A Justification of the KdV Approximation to First Order in the Case ofN-Soliton Water Waves in a Canal

1984 ◽  
Vol 15 (3) ◽  
pp. 468-489 ◽  
Author(s):  
Robert L. Sachs
Keyword(s):  
Water Waves ◽  
First Order

On a uniformly valid model for surface wave interaction

1993 ◽  
Vol 247 ◽  
pp. 589-601 ◽  
Author(s):  
Yehuda Agnon
Keyword(s):  
Surface Wave ◽  
Water Waves ◽  
Wave Train ◽  
Wave Interaction ◽  
Shallow Water Waves ◽  
First Order ◽  
Near Resonance ◽  
Wave Trains ◽  
Velocity Changes ◽  
Forced Waves

Nonlinear interaction of surface wave trains is studied. Zakharov's kernel is extended to include the vicinity of trio resonance. The forced wave amplitude and the wave velocity changes are then first order rather than second order. The model is applied to remove near-resonance singularities in expressions for the change of speed of one wave train in the presence of another. New results for Wilton ripples and the drift current and setdown in shallow water waves are readily derived. The ideas are applied to the derivation of forced waves in the vicinity of quartet and quintet resonance in an evolving wave field.

On the scattering of water waves by a submerged slender barrier

1998 ◽  
Vol 40 (2) ◽  
pp. 171-189
Author(s):  
P. K. Kundu ◽  
N. K. Saha
Keyword(s):  
Finite Length ◽  
Water Waves ◽  
Vertical Plate ◽  
Perturbation Technique ◽  
Reflection And Transmission Coefficients ◽  
Reflection And Transmission ◽  
First Order ◽  
Transmission Coefficients ◽  
Special Cases ◽  
Analytical Expressions

AbstractAn approximate analysis, based on the standard perturbation technique, is described in this paper to find the corrections, up to first order to the reflection and transmission coefficients for the scattering of water waves by a submerged slender barrier, of finite length, in deep water. Analytical expressions for these corrections for a submerged nearly vertical plate as well as for a submerged vertically symmetric slender barrier of finite length are also deduced, as special cases, and identified with the known results. It is verified, analytically, that there is no first order correction to the transmitted wave at any frequency for a submerged nearly vertical plate. Computations for the reflection and transmission coefficients up to O(ε), where ε is a small dimensionless quantity, are also performed and presented in the form of both graphs and tables.

scholarly journals New Exact Superposition Solutions to KdV2 Equation

2018 ◽  
Vol 2018 ◽  
pp. 1-9 ◽  
Author(s):  
Piotr Rozmej ◽  
Anna Karczewska
Keyword(s):  
Exact Solutions ◽  
Water Waves ◽  
Kdv Equation ◽  
Order Approximation ◽  
Elliptic Functions ◽  
Periodic Functions ◽  
Shallow Water Waves ◽  
First Order ◽  
New Exact Solutions ◽  
First Order Approximation

New exact solutions to the KdV2 equation (also known as the extended KdV equation) are constructed. The KdV2 equation is a second-order approximation of the set of Boussinesq’s equations for shallow water waves which in first-order approximation yields KdV. The exact solutionsA/2dn2[B(x-vt),m]±m cn[B(x-vt),m]dn[B(x-vt),m]+Din the form of periodic functions found in the paper complement other forms of exact solutions to KdV2 obtained earlier, that is, the solitonic ones and periodic ones given by singlecn2ordn2Jacobi elliptic functions.

scholarly journals Reflection of water waves in the presence of surface tension by a nearly vertical porous wall

1998 ◽  
Vol 39 (3) ◽  
pp. 308-317 ◽  
Author(s):  
A. Chakrabarti ◽  
T. Sahoo
Keyword(s):  
Surface Tension ◽  
Water Waves ◽  
Vertical Wall ◽  
Porous Wall ◽  
Finite Depth ◽  
Regularity Property ◽  
Positive Real ◽  
First Order ◽  
The Subject ◽  
First Order Correction

AbstractIn the present paper the problem of reflection of water waves by a nearly vertical porous wall in the presence of surface tension has been investigated. A perturbational approach for the first-order correction has been employed as compared with the corresponding vertical wall problem. A mixed Fourier transform together with the regularity property of the transformed function along the positive real axis has been used to obtain the potential functions along with the reflection coefficients up to first order. Whilst the problem of water of infinite depth is the subject matter of the present paper, a similar approach is applicable to problems associated with water of finite depth.

Bottom pressure distribution due to wave scattering near a submerged obstacle

2012 ◽  
Vol 702 ◽  
pp. 444-459 ◽  
Author(s):  
Julien Touboul ◽  
Vincent Rey
Keyword(s):  
Pressure Distribution ◽  
Deep Water ◽  
Water Waves ◽  
Dynamic Pressure ◽  
Finite Amplitude ◽  
Second Order ◽  
Local Modes ◽  
First Order ◽  
Pressure Zone ◽  
Water Conditions

AbstractThe dynamic pressure distribution on the bottom of a wave flume, due to the interaction of water waves with a submerged structure, is investigated experimentally and analytically, for both first- and second-order gravity waves of finite amplitude. The dynamic pressure excess is found to be very important, even for incoming waves propagating in deep water conditions. In this depth condition, a high pressure zone, thirty times larger than the dynamic pressure excess expected in the absence of the obstacle, is found in its vicinity. On the other hand, a low pressure zone is observed in the vicinity of the submerged obstacle for incoming waves propagating in smaller depth conditions. In any case, pressure gradients remain important. The second-order disturbance is found to be larger than first order in deep water conditions, for some specific conditions and locations. This result is interpreted in terms of nonlinear coupling of first-order components, including local modes.

