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Scattering of flexural-gravity waves by a crack in a floating ice sheet due to mode conversion during blocking

Published online by Cambridge University Press:  06 April 2021

S.C. Barman Open the ORCID record for S.C. Barman [Opens in a new window] ,
S. Das Open the ORCID record for S. Das [Opens in a new window] ,
T. Sahoo Open the ORCID record for T. Sahoo [Opens in a new window]  and
M.H. Meylan Open the ORCID record for M.H. Meylan [Opens in a new window]
S.C. Barman
Affiliation:
Department of Ocean Engineering and Naval Architecture, Indian Institute of Technology Kharagpur, Kharagpur721302, India
S. Das*
Affiliation:
Mathematical and Computational Sciences Division, Institute of Advanced Study in Science and Technology, Guwahati781035, India
T. Sahoo
Affiliation:
Department of Ocean Engineering and Naval Architecture, Indian Institute of Technology Kharagpur, Kharagpur721302, India
M.H. Meylan
Affiliation:
School of Mathematical and Physical Sciences, University of Newcastle, NSW2308, Australia
*
†Email address for correspondence: [email protected]
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Abstract

The scattering of flexural-gravity waves in a thin floating plate is investigated in the presence of compression. In this case, wave blocking occurs, which is associated with both a zero in the group velocity and coalition of two or more roots of the related dispersion relation. There exists a region in the frequency space in which there are three real roots of the dispersion equation and hence three propagating modes. This multiplicity leads to mode conversion when scattering occurs. In one of these modes, the energy propagation direction is opposite to the wavenumber, making enforcement of the Sommerfeld radiation condition challenging. The focus here is on a canonical problem in flexural-gravity wave scattering, the scattering of waves by a crack. Formulae are developed that apply uniformly at all frequencies, including through the blocking frequencies. This solution is developed by tracking the movement of the dispersion relation roots carefully in the complex plane. The mode conversion is verified by the scattering matrix of the process and through an energy identity. This energy identity for the case of more than one progressive modes is established using Green's theorem and later applied in the scattering matrix to identify the incident and transmitted waves in the scattering process and derive the radiation condition. Appropriate scaling of the reflection and transmission coefficients are provided with the energy identity. The solution method is illustrated with numerical examples.

JFM classification

Waves/Free-surface Flows: Wave scattering Geophysical and Geological Flows: Ice sheets
Type
JFM Papers
Information
Journal of Fluid Mechanics , Volume 916 , 10 June 2021 , A11
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Copyright
© The Author(s), 2021. Published by Cambridge University Press

