scholarly journals Transition from Discrete to Continuous Media: The Impact of Symmetry Changes on Asymptotic Behavior of Waves

Symmetry ◽  
2021 ◽  
Vol 13 (6) ◽  
pp. 1008
Author(s):  
Igor Andrianov ◽  
Steve Koblik ◽  
Galina Starushenko
Keyword(s):  
Asymptotic Behavior ◽  
Analytical Solution ◽  
Wave Motion ◽  
Initial Value ◽  
Phase And Group Velocities ◽  
The Impact ◽  
Group Velocities ◽  
Continuous Approximations ◽  
Initial Motion ◽  
Discrete Medium

This paper is devoted to comparing the asymptotics of a solution, describing the wave motion of a discrete lattice and its continuous approximations. The transition from a discrete medium to a continuous one changes the symmetry of the system. The influence of this change on the asymptotic behavior of waves is of great interest. For the discrete case, Schrödinger’s analytical solution of the initial-value problem for the Lagrange lattice is used. Various continuous approximations are proposed to approximate the lattice. They are based on Debye’s concept of quasicontinuum. The asymptotics of the initial motion and the behavior of the systems in the vicinity of the quasifront and at large times are compared. The approximations of phase and group velocities is analyzed. The merits and limitations of the described approaches are discussed.

On the Influence of External Fields on the Acoustic Characteristics of Phononic Crystals

2016 ◽  
Author(s):  
Stephan Rudykh ◽  
Pavel Galich
Keyword(s):  
Analytical Solution ◽  
Elastic Wave ◽  
Small Amplitude ◽  
Phononic Crystals ◽  
Acoustic Characteristic ◽  
External Fields ◽  
Acoustic Characteristics ◽  
Deformed State ◽  
Phase And Group Velocities ◽  
Group Velocities

In this work, we consider elastic wave prorogation in deformable microstructured materials. In particular, we focus on the influence of externally applied deformation on the acoustic characteristic of periodic layered phononic crystals. Based on an analytical solution for the finitely deformed layered media, an analysis of the superimposed on the finitely deformed state small amplitude motions is performed. The analysis provides the information about the important acoustic characteristics such as phase and group velocities, illustrated by the slowness and energy curves. The influence of deformation on these characteristics of phononic crystals is investigated, showing the strong tunability of the phononic crystals by applied deformations.

COVID-19: Time-Dependent Effective Reproduction Number and Sub-notification Effect Estimation Modeling (Preprint)

2020 ◽  
Author(s):  
Eduardo Atem De Carvalho ◽  
Rogerio Atem De Carvalho
Keyword(s):  
New York ◽  
Reproduction Number ◽  
Moving Average ◽  
Real Data ◽  
Time Dependent ◽  
Initial Value ◽  
Health Authorities ◽  
Reproduction Numbers ◽  
Effect Estimation ◽  
The Impact

BACKGROUND Since the beginning of the COVID-19 pandemic, researchers and health authorities have sought to identify the different parameters that govern their infection and death cycles, in order to be able to make better decisions. In particular, a series of reproduction number estimation models have been presented, with different practical results. OBJECTIVE This article aims to present an effective and efficient model for estimating the Reproduction Number and to discuss the impacts of sub-notification on these calculations. METHODS The concept of Moving Average Method with Initial value (MAMI) is used, as well as a model for Rt, the Reproduction Number, is derived from experimental data. The models are applied to real data and their performance is presented. RESULTS Analyses on Rt and sub-notification effects for Germany, Italy, Sweden, United Kingdom, South Korea, and the State of New York are presented to show the performance of the methods here introduced. CONCLUSIONS We show that, with relatively simple mathematical tools, it is possible to obtain reliable values for time-dependent, incubation period-independent Reproduction Numbers (Rt). We also demonstrate that the impact of sub-notification is relatively low, after the initial phase of the epidemic cycle has passed.

scholarly journals On the sub–diffusion fractional initial value problem with time variable order

2021 ◽  
Vol 10 (1) ◽  
pp. 1301-1315
Author(s):  
Eduardo Cuesta ◽  
Mokhtar Kirane ◽  
Ahmed Alsaedi ◽  
Bashir Ahmad
Keyword(s):  
Asymptotic Behavior ◽  
Fractional Derivative ◽  
Initial Value Problem ◽  
Regularity Result ◽  
Time Variable ◽  
Variable Order ◽  
Initial Value ◽  
Integration By Parts ◽  
Integration By Parts Formula ◽  
Variable Order Fractional Derivative

Abstract We consider a fractional derivative with order varying in time. Then, we derive for it a Leibniz' inequality and an integration by parts formula. We also study an initial value problem with our time variable order fractional derivative and present a regularity result for it, and a study on the asymptotic behavior.

