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 Transformation of Long Waves in a Canal of Variable Section

2016 ◽  
Vol 63 (1) ◽  
pp. 3-18
Author(s):  
Kazimierz Szmidt ◽  
Benedykt Hedzielski
Keyword(s):  
Water Waves ◽  
Fluid Motion ◽  
Finite Domain ◽  
Infinite Domain ◽  
Initial Value ◽  
One Dimensional ◽  
Starting Point ◽  
Transient Solutions ◽  
Variable Section ◽  
Laboratory Flume

Abstract The paper deals with long water waves propagating in a straight canal of constant depth and variable section. In the formulation of this problem, a simplified, one-dimensional model is considered that is based on the assumption of a “columnar” fluid motion. To this end, a system of material coordinates is employed as independent variables in the description of this phenomenon. The main attention is focused on transient solutions corresponding to a fluid motion starting from rest.With respect to the initial value problem considered,we confine our attention to a finite domain fluid motion induced by a piston-type generator placed at the beginning of the canal. For a finite elapse of time, measured from the starting point, the solution in the finite fluid area mimics a solution within an infinite domain, inherent for wave propagation problems. The main goal of our investigations is to describe the evolution of the free surface (the wave height) at the smallest section of the canal. Numerical examples are provided to illustrate the model formulation developed in this paper. The accuracy of this approximate description is assessed by comparing its results with data obtained in hydraulic experiments performed in a laboratory flume.

scholarly journals Vibrations of a Horizontal Elastic Band Plate Submerged in Fluid of Constant Depth

2016 ◽  
Vol 63 (2-3) ◽  
pp. 191-213
Author(s):  
Kazimierz Szmidt ◽  
Benedykt Hedzielski
Keyword(s):  
Water Waves ◽  
Fluid Pressure ◽  
Free Vibrations ◽  
Forced Vibrations ◽  
Thin Elastic Plate ◽  
Constant Depth ◽  
Infinite Domain ◽  
Elastic Band ◽  
Infinite Layer ◽  
Pressure Load

AbstractThe paper deals with free and forced vibrations of a horizontal thin elastic plate submerged in an infinite layer of fluid of constant depth. In free vibrations, the pressure load on the plate results from assumed displacements of the plate. In forced vibrations, the fluid pressure is mainly induced by water waves arriving at the plate. In both cases, we have a coupled problem of hydrodynamics in which the plate and fluid motions are coupled through boundary conditions at the plate surface. At the same time, the pressure load on the plate depends on the gap between the plate and the fluid bottom. The motion of the plate is accompanied by the fluid motion. This leads to the so-called co-vibrating mass of fluid, which strongly changes the eigenfrequencies of the plate. In formulation of this problem, a linear theory of small deflections of the plate is employed. In order to calculate the fluid pressure, a solution of Laplace’s equation is constructed in the doubly connected infinite fluid domain. To this end, this infinite domain is divided into sub-domains of simple geometry, and the solution of the problem equation is constructed separately for each of these domains. Numerical experiments are conducted to illustrate the formulation developed in this paper.

scholarly journals Flow-induced Vibrations of a Horizontal Elastic Band Plate Submerged in Fluid of Finite Depth

2019 ◽  
Vol 66 (3-4) ◽  
pp. 101-130
Author(s):  
Kazimierz Szmidt ◽  
Benedykt Hedzielski
Keyword(s):  
Boundary Conditions ◽  
Water Waves ◽  
Fluid Pressure ◽  
Plate Motion ◽  
Thin Elastic Plate ◽  
Infinite Domain ◽  
Elastic Band ◽  
Pressure Load ◽  
Coupled Problem ◽  
Starting Point

AbstractThe paper deals with forced vibrations of a horizontal thin elastic plate submerged in a semi-infinite layer of fluid of constant depth. The pressure load on this plate is induced by water waves arriving at the plate. This load is accompanied by pressure resulting from the motion of the plate. The plate and fluid motions depend on boundary conditions, and, in particular, the pressure load depends on the width of the gap between the plate and the bottom. In theoretical description of the phenomenon, we deal with a coupled problem of hydrodynamics in which the plate and fluid motions are coupled through boundary conditions at the plate surfaces. The main attention is focused on transient solutions of the problem, which correspond to fluid (and plate) motion starting from rest. In formulation of this problem, a linear theory of small deflections of the plate is employed. In order to calculate the fluid pressure, a solution of Laplace’s equation is constructed in a doubly connected fluid domain. 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 the finite fluid area mimics a solution within an infinite domain, inherent for wave propagation problems. Because of the complicated structure of boundary conditions of the coupled problem considered, the fluid domain is divided into sub-domains of simple geometry, and the solutions of the problem equations are constructed separately in each of these domains. Numerical experiments have been conducted to illustrate the formulation developed in this paper.

