Abstract
In this paper, a method to construct oblique wave-free potentials in the linearised theory of water waves for water with uniform finite depth is presented in a systematic manner. The water has either a free surface or an ice-cover modelled as a thin elastic plate. For the case of free surface, the effect of surface tension may be neglected or taken into account. Here, the wave-free potentials are singular solutions of the modified Helmholtz equation, having singularity at a point in the fluid region and they satisfy the conditions at the upper surface and the bottom of water region and decay rapidly away from the point of singularity. These are useful in obtaining solutions to oblique water wave problems involving bodies with circular cross-sections such as long horizontal cylinders submerged or half-immersed in water of uniform finite depth with a free surface or an ice-cover modelled as a floating elastic plate. Finally, the forms of the upper surface related to the wave-free potentials constructed here are depicted graphically in a number of figures to visualize the wave motion. The results for non-oblique wave-free potentials and the upper surface wave-free potentials are obtained. The wave-free potentials constructed here will be useful in the mathematical study of water wave problems involving infinitely long horizontal cylinders, either half-immersed or completely immersed in water.
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References
Athanassonlis GA (1984). An expansion theorem for water wave potentials. Journal of Engineering Mathematics, 18, 181–194.
Bolton WE, Ursell F (1973). The wave force on an infinitely long circular cylinder in an oblique sea. Journal of Fluid Mechanics, 57, 241–256.
Das D, Mandal BN (2010). Construction of wave-free potentials in linearized theory of water waves. Journal of Marine Science and Application, 9, 347–354. DOI: 10.1007/s11804-010-1019-0
Dhillon H, Mandal BN (2013). Three dimensional wave-free potentials in the theory of water waves. ANZIAM J., 55, 175–195. DOI: 10.1017/S1446181113000357
Fox C, Squire VA (1994). On the oblique reflection and transmission of ocean waves from shore fast sea ice. Philosophical Transaction of Royal Society, 347, 185–218.
Gayen R, Mandal BN (2006). Motion due to fundamental singularities in finite depth water with an elastic solid cover. Fluid Dynamics Research, 38, 224–240. DOI:10.1016/j.fluiddyn.2005.12.001
Athanassonlis GA (1984). An expansion theorem for water wave potentials. Journal of Engineering Mathematics, 18, 181–194.
Bolton WE, Ursell F (1973). The wave force on an infinitely long circular cylinder in an oblique sea. Journal of Fluid Mechanics, 57, 241–256.
Das D, Mandal BN (2010). Construction of wave-free potentials in linearized theory of water waves. Journal of Marine Science and Application, 9, 347–354. DOI: 10.1007/s11804-010-1019-0
Dhillon H, Mandal BN (2013). Three dimensional wave-free potentials in the theory of water waves. ANZIAM J., 55, 175–195. DOI: 10.1017/S1446181113000357
Fox C, Squire VA (1994). On the oblique reflection and transmission of ocean waves from shore fast sea ice. Philosophical Transaction of Royal Society, 347, 185–218.
Gayen R, Mandal BN (2006). Motion due to fundamental singularities in finite depth water with an elastic solid cover. Fluid Dynamics Research, 38, 224–240. DOI:10.1016/j.fluiddyn.2005.12.001Gradstein IS
Ryzhik IM (1980). Table of integrals, series and products. Academic Press Inc, Burlington, 711–747.
Lamb H (1932). Hydrodynamics. Cambridge University Press London, 1–15, 351–400.
Linton CM, McIver P (2001). Handbook of mathematical techniques for wave structure introductions. Chapman and Hall, CRC Boca Raton, Appendix-B, 247–270.
Mandal BN, Goswami SK (1984). Scattering of surface waves obliquely incident on a fixed half-immersed circular cylinder. Mathematical Proceeding of Cambridge Philosophical Society, 96, 359–369.
Mandal BN, Das D (2010). Construction of wave-free potentials in linearized theory of water waves in uniform finite depth water. Review Bulletin of the Calcutta Mathematical Society, 18, 173–184.
Rhodes-Robinson PF (1970). Fundamental singularities in the theory of water waves with surface tension. Bulletin of the Australian Mathematical Society, 2, 317–333.
Stoker JJ (1957). Water wave: The mathematical theory with application. Pure and Applied Mathematics. Interscience Publishers, New York.
Thorne RC (1953). Multipole expansions in the theory of surface waves. Proceedings of Cambridge Philosophical Society, 49, 707–716.
Ursell F (1949). On the heaving motion of a circular cylinder on the surface of a fluid. Quarterly Journal of Applied Mathematics, 2, 218–231.
Ursell F (1950). Surface waves on deep water in the presence of a submerged cylinder I, II. Mathematical Proceeding of Cambridge Philosophical Society, 46,141-152, 153–158.
Ursell F (1961). The transmission of surface waves under surface obstacles. Mathematical Proceeding of Cambridge Philosophical Society, 57, 638–663.
Ursell F (1968). The expansion of water wave potentials at great distances. Mathematical Proceeding of Cambridge Philosophical Society, 64, 811–826.
Weahausen JV, Laitone EV (1960). Surface waves. In: Encyclopedia of Physics. Springer, Berlin, 9, 446–478.
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Maiti, R., Basu, U. & Mandal, B.N. Oblique wave-free potentials for water waves in constant finite depth. J. Marine. Sci. Appl. 14, 126–137 (2015). https://doi.org/10.1007/s11804-015-1308-8
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DOI: https://doi.org/10.1007/s11804-015-1308-8