scholarly journals Structural Changes in French VF Treatment Wetland Porous Media during the Rest Period: An Ex Situ Study Using X-ray Tomography

Water ◽  
2021 ◽  
Vol 13 (3) ◽  
pp. 389
Author(s):  
German Dario Martinez-Carvajal ◽  
Laurent Oxarango ◽  
Jérôme Adrien ◽  
Pascal Molle ◽  
Nicolas Forquet
Keyword(s):  
Porous Media ◽  
Structural Changes ◽  
Rest Period ◽  
Treatment Wetlands ◽  
Ex Situ ◽  
Initial Value ◽  
Treatment Wetland ◽  
X Ray ◽  
The Impact ◽  
Deposit Layer

Clogging constitutes a major operational issue for treatment wetlands. The rest period is a key feature of French Vertical Flow (VF) treatment wetlands and serves to mitigate clogging. An ex-situ drying experiment was performed to mimic the rest period and record structural changes in the porous media using X-ray Computed Tomography (CT). Samples containing the deposit and gravel layers of a first stage French VF treatment wetland were extracted and left to dry in a control environment. Based on CT scans, three phases were identified (voids, biosolids, and gravels). The impact of the rest period was assessed by means of different pore-scale variables. Ultimately, the volume of biosolids had reduced to 58% of its initial value, the deposit layer thickness dropped to 68% of its initial value, and the void/biosolid specific surface area ratio increased from a minimum value of 1.1 to a maximum of 4.2. Cracks greater than 3 mm developed at the uppermost part of the deposit layer, while, in the gravel layer, the rise in void volume corresponds to pores smaller than 2 mm in diameter. Lastly, the air-filled microporosity is estimated to have increased by 0.11 v/v.

Asymptotic derivation of a refined equation for an elastic beam resting on a Winkler foundation

2021 ◽  
pp. 108128652110238
Author(s):  
Barış Erbaş ◽  
Julius Kaplunov ◽  
Isaac Elishakoff
Keyword(s):  
Ad Hoc ◽  
Moving Load ◽  
Low Frequency ◽  
Elastic Beam ◽  
Winkler Foundation ◽  
One Dimensional ◽  
Beam Equations ◽  
Dimensional Equation ◽  
Phase And Group Velocities ◽  
Group Velocities

A two-dimensional mixed problem for a thin elastic strip resting on a Winkler foundation is considered within the framework of plane stress setup. The relative stiffness of the foundation is supposed to be small to ensure low-frequency vibrations. Asymptotic analysis at a higher order results in a one-dimensional equation of bending motion refining numerous ad hoc developments starting from Timoshenko-type beam equations. Two-term expansions through the foundation stiffness are presented for phase and group velocities, as well as for the critical velocity of a moving load. In addition, the formula for the longitudinal displacements of the beam due to its transverse compression is derived.

Compressional‐wave velocities in attenuating media: A laboratory physical model study

Geophysics ◽  
2000 ◽  
Vol 65 (4) ◽  
pp. 1162-1167 ◽  
Author(s):  
Joseph B. Molyneux ◽  
Douglas R. Schmitt
Keyword(s):  
Phase Velocity ◽  
Group Velocity ◽  
Wave Velocity ◽  
Propagation Distance ◽  
Compressional Wave ◽  
Arrival Times ◽  
Wave Velocities ◽  
Amplitude Maximum ◽  
Phase And Group Velocities ◽  
Group Velocities

Elastic‐wave velocities are often determined by picking the time of a certain feature of a propagating pulse, such as the first amplitude maximum. However, attenuation and dispersion conspire to change the shape of a propagating wave, making determination of a physically meaningful velocity problematic. As a consequence, the velocities so determined are not necessarily representative of the material’s intrinsic wave phase and group velocities. These phase and group velocities are found experimentally in a highly attenuating medium consisting of glycerol‐saturated, unconsolidated, random packs of glass beads and quartz sand. Our results show that the quality factor Q varies between 2 and 6 over the useful frequency band in these experiments from ∼200 to 600 kHz. The fundamental velocities are compared to more common and simple velocity estimates. In general, the simpler methods estimate the group velocity at the predominant frequency with a 3% discrepancy but are in poor agreement with the corresponding phase velocity. Wave velocities determined from the time at which the pulse is first detected (signal velocity) differ from the predominant group velocity by up to 12%. At best, the onset wave velocity arguably provides a lower bound for the high‐frequency limit of the phase velocity in a material where wave velocity increases with frequency. Each method of time picking, however, is self‐consistent, as indicated by the high quality of linear regressions of observed arrival times versus propagation distance.

