Water wave scattering by a step of arbitrary profile

2000 ◽  
Vol 411 ◽  
pp. 131-164 ◽  
Author(s):  
R. PORTER ◽  
D. PORTER
Keyword(s):  
Water Waves ◽  
Trial Function ◽  
Fluid Velocity ◽  
Equation System ◽  
Constant Depth ◽  
Incoming Wave ◽  
Water Wave Scattering ◽  
Term Trial ◽  
Judicious Choice ◽  
Fluid Behaviour

The two-dimensional scattering of water waves over a finite region of arbitrarily varying topography linking two semi-infinite regions of constant depth is considered. Unlike many approaches to this problem, the formulation employed is exact in the context of linear theory, utilizing simple combinations of Green's functions appropriate to water of constant depth and the Cauchy–Riemann equations to derive a system of coupled integral equations for components of the fluid velocity at certain locations. Two cases arise, depending on whether the deepest point of the topography does or does not lie below the lower of the semi-infinite horizontal bed sections. In each, the reflected and transmitted wave amplitudes are related to the incoming wave amplitudes by a scattering matrix which is defined in terms of inner products involving the solution of the corresponding integral equation system.This solution is approximated by using the variational method in conjunction with a judicious choice of trial function which correctly models the fluid behaviour at the free surface and near the joins of the varying topography with the constant-depth sections, which may not be smooth. The numerical results are remarkably accurate, with just a two-term trial function giving three decimal places of accuracy in the reflection and transmission coefficents in most cases, whilst increasing the number of terms in the trial function results in rapid convergence. The method is applied to a range of examples.

Nonlinear Water Waves in the Presence of Submerged Elliptic Cylinder

2004 ◽  
Author(s):  
Nikolai I. Makarenko
Keyword(s):  
Free Surface ◽  
Water Waves ◽  
Boundary Integral Equations ◽  
Fluid Velocity ◽  
Elliptic Cylinder ◽  
Small Time ◽  
Boundary Integral ◽  
Nonlinear Water Waves ◽  
Wave Elevation ◽  
Boundary Equations

The fully nonlinear problem on unsteady two-dimensional water waves generated by elliptic cylinder, that is horizontally submerged beneath a free surface, is considered. An analytical boundary integral equations method using a version of Milne-Thomson transformation is developed. Boundary equations (the BEq system) determine immediately exact wave elevation and fluid velocity at free surface. Small-time solution expansion is obtained in the case of accelerated cylinder starting from rest.

Cancellation of Non-Harmonic Waves Using a Cycloidal Turbine

2011 ◽  
Author(s):  
John T. Imamura ◽  
Stefan G. Siegel ◽  
Casey Fagley ◽  
Tom McLaughlin
Keyword(s):  
Deep Water ◽  
Water Waves ◽  
Linear Time ◽  
Harmonic Wave ◽  
Point Vortex ◽  
Harmonic Waves ◽  
Incoming Wave ◽  
Response Characteristics ◽  
Depth Water ◽  
Multi Component System

We computationally investigate the ability of a cycloidal turbine to cancel two-dimensional non-harmonic waves in deep water. A cycloidal turbine employs the same geometry as the well established Cycloidal or Voith-Schneider Propeller. It consists of a shaft and one or more hydrofoils that are attached eccentrically to the main shaft and can be independently adjusted in pitch angle as the cycloidal turbine rotates. We simulate the cycloidal turbine interaction with incoming waves by viewing the turbine as a wave generator superimposed with the incoming flow. The generated waves ideally are 180° out of phase and cancel the incoming wave downstream of the turbine. The upstream wave is very small as generation of single-sided waves is a characteristic of the cycloidal turbine as has been shown in prior work. The superposition of the incoming wave and generated wave is investigated in the far-field and we model the hydrofoil as a point vortex. This model has previously been used to successfully terminate regular deep water waves as well as intermediate depth water waves. We explore the ability of this model to cancel non-harmonic waves. Near complete cancellation is possible for a non-harmonic wave with components designed to match those generated by the cycloidal turbine for specified parameters. Cancellation of a specific wave component of a multi-component system is also shown. Also, step changes in the device operating parameters of circulation strength, rotation rate, and submergence depth are explored to give insight to the cycloidal turbine response characteristics and adaptability to changes in incoming waves. Based on these studies a linear, time-invarient (LTI) model is developed which captures the steady state wave frequency response. Such a model can be used for control development in future efforts to efficiently cancel more complex incoming waves.

