scholarly journals Asymptotics of sloshing eigenvalues for a triangular prism

2021 ◽  
pp. 1-33
Author(s):  
JULIEN MAYRAND ◽  
CHARLES SENÉCAL ◽  
SIMON ST–AMANT
Keyword(s):  
Asymptotic Expansion ◽  
Surface Waves ◽  
Counting Function ◽  
Three Dimensional ◽  
Triangular Prism ◽  
Edge Waves ◽  
Counting Problem ◽  
Sloping Beach

Abstract We consider the three-dimensional sloshing problem on a triangular prism whose angles with the sloshing surface are of the form ${\pi}/{2q}$ , where q is an integer. We are interested in finding a two-term asymptotic expansion of the eigenvalue counting function. When both angles are ${\pi}/{4}$ , we compute the exact value of the second term. As for the general case, we conjecture an asymptotic expansion by constructing quasimodes for the problem and computing the counting function of the related quasi-eigenvalues. These quasimodes come from solutions of the sloping beach problem and correspond to two kinds of waves, edge waves and surface waves. We show that the quasi-eigenvalues are exponentially close to real eigenvalues of the sloshing problem. The asymptotic expansion of their counting function is closely related to a lattice counting problem inside a perturbed ellipse where the perturbation is in a sense random. The contribution of the angles can then be detected through that perturbation.

The generation of surface waves over a sloping beach by an oscillating line source

1976 ◽  
Vol 79 (3) ◽  
pp. 573-585 ◽  
Author(s):  
Clare A. N. Morris
Keyword(s):  
Free Surface ◽  
Surface Waves ◽  
Line Source ◽  
Short Wave ◽  
Long Waves ◽  
Edge Waves ◽  
Wave Regime ◽  
Wave Solutions ◽  
Sloping Beach

AbstractA line source whose strength varies sinusoidally with time and also with the co-ordinate measured along its length is situated parallel to the shoreline of a beach of angle ¼π0. Both long-and short-wave solutions are found. It is shown that for certain positions of the source, long waves are not radiated to infinity, while in the short-wave regime, the solutions take the form of edge-waves, with resonances occurring at certain wavenumbers. Computations of the free-surface contours are presented for a range of wavenumbers.

scholarly journals A Numerical Approach for Calculation of Characteristics of Edge Waves in Three-Dimensional Plates

2020 ◽  
pp. 2050014
Author(s):  
Wenbo Duan ◽  
Ray Kirby
Keyword(s):  
Numerical Model ◽  
Nondestructive Testing ◽  
Surface Waves ◽  
Three Dimensional ◽  
Love Waves ◽  
Mode Shapes ◽  
Finite Width ◽  
Two Dimensional ◽  
Edge Waves ◽  
Rayleigh And Love Waves

Surface waves have been extensively studied in earthquake seismology. Surface waves are trapped near an infinitely large surface. The displacements decay exponentially with depth. These waves are also named Rayleigh and Love waves. Surface waves are also used for nondestructive testing of surface defects. Similar waves exist in finite width three-dimensional plates. In this case, displacements are no longer constant in the direction perpendicular to the wave propagation plane. Wave energy could still be trapped near the edge of the three-dimensional plate, and hence the term edge waves. These waves are thus different to the two-dimensional Rayleigh and Love waves. This paper presents a numerical model to study dispersion properties of edge waves in plates. A two-dimensional semi-analytical finite element method is developed, and the problem is closed by a perfectly matched layer adjacent to the edge. The numerical model is validated by comparing with available analytical and numerical solutions in the literature. On this basis, higher order edge waves and mode shapes are presented for a three-dimensional plate. The characteristics of the presented edge wave modes could be used in nondestructive testing applications.

