# Propagation of obliquely incident water waves over a trench

TL;DR: In this paper, the diffraction of obliquely incident surface waves by an asymmetric trench is investigated using linearized potential theory and a numerical solution is constructed by matching particular solutions for each subregion of constant depth along vertical boundaries; the resulting matrix equation is solved numerically.

Abstract: The diffraction of obliquely incident surface waves by an asymmetric trench is investigated using linearized potential theory. A numerical solution is constructed by matching particular solutions for each subregion of constant depth along vertical boundaries ; the resulting matrix equation is solved numerically. Several cases where the trench-parallel wavenumber component in the incident-wave region exceeds the wavenumber for freely propagating waves in the trench are investigated and are found to result in large reductions in wave transmission ; however, reflection is not total owing to the finiteness of the obstacle. Results for one case are compared with data obtained from a small-scale wave-tank experiment. An approximate solution based on plane-wave modes is derived and compared with the numerical solution and, in the long-wave limit, with a previous analytic solution. 1. Introduction The problem of the diffraction of incident waves by a finite obstacle in an otherwise infinite and uniform domain remains of general interest in linear wave theory. Several geometries of interest can be schematized by domains divided into separate regions by vertical geometrical boundaries, with the fluid depth being constant in each subdomain. Representative two-dimensional problems, with the boundary geometry uniform in the direction normal to the plane of interest, include those of elevated rectangular sills, fixed or floating rectangular obstacles at the water surface, and submerged trenches. The approach to the solution of problems of this type has typically been to construct solutions for each constant-depth subdomain in terms of eigenfunction expansions of the velocity potential ; the solutions are then matched at the vertical boundaries, resulting in sets of linear integral equations which must be truncated to a finite number of terms and solved numerically. One of the earliest solutions of this type was given by Takano (1960), who studied the cases of normal wave incidence on an elevated sill and fixed obstacle at the surface. In this study, we employ a modification of Takano’s method, discussed in $3. Newman (19653) also employed an integral-equation approach to study reflection and transmission of waves normally incident on a single step between finite- and infinite-depth regions. A variational approach, developed by Schwinger to study discontinuitiesin waveguides (see Schwinger & Saxon 1968) has been used by Miles (1967), to study Newman’s single-step problem, and by Mei & Black (1969), who studied the symmetric elevated sill and a floating rectangular cylinder. Lassiter (1 972), using the variational approach, studied waves normally incident on a rectangular trench where the water depths before and after the trench are constant but not necessarily equal, referred to here as the asymmetric case. Lee &

##### Citations

More filters

••

TL;DR: In this paper, a modification of refraction-diffraction equation is developed for waves propagated over a bed consisting of substantial variations in water depth, where the Galerkin-Eigenfunction Method is used to determine all terms in the wave equation.

Abstract: A modification of refraction-diffraction equation is developed for waves propagated over a bed consisting of substantial variations in water depth. The Galerkin-Eigenfunction Method is used to determine all terms in the wave equation. The resulting equation includes higher order terms of the bottom slope and the term proportional to the bottom curvature, as well as the evanescent modes. The theory is verified against other theoretical and experimental results related to waves propagating over a patch of sinusoidal ripples. Numerical examples demonstrate the practical applicability of the extended refraction-diffraction equation for undulating bottom of steeper slope.

238 citations

••

TL;DR: In this paper, the authors expressed the "mild-slope" equation in the form of a pair of first-order equations, which constitute a hyperbolic system, without the loss of the reflected wave.

Abstract: The “mild-slope” equation which describes wave propagation in shoaling water is normally expressed in an elliptic form. The resulting computational effort involved in the solution of the boundary value problem renders the method suitable only for small sea areas. The parabolic approximation to this equation considerably reduces the computation involved but must omit the reflected wave. Hence this method is not suited to the modelling of harbour systems or areas near to sea walls where reflections are considerable. This paper expresses the “mild-slope” equation in the form of a pair of first-order equations, which constitute a hyperbolic system, without the loss of the reflected wave. A finite-difference numerical scheme is described for the efficient solution of the equations which includes boundaries of arbitrary reflecting power.

153 citations

••

TL;DR: In this article, higher-order Bragg resonant interactions between linear gravity waves and doubly sinusoidal beds have been observed by making very precise measurements in a wave tank.

Abstract: Experiments are described which demonstrate higher-order Bragg resonant interactions between linear gravity waves and doubly sinusoidal beds. These higher-order effects, which include harmonic and subharmonic Bragg reflections, have been observed by making very precise measurements in a wave tank. Subharmonic reflection was found to be very large, even for small bottom undulation amplitudes. The experimental data are compared with the predictions of a numerical model based on the full potential theory of linear waves.

128 citations

••

TL;DR: In this paper, experiments on the propagation of linear and weakly nonlinear gravity waves over a rectangular submerged bar were undertaken through very careful measurements in a wave tank, and the effects arising from the finite amplitude of the surface wave and those coming from the generation of vortices around bar edges were examined.

Abstract: Experiments on the propagation of linear and weakly nonlinear gravity waves over a rectangular submerged bar were undertaken through very careful measurements in a wave tank. Effects arising from the finite amplitude of the surface wave and those coming from the generation of vortices around bar edges were examined. Experimental data are compared with results of two theoretical models. The first model was derived from Takano (1960) and Kirby & Dalrymple's (1983) work and the second model was developed by Devillard, Dunlop & Souillard (1988) using the renormalized transfer matrix introduced by Miles (1967).

