Refraction and Diffraction of Nonlinear Waves in Coastal Waters by the Level I Green-Naghdi Equations

2003 ◽  
Author(s):  
R. C. Ertekin ◽  
H. Sundararaghavan
Keyword(s):  
Coastal Waters ◽  
Water Waves ◽  
Wave Equations ◽  
Diffraction Problem ◽  
Initial Boundary ◽  
Initial Boundary Value Problem ◽  
Nonlinear Wave Propagation ◽  
Shallow Water Wave Equations ◽  
Level I ◽  
Initial Boundary Value

The Green-Naghdi (GN) equations are shallow-water wave equations which can be solved to predict the nonlinear and dispersive effects of water waves propagating in coastal waters. In this study, we formulate the Level I GN equations for variable bathymetry in 3-D, and numerically model the refraction-diffraction problem as an initial-boundary-value problem. A finite-difference model, in conjunction with elliptic grid-generation, is developed to solve the GN equations. Nonlinear wave propagation over a varying bathymetry is presented for solitary and cnoidal waves. A number of benchmark cases have been tested and compared with the available theoretical predictions. The results are presented and discussed to reveal the accuracy of the present model.

scholarly journals Existence and decay of finite energy solutions for semilinear dissipative wave equations in time-dependent domains

2020 ◽  
Vol 40 (6) ◽  
pp. 725-736
Author(s):  
Mitsuhiro Nakao
Keyword(s):  
Existence And Uniqueness ◽  
Wave Equations ◽  
Initial Boundary ◽  
Initial Boundary Value Problem ◽  
Finite Energy ◽  
Time Dependent ◽  
Energy Solution ◽  
Noncylindrical Domain ◽  
Finite Energy Solution ◽  
Initial Boundary Value

We consider the initial-boundary value problem for semilinear dissipative wave equations in noncylindrical domain \(\bigcup_{0\leq t \lt\infty} \Omega(t)\times\{t\} \subset \mathbb{R}^N\times \mathbb{R}\). We are interested in finite energy solution. We derive an exponential decay of the energy in the case \(\Omega(t)\) is bounded in \(\mathbb{R}^N\) and the estimate \[\int\limits_0^{\infty} E(t)dt \leq C(E(0),\|u(0)\|)< \infty\] in the case \(\Omega(t)\) is unbounded. Existence and uniqueness of finite energy solution are also proved.

Stability for a nonlinear coupled system of elasticity and thermoelasticity with second sound

2014 ◽  
Vol 13 (01) ◽  
pp. 45-75 ◽  
Author(s):  
Yi-Ping Meng ◽  
Ya-Guang Wang
Keyword(s):  
Wave Equations ◽  
Coupled System ◽  
Initial Boundary ◽  
Initial Boundary Value Problem ◽  
Middle Part ◽  
Second Sound ◽  
Qualitative Properties ◽  
Thermal Change ◽  
Nonlinear Coupled System ◽  
Initial Boundary Value

In this paper, we study the qualitative properties of solutions to a nonlinear system describing the motion of a bar in which the middle part is sensitive to the thermal change, while the outer parts are insensible. By the energy method, we show that the initial boundary value problem for this coupled system of wave equations and thermoelastic equations with second sound in one space variable is well-posed globally in time, and it is also stable exponentially as the time goes to infinity when the wave speed of the outer parts is properly large, under certain restrictions on the initial data and the growth rate of the nonlinear terms.

scholarly journals Global Existence and Asymptotic Behavior of Solutions for a Class of Nonlinear Degenerate Wave Equations

2007 ◽  
Vol 2007 ◽  
pp. 1-9 ◽  
Author(s):  
Yaojun Ye
Keyword(s):  
Asymptotic Behavior ◽  
Wave Equations ◽  
Decay Estimate ◽  
Global Solutions ◽  
Initial Boundary ◽  
Initial Boundary Value Problem ◽  
Asymptotic Behavior Of Solutions ◽  
Potential Well ◽  
Initial Boundary Value ◽  
Nonlinear Degenerate

This paper studies the existence of global solutions to the initial-boundary value problem for some nonlinear degenerate wave equations by means of compactness method and the potential well idea. Meanwhile, we investigate the decay estimate of the energy of the global solutions to this problem by using a difference inequality.

Global well-posedness for a class of fourth-order nonlinear strongly damped wave equations

2019 ◽  
Vol 0 (0) ◽  
Author(s):  
Wei Lian ◽  
Vicenţiu D. Rădulescu ◽  
Runzhang Xu ◽  
Yanbing Yang ◽  
Nan Zhao
Keyword(s):  
Fourth Order ◽  
Wave Equations ◽  
Initial Boundary ◽  
Initial Boundary Value Problem ◽  
Point Theory ◽  
Initial Energy ◽  
Damping Term ◽  
Blowup Of Solutions ◽  
Positive Initial Energy ◽  
Initial Boundary Value

Abstract In this paper, we consider the initial boundary value problem for a class of fourth-order wave equations with strong damping term, nonlinear weak damping term, strain term and nonlinear source term in polynomial form. First, the local solution is obtained by using fix point theory. Then, by constructing the potential well structure frame, we get the global existence, asymptotic behavior and blowup of solutions for the subcritical initial energy and critical initial energy respectively. Ultimately, we prove the blowup in finite time of solutions for the arbitrarily positive initial energy case.

A mathematical model for long waves generated by wavemakers in non-linear dispersive systems

1973 ◽  
Vol 73 (2) ◽  
pp. 391-405 ◽  
Author(s):  
J. L. Bona ◽  
P. J. Bryant
Keyword(s):  
Mathematical Model ◽  
Initial Data ◽  
Water Waves ◽  
Open Channel ◽  
Initial Boundary ◽  
Initial Boundary Value Problem ◽  
Finite Amplitude ◽  
Mathematical Properties ◽  
Image Position ◽  
Initial Boundary Value

An initial-boundary-value problem for the equationis considered for x, t ≥ 0. This system is a model for long water waves of small but finite amplitude, generated in a uniform open channel by a wavemaker at one end. It is shown that, in contrast to an alternative, more familiar model using the Korteweg–deVries equation, the solution of (a) has good mathematical properties: in particular, the problem is well set in Hadamard's classical sense that solutions corresponding to given initial data exist, are unique, and depend continuously on the specified data.

Non-existence of global solutions for a class of wave equations with nonlinear damping and source terms

2011 ◽  
Vol 141 (4) ◽  
pp. 865-880 ◽  
Author(s):  
Shun-Tang Wu
Keyword(s):  
Wave Equations ◽  
Global Solutions ◽  
Initial Boundary ◽  
Initial Boundary Value Problem ◽  
Nonlinear Damping ◽  
Critical Value ◽  
Initial Energy ◽  
Source Terms ◽  
Initial Boundary Value ◽  
Damping And Source Terms

An initial–boundary-value problem for a class of wave equations with nonlinear damping and source terms in a bounded domain is considered. We establish the non-existence result of global solutions with the initial energy controlled above by a critical value via the method introduced in a work by Autuori et al. in 2010. This improves the 2009 result of Liu and Wang.

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