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References

REFERENCES

Abdolali, A., Kadri, U., Parsons, W. & Kirby, J.T. 2018 On the propagation of acoustic-gravity waves under elastic ice sheets. J. Fluid Mech. 837, 640–656.CrossRefGoogle Scholar
Barrett, M.D. & Squire, V.A. 1996 Ice-coupled wave propagation across an abrupt change in ice rigidity, density, or thickness. J. Geophys. Res.: Oceans 101 (C9), 20825–20832.CrossRefGoogle Scholar
Bukatov, A.E. 1980 Influence of a longitudinally compressed elastic plate on the nonstationary wave motion of a homogeneous liquid. Fluid Dyn. 15 (5), 687–693.CrossRefGoogle Scholar
Bukatov, A.E. & Zav'yalov, D.D. 1995 Impingement of surface waves on the edge of compressed ice. Fluid Dyn. 30 (3), 435–440.CrossRefGoogle Scholar
Collins, C.O., Rogers, W.E. & Lund, B. 2017 An investigation into the dispersion of ocean surface waves in sea ice. Ocean Dyn. 67 (2), 263–280.CrossRefGoogle Scholar
Das, S., Kar, P., Sahoo, T. & Meylan, M.H. 2018 a Flexural-gravity wave motion in the presence of shear current: wave blocking and negative energy waves. Phys. Fluids 30 (10), 106606.CrossRefGoogle Scholar
Das, S., Sahoo, T. & Meylan, M.H. 2018 b Dynamics of flexural gravity waves: from sea ice to hawking radiation and analogue gravity. Proc. R. Soc. Lond. A 474 (2209), 20170223.Google ScholarPubMed
Das, S., Sahoo, T. & Meylan, M.H. 2018 c Flexural-gravity wave dynamics in two–layer fluid: blocking and dead water analogue. J. Fluid Mech. 854, 121–145.CrossRefGoogle Scholar
Davys, J.W., Hosking, R.J. & Sneyd, A.D. 1985 Waves due to a steadily moving source on a floating ice plate. J. Fluid Mech. 158, 269–287.CrossRefGoogle Scholar
Evans, D.V. & Porter, R. 2003 Wave scattering by narrow cracks in ice sheets floating on water of finite depth. J. Fluid Mech. 484, 143–165.CrossRefGoogle Scholar
Fox, C. & Squire, V.A. 1990 Reflection and transmission characteristics at the edge of shore fast sea ice. J. Geophys. Res.: Oceans 95 (C7), 11629–11639.CrossRefGoogle Scholar
Fox, C. & Squire, V.A. 1994 On the oblique reflexion and transmission of ocean waves at shore fast sea ice. Phil. Trans. R. Soc. Lond. A 347 (1682), 185–218.Google Scholar
Havelock, T.H. 1929 Lix. Forced surface-waves on water. Lond. Edinb. Dub. Phil. Mag. J. Sci. 8 (51), 569–576.CrossRefGoogle Scholar
Karmakar, D. & Sahoo, T. 2005 Scattering of waves by articulated floating elastic plates in water of infinite depth. Mar. Struct. 18 (5–6), 451–471.CrossRefGoogle Scholar
Kerr, A.D. 1983 The critical velocities of a load moving on a floating ice plate that is subjected to in-plane forces. Cold. Reg. Sci. Technol. 6 (3), 267–274.CrossRefGoogle Scholar
Kohout, A.L. & Meylan, M.H. 2009 Wave scattering by multiple floating elastic plates with spring or hinged boundary conditions. Mar. Struct. 22 (4), 712–729.CrossRefGoogle Scholar
Korobkin, A.A., Malenica, S. & Khabakhpasheva, T. 2018 Interaction of flexural-gravity waves in ice cover with vertical walls. Phil. Trans. R. Soc. Lond. A 376 (2129), 20170347.Google ScholarPubMed
Kouzov, D.P. 1963 Diffraction of a plane hydro-acoustic wave at a crack in an elastic plate. Z. Angew. Math. Mech. 27 (6), 1593–1601.CrossRefGoogle Scholar
Lawrie, J.B. 2007 On eigenfunction expansions associated with wave propagation along ducts with wave-bearing boundaries. IMA J. Appl. Maths 72 (3), 376–394.CrossRefGoogle Scholar
Lawrie, J.B. 2009 Orthogonality relations for fluid-structural waves in a three-dimensional, rectangular duct with flexible walls. Proc. R. Soc. Lond. A 465 (2108), 2347–2367.Google Scholar
