Skip to main content

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

  • Nikolay Kuznetsov (a1) and Oleg Motygin (a1)

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.

Corresponding author
Email address for correspondence:
Hide All
1. Abramowitz, M. & Stegun, I. A. 1965 Handbook of Mathematical Functions. Dover.
2. Fitzgerald, C. J. & McIver, P. 2010 Passive trapped modes in the water-wave problem for a floating structure. J. Fluid Mech. 657, 456477.
3. Fox, D. W. & Kuttler, J. R. 1983 Sloshing frequencies. Z. Angew. Math. Phys. 34, 668696.
4. Gilbarg, D. & Trudinger, N. S. 1983 Elliptic Partial Differential Equations of Second Order. Springer.
5. John, F. 1949 On the motion of floating bodies, I. Commun. Pure Appl. Maths 2, 1357.
6. John, F. 1950 On the motion of floating bodies, II. Commun. Pure Appl. Maths 3, 45101.
7. John, F. 1955 Plane Waves and Spherical Means. Interscience.
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), 985995.
9. Kuznetsov, N., Maz’ya, V. & Vainberg, B. 2002 Linear Water Waves: A Mathematical Approach. Cambridge University Press.
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, 616627.
11. Linton, C. M. & McIver, P. 2007 Embedded trapped modes in water waves and acoustics. Wave Motion 45, 1629.
12. McIver, M. 1996 An example of non-uniqueness in the two-dimensional linear water wave problem. J. Fluid Mech. 315, 257266.
13. McIver, P. & McIver, M. 2007 Motion trapping structures in the three-dimensional water-wave problem. J. Engng Maths 58, 6775.
14. McIver, P. & Newman, J. N. 2003 Trapping structures in the three-dimensional water-wave problem. J. Fluid Mech. 484, 283301.
15. Mei, C. C., Stiassnie, M. & Yue, D. K.-P. 2005 Theory and Applications of Ocean Surface Waves, Part 1, Linear Aspects . World Scientific.
16. Motygin, O. V. 2011 Trapped modes in a linear problem of the theory of surface water waves. J. Math. Sci. 173 (6), 717736.
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:
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, 36133632.
19. Neményi, P. F. 1951 Recent developments in inverse and semi-inverse methods in the mechanics of continua. Adv. Appl. Mech. 2, 123148.
20. Prudnikov, A. P., Brychkov, Yu. A. & Marichev, O. I. 1986 Integrals and Series, Vol. 2, Special Functions . Gordon & Breach.
21. Watson, G. N. 1944 A Treatise on the Theory of Bessel Functions. Cambridge University Press.
Recommend this journal

Email your librarian or administrator to recommend adding this journal to your organisation's collection.

Journal of Fluid Mechanics
  • ISSN: 0022-1120
  • EISSN: 1469-7645
  • URL: /core/journals/journal-of-fluid-mechanics
Please enter your name
Please enter a valid email address
Who would you like to send this to? *

JFM classification


Full text views

Total number of HTML views: 0
Total number of PDF views: 0 *
Loading metrics...

Abstract views

Total abstract views: 0 *
Loading metrics...

* Views captured on Cambridge Core between <date>. This data will be updated every 24 hours.

Usage data cannot currently be displayed