Large-scale modulations of edge waves

1983 ◽  
Vol 132 ◽  
pp. 197-208 ◽  
Author(s):  
T. R. Akylas
Keyword(s):  
Large Scale ◽  
Three Dimensional ◽  
Physical Significance ◽  
Spatial Evolution ◽  
Edge Waves ◽  
Weakly Nonlinear ◽  
Resonant Interactions ◽  
Envelope Solitons ◽  
Temporal And Spatial ◽  
Sloping Beach

The temporal and spatial evolution of large-scale modulations of weakly nonlinear edge waves on a uniformly sloping beach is studied using the full water-wave formulation for beach angles α = π/2N. Equations governing the evolution of envelopes of edge waves, excited by resonant interactions with incident wavetrains, are derived. It is deduced that a uniform train of free periodic edge waves is always unstable to large-scale variations, so that envelope solitons will develop; the resulting three-dimensional solitons are described in detail. In addition, it is shown that steady-state standing subharmonic edge waves, excited by incident wavetrains on a long, mildly sloping beach, can be unstable to large-scale modulations. The possible physical significance of these findings is discussed.

Three-dimensional numerical simulation of wave-induced scour around a pile on a sloping beach

2021 ◽  
Vol 233 ◽  
pp. 109174
Author(s):  
Jinzhao Li ◽  
David R. Fuhrman ◽  
Xuan Kong ◽  
Mingxiao Xie ◽  
Yilin Yang
Keyword(s):  
Numerical Simulation ◽  
Three Dimensional ◽  
Sloping Beach ◽  
Wave Induced

scholarly journals Development and Validation of Quasi-Eulerian Mean Three-Dimensional Equations of Motion Using the Generalized Lagrangian Mean Method

2021 ◽  
Vol 9 (1) ◽  
pp. 76
Author(s):  
Duoc Nguyen ◽  
Niels Jacobsen ◽  
Dano Roelvink
Keyword(s):  
Surface Waves ◽  
Deep Water ◽  
Surf Zone ◽  
Equations Of Motion ◽  
Three Dimensional ◽  
Breaking Waves ◽  
Successful Implementation ◽  
Random Waves ◽  
Mean Motion ◽  
Generalized Lagrangian

This study aims at developing a new set of equations of mean motion in the presence of surface waves, which is practically applicable from deep water to the coastal zone, estuaries, and outflow areas. The generalized Lagrangian mean (GLM) method is employed to derive a set of quasi-Eulerian mean three-dimensional equations of motion, where effects of the waves are included through source terms. The obtained equations are expressed to the second-order of wave amplitude. Whereas the classical Eulerian-mean equations of motion are only applicable below the wave trough, the new equations are valid until the mean water surface even in the presence of finite-amplitude surface waves. A two-dimensional numerical model (2DV model) is developed to validate the new set of equations of motion. The 2DV model passes the test of steady monochromatic waves propagating over a slope without dissipation (adiabatic condition). This is a primary test for equations of mean motion with a known analytical solution. In addition to this, experimental data for the interaction between random waves and a mean current in both non-breaking and breaking waves are employed to validate the 2DV model. As shown by this successful implementation and validation, the implementation of these equations in any 3D model code is straightforward and may be expected to provide consistent results from deep water to the surf zone, under both weak and strong ambient currents.

Guided Surface Waves on an Elastic Half Space

1971 ◽  
Vol 38 (4) ◽  
pp. 899-905 ◽  
Author(s):  
L. B. Freund
Keyword(s):  
Wave Propagation ◽  
Surface Wave ◽  
Surface Waves ◽  
Half Space ◽  
Body Wave ◽  
Three Dimensional ◽  
Elastic Half Space ◽  
Normal Displacement ◽  
Rigid Barrier ◽  
Wave Disturbances

Three-dimensional wave propagation in an elastic half space is considered. The half space is traction free on half its boundary, while the remaining part of the boundary is free of shear traction and is constrained against normal displacement by a smooth, rigid barrier. A time-harmonic surface wave, traveling on the traction free part of the surface, is obliquely incident on the edge of the barrier. The amplitude and the phase of the resulting reflected surface wave are determined by means of Laplace transform methods and the Wiener-Hopf technique. Wave propagation in an elastic half space in contact with two rigid, smooth barriers is then considered. The barriers are arranged so that a strip on the surface of uniform width is traction free, which forms a wave guide for surface waves. Results of the surface wave reflection problem are then used to geometrically construct dispersion relations for the propagation of unattenuated guided surface waves in the guiding structure. The rate of decay of body wave disturbances, localized near the edges of the guide, is discussed.

