Water waves generated by disturbances at an ice cover

This paper is concerned with two-dimensional unsteady motion of water waves generated by an initial disturbance created at an ice sheet covering the water. The ice cover is modeled as a thin elastic plate. Using linear theory, the problem is formulated as an initial value problem for the velocity potential describing the motion in the liquid. In the mathematical analysis, the Laplace and Fourier transform techniques have been utilized to obtain the depression of the ice-covered surface in the form of an infinite integral. For the special case of initial disturbance concentrated at the origin, taken on the ice cover, this integral is evaluated asymptotically by the method of a stationary phase for a long time and large distance from the origin. The form of the ice-covered surface is graphically depicted for two types of initial disturbances.

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Diffraction of water waves by a submerged vertical plate

Journal of Fluid Mechanics ◽

10.1017/s0022112070000253 ◽

1970 ◽

Vol 40 (3) ◽

pp. 433-451 ◽

Cited By ~ 84

Author(s):

D. V. Evans

Keyword(s):

Free Surface ◽

Boundary Value Problem ◽

Incident Wave ◽

Water Waves ◽

Vertical Plate ◽

Velocity Potential ◽

Wave Train ◽

Plane Waves ◽

Two Dimensional ◽

Special Case

A thin vertical plate makes small, simple harmonic rolling oscillations beneath the surface of an incompressible, irrotational liquid. The plate is assumed to be so wide that the resulting equations may be regarded as two-dimensional. In addition, a train of plane waves of frequency equal to the frequency of oscillation of the plate, is normally incident on the plate. The resulting linearized boundary-value problem is solved in closed form for the velocity potential everywhere in the fluid and on the plate. Expressions are derived for the first- and second-order forces and moments on the plate, and for the wave amplitudes at a large distance either side of the plate. Numerical results are obtained for the case of the plate held fixed in an incident wave-train. It is shown how these results, in the special case when the plate intersects the free surface, agree, with one exception, with results obtained by Ursell (1947) and Haskind (1959) for this problem.

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Generation of Surface Waves Due to Initial Axisymmetric Surface Disturbance in Water with a Porous Bottom

International Journal of Applied Mechanics and Engineering ◽

10.2478/ijame-2019-0039 ◽

2019 ◽

Vol 24 (3) ◽

pp. 625-644 ◽

Cited By ~ 1

Author(s):

P. Kundu ◽

B.N. Mandal

Keyword(s):

Free Surface ◽

Asymptotic Form ◽

Velocity Potential ◽

Hankel Transform ◽

Initial Surface ◽

Initial Value ◽

Poisson Problem ◽

Different Types ◽

Surface Disturbance ◽

Infinite Integral

Abstract A two-dimensional Cauchy Poisson problem for water with a porous bottom generated by an axisymmetric initial surface disturbance is investigated here. The problem is formulated as an initial value problem for the velocity potential describing the motion in the fluid. The Laplace and Hankel transform techniques have been used in the mathematical analysis to obtain the form of the free surface in terms of a multiple infinite integral. This integral is then evaluated asymptotically by the method of stationary phase. The asymptotic form of the free surface is depicted graphically in a number of figures for different values of the porosity parameter and for different types of initial disturbances.

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Integrable Models

10.1093/oso/9780198805021.003.0009 ◽

2018 ◽

Author(s):

S. G. Rajeev

Keyword(s):

Fluid Mechanics ◽

Soliton Solution ◽

Water Waves ◽

Heisenberg Model ◽

Kdv Equation ◽

Short Review ◽

Lax Pair ◽

Shallow Water Waves ◽

Initial Value ◽

The Kdv Equation

Some exceptional situations in fluid mechanics can be modeled by equations that are analytically solvable. The most famous example is the Korteweg–de Vries (KdV) equation for shallow water waves in a channel. The exact soliton solution of this equation is derived. The Lax pair formalism for solving the general initial value problem is outlined. Two hamiltonian formalisms for the KdV equation (Fadeev–Zakharov and Magri) are explained. Then a short review of the geometry of curves (Frenet–Serret equations) is given. They are used to derive a remarkably simple equation for the propagation of a kink along a vortex filament. This equation of Hasimoto has surprising connections to the nonlinear Schrödinger equation and to the Heisenberg model of ferromagnetism. An exact soliton solution is found.

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On global dynamics of 2D convective Cahn–Hilliard equation

Boundary Value Problems ◽

10.1186/s13661-020-01477-3 ◽

2020 ◽

Vol 2020 (1) ◽

Author(s):

Xiaopeng Zhao

Keyword(s):

Initial Boundary ◽

Initial Boundary Value Problem ◽

Global Dynamics ◽

Time Behavior ◽

Long Time Behavior ◽

Initial Value ◽

Hilliard Equation ◽

Long Time ◽

Cahn Hilliard Equation ◽

Initial Boundary Value

AbstractIn this paper, we study the long time behavior of solution for the initial-boundary value problem of convective Cahn–Hilliard equation in a 2D case. We show that the equation has a global attractor in $H^{4}(\Omega )$ H 4 ( Ω ) when the initial value belongs to $H^{1}(\Omega )$ H 1 ( Ω ) .