Adaptive Predictive Based on Equal-Dimension and New Information for the Hydraulic Mechanism of Wave Motion Compensating Platform

2010 ◽  
Vol 20-23 ◽  
pp. 236-242 ◽  
Author(s):  
Zhi Gang Zeng ◽  
Guo Hua Chen
Keyword(s):  
Predictive Control ◽  
Wave Motion ◽  
Control Policy ◽  
Least Square ◽  
Problem Solution ◽  
New Information ◽  
Hydraulic Mechanism ◽  
Heave Motion ◽  
Adaptive Predictive Control ◽  
The Impact

A wave motion compensating platform has the function of compensating the ship’s generalized heave motion (a coupling result of roll, pitch and heave). It can decrease the impact of ship motion on some sea works and equipments. The hydraulic mechanism of platform system has the characteristics of nonlinear and big inertia. In order to compensate generalized heave motion effectively, an adaptive predictive control policy is used for controlling the hydraulic mechanism. Based on equal-dimension and new information, an automation regressive model can get adaptive multi-step prediction. The model parameter estimation based on the least square algorithm is easy to blow up and be unstable when the system has random noise. To improve the problem solution, a damped recursive least square algorithm is proposed to estimate the parameters on line. For the short regulation time, strong anti-disturbance ability and great robustness, a nonlinear PID controller whose gain parameters vary with errors is suitable for controlling the hydraulic mechanism. Using the collected experimental data, the simulations suggest that adopting the above adaptive predictive control policy to control hydraulic mechanism is able to decrease the generalized heave amplitude of wave motion compensating platform.

Errors Introduced in Estimation of Surface Wave Phase and Group Velocities Due to Wrong Assumptions: An Assessment Using a Simple Model For Love Wave

2021 ◽  
Author(s):  
Akash Kharita ◽  
Sagarika Mukhopadhyay
Keyword(s):  
Surface Wave ◽  
Surface Waves ◽  
Love Wave ◽  
Arrival Time ◽  
Epicentral Distance ◽  
Horizontal Distance ◽  
Wave Analysis ◽  
Time Period ◽  
Phase And Group Velocities ◽  
Group Velocities

<p>The surface wave phase and group velocities are estimated by dividing the epicentral distance by phase and group travel times respectively in all the available methods, this is based on the assumptions that (1) surface waves originate at the epicentre and (2) the travel time of the particular group or phase of the surface wave is equal to its arrival time to the station minus the origin time of the causative earthquake; However, both assumptions are wrong since surface waves generate at some horizontal distance away from the epicentre. We calculated the actual horizontal distance from the focus at which they generate and assessed the errors caused in the estimation of group and phase velocities by the aforementioned assumptions in a simple isotropic single layered homogeneous half space crustal model using the example of the fundamental mode Love wave. We took the receiver locations in the epicentral distance range of 100-1000 km, as used in the regional surface wave analysis, varied the source depth from 0 to 35 Km with a step size of 5 km and did the forward modelling to calculate the arrival time of Love wave phases at each receiver location. The phase and group velocities are then estimated using the above assumptions and are compared with the actual values of the velocities given by Love wave dispersion equation. We observed that the velocities are underestimated and the errors are found to be; decreasing linearly with focal depth, decreasing inversely with the epicentral distance and increasing parabolically with the time period. We also derived empirical formulas using MATLAB curve fitting toolbox that will give percentage errors for any realistic combination of epicentral distance, time period and depths of earthquake and thickness of layer in this model. The errors are found to be more than 5% for all epicentral distances lesser than 500 km, for all focal depths and time periods indicating that it is not safe to do regional surface wave analysis for epicentral distances lesser than 500 km without incurring significant errors. To the best of our knowledge, the study is first of its kind in assessing such errors.</p>

Free spheroidal vibrations of the earth at very long periods, Part I— Calculation of periods for several earth models

1963 ◽  
Vol 53 (3) ◽  
pp. 483-501 ◽  
Author(s):  
Leonard E. Alsop
Keyword(s):  
Inner Core ◽  
Computer Programs ◽  
Free Vibrations ◽  
Stoneley Waves ◽  
The Core ◽  
The Earth ◽  
Earth Models ◽  
Shear Modes ◽  
Phase And Group Velocities ◽  
Group Velocities

Abstract Periods of free vibrations of the spheroidal type have been calculated numerically on an IBM 7090 for the fundamental and first two shear modes for periods greater than about two hundred seconds. Calculations were made for four different earth models. Phase and group velocities were also computed and are tabulated herein for the first two shear modes. The behavior of particle motions for different modes is discussed. In particular, particle motions for the two shear modes indicate that they behave in some period ranges like Stoneley waves tied to the core-mantle interface. Calculations have been made also for a model which presumes a solid inner core and will be discussed in Part II. The two computer programs which were made for these calculations are described briefly.

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