# wAVE SCATTERING BY SUBMERGED ELLIPTICAL DISK

01 Jan 1996-Journal of Waterway Port Coastal and Ocean Engineering-asce (American Society of Civil Engineers)-Vol. 122, Iss: 1, pp 38-45

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.

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TL;DR: In this paper, it was shown that the added-mass coefficient becomes negative for a range of frequencies when the disc is sufficiently close to the free surface of deep water, and the resulting integral equation is a generalization of Love's integral equation for the electrostatic field of a parallel-plate capacitor.

Abstract: A thin rigid plate is submerged beneath the free surface of deep water. The plate performs small-amplitude oscillations. The problem of calculating the radiated waves can be reduced to solving a hypersingular boundary integral equation. In the special case of a horizontal circular plate, this equation can be reduced further to one-dimensional Fredholm integral equations of the second kind. If the plate is heaving, the problem becomes axisymmetric, and the resulting integral equation has a very simple structure; it is a generalization of Love's integral equation for the electrostatic field of a parallel-plate capacitor. Numerical solutions of the new integral equation are presented. It is found that the added-mass coefficient becomes negative for a range of frequencies when the disc is sufficiently close to the free surface.

82 citations

### Cites methods from "wAVE SCATTERING BY SUBMERGED ELLIPT..."

...Previous work on submerged plates in three dimensions is limited to the use of matched eigenfunction expansions in water of finite depth: this method was used by Yu & Chwang (1993) for a horizontal circular plate and by Zhang & Williams (1996) for a horizontal elliptical plate....

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TL;DR: In this paper, the hydrodynamic performance of a modified two-layer horizontal-plate breakwater is examined by means of the matched eigenfunction expansion method, and a linear analytical solution is developed for the interaction of water waves with the structure.

79 citations

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TL;DR: In this paper, a wave scattering of a solitary wave traveling over a submerged horizontal plate was studied, and the flow underneath the plate was found to behave almost like a plug flow, driven by the time-dependent, spatially uniform pressure gradient between the two openings.

Abstract: Wave scattering of a solitary wave traveling over a submerged horizontal plate was studied. Experiments for normal incidence were conducted in a wave flume, with a horizontal plate suspended at two different depths in the middle of the flume. Gauge pressures above and underneath the plate, surface elevations on and near the plate, and flow velocities at three representative fields of view were measured. The flow underneath the plate was found to behave almost like a plug flow, driven by the time-dependent, spatially uniform pressure gradient between the two openings. Complex vortices formed near the two edges of the plate as the plate acted like a flow divider. A numerical model based on two-dimensional Navier-Stokes equations was used to confirm the main features captured by the experimental measurements. Analytical solutions based on the linear long wave theory, which admits a soliton-like impulse wave solution, were also derived. The linear theory was applicable for obliquely incident impulse w...

45 citations

### Cites methods from "wAVE SCATTERING BY SUBMERGED ELLIPT..."

...Wave scattering by a submerged circular disk was analytically studied by Yu and Chwang (1993) and Zhang and Williams (1996)....

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TL;DR: In this article, the authors considered the three-dimensional interaction between water waves and a submerged disc, in deep water, and reduced the problem to a hypersingular integral equation over the surface of the disc.

38 citations

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TL;DR: In this paper, an analytic solution for the total velocity potential for an arbitrary body of the array and accordingly, to express the hydrodynamic pressure, the exciting forces and the wave elevation in compact analytic closed-form.

29 citations

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TL;DR: In this article, a horizontally submerged circular disk is analyzed by the linear potential wave theory, and a simple approach is employed to match these harmonic expressions at the common boundary of the subregions.

Abstract: Wave scattering over a horizontally submerged circular disk is analyzed by the linear potential wave theory. By means of the method of separation of variables, harmonic expressions of the velocity potential in terms of unknown constants are obtained in three subregions of the domain occupied by the fluid. A simple approach is employed to match these harmonic expressions at the common boundary of the subregions. The velocity potential is thus obtained, and consequently, the water‐surface elevation is determined. Variation of the wave height around the submerged disk versus the dimensionless disk radius, the relative water depth, and the submerged depth ratio of the disk to the total water depth is presented and discussed. The results show an effect of wave focusing around the rear of the disk. It is also noted that shallower‐water waves undergo more significant scattering when the ratio of the disk radius to the incident wavelength is fixed. The pattern of the wave‐height distribution is found to be closel...

58 citations

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TL;DR: In this paper, two approximate methods are presented for the calculation of the wave induced forces and moments on a vertical, surface-piercing cylinder of elliptic cross section, which provide a substantial reduction in computational effort when Compared with the exact solution which involves the numerical evaluation of Mathieu functions.

Abstract: Two approximate methods are presented for the calculation of the wave induced forces and moments on a vertical, surface‐piercing cylinder of elliptic cross section. Both methods provide a substantial reduction in computational effort when Compared with the exact solution which involves the numerical evaluation of Mathieu functions. One method involves the expansion of the exact expressions for the forces and moments for small values of the elliptic eccentricity parameter. The second method is based on Green's theorem and gives rise to an integral equation for the fluid velocity potential on the cylinder surface. Numerical results are presented for a range of relevant parameters and show excellent agreement with the computed values of the exact solution.

45 citations

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TL;DR: In this article, the mathematical solution to the diffraction problem is obtained on the basis of the linearized long-wave approximation for a shiplike body with beam-to-length ratio approximately equal to 0.2.

Abstract: Exciting forces and moments due to plane incident waves on a stationary platform are studied in the report. The platform is a vertical cylinder with a finite draft and elliptical cross section. The mathematical solution to the diffraction problem is obtained on the basis of the linearized long-wave approximation. Numerical results via Mathieu functions are presented for a shiplike body with beam-to-length ratio approximately equal to 0.2. Various draft-to-depth ratios and angles of incidence are considered. Results have been checked with the limiting case of a circular cylinder for the long-wavelength range. Aside from its own practical interest, the present theory provides a basis for comparison with other approximate theories of slender-body type and serves as a prelude to the corresponding calculations for arbitrary wavelengths. (Author)

38 citations

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TL;DR: Algor i thm 352 is a package of double-precision FORTRAN rout ines which consists of the following p r imary Rout ines : MFCVAL, M A T H, and BESSEL.

37 citations