A Stability Property of Stochastic Vibration

2007 ◽  
Vol 44 (02) ◽  
pp. 444-457
Author(s):  
M. Elshamy
Keyword(s):  
Wave Equation ◽  
Random Fields ◽  
Stability Property ◽  
Space Variable ◽  
Time Variable ◽  
Initial Condition ◽  
Brownian Sheet ◽  
Random Forcing ◽  
Continuity Properties ◽  
Stochastic Wave Equation

Let u(t,x) be the displacement at time t of a point x on a string; the time variable t varies in the interval I≔[0,T] and the space variable x varies in the interval J≔[0,L], where T and L are fixed positive constants. The displacement u(t,x) is the solution to a stochastic wave equation. Two forms of random excitations are considered, a white noise in the initial condition and a nonlinear random forcing which involves the formal derivative of a Brownian sheet. In this article, we consider the continuity properties of solutions to this equation. Smoothness characteristics of these random fields, in terms of Hölder continuity, are also investigated.

scholarly journals A Stability Property of Stochastic Vibration

2007 ◽  
Vol 44 (02) ◽  
pp. 444-457
Author(s):  
M. Elshamy
Keyword(s):  
Wave Equation ◽  
Random Fields ◽  
Stability Property ◽  
Space Variable ◽  
Time Variable ◽  
Initial Condition ◽  
Brownian Sheet ◽  
Random Forcing ◽  
Continuity Properties ◽  
Stochastic Wave Equation

Let u(t,x) be the displacement at time t of a point x on a string; the time variable t varies in the interval I≔[0,T] and the space variable x varies in the interval J≔[0,L], where T and L are fixed positive constants. The displacement u(t,x) is the solution to a stochastic wave equation. Two forms of random excitations are considered, a white noise in the initial condition and a nonlinear random forcing which involves the formal derivative of a Brownian sheet. In this article, we consider the continuity properties of solutions to this equation. Smoothness characteristics of these random fields, in terms of Hölder continuity, are also investigated.

scholarly journals A Stability Property of Stochastic Vibration

2007 ◽  
Vol 44 (2) ◽  
pp. 444-457
Author(s):  
M. Elshamy
Keyword(s):  
Wave Equation ◽  
Random Fields ◽  
Stability Property ◽  
Space Variable ◽  
Time Variable ◽  
Initial Condition ◽  
Brownian Sheet ◽  
Random Forcing ◽  
Continuity Properties ◽  
Stochastic Wave Equation

Let u(t,x) be the displacement at time t of a point x on a string; the time variable t varies in the interval I≔[0,T] and the space variable x varies in the interval J≔[0,L], where T and L are fixed positive constants. The displacement u(t,x) is the solution to a stochastic wave equation. Two forms of random excitations are considered, a white noise in the initial condition and a nonlinear random forcing which involves the formal derivative of a Brownian sheet. In this article, we consider the continuity properties of solutions to this equation. Smoothness characteristics of these random fields, in terms of Hölder continuity, are also investigated.

Stochastic models of damped vibrations

1996 ◽  
Vol 33 (4) ◽  
pp. 1159-1168 ◽  
Author(s):  
Maged Elshamy
Keyword(s):  
Random Fields ◽  
Space Variable ◽  
Displacement Function ◽  
Time Variable ◽  
Stochastic Disturbances ◽  
Large Deviations Principle ◽  
Stochastic Perturbations ◽  
Random Forcing ◽  
Non Gaussian ◽  
Random Fluctuations

In this article we study stochastic perturbations of partial differential equations describing forced-damped vibrations of a string. Two models of such stochastic disturbances are considered; one is triggered by an initial white noise, and the other is in the form of non-Gaussian random forcing. Let uε (t, x) be the displacement at time t of a point x on a string, where the time variable t ≧ 0, and the space variable . The small parameter ε controls the intensity of the random fluctuations. The random fields uε (t, x) are shown to satisfy a large deviations principle, and the random deviations of the unperturbed displacement function are analyzed as the noise parameter ε tends to zero.

Stochastic models of damped vibrations

1996 ◽  
Vol 33 (04) ◽  
pp. 1159-1168 ◽  
Author(s):  
Maged Elshamy
Keyword(s):  
Random Fields ◽  
Space Variable ◽  
Displacement Function ◽  
Time Variable ◽  
Stochastic Disturbances ◽  
Large Deviations Principle ◽  
Stochastic Perturbations ◽  
Random Forcing ◽  
Non Gaussian ◽  
Random Fluctuations

In this article we study stochastic perturbations of partial differential equations describing forced-damped vibrations of a string. Two models of such stochastic disturbances are considered; one is triggered by an initial white noise, and the other is in the form of non-Gaussian random forcing. Let uε (t, x) be the displacement at time t of a point x on a string, where the time variable t ≧ 0, and the space variable . The small parameter ε controls the intensity of the random fluctuations. The random fields uε (t, x) are shown to satisfy a large deviations principle, and the random deviations of the unperturbed displacement function are analyzed as the noise parameter ε tends to zero.