A New Methodology for Generation of Solitary Water Waves in Laboratory

2020 ◽  
Author(s):  
Xiao Liu ◽  
Yong Liu
Keyword(s):  
Solitary Wave ◽  
Wave Height ◽  
Water Waves ◽  
Particle Image ◽  
Fluid Velocity ◽  
Simple System ◽  
Wave Surface ◽  
Wave Flume ◽  
Image Velocimetry ◽  
Laboratory Flume

Abstract In this article, a very simple system based on the enhanced dam-break flows was proposed and implemented to generate solitary wave with larger relative wave height (the ratio of wave height to water depth) in a laboratory flume. The experimental results showed that stable waves with the solitary wave profiles were successfully generated in the wave flume. The wave surface elevations were recorded by a series of wave gauges, and the fluid velocity field of the solitary wave was measured by Particle Image Velocimetry (PIV) system. The measurements of solitary wave profile, celerity and horizontal fluid velocity were also compared with the predictions by three different solitary wave theories. Results demonstrated that the present simple system was reliable and effective for the generation of solitary waves in laboratory.

scholarly journals Far-Field Characteristics of Linear Water Waves Generated by a Submerged Landslide over a Flat Seabed

2020 ◽  
Vol 8 (3) ◽  
pp. 196
Author(s):  
Haixiao Jing ◽  
Yanyan Gao ◽  
Changgen Liu ◽  
Jingming Hou
Keyword(s):  
Shallow Water ◽  
Water Depth ◽  
Water Waves ◽  
Wave Model ◽  
Linear Wave ◽  
Far Field ◽  
Constant Depth ◽  
Low Froude Number ◽  
Dispersive Model ◽  
Wave Models

Understanding the propagation of landslide-generated water waves is of great help against tsunami hazards. In order to investigate the effects of landslide shapes on the far-field leading wave generated by a submerged landslide at a constant depth, three linear wave models with different degrees of dispersive properties are employed in this study. The linear fully dispersive model is then validated by comparing the results against the experimental data available for landslides with a low Froude number. Three simplified shapes of landslides with the same volume, which are unnatural for a body of incoherent material, are used to investigate the effects of landslide shapes on the far-field properties of the generated leading wave over a flat seabed. The results show that the far-field leading crest over a constant depth is independent of the exact landslide shape and is invalid at a shallow water depth. Therefore, the most popular non-dispersive model (also called the shallow water wave model) cannot be used to reproduce the phenomenon. The weakly dispersive wave model can predict this phenomenon well. If only the leading wave is considered, this model is accurate up to at least μ = h0/Lc = 0.6, where h0 is the water depth and Lc denotes the characteristic length of the landslide.

scholarly journals Nonlinear Interactions between Gravity Waves in Water of Constant Depth

2015 ◽  
Vol 62 (1-2) ◽  
pp. 3-25
Author(s):  
Kazimierz Szmidt ◽  
Benedykt Hedzielski
Keyword(s):  
Water Waves ◽  
Fluid Motion ◽  
Power Series Expansion ◽  
Nonlinear Boundary Conditions ◽  
Constant Depth ◽  
Infinite Domain ◽  
Starting Point ◽  
Transient Solutions ◽  
Nonlinear Character ◽  
Approximate Formulation

AbstractThe paper deals with interactions between water waves propagating in fluid of constant depth. In formulation of this problem, a nonlinear character of these interactions is taken into account. In particular, in order to simplify a solution to nonlinear boundary conditions at the free surface, a system of material coordinates is employed as independent variables in the description of the phenomenon. The main attention is focused on the transient solutions corresponding to fluid motion starting from rest. With respect to the initial value problem considered, we confine our attention to a finite fluid domain. For a finite elapse of time, measured from the starting point, the solution in a finite fluid area mimics a solution within an infinite domain, inherent for wave propagation problems. Because of the complicated structure of equations describing nonlinear waves, an approximate formulation is considered, which is based on a power series expansion of dependent variables with respect to a small parameter. Such a solution is assumed to be accurate in describing the main features of the phenomenon. Numerical experiments are conducted to illustrate the approximate formulation developed in this paper.

