Water wave scattering by a nearly circular cylinder submerged beneath an ice-cover

Abstract

Assuming linear theory, the two-dimensional problem of water wave scattering by a horizontal nearly circular cylinder submerged in infinitely deep water with an ice cover modeled as a thin-elastic plate floating on water, is investigated here. The cross-section of the nearly circular cylinder is taken as r=a(1+δC(θ)), where a is the radius of the corresponding circular cross-section of the cylinder, δ is a measure of small departure of the cross-section of the cylinder from its circularity and C(θ) is the shape function. Using a simplified perturbation technique the problem is reduced to two independent boundary value problems up to first order in δ. The first one corresponds to water wave scattering by a circular cylinder submerged in water with an ice-cover, while the second problem describes wave radiation by a submerged circular cylinder and involves first order correction to the reflection and transmission coefficients. The corrections are obtained in terms of integrals involving the shape function. Assuming a general Fourier expansion of the shape function, these corrections are evaluated approximately. It is well known that normally incident wave trains experience no reflection by a circular cylinder submerged in infinitely deep water with an ice cover. It is shown here that the reflection coefficient also vanishes up to first order for some particular choice of the shape function representing a nearly circular cylinder. For these cases, full transmission occurs, only change is in its phase which is depicted graphically against the wave number in a number of figures and appropriate conclusions are drawn.

Keywords

water wave scattering / ice-cover / nearly circular cylinder / shape function / reflection coefficient / transmission coefficient

References

[1] Das D, Mandal BN. Oblique wave scattering by a circular cylinder submerged beneath an ice-cover. International Journal of Engineering Science, 2006, 44(3–4): 166-179

[2] Das D, Mandal BN. Water wave scattering by a circular cylinder submerged in water with an ice cover. Indian Journal of Pure and Applied Mathematics, 2008, 39(4): 299-315

[3] Dean WR. On the reflexion of surface waves by a submerged circular cylinder. Mathematical Proceedings of the Cambridge Philosophical Society, 1948, 44(4): 957-983

[4] Friis A, Grue J, Palm E. Miloh T. Application of Fourier transform to the second order 2D wave diffraction problem. M. P. Tulin’s Festschrift: Mathematical Approaches in Hydrodynamics., 1991, Philadelphia, USA: SIAM, 209-227

[5] Fox C, Squire VA. On the oblique reflection and transmission of ocean waves at shore fast sea ice. Proceedings of the Royal Society of London Series A, 1994, 347: 185-218

[6] Levine H. Scattering of surface waves by a submerged circular cylinder. Journal of Mathematical Physics, 1965, 6(8): 1231-1243

[7] Mandal BN, Banerjea S. Scattering of water waves by a submerged nearly circular cylinder. J. Austral. Math. Soc. Ser. B, 1994, 36(3): 372-380

[8] Maiti P, Mandal BN. Water wave scattering by bottom undulations in an ice-covered two-layer fluid. Applied Ocean Research, 2008, 30(4): 264-272

[9] Ogilvie TP. First and second order forces on a cylinder submerged under a free surface. Journal of Fluid Mechanics, 1963, 16(3): 451-472

[10] Porter R. Surface wave scattering by submerged cylinders of arbitrary cross-section. Proceedings of the Royal Society A, 2002, 458(2019): 581-606

[11] Ursell F. Surface wave on deep water in the presence of submerged circular cylinder I and II. Mathematical Proceedings of the Cambridge Philosophical Society, 1950, 46(1): 141-155