An experimental study of the pressure variations in standing water waves

1951 ◽  
Vol 206 (1086) ◽  
pp. 424-435 ◽  
Keyword(s):  
Experimental Study ◽  
Surface Waves ◽  
Standing Wave ◽  
Water Waves ◽  
Wave Length ◽  
Second Order ◽  
First Order ◽  
Standing Water ◽  
Plane Barrier ◽  
Progressive Waves

Although the first-order pressure variations in surface waves on water are known to decrease exponentially downwards, it has recently been shown theoretically that in a standing wave there should be some second-order terms which are unattenuated with depth. The present paper describes experiments which verify the existence of pressure variations of this type in waves of period 0·45 to 0·50 sec. When the motion consists of two progressive waves of equal wave-length travelling in opposite directions, the amplitude of the unattenuated pressure variations is found to be proportional to the product of the wave amplitudes. This property is used to determine the coefficient of reflexion from a sloping plane barrier.

scholarly journals Scattering of water waves by a submerged nearly circular cylinder

1995 ◽  
Vol 36 (3) ◽  
pp. 372-380 ◽  
Author(s):  
B. N. Mandal ◽  
Sudeshna Banerjea
Keyword(s):  
Reflection Coefficient ◽  
Circular Cylinder ◽  
Cross Section ◽  
Water Waves ◽  
Wave Train ◽  
Integral Theorem ◽  
Water Wave Scattering ◽  
First Order ◽  
Transmission Coefficients ◽  
Surface Water Waves

AbstractThe problem of scattering of surface water waves by a horizontal circular cylinder totally submerged in deep water is well studied in the literature within the framework of linearised theory with the remarkable conclusion that a normally incident wave train experiences no reflection. However, if the cross-section of the cylinder is not circular then it experiences reflection in general. The present paper studies the case when the cylinder is not quite circular and derives expressions for reflection and transmission coefficients correct to order ∈, where ∈ is a measure of small departure of the cylinder cross-section from circularity. A simplified perturbation analysis is employed to derive two independent boundary value problems (BVP) up to first order in ∈. The first BVP corresponds to the problem of water wave scattering by a submerged circular cylinder. The reflection coefficient up to first order and the first order correction to the transmission coefficient arise in the second BVP in a natural way and are obtained by a suitable use of Green' integral theorem without solving the second BVP. Assuming a Fourier expansion of the shape function, these are evaluated approximately. It is noticed that for some particular shapes of the cylinder, these vanish. Also, the numerical results for the transmission coefficients up to first order for a nearly circular cylinder for which the reflection coefficients up to first order vanish, are given in tabular form. It is observed that for many other smooth cylinders, the result for a circular cylinder that the reflection coefficient vanishes, is also approximately valid.

scholarly journals On High Order ADER Discontinuous Galerkin Schemes for First Order Hyperbolic Reformulations of Nonlinear Dispersive Systems

2021 ◽  
Vol 87 (2) ◽  
Author(s):  
Saray Busto ◽  
Michael Dumbser ◽  
Cipriano Escalante ◽  
Nicolas Favrie ◽  
Sergey Gavrilyuk
Keyword(s):  
Schrödinger Equation ◽  
Discontinuous Galerkin ◽  
Water Waves ◽  
Schrodinger Equation ◽  
High Order ◽  
First Order ◽  
Fully Discrete ◽  
New Class ◽  
Dispersive Systems ◽  
One Step

AbstractThis paper is on arbitrary high order fully discrete one-step ADER discontinuous Galerkin schemes with subcell finite volume limiters applied to a new class of first order hyperbolic reformulations of nonlinear dispersive systems based on an extended Lagrangian approach introduced by Dhaouadi et al. (Stud Appl Math 207:1–20, 2018), Favrie and Gavrilyuk (Nonlinearity 30:2718–2736, 2017). We consider the hyperbolic reformulations of two different nonlinear dispersive systems, namely the Serre–Green–Naghdi model of dispersive water waves and the defocusing nonlinear Schrödinger equation. The first order hyperbolic reformulation of the Schrödinger equation is endowed with a curl involution constraint that needs to be properly accounted for in multiple space dimensions. We show that the original model proposed in Dhaouadi et al. (2018) is only weakly hyperbolic in the multi-dimensional case and that strong hyperbolicity can be restored at the aid of a novel thermodynamically compatible GLM curl cleaning approach that accounts for the curl involution constraint in the PDE system. We show one and two-dimensional numerical results applied to both systems and compare them with available exact, numerical and experimental reference solutions whenever possible.

scholarly journals Selected Problems of Sensitivity and Reliability of a Jack-Up Platform

2018 ◽  
Vol 25 (1) ◽  
pp. 77-84
Author(s):  
Bogdan Rozmarynowski ◽  
Tomasz Mikulski
Keyword(s):  
Water Waves ◽  
Structural Reliability ◽  
Structural Parameters ◽  
Probability Of Failure ◽  
First Order ◽  
Numerical Studies ◽  
Sensitivity Approach ◽  
The Baltic ◽  
Time Invariant ◽  
Jack Up

Abstract The paper deals with sensitivity and reliability applications to numerical studies of an off-shore platform model. Structural parameters and sea conditions are referred to the Baltic jack-up drilling platform. The sudy aims at the influence of particular basic variables on static and dynamic response as well as the probability of failure due to water waves and wind loads. The paper presents the sensitivity approach to a generalized eigenvalue problem and evaluation of the performace functions. The first order time-invariant problems of structural reliability analysis are under concern.

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