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Time-dependent water-wave scattering by arrays of cylinders and the approximation of near trapping

Published online by Cambridge University Press:  17 July 2009

MICHAEL H. MEYLAN  and
RODNEY EATOCK TAYLOR
MICHAEL H. MEYLAN*
Affiliation:
Department of Mathematics, The University of Auckland, New Zealand
RODNEY EATOCK TAYLOR
Affiliation:
Department of Engineering Science, Oxford University, UK
*
†Email address for correspondence: meylan@math.auckland.ac.nz
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Abstract

We consider the solution in the time domain of water-wave scattering by arrays of bottom-mounted cylinders. It has already been shown that near trapping occurs for certain arrangements of cylinders and we are especially focused on this phenomenon. We begin with the well-known single-frequency solution to the problem of a group of cylinders, and the extension of this solution to complex frequencies. It has been shown that singularities (scattering frequencies or resonances) occur for certain values of the complex frequency and these singularities are associated with the near-trapped mode. We show that it is possible to approximate the solution near these singularities, and produce a modal shape which is associated with the near-trapped mode. We then consider the time-dependent problem, beginning with the well-known incident plane wave packet solution. We also show how the problem of an arbitrary initial displacement can be found using the single-frequency solutions. This latter result relies on a special inner product which gives a generalized eigenfunction expansion (because the operator has a continuous spectrum). We then consider the approximation of the time-dependent motion using special mode shapes associated with the scattering frequencies. This approximation relies on the scattering frequencies lying close to the real axis. We present numerical results which show that this approximation is accurate for sufficiently large time.

JFM classification

Waves/Free-surface Flows: Wave-structure interactions Waves/Free-surface Flows: Wave scattering
Type
Papers
Information
Journal of Fluid Mechanics , Volume 631 , 25 July 2009 , pp. 103 - 125
Copyright
Copyright © Cambridge University Press 2009

