# Scattering and Radiation of Gravity Waves by an Elliptical Cylinder.

01 Aug 1971-

TL;DR: Theoretical results of scattering and radiation of simple-harmonic gravity waves by a vertical cylinder of elliptical cross-section are worked out in this article, where the question of flow separation is not treated.

Abstract: : Theoretical results of scattering and radiation of simple-harmonic gravity waves by a vertical cylinder of elliptical cross-section are worked out in this report. The cylinder extends from the free surface to the bottom of the sea of constant depth. Linearized theory for small amplitude waves is adopted. The question of flow separation is not treated. The mathematical solution leads to Mathieu's equations and the computer program by Clemm is used. Physical quantities calculated include the wave forces and moments in various directions on, and the scattering amplitude around, a stationary cylinder due to a plane incident wavetrain. Also calculated are the damping coefficients representing radiation energy losses due to various modes of oscillation of a cylinder in the absence of incident waves. By the method of images the scattering of plane incident wave by a semi-elliptical peninsula is studied. Extensive numerical results are presented for various degrees of ellipticity, angles of incidence, and for wave-lengths ranging from very long to comparable to the horizontal dimensions of the cylinder. These results should provide useful information for the design of large ocean structures. (Author)

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TL;DR: In this article, the authors derived a relation for the fluid motion through thin porous structures in addition to the conventional governing equation and boundary conditions for small-amplitude waves in ideal fluids.

Abstract: Diffraction of water waves by porous breakwaters is studied based on the linear potential wave theory. The formulation of the problem includes a newly derived relation for the fluid motion through thin porous structures in addition to the conventional governing equation and boundary conditions for small-amplitude waves in ideal fluids. The porous boundary condition, indirectly verified by collected experimental data, is obtained by assuming that the flow within the porous medium is governed by a convection-neglected and porous-effect-modeled Euler equation. A vertically two-dimensional problem with long-crested waves propagating in the normal direction of an infinite porous wall is first solved and the solution is compared with available experimental data. The wave diffraction by a semiinfinite porous wall is then studied by the boundary-layer method, in which the outer approximation is formulated by virtue of the reduced two-dimensional solution. It is demonstrated that neglect of the inertial effect of the porous medium leads to an overestimate of the functional performance of a porous breakwater.

280 citations

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TL;DR: In this article, a finite element model for the solution of Helmholtz problems at higher frequencies is described, which offers the possibility of computing many wavelengths in a single finite element.

Abstract: This paper describes a finite element model for the solution of Helmholtz problems at higher frequencies that offers the possibility of computing many wavelengths in a single finite element. The approach is based on partition of unity isoparametric elements. At each finite element node the potential is expanded in a discrete series of planar waves, each propagating at a specified angle. These angles can be uniformly distributed or may be carefully chosen. They can also be the same for all nodes of the studied mesh or may vary from one node to another. The implemented approach is used to solve a few practical problems such as the diffraction of plane waves by cylinders and spheres. The wave number is increased and the mesh remains unchanged until a single finite element contains many wavelengths in each spatial direction and therefore the dimension of the whole problem is greatly reduced. Issues related to the integration and the conditioning are also discussed.

131 citations

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TL;DR: The diffraction of small-amplitude surface waves by a horizontally submerged disk of elliptic cross section located at a finite depth beneath the free surface is investigated analytically in this article.

Abstract: The diffraction of small-amplitude surface waves by a horizontally submerged disk of elliptic cross section located at a finite depth beneath the free surface is investigated analytically. The fluid domain is divided into three regions, two internal regions, one above and one beneath the disk, and an external region extending to infinity in the horizontal plane. The theoretical formulation leads to solutions for the fluid velocity potentials in each region in terms of series of Mathieu and modified Mathieu functions of real argument. Numerical results are presented for the wave-induced forces and moments, and the variation of water surface elevation in the vicinity of the disk for a range of wave and structural parameters. In particular, the results for the hydrodynamic loads show significant differences from the corresponding estimates for a circular disk, while the results for the water surface elevation clearly show the effect of wave focusing around the rear of the disk.

25 citations

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TL;DR: In this paper, the diffraction of small-amplitude surface waves by floating and submerged stationary elliptical breakwaters in water of arbitrary uniform depth is investigated analytically, and the theoretical formulation leads to solutions for the fluid velocity potential in terms of series of Mathieu and modified Mathieu functions of real argument.

22 citations

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TL;DR: In this article, the theory for an improved wave element for wave diffraction is presented, which is based on the original wave element presented by Bettess, Zienkiewicz and others.

20 citations