplitude motion of the mechanical system that consists of water and a body freely floating in it; water occupies a half-space, whereas the body is either surface-piercing or totally submerged. As a mathematical model of this coupled motion, we consider a spectral problem (the spectral parameter is the frequency of oscillations), for which the following results are obtained. The total energy of the water motion is finite and the equipartition of energy holds for the whole system. For any value of frequency, infinitely many eigensolutions are constructed and each of them consists of a non-trivial velocity potential and the zero vector describing the motion of the body; the latter means that trapping bodies (infinitely many of them are found) are motionless although they float freely. They are surface-piercing, have axisymmetric submerged parts and are obtained by virtue of the so-called semi-inverse procedure. We also prove that certain restrictions on the body geometry (which are violated for the constructed trapping bodies) guarantee that the problem has only a trivial solution for frequencies that are sufficiently large being measured in terms of a certain dimensionless quantity.

JFM classification

Waves/Free-surface Flows: Surface gravity waves
Type
Papers
Information
Journal of Fluid Mechanics , Volume 703 , 25 July 2012 , pp. 142 - 162
Copyright
Copyright © Cambridge University Press 2012

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

1. Abramowitz, M. & Stegun, I. A. 1965 Handbook of Mathematical Functions. Dover.Google Scholar
2. Fitzgerald, C. J. & McIver, P. 2010 Passive trapped modes in the water-wave problem for a floating structure. J. Fluid Mech. 657, 456–477.Google Scholar
3. Fox, D. W. & Kuttler, J. R. 1983 Sloshing frequencies. Z. Angew. Math. Phys. 34, 668–696.Google Scholar
4. Gilbarg, D. & Trudinger, N. S. 1983 Elliptic Partial Differential Equations of Second Order. Springer.Google Scholar
5. John, F. 1949 On the motion of floating bodies, I. Commun. Pure Appl. Maths 2, 13–57.CrossRefGoogle Scholar
6. John, F. 1950 On the motion of floating bodies, II. Commun. Pure Appl. Maths 3, 45–101.Google Scholar
7. John, F. 1955 Plane Waves and Spherical Means. Interscience.Google Scholar
8. Kuznetsov, N. 2011 On the problem of time-harmonic water waves in the presence of a freely-floating structure. St. Petersburg Math. J. 22 (6), 985–995.CrossRefGoogle Scholar
9. Kuznetsov, N., Maz’ya, V. & Vainberg, B. 2002 Linear Water Waves: A Mathematical Approach. Cambridge University Press.Google Scholar
10. Kuznetsov, N. & Motygin, O. 2011 On the coupled time-harmonic motion of water and a body freely floating in it. J. Fluid Mech. 679, 616–627.Google Scholar
11. Linton, C. M. & McIver, P. 2007 Embedded trapped modes in water waves and acoustics. Wave Motion 45, 16–29.CrossRefGoogle Scholar
12. McIver, M. 1996 An example of non-uniqueness in the two-dimensional linear water wave problem. J. Fluid Mech. 315, 257–266.Google Scholar
13. McIver, P. & McIver, M. 2007 Motion trapping structures in the three-dimensional water-wave problem. J. Engng Maths 58, 67–75.CrossRefGoogle Scholar
14. McIver, P. & Newman, J. N. 2003 Trapping structures in the three-dimensional water-wave problem. J. Fluid Mech. 484, 283–301.Google Scholar
15. Mei, C. C., Stiassnie, M. & Yue, D. K.-P. 2005 Theory and Applications of Ocean Surface Waves, Part 1, Linear Aspects . World Scientific.Google Scholar
16. Motygin, O. V. 2011 Trapped modes in a linear problem of the theory of surface water waves. J. Math. Sci. 173 (6), 717–736.Google Scholar
17. Motygin, O. & Kuznetsov, N. 1998 Non-uniqueness in the water-wave problem: an example violating the inside John condition. In Proceedings of 13th Workshop on Water Waves and Floating Bodies, Alpen aan den Rijn, 29 March–1 April 1998 (ed. A. Hermans), pp. 107–110. Available at: http://www.iwwwfb.org/Abstracts/iwwwfb13/iwwwfb13_28.pdf.Google Scholar
18. Nazarov, S. A. & Videman, J. H. 2011 Trapping of water waves by freely floating structures in a channel. Proc. R. Soc. Lond. A 467, 3613–3632.Google Scholar
19. Neményi, P. F. 1951 Recent developments in inverse and semi-inverse methods in the mechanics of continua. Adv. Appl. Mech. 2, 123–148.Google Scholar
20. Prudnikov, A. P., Brychkov, Yu. A. & Marichev, O. I. 1986 Integrals and Series, Vol. 2, Special Functions . Gordon & Breach.Google Scholar
21. Watson, G. N. 1944 A Treatise on the Theory of Bessel Functions. Cambridge University Press.Google Scholar