Construction of wave-free potential in the linearized theory of water waves

Abstract

Various water wave problems involving an infinitely long horizontal cylinder floating on the surface water were investigated in the literature of linearized theory of water waves employing a general multipole expansion for the wave potential. 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 (for non-oblique waves in two dimensions) or two-dimensional Helmholz equation (for oblique waves) satisfying the free surface condition and decaying rapidly away from the point of singularity. The method of constructing these wave-free potentials is presented here in a systematic manner for a number of situations such as deep water with a free surface, neglecting or taking into account the effect of surface tension, or with an ice-cover modelled as a thin elastic plate floating on water.

Keywords

wave-free potential / free surface / surface tension / ice-cover / Laplace equation / Helmholz equation

References

[1] Athanassonlis G.A. An expansion theorem for water-wave potentials. Journal of Engineering Mathematics, 1984, 18: 181-194

[2] Bolton W.E., Ursell F. The wave force on an infinite long circular cylinder in an oblique sea. Journal of Fluid Mechanics, 1973, 57: 241-256

[3] Das D., Mandal B.N. Wave scattering by a circular cylinder half-immersed in water with an ice-cover. International Journal of Engineering Sciences, 2009, 47: 463-474

[4] Gradshteyn I.S., Ryzhik I.M. Table of Integrals, Series and Products, 1980, Burlington: Academic Press Inc.

[5] Havelock T.H. Waves due to a floating sphere making periodic heaving oscillations. Proceedings of Royal Society London, London, 1955, A231: 1-7

[6] Linton CM, McIver P (2001). Handbook of mathematical techniques for wave/structure introductions. Chapman and Hall/CRC Boca Raton, 247–270.

[7] Mandal B.N., Goswami S.K. Scattering of surface waves obliquely incident on a fixed half-immersed circular cylinder. Mathematical Proceedings of Cambridge Philosophical Society, Cambridge, 1984, 96: 359-369

[8] Rhodes-Robinson P.F. Fundamental singularities in the theory of water waves with surface tension. Bulletin of Australian Mathematical Society, 1970, 2: 317-333

[9] Thorne R.C. Multipole expansions in the theory of surfa-ce waves. Proceedings of Cambridge Philosophical Society, Cambridge, 1953, 49: 707-716

[10] Ursell F. On the heaving motion of a circular cylinder on the surface of a fluid. Quarterly Journal of Mechanics and Applied Mathematics, 1949, 2: 218-231

[11] Ursell F. The transmission of surface waves under surface obstacles. Proceedings of Cambridge Philosophical Society, Cambridge, 1961, 57: 638-663

[12] Ursell F. Slender oscillating ships at zero forward speed. Journal of Fluid Mechanics, 1961, 14: 496-516

[13] Ursell F. The expansion of water wave potentials at great distances. Proceedings of Cambridge Philosophical Society, Cambridge, 1968, 64: 811-826