Hostname: page-component-6bf8c574d5-b4m5d Total loading time: 0 Render date: 2025-03-01T13:25:10.317Z Has data issue: false hasContentIssue false
  • English
  • Français

Generalized eigenfunction method for floating bodies

Published online by Cambridge University Press:  14 January 2011

COLM J. FITZGERALD  and
MICHAEL H. MEYLAN
COLM J. FITZGERALD
Affiliation:
Department of Mathematics, The University of Auckland, Private Bag 92019, Auckland 1142, New Zealand
MICHAEL H. MEYLAN*
Affiliation:
Department of Mathematics, The University of Auckland, Private Bag 92019, Auckland 1142, New Zealand
*
†Email address for correspondence: meylan@math.auckland.ac.nz
Get access Rights & Permissions [Opens in a new window]

Abstract

We consider the time domain problem of a floating body in two dimensions, constrained to move in heave and pitch only, subject to the linear equations of water waves. We show that using the acceleration potential, we can write the equations of motion as an abstract wave equation. From this we derive a generalized eigenfunction solution in which the time domain problem is solved using the frequency-domain solutions. We present numerical results for two simple cases and compare our results with an alternative time domain method.

JFM classification

Waves/Free-surface Flows: Surface gravity waves Waves/Free-surface Flows: Wave scattering
Type
Papers
Information
Journal of Fluid Mechanics , Volume 667 , 25 January 2011 , pp. 544 - 554
Copyright
Copyright © Cambridge University Press 2011

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

REFERENCES

Beale, J. T. 1977 Eigenfunction expansions for objects floating in an open sea. Commun. Pure Appl. Maths 30, 283–313.CrossRefGoogle Scholar
Clément, A. H. 1998 An ordinary differential equation for the green function of time-domain free-surface hydrodynamics. J. Engng Math. 33, 201–217.CrossRefGoogle Scholar
Cummins, W. E. 1962 The impulse response function and ship motions. Schiffstechnik 9, 101–109.Google Scholar
Friedman, A. & Shinbrot, M. 1967 The initial value problem for the linearized equations of water waves. J. Math. Mech. 17 (2), 107–180.Google Scholar
Hazard, C. & Lenoir, M. 2002 Surface water waves. In Scattering (ed. Pike, R. & Sabatier, P.), pp. 618–636. Academic Press.CrossRefGoogle Scholar
Hazard, C. & Loret, F. 2007 Generalized eigenfunction expansions for scattering problems with an application to water waves. Proc. R. Soc. Edin. A 137, 995–1035.CrossRefGoogle Scholar
Hazard, C. & Meylan, M. H. 2007 Spectral theory for a two-dimensional elastic thin plate floating on water of finite depth. SIAM J. Appl. Math. 68 (3), 629–647.CrossRefGoogle Scholar
Ikebe, T. 1960 Eigenfunction expansions associated with the Schroedinger operators and their applications to scattering theory. Arch. Rat. Mech. Anal. 5, 1–34.CrossRefGoogle Scholar
Maskell, S. & Ursell, F. 1970 The transient motion of a floating body. J. Fluid Mech. 44, 303–313.CrossRefGoogle Scholar
McIver, P., McIver, M. & Zhang, J. 2003 Excitation of trapped water waves by the forced motion of structures. J. Fluid Mech. 494, 141–162.CrossRefGoogle Scholar
Mei, C. C. 1989 The Applied Dynamics of Ocean Surface Waves. World Scientific.Google Scholar
Meylan, M. H. 2002 Spectral solution of time dependent shallow water hydroelasticity. J. Fluid Mech. 454, 387–402.CrossRefGoogle Scholar
Meylan, M. H. 2009 Generalized eigenfunction expansion for linear water-waves. J. Fluid Mech. 632, 447–455.CrossRefGoogle Scholar
Meylan, M. H. & Eatock Taylor, R. 2009 Time-dependent water-wave scattering by arrays of cylinders and the approximation of near trapping. J. Fluid Mech. 631, 103–125.CrossRefGoogle Scholar
Meylan, M. H. & Sturova, I. V. 2009 Time-dependent motion of a two-dimensional floating elastic plate. J. Fluids Struct. 25 (3), 445–460.CrossRefGoogle Scholar
Peter, M. & Meylan, M. H. 2010 A general spectral approach to the time-domain evolution of linear water waves impacting on a vertical elastic plate. SIAM J. Appl. Maths 70 (7), 2308–2328.CrossRefGoogle Scholar
Povzner, A. Y. 1953 On the expansions of arbitrary functions in terms of the eigenfunctions of the operator −Δu + cu (in Russian). Mat. Sbornik 32 (74), 109–156.Google Scholar
Ursell, F. 1964 The decay of the free motion of a floating body. J. Fluid Mech. 19, 303–319.CrossRefGoogle Scholar
Wehausen, J. & Laitone, E. 1960 Surface waves. In Fluid Dynamics III (ed. Flügge, S. & Truesdell, C.), Handbuch der Physik, vol. 9, chapter 3, pp. 446–778. Springer.Google Scholar
Wilcox, C. H. 1975 Scattering Theory for the d'Alembert Equation in Exterior Domains. Springer.CrossRefGoogle Scholar

Fitzgerald and Meylan supplementary material

Movie 1. The solution for a circular body half immersed for the times shown. At t = 0 the heave ammplitude is 0.5 and all other displacements and velocities are zero.

Download Fitzgerald and Meylan supplementary material(Video)
Video 626.7 KB

Fitzgerald and Meylan supplementary material

Movie 2. The solution for a dock of negligible submergence for the times shown. At t = 0 the heave amplitude is 0.5 and the pitch amplitude is π / 16 and all other displacements and velocities are zero.

Download Fitzgerald and Meylan supplementary material(Video)
Video 802.8 KB

Fitzgerald and Meylan supplementary material

Movie 3. The solution for a circular body half immersed for the times shown. At t = 0 the heave amplitude is 0.5, the surface displacement is exp(-4(x+3)2) and all velocities are zero.

Download Fitzgerald and Meylan supplementary material(Video)
Video 882.7 KB

Fitzgerald and Meylan supplementary material

Movie 4. The solution for a dock of negligible submergence for the times shown. At t = 0 the heave amplitude is 0.5 and the pitch amplitude is π / 16, the surface displacement is exp(-4(x+3)2) and all velocities are zero.

Download Fitzgerald and Meylan supplementary material(Video)
Video 1 MB