On the Stability of Evolution Galerkin Schemes Applied to a Two‐Dimensional Wave Equation System

Boundary control of two-dimensional wave equation on the rectangle is considered in this paper. Boundary controllers are designed through backstepping method. Stabilization of the closed system is obtained under the controllers.

Download Full-text

Construction and optimization techniques for high order schemes for the two-dimensional wave equation

10.1121/1.4799230 ◽

2013 ◽

Cited By ~ 2

Author(s):

Stefan Bilbao ◽

Brian Hamilton

Keyword(s):

Wave Equation ◽

High Order ◽

Optimization Techniques ◽

Two Dimensional ◽

High Order Schemes ◽

Dimensional Wave

Download Full-text

Stability results for an elastic–viscoelastic wave equation with localized Kelvin–Voigt damping and with an internal or boundary time delay

Asymptotic Analysis ◽

10.3233/asy-201649 ◽

2020 ◽

pp. 1-57

Author(s):

Mouhammad Ghader ◽

Rayan Nasser ◽

Ali Wehbe

Keyword(s):

Wave Equation ◽

Time Delay ◽

Frequency Domain ◽

Viscoelastic Damping ◽

One Dimensional ◽

Viscoelastic Wave ◽

Smooth Coefficients ◽

The Stability ◽

Stability Results ◽

Dimensional Wave

We investigate the stability of a one-dimensional wave equation with non smooth localized internal viscoelastic damping of Kelvin–Voigt type and with boundary or localized internal delay feedback. The main novelty in this paper is that the Kelvin–Voigt and the delay damping are both localized via non smooth coefficients. Under sufficient assumptions, in the case that the Kelvin–Voigt damping is localized faraway from the tip and the wave is subjected to a boundary delay feedback, we prove that the energy of the system decays polynomially of type t − 4 . However, an exponential decay of the energy of the system is established provided that the Kelvin–Voigt damping is localized near a part of the boundary and a time delay damping acts on the second boundary. While, when the Kelvin–Voigt and the internal delay damping are both localized via non smooth coefficients near the boundary, under sufficient assumptions, using frequency domain arguments combined with piecewise multiplier techniques, we prove that the energy of the system decays polynomially of type t − 4 . Otherwise, if the above assumptions are not true, we establish instability results.

Download Full-text

A study of the stability of dynamically stable breakwaters

Canadian Journal of Civil Engineering ◽

10.1139/l91-113 ◽

1991 ◽

Vol 18 (6) ◽

pp. 916-925 ◽

Cited By ~ 14

Author(s):

K. R. Hall ◽

Joseph S. Kao

Keyword(s):

Research Laboratory ◽

Coastal Engineering ◽

Irregular Waves ◽

Two Dimensional ◽

Uniformity Coefficient ◽

Engineering Research ◽

Hydraulic Models ◽

Wave Flume ◽

The Stability ◽

Dimensional Wave

The effect of gradation of armour stones and the amount of rounded stones in the armour on dynamically stable breakwaters was assessed in a two-dimensional wave flume. A total of 52 series of tests were undertaken at the Coastal Engineering Research Laboratory of Queen's University, Kingston, Canada using irregular waves. Profiles of the structure during the various stages of reshaping were measured using a semiautomatic profiler developed for this study. Four gradations of armour stones were used, giving a range in uniformity coefficient of 1.35–5.4. The volume of stones and the initial berm width required for the development of a stable profile, along with the extent to which the toe of the structure progressed seaward, were chosen as representative parameters of the reshaped breakwater. The results indicated that the toe width formed as a result of reshaping and the area of stones required for reshaping were dependent on the gradation of the armour stones. The initial berm width required for reshaping was also found to be dependent on the gradation and the percentage of rounded stones in the armour. Key words: breakwaters, dynamic stability, hydraulic models, stability, armour stones.

Download Full-text

The development of three-dimensional wave-packets in unbounded parallel flows

Journal of Fluid Mechanics ◽

10.1017/s0022112068001047 ◽

1968 ◽

Vol 32 (4) ◽

pp. 801-808 ◽

Cited By ~ 20

Author(s):

M. Gaster ◽

A. Davey

Keyword(s):

Wave Packet ◽

Three Dimensional ◽

Two Dimensional ◽

Mean Flow ◽

Physical Plane ◽

Parallel Flows ◽

Flow Velocity Profile ◽

The Stability ◽

Special Case ◽

Dimensional Wave

In this paper we examine the stability of a two-dimensional wake profile of the form u(y) = U∞(1 – r e-sy2) with respect to a pulsed disturbance at a point in the fluid. The disturbed flow forms an expanding wave packet which is convected downstream. Far downstream, where asymptotic expansions are valid, the motion at any point in the wave packet is described by a particular three-dimensional wave having complex wave-numbers. In the special case of very unstable flows, where viscosity does not have a significant influence, it is possible to evaluate the three-dimensional eigenvalues in terms of two-dimensional ones using the inviscid form of Squire's transformation. In this way each point in the physical plane can be linked to a particular two-dimensional wave growing in both space and time by simple algebraic expressions which are independent of the mean flow velocity profile. Computed eigenvalues for the wake profile are used in these relations to find the behaviour of the wave packet in the physical plane.

Download Full-text