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A Multiple Interval Chebyshev-Gauss-Lobatto Collocation Method for Ordinary Differential Equations

Numerical Mathematics Theory Methods and Applications ◽

10.4208/nmtma.2016.m1429 ◽

2016 ◽

Vol 9 (4) ◽

pp. 619-639 ◽

Cited By ~ 3

Author(s):

Zhong-Qing Wang ◽

Jun Mu

Keyword(s):

Differential Equations ◽

Ordinary Differential Equations ◽

Collocation Method ◽

Initial Value Problems ◽

Numerical Experiments ◽

High Order Accuracy ◽

Nonlinear Ordinary Differential Equations ◽

Initial Value ◽

Order Accuracy ◽

Long Time

AbstractWe introduce a multiple interval Chebyshev-Gauss-Lobatto spectral collocation method for the initial value problems of the nonlinear ordinary differential equations (ODES). This method is easy to implement and possesses the high order accuracy. In addition, it is very stable and suitable for long time calculations. We also obtain thehp-version bound on the numerical error of the multiple interval collocation method underH1-norm. Numerical experiments confirm the theoretical expectations.

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A conservative fully-discrete numerical method for the regularised shallow water wave equations

10.26686/wgtn.14610507.v1 ◽

2021 ◽

Author(s):

Dimitrios Mitsotakis ◽

Hendrik Ranocha ◽

David I Ketcheson ◽

Endre Süli

Keyword(s):

Finite Element ◽

Shallow Water ◽

Numerical Method ◽

Solitary Waves ◽

Water Waves ◽

Wave Equations ◽

Mixed Finite Element ◽

Boussinesq System ◽

Fully Discrete ◽

Long Time

The paper proposes a new, conservative fully-discrete scheme for the numerical solution of the regularised shallow water Boussinesq system of equations in the cases of periodic and reflective boundary conditions. The particular system is one of a class of equations derived recently and can be used in practical simulations to describe the propagation of weakly nonlinear and weakly dispersive long water waves, such as tsunamis. Studies of small-amplitude long waves usually require long-time simulations in order to investigate scenarios such as the overtaking collision of two solitary waves or the propagation of transoceanic tsunamis. For long-time simulations of non-dissipative waves such as solitary waves, the preservation of the total energy by the numerical method can be crucial in the quality of the approximation. The new conservative fully-discrete method consists of a Galerkin finite element method for spatial semidiscretisation and an explicit relaxation Runge--Kutta scheme for integration in time. The Galerkin method is expressed and implemented in the framework of mixed finite element methods. The paper provides an extended experimental study of the accuracy and convergence properties of the new numerical method. The experiments reveal a new convergence pattern compared to standard Galerkin methods.

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Reflexion of water waves by a permeable barrier

Journal of Fluid Mechanics ◽

10.1017/s0022112079001385 ◽

1979 ◽

Vol 95 (1) ◽

pp. 141-157 ◽

Cited By ~ 36

Author(s):

C. Macaskill

Keyword(s):

Water Waves ◽

Water Wave ◽

Finite Depth ◽

Infinite Depth ◽

Permeable Barrier ◽

Impermeable Barrier ◽

Linearized Problem ◽

Special Case ◽

Thin Barrier ◽

Standard Techniques

The linearized problem of water-wave reflexion by a thin barrier of arbitrary permeability is considered with the restriction that the flow be two-dimensional. The formulation includes the special case of transmission through one or more gaps in an otherwise impermeable barrier. The general problem is reduced to a set of integral equations using standard techniques. These equations are then solved using a special decomposition of the finite depth source potential which allows accurate solutions to be obtained economically. A representative range of solutions is obtained numerically for both finite and infinite depth problems.

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Numerical Calculation of the Wave Integrals in the Linearized Theory of Water Waves

Journal of Ship Research ◽

10.5957/jsr.1977.21.1.1 ◽

1977 ◽

Vol 21 (01) ◽

pp. 1-10 ◽

Cited By ~ 1

Author(s):

Hung-Tao Shen ◽

Cesar Farell

Keyword(s):

Water Waves ◽

Velocity Potential ◽

Three Dimensional ◽

Three Dimensional Flow ◽

Integral Term ◽

Linearized Theory ◽

Single Integral ◽

Numerical Computing ◽

Complex Exponential ◽

Derivatives Of

A method for the numerical evaluation of the derivatives of the linearized velocity potential for three-dimensional flow past a unit source submerged in a uniform stream is presented together with a discussion of existing techniques. It is shown in particular that calculation of the double integral term in these functions can be efficiently accomplished in terms of a single integral with the integrand expressed in terms of the complex exponential integral, for which numerical computing techniques are available.

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