106 citations

••

TL;DR: In this article, a consistent coupled-mode model was applied to the diffraction of small-amplitude water waves from localized three-dimensional scatterers lying over a parallel-contour bathymetry.

Abstract: A consistent coupled-mode model recently developed by Athanassoulis and Belibassakis [1] , is generalized in 2+1 dimensions and applied to the diffraction of small-amplitude water waves from localized three-dimensional scatterers lying over a parallel-contour bathymetry. The wave field is decomposed into an incident field, carrying out the effects of the background bathymetry, and a diffraction field, with forcing restricted on the surface of the localized scatterer(s). The vertical distribution of the wave potential is represented by a uniformly convergent local-mode series containing, except of the ususal propagating and evanescent modes, an additional mode, accounting for the sloping bottom boundary condition. By applying a variational principle, the problem is reduced to a coupled-mode system of differential equations in the horizontal space. To treat the unbounded domain, the Berenger perfectly matched layer model is optimized and used as an absorbing boundary condition. Computed results are compared with other simpler models and verified against experimental data. The inclusion of the sloping-bottom mode in the representation substantially accelerates its convergence, and thus, a few modes are enough to obtain accurately the wave potential and velocity up to and including the boundaries, even in steep bathymetry regions. The present method provides high-quality information concerning the pressure and the tangential velocity at the bottom, useful for the study of oscillating bottom boundary layer, sea-bed movement and sediment transport studies.

91 citations

##### References

More filters

[...]

TL;DR: In this paper, the evanescent field structure over the wave front, as represented by equiphase planes, is identified as one of the most important and easily recognizable forms of surface wave.

Abstract: This paper calls attention to some of the most important and easily recognizable forms of surface wave, pointing out that their essential common characteristic is the evanescent field structure over the wave front, as represented by equiphase planes. The problems of launching and supporting surface waves must, in general, be distinguished from one another and it does not necessarily follow that because a particular surface is capable of supporting a surface wave that a given aperture distribution of radiation, e.g. a vertical dipole, can excite such a wave. The paper concludes with a discussion of the behavior of surface waves and their applications.

1,244 citations

### Additional excerpts

...Kreisel (1949) presented a general analysis for obstacles of finite extent and suitable geometry which proceeds by mapping the two-dimensional fluid domain into a rectangular strip ; the velocity potential is then obtained by solving a linear integral equation by iteration....

[...]

••

TL;DR: In this paper, the scattering of infinitesimal surface waves normally incident on a rectangular obstacle in a channel of finite depth is considered and a variational formulation is used as the basis of numerical computations.

Abstract: The scattering of infinitesimal surface waves normally incident on a rectangular obstacle in a channel of finite depth is considered. A variational formulation is used as the basis of numerical computations. Scattering properties for bottom and surface obstacles of various proportions, including thin barriers and surface docks, are presented. Comparison with experimental and theoretical results by other investigators is also made.

291 citations

••

TL;DR: In this article, the reflection of a plane wave of sound at an interface between two perfect fluids is considered, and it is found that, in addition to total reflection in some range of angles of incidence at all finite, relative speeds, there exists the possibility of a reflection coefficient exceeding unity for sufficiently high, supersonic speeds.

Abstract: The reflection of a plane wave of sound at an interface between two perfect fluids is considered. Previous analyses of Rudnick, Keller, and Franken and Ingard are found in error as a result of improper boundary conditions. It is found that, in addition to the possibility of total reflection in some range of angles of incidence at all finite, relative speeds, there exists the possibility of a reflection coefficient exceeding unity for sufficiently high, supersonic speeds; in particular, resonance may occur at one or more angles of incidence. The question of stability of the vortex sheet separating the two fluids also is discussed.

215 citations

••

TL;DR: In this paper, a formal solution to the initial value problem for a plane vortex sheet in an inviscid fluid is obtained by transform methods, and the eigenvalue problem is investigated and the stability criterion determined.

Abstract: A formal solution to the initial value problem for a plane vortex sheet in an inviscid fluid is obtained by transform methods. The eigenvalue problem is investigated and the stability criterion determined. This criterion is found to be in agreement with that obtained previously by Landau (1944), Hatanaka (1949), and Pai (1954), all of whom had included spurious eigenvalues in their analyses. It is also established that supersonic disturbances may be unstable; related investigations in hydrodynamic stability have conjectured on this possibility, but the vortex sheet appears to afford the first definite example. Finally, an asymptotic approximation is developed for the displacement of a vortex sheet following a suddenly imposed, spatially periodic velocity.

187 citations

••

TL;DR: In this article, a plane-wave and variational approximation to the magnitude of the transmission and reflexion coefficients for all wavelengths is presented, but the corresponding approximation for the reflexion coefficient is satisfactory only for rather long wavelengths.

Abstract: The diffraction of gravity waves at a discontinuity in depth is described by a scattering matrix that relates the asymptotic, plane-wave fields (each of which may contain waves travelling towards and away from the discontinuity) on the two sides of the discontinuity. Plane-wave and variational approximations for the elements of this scattering matrix are developed. These approximate results are tested by comparison with the more accurate results obtained by Newman for an infinite step. The plane-wave approximation to the magnitude of the transmission coefficient is within 5% of Newman's result for all wavelengths, but the corresponding approximation to the reflexion coefficient is satisfactory only for rather long wavelengths. The variational approximations to the complex transmission and reflexion coefficients agree with Newman's results, within the accuracy with which his graphs can be read, for all wavelengths. The variational approximations also are used to determine the effects of trapped modes on the resonant width of a shelf that terminates at a vertical cliff.

163 citations