scholarly journals On the Benjamin–Lighthill conjecture for water waves with vorticity

2017 ◽  
Vol 825 ◽  
pp. 961-1001 ◽  
Author(s):  
V. Kozlov ◽  
N. Kuznetsov ◽  
E. Lokharu
Keyword(s):  
Water Waves ◽  
Finite Depth ◽  
Two Dimensional ◽  
Constant Depth ◽  
Lipschitz Continuous ◽  
Vorticity Distribution ◽  
Inviscid Incompressible Fluid ◽  
Two Dimensional Flow ◽  
Wave Heights ◽  
Gravity Driven

We consider the nonlinear problem of steady gravity-driven waves on the free surface of a two-dimensional flow of an inviscid, incompressible fluid (say, water). The water motion is supposed to be rotational with a Lipschitz continuous vorticity distribution, whereas the flow of finite depth is assumed to be unidirectional. We verify the Benjamin–Lighthill conjecture for flows with values of Bernoulli’s constant close to the critical one. For this purpose it is shown that a set of near-critical waves consists only of Stokes and solitary waves provided their slopes are bounded by a constant. Moreover, the subset of waves with crests located on a fixed vertical is uniquely parametrised by the flow force, which varies between its values for the supercritical and subcritical shear flows of constant depth. There exists another parametrisation for this set; it involves wave heights varying between the constant depth of the subcritical shear flow and the height of a solitary wave.

scholarly journals Gaussian solitary waves to Boussinesq equation with dual dispersion and logarithmic nonlinearity

2018 ◽  
Vol 23 (6) ◽  
pp. 942-950 ◽  
Author(s):  
Anjan Biswasa ◽  
Mehmet Ekici ◽  
Abdullah Sonmezoglu
Keyword(s):  
Solitary Wave ◽  
Shallow Water ◽  
Solitary Waves ◽  
Water Waves ◽  
Trial Function ◽  
Boussinesq Equation ◽  
Solitary Wave Solutions ◽  
Shallow Water Waves ◽  
Wave Solutions ◽  
Extended Trial

This paper discusses shallow water waves that is modeled with Boussinesq equation that comes with dual dispersion and logarithmic nonlinearity. The extended trial function scheme retrieves exact Gaussian solitary wave solutions to the model.

Water Wave Scattering by Two Circular-Arc-Shaped Thin Plates With Non-Uniform Permeability

2019 ◽  
Author(s):  
R. Gayen ◽  
Sourav Gupta
Keyword(s):  
Approximate Solution ◽  
Integral Equations ◽  
Potential Function ◽  
Water Waves ◽  
Thin Plates ◽  
Integral Theorem ◽  
Water Wave Scattering ◽  
Circular Arc ◽  
Transmission Coefficients ◽  
Different Depths

Abstract The interaction of small amplitude water waves with a pair of circular-arc-shaped thin porous plates is studied under the assumption of linear theory. The plates are submerged at different depths in deep water and the permeability of the plates varies along the circumference of the plates. Applying Green’s integral theorem to the fundamental potential function and the scattered potential function and using the condition on the porous plates, the problem is reduced to that of solving a set of coupled integral equations for the potential difference functions across the plates. Two kernels of these integral equations are hypersingular and the other two kernels are regular. An expansion-cum-collocation method involving Chebyshev polynomials is employed to obtain the approximate solution of the integral equations. The numerical estimates for the reflection and the transmission coefficients are computed utilizing the approximate solution of the aforesaid integral equations. The numerical results for the reflection and the transmission coefficients are depicted graphically for several values of various parameters. Known results for dual symmetric circular-arc-shaped porous plates are recovered as special case.

