Comparison of the Effects of Density and Sound Speed Variation in the Propagation of Planar Transient Waves

1992 ◽  
Vol 114 (3) ◽  
pp. 409-414
Author(s):  
J. H. Ginsberg
Keyword(s):  
Differential Equations ◽  
Material Properties ◽  
Sound Speed ◽  
Heterogeneous Media ◽  
Numerical Procedure ◽  
Mean Value ◽  
The Other ◽  
Transient Waves ◽  
One Dimensional ◽  
Spatial Grid

When expressed in the form of characteristic differential equations, the laws governing propagation of linear one-dimensional waves through heterogeneous media show that the only properties of significance are the sound speed c and the acoustic impedance ρc, either of which may vary spatially. The former occurs in the differential equations governing the (curved) characteristics, while the latter appears in the differential equations governing the evolution of particle velocity and stress along the characteristics. The present study employs an inherently stable finite difference representation of the characteristic equations, in which the spatial grid is obtained by evaluating the intersections in space-time of constant time lines with comparable increments of the characteristic variables. The numerical procedure is used to follow the propagation of a single-lobe sine pulse in cases where only ρ or c fluctuates spatially about a mean value while the other property is constant, and compares those results to the case were both material properties vary. Nonconstancy of c is shown to cause temporal shifts in waveforms, while spatial variation of ρc causes attenuation and distortion of the waveform.

scholarly journals Algebraic approach for the one-dimensional Dirac–Dunkl oscillator

2020 ◽  
Vol 35 (31) ◽  
pp. 2050255
Author(s):  
D. Ojeda-Guillén ◽  
R. D. Mota ◽  
M. Salazar-Ramírez ◽  
V. D. Granados
Keyword(s):  
Energy Spectrum ◽  
Irreducible Representation ◽  
Differential Equations ◽  
Representation Theory ◽  
Algebraic Approach ◽  
The Other ◽  
One Dimensional ◽  
The One

We extend the (1 + 1)-dimensional Dirac–Moshinsky oscillator by changing the standard derivative by the Dunkl derivative. We demonstrate in a general way that for the Dirac–Dunkl oscillator be parity invariant, one of the spinor component must be even, and the other spinor component must be odd, and vice versa. We decouple the differential equations for each of the spinor component and introduce an appropriate su(1, 1) algebraic realization for the cases when one of these functions is even and the other function is odd. The eigenfunctions and the energy spectrum are obtained by using the su(1, 1) irreducible representation theory. Finally, by setting the Dunkl parameter to vanish, we show that our results reduce to those of the standard Dirac-Moshinsky oscillator.

Transient Waves in a Class of Random Heterogeneous Media

1991 ◽  
Vol 44 (11S) ◽  
pp. S199-S209 ◽  
Author(s):  
Martin Ostoja-Starzewski
Keyword(s):  
Heterogeneous Media ◽  
Signal To Noise Ratio ◽  
Markov Property ◽  
Short Review ◽  
High Signal ◽  
Signal To Noise ◽  
Transient Waves ◽  
One Dimensional ◽  
Nonlinear Elastic ◽  
Transient Dynamic Response

A stochastic method is developed for analysis of transient waves propagating in one-dimensional random granular-type media. The method is suited to study transient dynamic response of nonlinear microstructures with material randomess, of high signal-to-noise ratio, being present in constitutive moduli and grain lengths. It generalizes the classical solution techniques, based on the theory of characteristics, by taking advantage of the Markov property of the forward propagaing disturbances. Pulses propagating in bilinear elastic, nonlinear elastic, and linear-hysteretic media are studied. Additionally, a short review is given of an investigation of acceleration wavefronts making a transition into shocks in random nonlinear elastic/dissipative continua, where Markov property can again be exploited.

Solving One-Dimensional Partial Differential Equations

2021 ◽  
pp. 191-215
Keyword(s):  
Heat Transfer ◽  
Fluid Flow ◽  
Partial Differential Equations ◽  
Differential Equations ◽  
Boundary Conditions ◽  
Material Properties ◽  
Final Part ◽  
Temperature Dependent ◽  
Partial Differential ◽  
One Dimensional

This chapter describes the pdepe command, which is used to solve spatially one-dimensional partial differential equations (PDEs). It begins with a description of the standard forms of PDEs and its initial and boundary conditions that the pdepe solver uses. It is shown how various PDEs and boundary conditions can be represented in standard forms. Applications to the mechanics are presented in the final part of the chapter. They illustrate how to solve: heat transfer PDE with temperature dependent material properties, startup velocities of the fluid flow in a pipe, Burger's PDE, and coupled FitzHugh-Nagumo PDE.