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References

REFERENCES

Abramowitz, M. & Stegun, I. A. 1970 Handbook of Mathematical Functions. Dover.Google Scholar
Eatock Taylor, R. & Meylan, M. H. 2007 Theory of scattering frequencies applied to near-trapping by cylinders. In 22nd International Workshop on Water Waves and Floating Bodies, pp. 73–76, Plitvice, Croatia.Google Scholar
Eatock Taylor, R. Zang, J., Bai, W. & Walker, D. A. G. 2006 Transients in wave diffraction by cylinders and cylinder arrays. 21st International Workshop on Water Waves and Floating Bodies, pp. 37–40, Loughborough, UK.Google Scholar
Evans, D. V. & Porter, R. 1997 Near-trapping of waves by circular arrays of vertical cylinders. Appl. Ocean Res. 19 (2), 83–99.CrossRefGoogle Scholar
Hazard, C. & Lenoir, M. 1993 Determination of scattering frequencies for an elastic floating body. SIAM J. Math. Anal. 24 (4), 1458–1514.CrossRefGoogle Scholar
Hazard, C. & Lenoir, M. 2002 Surface water waves. In Scattering (ed. R. Pike & P. Sabatier), pp. 618–636. Academic.CrossRefGoogle Scholar
Hazard, C. & Loret, F. 2007 a Generalized eigenfunction expansions for scattering problems with an application to water waves. Proc. R. Soc. Edinburgh 137A, 995–1035.CrossRefGoogle Scholar
Hazard, C. & Loret, F. 2007 b The singularity expansion method applied to the transient motions of a floating elastic plate. Math. Modelling Numer. Anal. 41 (5), 925–943.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. Ration. Mech. Anal. 5, 1–34.CrossRefGoogle Scholar
Kagemoto, H. & Yue, D. K. P. 1986 Interactions among multiple three-dimensional bodies in water waves: an exact algebraic method. J. Fluid Mech. 166, 189–209.CrossRefGoogle Scholar
Linton, C. M. & Evans, D. V. 1990 The interaction of waves with arrays of vertical circular cylinders. J. Fluid Mech. 215, 549–569.CrossRefGoogle Scholar
Maskell, S. J. & Ursell, F. 1970 The transient motion of a foating body. J. Fluid Mech. 44, 303–313.CrossRefGoogle Scholar
McIver, M. 1996 An example of non-uniqueness in the two-dimensional water wave problem. J. Fluid Mech. 315, 257–266.CrossRefGoogle Scholar
McIver, M. 2000 Trapped modes supported by submerged obstacles. Proc. R. Soc. A 456, 1851–1860.CrossRefGoogle Scholar
McIver, P. & McIver, M. 1997 Excitation of trapped water waves by the forced motion of structures. J. Mech. Appl. Math. 50, 165–178.CrossRefGoogle Scholar
McIver, P. & McIver, M. 2006 Trapped modes in the water-wave problem for a freely-floating structure. J. Fluid Mech. 558, 53–67.CrossRefGoogle Scholar
Meylan, M. H 2002 Spectral solution of time dependent shallow water hydroelasticity. J. Fluid Mech. 454, 387–402.CrossRefGoogle Scholar
Meylan, M. H. & Gross, L. 2003 A parallel algorithm to find the zeros of a complex analytic function. ANZIAM J. 44 (E), E216–E234.Google Scholar
Povzner, A. Ya. 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
Steinberg, S. 1968 Meromorphic families of compact operators. Arch. Ration. Mech. Anal. 31, 72–379.CrossRefGoogle Scholar
Ursell, F. 1964 The decay of the free motion of a floating body. J. Fluid Mech. 19, 303–319.CrossRefGoogle Scholar
Wilcox, C. H. 1975 Scattering Theory for the d'Alembert Equation in Exterior Domains. Springer-Verlag.CrossRefGoogle Scholar
Yilmaz, O. & Incecik, A. 1998 Analytical solutions of the diffraction problem of a group of truncated vertical cylinders. Ocean Engng 25, 385–394.CrossRefGoogle Scholar

Meylan and Taylor supplementary movie

Movie 1. The near trapped mode for four cylinders associated with the scattering frequency at 2.7641 - 0.0122i.

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Video 490 KB

Meylan and Taylor supplementary movie

Movie 2. The near trapped mode for nine cylinders associated with the scattering frequency at 2.7114 - 0.0041i

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Video 818.7 KB

Meylan and Taylor supplementary movie

Movie 3. One of the near trapped modes for nine cylinders associated with the scattering frequency at 2.7635 - 0.0086i

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Video 779.8 KB

Meylan and Taylor supplementary movie

Movie 4. One of the near trapped modes for nine cylinders associated with the scattering frequency at 2.7635 - 0.0086i

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Video 721.9 KB

Meylan and Taylor supplementary movie

Movie 5. The near trapped mode for nine cylinders associated with the scattering frequency at 2.8284 - 0.0102i

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Video 840.2 KB

Meylan and Taylor supplementary movie

Movie 6. The true (a) and approximate (b) solution for four cylinders for a plane incident wave given by equation (4.7).

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Video 2.3 MB

Meylan and Taylor supplementary movie

Movie 7. The true (a) and approximate (b) solution for nine cylinders for a plane incident wave given by equation (4.7).

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Video 3.8 MB

Meylan and Taylor supplementary movie

Movie 8. The true (a) and approximate (b) solution for four cylinders for a plane incident wave given by equation (4.8). We do not start the approximate solution until t=0.

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Video 4.1 MB

Meylan and Taylor supplementary movie

Movie 9. The true (a) and approximate solution (b) for an incident displacement given by equation (4.10) for four cylinders.

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Video 2 MB

Meylan and Taylor supplementary movie

Movie 10. The true (a) and approximate solution (b) for an incident displacement given by equation (4.10) for nine cylinders.

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Video 2.1 MB