scholarly journals Detecting nonlinearity in run-up on a natural beach

2007 ◽  
Vol 14 (4) ◽  
pp. 385-393 ◽  
Author(s):  
K. R. Bryan ◽  
G. Coco
Keyword(s):  
Water Waves ◽  
Nonlinear Behaviour ◽  
Correlated Noise ◽  
Incoming Wave ◽  
Linear Interaction ◽  
Natural Systems ◽  
Relative Contribution ◽  
Cyclic Behaviour ◽  
Non Linear ◽  
Run Up

Abstract. Natural geophysical timeseries bear the signature of a number of complex, possibly inseparable, and generally unknown combination of linear, stable non-linear and chaotic processes. Quantifying the relative contribution of, in particular, the non-linear components will allow improved modelling and prediction of natural systems, or at least define some limitations on predictability. However, difficulties arise; for example, in cases where the series are naturally cyclic (e.g. water waves), it is most unclear how this cyclic behaviour impacts on the techniques commonly used to detect the nonlinear behaviour in other fields. Here a non-linear autoregressive forecasting technique which has had success in demonstrating nonlinearity in non-cyclical geophysical timeseries, is applied to a timeseries generated by videoing the waterline on a natural beach (run-up), which has some irregular oscillatory behaviour that is in part induced by the incoming wave field. In such cases, the deterministic shape of each run-up cycle has a strong influence on forecasting results, causing questionable results at small (within a cycle) prediction distances. However, the technique can clearly differentiate between random surrogate series and natural timeseries at larger prediction distances (greater than one cycle). Therefore it was possible to clearly identify nonlinearity in the relationship between observed run-up cycles in that a local autoregressive model was more adept at predicting run-up cycles than a global one. Results suggest that despite forcing from waves impacting on the beach, each run-up cycle evolves somewhat independently, depending on a non-linear interaction with previous run-up cycles. More generally, a key outcome of the study is that oscillatory data provide a similar challenge to differentiating chaotic signals from correlated noise in that the deterministic shape causes an additional source of autocorrelation which in turn influences the predictability at small forecasting distances.

On the stability of lumps and wave collapse in water waves

2008 ◽  
Vol 366 (1876) ◽  
pp. 2761-2774 ◽  
Author(s):  
T.R Akylas ◽  
Yeunwoo Cho
Keyword(s):  
Ground State ◽  
Water Waves ◽  
Initial Conditions ◽  
Finite Depth ◽  
Equation System ◽  
Capillary Waves ◽  
Mean Flow ◽  
Water Wave Problem ◽  
Petviashvili Equation ◽  
Wave Collapse

In the classical water-wave problem, fully localized nonlinear waves of permanent form, commonly referred to as lumps, are possible only if both gravity and surface tension are present. While much attention has been paid to shallow-water lumps, which are generalizations of Korteweg–de Vries solitary waves, the present study is concerned with a distinct class of gravity–capillary lumps recently found on water of finite or infinite depth. In the near linear limit, these lumps resemble locally confined wave packets with envelope and wave crests moving at the same speed, and they can be approximated in terms of a particular steady solution (ground state) of an elliptic equation system of the Benney–Roskes–Davey–Stewartson (BRDS) type, which governs the coupled evolution of the envelope along with the induced mean flow. According to the BRDS equations, however, initial conditions above a certain threshold develop a singularity in finite time, known as wave collapse, due to nonlinear focusing; the ground state, in fact, being exactly at the threshold for collapse suggests that the newly discovered lumps are unstable. In an effort to understand the role of this singularity in the dynamics of lumps, here we consider the fifth-order Kadomtsev–Petviashvili equation, a model for weakly nonlinear gravity–capillary waves on water of finite depth when the Bond number is close to one-third, which also admits lumps of the wave packet type. It is found that an exchange of stability occurs at a certain finite wave steepness, lumps being unstable below but stable above this critical value. As a result, a small-amplitude lump, which is linearly unstable and according to the BRDS equations would be prone to wave collapse, depending on the perturbation, either decays into dispersive waves or evolves into an oscillatory state near a finite-amplitude stable lump.

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