scholarly journals Measurements of sound-speed and density contrasts of zooplankton in Antarctic waters

2005 ◽  
Vol 62 (4) ◽  
pp. 818-831 ◽  
Author(s):  
D. Chu ◽  
P.H. Wiebe
Keyword(s):  
Standard Deviation ◽  
Material Properties ◽  
Sound Speed ◽  
Euphausia Superba ◽  
Mean Value ◽  
Zooplankton Biomass ◽  
Density Contrast ◽  
Depth Dependence ◽  
Type Size

Abstract Sound-speed and density contrasts (h and g, respectively), two important acoustic material properties, of live zooplankton were measured off the western Antarctic Peninsula during a Southern Ocean GLOBEC cruise conducted from 9 April to 21 May 2002. The work included in situ sound-speed contrast and shipboard density-contrast measurements. The temperature and pressure (depth) dependence of the sound-speed contrast of Euphausia superba and E. crystallorophias as well as that of some other zooplankton species were investigated. The size range of E. superba used in the measurements varied from about 20 mm to 57 mm, with mean length of 36.7 mm and standard deviation of 9.8 mm, which covered life stages from juvenile to adult. For E. superba, there was no statistically significant depth dependence, but there was a moderate dependence of sound-speed and density contrasts on the size of the animals. The measured sound-speed contrast varied between 1.018 and 1.044, with mean value 1.0279 and standard deviation 0.0084, while the measured density contrast varied between 1.007 and 1.036, with mean value 1.0241 and standard deviation 0.0082. For E. crystallorophias and Calanus there was a measurable depth dependence in sound-speed contrast. The in situ sound-speed contrasts for E. crystallorophias were 1.025 ± 0.004 to 1.029 ± 0.009. For Calanus, they were variable, with one set giving a value of 0.949 ± 0.001 and the other giving 1.013 ± 0.002. Shipboard measurements of other taxa/species also showed substantial variation in g and h. In general, values of g ranged from 0.9402 to 1.051 and h ranged from 0.949 to 1.096. The variation of the material properties is related to species, type, size, stage, and in some cases depth of occurrence. The uncertainty of the estimates of zooplankton biomass attributable to these variations in g and h can be quite large (more than 100 fold). Improvements in making biological inferences from acoustic data depend strongly on increased information about the material properties of zooplankton and the biological causes for their variation, as well as a knowledge of the species composition and abundance.

scholarly journals New oscillation criteria for second-order neutral differential equations with distributed deviating arguments

2021 ◽  
Vol 2021 (1) ◽  
Author(s):  
Osama Moaaz ◽  
Choonkil Park ◽  
Elmetwally M. Elabbasy ◽  
Waed Muhsin
Keyword(s):  
Differential Equations ◽  
Second Order ◽  
The Other ◽  
Riccati Transformation ◽  
Transformation Technique ◽  
Deviating Arguments ◽  
Neutral Differential Equations ◽  
Second Order Differential Equations ◽  
Continuous Delay ◽  
New Criteria

AbstractIn this work, we create new oscillation conditions for solutions of second-order differential equations with continuous delay. The new criteria were created based on Riccati transformation technique and comparison principles. Furthermore, we obtain iterative criteria that can be applied even when the other criteria fail. The results obtained in this paper improve and extend the relevant previous results as illustrated by examples.

scholarly journals Existence of weak solutions to SPDEs with fractional Laplacian and non-Lipschitz coefficients

2021 ◽  
Author(s):  
Shohei Nakajima
Keyword(s):  
Partial Differential Equations ◽  
Differential Equations ◽  
Weak Solutions ◽  
Stochastic Partial Differential Equations ◽  
Fractional Laplacian ◽  
Existence Of Solutions ◽  
Partial Differential ◽  
Continuity Theorem ◽  
Existence Of Weak Solutions ◽  
One Dimensional

AbstractWe prove existence of solutions and its properties for a one-dimensional stochastic partial differential equations with fractional Laplacian and non-Lipschitz coefficients. The method of proof is eatablished by Kolmogorov’s continuity theorem and tightness arguments.

scholarly journals Approximation of linear one dimensional partial differential equations including fractional derivative with non-singular kernel