Lawrie, J.B. & Abrahams, I.D. 1999 An orthogonality relation for a class of problems with high-order boundary conditions; applications in sound-structure interaction. Q. J. Mech. Appl. Maths 52 (2), 161–181.CrossRefGoogle Scholar
Lee, C.-H. & Newman, J.N. 2000 An assessment of hydroelasticity for very large hinged vessels. J. Fluids Struct. 14 (7), 957–970.CrossRefGoogle Scholar
Li, Z.F., Wu, G.X. & Ji, C.Y. 2018 a Interaction of wave with a body submerged below an ice sheet with multiple arbitrarily spaced cracks. Phys. Fluids 30 (5), 057107.CrossRefGoogle Scholar
Li, Z.F., Wu, G.X. & Ji, C.Y. 2018 b Wave radiation and diffraction by a circular cylinder submerged below an ice sheet with a crack. J. Fluid Mech. 845, 682–712.CrossRefGoogle Scholar
Liu, A.K. & Mollo-Christensen, E. 1988 Wave propagation in a solid ice pack. J. Phys. Oceanogr. 18 (11), 1702–1712.2.0.CO;2>CrossRefGoogle Scholar
Maïssa, P., Rousseaux, G. & Stepanyants, Y. 2016 Wave blocking phenomenon of surface waves on a shear flow with a constant vorticity. Phys. Fluids 28 (3), 032102.CrossRefGoogle Scholar
Manam, S.R., Bhattacharjee, J. & Sahoo, T. 2005 Expansion formulae in wave structure interaction problems. Proc. R. Soc. Lond. A 462 (2065), 263–287.Google Scholar
Mandal, S., Sahoo, T. & Chakrabarti, A. 2017 Characteristics of eigen-system for flexural gravity wave problems. Geophys. Astrophys. Fluid Dyn. 111 (4), 249–281.CrossRefGoogle Scholar
Meylan, M. & Squire, V.A. 1994 The response of ice floes to ocean waves. J. Geophys. Res.: Oceans 99 (C1), 891–900.CrossRefGoogle Scholar
Mondal, R., Mohanty, S.K. & Sahoo, T. 2013 Expansion formulae for wave structure interaction problems in three dimensions. IMA J. Appl. Maths 78 (2), 181–205.CrossRefGoogle Scholar
Ren, K., Wu, G.X. & Li, Z.F. 2020 Hydroelastic waves propagating in an ice-covered channel. J. Fluid Mech. 886, A18.CrossRefGoogle Scholar
Rhodes-Robinson, P.F. 1971 On the forced surface waves due to a vertical wave-maker in the presence of surface tension. Math. Proc. Cambridge 70 (2), 323–337.CrossRefGoogle Scholar
Rhodes-Robinson, P.F. 1979 On surface waves in the presence of immersed vertical boundaries. II. Q. J. Mech. Appl. Maths 32 (2), 125–133.CrossRefGoogle Scholar
Sahoo, T. 2012 Mathematical Techniques for Wave Interaction with Flexible Structures. CRC Press, Taylor and Francis Group.CrossRefGoogle Scholar
Sahoo, T., Yip, T.L. & Chwang, A.T. 2001 Scattering of surface waves by a semi-infinite floating elastic plate. Phys. Fluids 13 (11), 3215–3222.CrossRefGoogle Scholar
Schulkes, R.M.S.M., Hosking, R.J. & Sneyd, A.D. 1987 Waves due to a steadily moving source on a floating ice plate. Part 2. J. Fluid Mech. 180, 297–318.CrossRefGoogle Scholar
Shi, Y.Y., Li, Z.F. & Wu, G.X. 2019 Interaction of wave with multiple wide polynyas. Phys. Fluids 31 (6), 067111.Google Scholar
Squire, V.A. 2018 A fresh look at how ocean waves and sea ice interact. Phil. Trans. R. Soc. Lond. A 376 (2129), 20170342.Google Scholar
Squire, V.A. 2020 Ocean wave interactions with sea ice: a reappraisal. Annu. Rev. Fluid Mech. 52, 37–60.CrossRefGoogle Scholar
Squire, V.A. & Dixon, T.W. 2000 An analytic model for wave propagation across a crack in an ice sheet. Intl J. Offshore Polar Engng 10 (03), ISOPE-00-10-3-173.Google Scholar
Squire, V.A., Hosking, R.J., Kerr, A.D. & Langhorne, P.J. 2012 Moving Loads on Ice Plates, vol. 45. Springer Science & Business Media.Google Scholar
Williams, T.D.C. 2006 Reflections on ice: scattering of flexural gravity waves by irregularities in arctic and antarctic ice sheets. PhD thesis, University of Otago.Google Scholar
Xia, D., Kim, J.W. & Ertekin, R.C. 2000 On the hydroelastic behavior of two-dimensional articulated plates. Mar. Struct. 13 (4–5), 261–278.CrossRefGoogle Scholar