Abstract
There is a large class of problems in the field of fluid structure interaction where higher-order boundary conditions arise for a second-order partial differential equation. Various methods are being used to tackle these kind of mixed boundary-value problems associated with the Laplace’s equation (or Helmholtz equation) arising in the study of waves propagating through solids or fluids. One of the widely used methods in wave structure interaction is the multipole expansion method. This expansion involves a general combination of a regular wave, a wave source, a wave dipole and a regular wave-free part. The wave-free part can be further expanded in terms of wave-free multipoles which are termed as wave-free potentials. These are singular solutions of Laplace’s equation or two-dimensional Helmholz equation. Construction of these wave-free potentials and multipoles are presented here in a systematic manner for a number of situations such as two-dimensional non-oblique and oblique waves, three dimensional waves in two-layer fluid with free surface condition with higher order partial derivative are considered. In particular, these are obtained taking into account of the effect of the presence of surface tension at the free surface and also in the presence of an ice-cover modelled as a thin elastic plate. Also for limiting case, it can be shown that the multipoles and wave-free potential functions go over to the single layer multipoles and wave-free potential.
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References
Athanassonlis GA (1984). An expansion theorem for water-wave potentials. Journal of Engineering Mathematics, 18(3), 181–194. DOI: 10.1007/BF00039187
Bolton WE, Ursell F (1973). The wave force on an infinite long circular cylinder in an oblique sea. Journal of Fluid Mechanics, 57(2), 241–256.
Cadby JR, Linton CM (2000). Three-dimensional water wave scattering in two-layer fluids. Journal of Fluid Mechanics, 423, 155–173. DOI: 10.1017/S0022112000002007
Chakrabarti A (2000). On the solution of the problem of scattering of surface water waves of the edge of an ice-cover. Proc. R. Soc. Lond. A, 456, 1087–1099.
Das D, Mandal BN (2006). Oblique wave scattering by a circular cylinder submerged beneath an ice-cover. International Journal of Engineering Science, 44(3–4), 166–179. DOI: 10.1016/j.ijengsci.2006.01.001
Das D, Mandal BN (2007). Wave scattering by a horizontal circular cylinder in a two-layer fluid with an ice-cover. International Journal of Engineering Science, 45(10), 842–872. DOI: 10.1016/j.ijengsci.2007.05.008
Das D, Mandal BN (2008). Water wave radiation by a sphere submerged in water with an ice-cover. Archive of Applied Mechanics, 78(8), 649–661. DOI: 10.1007/s00419-007-0186-1
Das D, Mandal BN (2009). Wave scattering by a circular cylinder half-immersed in water with an ice-cover. International Journal of Engineering Science, 47(3), 463–474. DOI: 10.1016/j.ijengsci.2008.10.001
Das D, Mandal BN (2010a). Construction of wave-free potentials in linearized theory of water waves. Journal of Marine Science and Application, 9(4), 347–354. DOI: 10.1007/s11804-010-1019-0
Das D, Mandal BN (2010b). Wave radiation by a sphere submerged in a two-layer ocean with an ice-cover. Applied Ocean Research, 32(3), 358-366. DOI: 10.1016/j.apor.2009.11.002
Das D, Mandal BN, Chakrabarti A (2008). Energy identities in water wave theory for free-surface boundary condition with higher-order derivatives. Fluid Dynamics Research, 40(4), 253–272. DOI: 10.1016/j.fluiddyn.2007.10.002
Dhillon H, Mandal BN (2013). Three dimensional wave-free potentials in the theory of water waves. The ANZIAM Journal, 55(2), 175–195. DOI: 10.1017/S1446181113000357
Evans DV, Porter R (2003). Wave scattering by narrow cracks in ice-sheets floating on water of finite depth. Journal of Fluid Mechanics, 484, 143–165. DOI: 10.1017/S002211200300435X
Havelock TH (1995). Waves due to a floating sphere making periodic heaving oscillations. Proc. R. Soc. Lond. A, 231, 1–7. DOI: 10.1098/rspa.1955.0152
Kassem SE (1982). Multipole expansions for two superposed fluids, each of finite depth. Mathematical Proceedings of the Cambridge Philosophical Society, 91(2), 323–329.
Landau LD, Lifshitz EM (1959). Theory of elasticity. Pergamon Press, London, 45–50.
Linton CM, Cadby JR (2002). Scattering of oblique waves in a two-layer fluid. Journal of Fluid Mechanics, 461, 343–364. DOI: 10.1017/S002211200200842X
Linton CM, McIver M (1995). The interaction of waves with horizontal cylinders in two-layer fluids. Journal of Fluid Mechanics, 304, 213–229. DOI: 10.1017/S002211209500440X
Linton CM, McIver P (2001). Handbook of mathematical techniques for wave/structure introductions. Chapman and Hall/CRC Press, London, 243–268.
Manam SR, Bhattacharjee J, Sahoo T (2006). Expansion formula in wave structure interaction problems. Proc. R. Soc. A, 462, 263–287.
Mandal BN, Das D (2010). Construction of wave-free potentials in linearized theory of water waves in uniform finite depth water. Rev. Bull. Cal. Math. Soc., 18(2), 173–184.
Mandal BN, Goswami SK (1984). Scattering of surface waves obliquely incident on a fixed half-immersed circular cylinder. Mathematical Proceedings of the Cambridge Philosophical Society, 96(2), 359–369. DOI: 10.1017/S0305004100062265
Taylor RE, Hu CS (1991). Multipole expansions for wave diffraction and radiation in deep water. Ocean Engineering, 18(3), 191–224. DOI: 10.1016/0029-8018(91)90002-8
Thorne RC (1953). Multipole expansions in the theory of surface waves. Mathematical Proceedings of the Cambridge Philosophical Society, 49(4), 707–716. DOI: 10.1017/S0305004100028905
Ursell F (1949). On the heaving motion of a circular cylinder on the surface of a fluid. Q. J. Mech. Appl. Math., 2(2), 218–231 DOI: 10.1093/qjmam/2.2.218
Ursell F (1961a). The transmission of surface waves under surface obstacles. Proc. Camb. Philos. Soc., 57, 638–663.
Ursell F (1961b). Slender oscillating ships at zero forward speed. Journal of Fluid Mechanics, 14, 496–516.
Ursell F (1968). The expansion of water wave potentials at great distances. Proc. Camb. Phill. Soc., 64, 811–826.
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Das, D. Construction of wave-free potentials and multipoles in a two-layer fluid having free-surface boundary condition with higher-order derivatives. J. Marine. Sci. Appl. 14, 270–282 (2015). https://doi.org/10.1007/s11804-015-1321-y
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DOI: https://doi.org/10.1007/s11804-015-1321-y