Internal-inertia waves in a fluid of variable depth

1973 ◽  
Vol 73 (1) ◽  
pp. 205-213 ◽  
Author(s):  
W. D. McKee
Keyword(s):  
Exact Solutions ◽  
Surface Waves ◽  
Internal Waves ◽  
Vertical Velocity ◽  
General Problem ◽  
Three Dimensional ◽  
Variable Depth ◽  
Internal Inertia ◽  
Perturbation Pressure ◽  
Rotating Stratified Fluid

AbstractWaves in a rotating, stratified fluid of variable depth are considered. The perturbation pressure is used throughout as the dependent variable. This proves to have some advantages over the use of the vertical velocity. Some previous three-dimensional solutions for internal waves in a wedge are shown to be incorrect and the correct solutions presented. A WKB analysis is then performed for the general problem and the results compared with the exact solutions for a wedge. The WKB solution is also applied to long surface waves on a rotating ocean.

A Numerical Study of Three-Dimensional Natural Convective Heat Transfer From a Plate With a “Wavy” Surface

2001 ◽  
Author(s):  
Patrick H. Oosthuizen ◽  
Matt Garrett
Keyword(s):  
Heat Transfer ◽  
Prandtl Number ◽  
Surface Waves ◽  
Heat Transfer Rate ◽  
Transfer Rate ◽  
Three Dimensional ◽  
Wavy Surface ◽  
Surface Waviness ◽  
Dimensionless Amplitude ◽  
The Mean

Abstract Natural convective heat transfer from a wide isothermal plate which has a “wavy” surface, i.e., has a surface which periodically rises and falls, has been numerically studied. The surface waves run parallel to the direction of flow over the surface and have a relatively small amplitude. Two types of wavy surface have been considered here — saw-tooth and sinusoidal. Surfaces of the type considered are approximate models of situations that occur in certain window covering applications, for example, and are also sometimes used to try to enhance the heat transfer rate from the surface. The flow has been assumed to be laminar. Because the surface waves are parallel to the direction of flow, the flow over the surface will be three-dimensional. Fluid properties have been assumed constant except for the density change with temperature that gives rise to the buoyancy forces, this being treated by means of the Boussinesq type approximation. The governing equations have been written in dimensionless form, the height of the surface being used as the characteristic length scale and the temperature difference between the surface temperature and the temperature of the fluid far from the plate being used as the characteristic temperature. The dimensionless equations have been solved using a finite-element method. Although the flow is three-dimensional because the surface waves are all assumed to have the same shape, the flow over each surface thus being the same, and it was only necessary to solve for the flow over one of the surface waves. The solution has the following parameters: the Grashof number based on the height, the Prandtl number, the dimensionless amplitude of the surface waviness, the dimensionless pitch of the surface waviness, and the form of the surface waviness (saw-tooth or sinusoidal). Results have been obtained for a Prandtl number of 0.7 for Grashof numbers up to 106. The effects of Grashof number, dimensionless amplitude and dimensionless pitch on the mean heat transfer rate have been studied. It is convenient to introduce two mean heat transfer rates, one based on the total surface area and the other based on the projected frontal area of the surface. A comparison of the values of these quantities gives a measure of the effectiveness of the surface waviness in increasing the mean heat transfer rate. The results show that while surface waviness increases the heat transfer rate based on the frontal area, the modifications of the flow produced by the surface waves are such that the increase in heat transfer rate is less than the increase in surface area.

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