Initial Propagation Direction Research Articles

Radar Method in a Uniformly Accelerated Reference Frame

The purpose of this work is to generalize the radar method known for the inertial frame of reference to the case of a uniformly accelerated frame of reference.The derivation of the corresponding formulas is based on the standard for the theory of relativity metric of a uniformly accelerated Möller frame of reference without applying any space-time transformation between some auxiliary inertial frame and the accelerated frame. To solve the problem of determining the trajectory of a light beam, depending on the initial direction of propagation, Fermat’s principle is used. To calculate the flight time of a photon to an object, knowing its coordinates, the condition of the light-likeness of the interval for the propagation of light is additionally introduced.The resulting trajectory of the light particle is an arc of a circle. For a small area near the source, the photon trajectory coincides with the parabolic trajectory of a classical corpuscle. An equation has been derived for the direction in which the radio signal is sent. The actual location of the object is not in the direction of the initial motion of the photon, but somewhat lower. The value of the angle of gravitational refraction for a closely spaced resting object is calculated. The further the object is in the “horizontal” direction, the greater the angle of refraction. The flight time of the light signal to the object is found. The signal emitted in the direction that forms an acute angle with the direction of acceleration leads the radio signal in the inertial frame of reference. Therefore, for a close object located above the radiation source, the calculated Shapiro delay time is negative. The coordinates of the remote object are also calculated.The totality of the obtained equalities completely determines the radar method. The resulting equalities, perhaps, allow for experimental verification.

Quantitative assessment of a beam hole scatter correction in industrial computed tomography

Abstract In industrial Computed Tomography (CT), scattered radiation causes a loss of quality of the reconstructed 3D volume. Scatter is caused by photon-matter interaction, whereby a photon is deflected from its initial propagation direction. The most prominent contributor to scattered radiation in the scope of photon energies occurring in industrial CT is Compton Scattering. With the used Beam Hole Array (BHA) technology for scatter correction, scatter intensities are determined by subtracting the primary intensities, which are gathered behind small apertures from the total intensities, which are gathered in the absence of the apertures. Intensities measured behind the apertures are assumed to be scatter free, since the transversal components of the radiation are strongly reduced. The scatter correction is performed by subtracting scatter intensities from primary intensities in the projection image. Improvements are evaluated for a newly developed test specimen considering image quality metrics and dimensional accuracy. The contrast could be improved especially for small structures. Cupping artifacts that express as systematic offset of grey values rising with the penetration length were reduced. Measurement deviations towards tactile references were decreased.

Numerical Research on the Effect of the Initial Parameters of a CME Flux-rope Model on Simulation Results. II. Different Locations of Observers

Abstract In numerical studies of the initiation and propagation of coronal mass ejections (CMEs), it has been proven that the shape, size, and plasma parameters of CMEs could significantly affect simulation results and subsequent space weather predictions. In our previous research, we proposed a new way to initiate a CME based on the graduated cylindrical shell model, and studied the effect of different initial parameters of CMEs on the simulation results when the observer is aligned with the initial propagation direction of the CME. In this paper, we investigate the influence of the different initial parameters of CMEs on simulation results at the observational points with different longitudes and latitudes. Our results indicate that as long as the initial mass of the CME remains unchanged, the initial geometric thickness will have a different influence in the latitudinal and longitudinal directions. The deflection of the CMEs always occurs in both latitudinal and longitudinal directions, when the CMEs interact with the background solar wind structures, such as the corotating interaction region, in the heliosphere.

Numerical Research on the Effect of the Initial Parameters of a CME Flux-rope Model on Simulation Results

Abstract Coronal mass ejections (CMEs) are the major drivers of space weather, and an accurate modeling of their initialization and propagation up to 1 au and beyond is an important issue for space weather research and forecasts. In this research, we use the newly developed three-dimensional (3D) flux-rope CME initialization model and 3D IN (interplanetary)-TVD MHD model to study the effect of different CME initial parameters on simulation outputs. The initial CME flux model is established based on the graduated cylindrical shell model. In order to test the influence of the CME initial parameters on the simulation results, we try to run several simulations with different CME initial parameters, then investigate the outputs in interplanetary space. Here, we focus only on cases in which observers are located in the same initial direction of propagation of the CME. Our analysis shows that the parameters specifying the CME initialization in the model, including the initial density, the thickness of CME flux tube, initial mass, and initial magnetic field, have different effects on the simulation results for observers near the Earth and Mars, and on the process of propagation of the CME in interplanetary space. This confirms the important role played by details of the initial implementation of geometric and physical parameters on space weather research and forecasts.