Randomly perturbed vibrations

1995 ◽  
Vol 32 (02) ◽  
pp. 417-428 ◽  
Author(s):  
M. Elshamy
Keyword(s):  
Dirichlet Boundary ◽  
Space Variable ◽  
Initial Conditions ◽  
Linear Wave ◽  
Stochastic Perturbations ◽  
Uniform Topology ◽  
Linear Wave Equation ◽  
Random Forcing ◽  
Continuity Properties ◽  
Positive Time

Let u ε(t, x) be the position at time t of a point x on a string, where the time variable t varies in an interval I: = [0, T], T is a fixed positive time, and the space variable x varies in an interval J. The string is performing forced vibrations and also under the influence of small stochastic perturbations of intensity ε. We consider two kinds of random perturbations, one in the form of initial white noise, and the other is a nonlinear random forcing which involves the formal derivative of a Brownian sheet. When J has finite endpoints, a Dirichlet boundary condition is imposed for the solutions of the resulting non-linear wave equation. Assuming that the initial conditions are of sufficient regularity, we analyze the deviations u ε(t, x) from u 0(t, x), the unperturbed position function, as the intensity of perturbation ε ↓ 0 in the uniform topology. We also discuss some continuity properties of the realization of the solutions u ε(t, x).

Randomly perturbed vibrations

1995 ◽  
Vol 32 (2) ◽  
pp. 417-428 ◽  
Author(s):  
M. Elshamy
Keyword(s):  
Dirichlet Boundary ◽  
Space Variable ◽  
Initial Conditions ◽  
Linear Wave ◽  
Stochastic Perturbations ◽  
Uniform Topology ◽  
Linear Wave Equation ◽  
Random Forcing ◽  
Continuity Properties ◽  
Positive Time

Let uε(t, x) be the position at time t of a point x on a string, where the time variable t varies in an interval I: = [0, T], T is a fixed positive time, and the space variable x varies in an interval J. The string is performing forced vibrations and also under the influence of small stochastic perturbations of intensity ε. We consider two kinds of random perturbations, one in the form of initial white noise, and the other is a nonlinear random forcing which involves the formal derivative of a Brownian sheet. When J has finite endpoints, a Dirichlet boundary condition is imposed for the solutions of the resulting non-linear wave equation. Assuming that the initial conditions are of sufficient regularity, we analyze the deviations uε(t, x) from u0(t, x), the unperturbed position function, as the intensity of perturbation ε ↓ 0 in the uniform topology. We also discuss some continuity properties of the realization of the solutions uε(t, x).

scholarly journals Depth-Extrapolation-Based True-Amplitude Full-Wave-Equation Migration from Topography

2021 ◽  
Vol 11 (7) ◽  
pp. 3010
Author(s):  
Hao Liu ◽  
Xuewei Liu
Keyword(s):  
Wave Equation ◽  
Partial Derivative ◽  
Model Experiment ◽  
Surface Velocity ◽  
Seismic Exploration ◽  
Initial Condition ◽  
Reverse Time ◽  
Reverse Time Migration ◽  
Full Wave ◽  
Time Migration

The lack of an initial condition is one of the major challenges in full-wave-equation depth extrapolation. This initial condition is the vertical partial derivative of the surface wavefield and cannot be provided by the conventional seismic acquisition system. The traditional solution is to use the wavefield value of the surface to calculate the vertical partial derivative by assuming that the surface velocity is constant. However, for seismic exploration on land, the surface velocity is often not uniform. To solve this problem, we propose a new method for calculating the vertical partial derivative from the surface wavefield without making any assumptions about the surface conditions. Based on the calculated derivative, we implemented a depth-extrapolation-based full-wave-equation migration from topography using the direct downward continuation. We tested the imaging performance of our proposed method with several experiments. The results of the Marmousi model experiment show that our proposed method is superior to the conventional reverse time migration (RTM) algorithm in terms of imaging accuracy and amplitude-preserving performance at medium and deep depths. In the Canadian Foothills model experiment, we proved that our method can still accurately image complex structures and maintain amplitude under topographic scenario.

scholarly journals Сlassical solution of the mixed problem for the one-dimensional wave equation with the nonsmooth second initial condition

2020 ◽  
Vol 64 (6) ◽  
pp. 657-662
Author(s):  
V. I. Korzyuk ◽  
J. V. Rudzko
Keyword(s):  
Boundary Condition ◽  
Wave Equation ◽  
Mixed Problem ◽  
Method Of Characteristics ◽  
Smooth Boundary ◽  
Analytical Form ◽  
Initial Condition ◽  
One Dimensional ◽  
The One ◽  
Dimensional Wave

In this article, we study the classical solution of the mixed problem in a quarter of a plane and a half-plane for a one-dimensional wave equation. On the bottom of the boundary, Cauchy conditions are specified, and the second of them has a discontinuity of the first kind at one point. Smooth boundary condition is set at the side boundary. The solution is built using the method of characteristics in an explicit analytical form. Uniqueness is proved and conditions are established under which a piecewise-smooth solution exists. The problem with linking conditions is considered.

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