2021 ◽  
Vol 2021 (1) ◽  
Author(s):  
Raheel Kamal ◽  
Kamran ◽  
Gul Rahmat ◽  
Ali Ahmadian ◽  
Noreen Izza Arshad ◽  
...  
Keyword(s):  
Partial Differential Equations ◽  
Differential Equations ◽  
Laplace Transform ◽  
Fractional Derivative ◽  
Meshless Method ◽  
Inverse Laplace Transform ◽  
Contour Integral ◽  
Partial Differential ◽  
One Dimensional ◽  
The Laplace Transform

AbstractIn this article we propose a hybrid method based on a local meshless method and the Laplace transform for approximating the solution of linear one dimensional partial differential equations in the sense of the Caputo–Fabrizio fractional derivative. In our numerical scheme the Laplace transform is used to avoid the time stepping procedure, and the local meshless method is used to produce sparse differentiation matrices and avoid the ill conditioning issues resulting in global meshless methods. Our numerical method comprises three steps. In the first step we transform the given equation to an equivalent time independent equation. Secondly the reduced equation is solved via a local meshless method. Finally, the solution of the original equation is obtained via the inverse Laplace transform by representing it as a contour integral in the complex left half plane. The contour integral is then approximated using the trapezoidal rule. The stability and convergence of the method are discussed. The efficiency, efficacy, and accuracy of the proposed method are assessed using four different problems. Numerical approximations of these problems are obtained and validated against exact solutions. The obtained results show that the proposed method can solve such types of problems efficiently.

scholarly journals Assessing a Linear Nanosystem's Limiting Reliability from its Components

2008 ◽  
Vol 45 (03) ◽  
pp. 879-887 ◽  
Author(s):  
Nader Ebrahimi
Keyword(s):  
Present State ◽  
Size Range ◽  
Two Dimensions ◽  
The Other ◽  
One Dimension ◽  
Present Moment ◽  
One Dimensional ◽  
The One ◽  
Fixed Moment ◽  
Nanosized Systems

Nanosystems are devices that are in the size range of a billionth of a meter (1 x 10-9) and therefore are built necessarily from individual atoms. The one-dimensional nanosystems or linear nanosystems cover all the nanosized systems which possess one dimension that exceeds the other two dimensions, i.e. extension over one dimension is predominant over the other two dimensions. Here only two of the dimensions have to be on the nanoscale (less than 100 nanometers). In this paper we consider the structural relationship between a linear nanosystem and its atoms acting as components of the nanosystem. Using such information, we then assess the nanosystem's limiting reliability which is, of course, probabilistic in nature. We consider the linear nanosystem at a fixed moment of time, say the present moment, and we assume that the present state of the linear nanosystem depends only on the present states of its atoms.

Influence of rotating magnetic field on Maxwell saturated ferrofluid flow over a heated stretching sheet with heat generation/absorption

2019 ◽  
Vol 20 (5) ◽  
pp. 502 ◽  
Author(s):  
Aaqib Majeed ◽  
Ahmed Zeeshan ◽  
Farzan Majeed Noori ◽  
Usman Masud
Keyword(s):  
Magnetic Field ◽  
Temperature Field ◽  
Velocity Field ◽  
Differential Equations ◽  
Heat Transport ◽  
Viscous Dissipation ◽  
Heat Generation ◽  
Fluid Motion ◽  
Numerical Procedure ◽  
The Impact

This article is focused on Maxwell ferromagnetic fluid and heat transport characteristics under the impact of magnetic field generated due to dipole field. The viscous dissipation and heat generation/absorption are also taken into account. Flow here is instigated by linearly stretchable surface, which is assumed to be permeable. Also description of magneto-thermo-mechanical (ferrohydrodynamic) interaction elaborates the fluid motion as compared to hydrodynamic case. Problem is modeled using continuity, momentum and heat transport equation. To implement the numerical procedure, firstly we transform the partial differential equations (PDEs) into ordinary differential equations (ODEs) by applying similarity approach, secondly resulting boundary value problem (BVP) is transformed into an initial value problem (IVP). Then resulting set of non-linear differentials equations is solved computationally with the aid of Runge–Kutta scheme with shooting algorithm using MATLAB. The flow situation is carried out by considering the influence of pertinent parameters namely ferro-hydrodynamic interaction parameter, Maxwell parameter, suction/injection and viscous dissipation on flow velocity field, temperature field, friction factor and heat transfer rate are deliberated via graphs. The present numerical values are associated with those available previously in the open literature for Newtonian fluid case (γ 1 = 0) to check the validity of the solution. It is inferred that interaction of magneto-thermo-mechanical is to slow down the fluid motion. We also witnessed that by considering the Maxwell and ferrohydrodynamic parameter there is decrement in velocity field whereas opposite behavior is noted for temperature field.

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