A new Boussinesq method for fully nonlinear waves from shallow to deep water

2002 ◽  
Vol 462 ◽  
pp. 1-30 ◽  
Author(s):  
P. A. MADSEN ◽  
H. B. BINGHAM ◽  
HUA LIU
Keyword(s):  
Laplace Equation ◽  
Infinite Series ◽  
Deep Water ◽  
Water Waves ◽  
Modulational Instability ◽  
Surface Boundary ◽  
Nonlinear Water Waves ◽  
Finite Series ◽  
Starting Point ◽  
Highly Nonlinear

A new method valid for highly dispersive and highly nonlinear water waves is presented. It combines a time-stepping of the exact surface boundary conditions with an approximate series expansion solution to the Laplace equation in the interior domain. The starting point is an exact solution to the Laplace equation given in terms of infinite series expansions from an arbitrary z-level. We replace the infinite series operators by finite series (Boussinesq-type) approximations involving up to fifth-derivative operators. The finite series are manipulated to incorporate Padé approximants providing the highest possible accuracy for a given number of terms. As a result, linear and nonlinear wave characteristics become very accurate up to wavenumbers as high as kh = 40, while the vertical variation of the velocity field becomes applicable for kh up to 12. These results represent a major improvement over existing Boussinesq-type formulations in the literature. A numerical model is developed in a single horizontal dimension and it is used to study phenomena such as solitary waves and their impact on vertical walls, modulational instability in deep water involving recurrence or frequency downshift, and shoaling of regular waves up to breaking in shallow water.

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 Effect of Capillarity on Fourth Order Nonlinear Evolution Equation for Two Stokes Wave Trains in Deep Water in the Presence of Air Flowing Over Water

2015 ◽  
Vol 20 (2) ◽  
pp. 267-282
Author(s):  
A.K. Dhar ◽  
J. Mondal
Keyword(s):  
Fourth Order ◽  
Deep Water ◽  
Water Waves ◽  
Nonlinear Evolution ◽  
Wave Train ◽  
Nonlinear Evolution Equations ◽  
Wave Number ◽  
Stokes Wave ◽  
Instability Wave ◽  
Starting Point

Abstract Fourth order nonlinear evolution equations, which are a good starting point for the study of nonlinear water waves, are derived for deep water surface capillary gravity waves in the presence of second waves in which air is blowing over water. Here it is assumed that the space variation of the amplitude takes place only in a direction along which the group velocity projection of the two waves overlap. A stability analysis is made for a uniform wave train in the presence of a second wave train. Graphs are plotted for the maximum growth rate of instability wave number at marginal stability and wave number separation of fastest growing sideband component against wave steepness. Significant improvements are noticed from the results obtained from the two coupled third order nonlinear Schrödinger equations.

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.

A Transient Infinite Element for Multi-Dimensional Acoustic Radiation

1995 ◽  
Author(s):  
R. Jeremy Astley
Keyword(s):  
Exact Solution ◽  
Full Range ◽  
Three Dimensional ◽  
Test Problems ◽  
Transient Pressure ◽  
Infinite Domain ◽  
Transient Wave ◽  
Exterior Region ◽  
Infinite Elements ◽  
Transient Solutions

Abstract A novel family of infinite “wave envelope” elements is proposed for the solution of transient wave problems in unbounded regions. The elements are formed by applying an inverse Fourier transformation to a discrete wave envelope model in the frequency domain. This gives a coupled system of second-order equations which are readily integrated in time to yield transient pressure histories at nodal points on the surface of the radiating body and — in retarded form — at discrete points within the infinite domain. The infinite elements formed in this way can be applied quite generally to two-dimensional and three-dimensional problems and are fully compatible with conventional finite acoustical elements. They can be used to model radiating bodies of arbitrary shape but are demonstrated in the current instance in application to test problems which involve sound fields generated by spherical surfaces excited from rest, the exterior region being modeled by finite and infinite elements with explicit transverse interpolation. The computed transient solutions obtained from this formulation are compared to analytic solutions and shown to yield accurate results over a full range of exciting frequencies. The utility of the method for problems which involve broadband excitation is confirmed by comparisons of computed and analytic surface impedances for the steady harmonic case. These indicate that the accuracy of the scheme is limited only by a requirement to match element order to the highest order multi-pole component present in the radiated field. That is to say, elements of radial order 1 give an exact solution for monopole fields at all frequencies; elements of order 2 give an exact solution for dipole fields; elements of order 3 give an exact solution for quadrupole fields and so on. Similar results are presented for the fully three dimensional case. These support the extension of this hypothesis to three dimensional transient solutions subject only to the normal limitations imposed by spatial resolution in the transverse direction.

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.

scholarly journals Surface and Internal Waves due to a Moving Load on a Very Large Floating Structure

2012 ◽  
Vol 2012 ◽  
pp. 1-14 ◽  
Author(s):  
Taro Kakinuma ◽  
Kei Yamashita ◽  
Keisuke Nakayama
Keyword(s):  
Thin Plate ◽  
Internal Waves ◽  
Water Waves ◽  
Moving Load ◽  
Fluid Motion ◽  
Point Load ◽  
Internal Mode ◽  
Floating Platform ◽  
Floating Structure ◽  
Very Large Floating Structure

Interaction of surface/internal water waves with a floating platform is discussed with nonlinearity of fluid motion and flexibility of oscillating structure. The set of governing equations based on a variational principle is applied to a one- or two-layer fluid interacting with a horizontally very large and elastic thin plate floating on the water surface. Calculation results of surface displacements are compared with the existing experimental data, where a tsunami, in terms of a solitary wave, propagates across one-layer water with a floating thin plate. We also simulate surface and internal waves due to a point load, such as an airplane, moving on a very large floating structure in shallow water. The wave height of the surface or internal mode is amplified when the velocity of moving point load is equal to the surface- or internal-mode celerity, respectively.

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