A Machine-Learning Model for Forecasting Internal Wave Propagation in the Andaman Sea

Internal waves (IWs) are broadly distributed globally and have significant impacts on offshore engineering and underwater navigation. The prediction of IW propagation is a challenging task because of the complex factors involved. In this study, a machine-learning model was developed to predict IW propagation in the Andaman Sea. The model is based on a back-propagation neural network trained by 1189 IW samples, including the crest length and the peak-to-peak distance of IWs, extracted from 123 Moderate-Resolution Imaging Spectroradiometer (MODIS) images and 33 Ocean Land Color Instrument (OLCI) images acquired from 2015 to 2019 and corresponding ocean environment parameters. Using the leading wave crest within an IW packet as input, we ran the model to forecast the IW locations and compare them with satellite observations. The average root-mean-square difference between the model-forecasted and satellite-observed IW leading crest after one tidal cycle was 3.21 km. The corresponding averaged correlation coefficient was 0.95 and the average Fréchet Distance was 11.46 km. We reiterated the model run over two tidal periods and obtained similar statistical results, indicating the robustness of forecasting IW packets. We find that reducing the time step helped to improve forecasting accuracy. The influence of input errors and seasonal variations on model results are discussed and an analysis shows that the initial propagation direction introduced to the model is necessary for cross-propagating IW patterns. Comparisons with the Korteweg-de Vries equation results show that the developed model has better performance and is more robust.

A theoretical model for the study of thermal fracture of functionally graded thermal barrier coatings with a system of edge and internal cracks

The problem of fracture of functionally graded coatings (FGCs) on a homogeneous substrate (a semi-infinite medium) is investigated under the influence of thermal and/or mechanical loads (e.g. a heat flux, residual thermal stresses caused by cooling-heating, tension). These loads reflect the most important cases, which arise during the exploitation of FGC structures. The FGC contains pre-existing systems of cracks, such as edge, internal and interface cracks. The mathematical description of the model is based on singular integral equations. The properties of the FGC are continuous functions of the thickness coordinate. Furthermore, the non-homogeneity of the functionally graded material is revealed in the form of corresponding inhomogeneous traction distributions on the surfaces of cracks. This method is approximate and used with the assumption, that the gradation of material properties of the FGC with the depth of the layer is not abrupt. The influence of residual stresses caused by temperature changes on ΔT (e.g. cooling from operating temperatures) is investigated in detail. Different crack patterns (which are reported in experiments and available in the literature) are studied by carrying out numerical experiments with respect to stress intensity factors and fracture angles (a deviation of cracks from the initial direction of propagation). The proposed model in combination with a detailed parametric analysis can help to optimize FGCs in order to improve the fracture resistance of FGC/homogeneous structures.

Anisotropic phononic crystal structure with low-frequency bandgap and heat flux manipulation

This study presents a two-dimensional phononic crystal with heat flux manipulation and wide bandgaps of out-of-plane modes within the low-frequency range. The anisotropic matrix made of spiral-multilayered materials with different thermal conductivities, and the coating layer inserted with metal are designed for heat flux manipulation. Rubber-coated metal cylinders are periodically embedded in the anisotropic matrix to obtain the low-frequency bandgaps of out-of-plane modes. Numerical simulation is carried out to validate the heat and elastic characteristics of the spiral-multilayered anisotropic structure and reveal the effects of the laying angle and temperature on the bandgaps. Subsequently, a spiral-multilayered plate with periodic structures is studied, which shows an obvious vibration attenuation in the frequency ranges of the bandgaps and a deflected heat flux from the initial propagation direction. In the experimental investigation, the multi-phase spiral-multilayered anisotropic plate is simplified to a single-phase anisotropic plate made of aluminum. The characteristics of this type of anisotropic phononic crystal structure may pave the way for the design of a new kind of thermo-acoustic metamaterial serving in combined thermal and acoustic environments.

Normal-mode-based theory of collisionless plasma waves

The Van Kampen normal-mode method is applied in a comprehensive study of the linear wave perturbations of a homogeneous, magnetized and finite-temperature plasma, described by the collisionless Vlasov–Maxwell system in its non-relativistic version. The analysis considers a stable, Maxwellian background, but is otherwise completely general in that it allows for arbitrary wave propagation direction relative to the equilibrium magnetic field, multiple plasma species and general polarization states of the perturbed electromagnetic fields. A convenient formulation is introduced whereby the generator of the time advance is a Hermitian operator, analogous to the Hamiltonian in the Schrödinger equation of quantum mechanics. This guarantees a real frequency spectrum and complete bases of normal modes. Expansions in these normal-mode bases yield immediately the solutions of initial-value problems for general initial conditions. With standard initial conditions and propagation direction parallel to the equilibrium magnetic field, all the familiar results obtained following Landau’s Laplace transform approach are recovered. Considering such parallel propagation, the present work shows also explicitly and provides an example of how to construct special initial conditions that result in different, damped but otherwise arbitrarily prescribed time variations of the macroscopic variables. The known dispersion relations for perpendicular propagation are also recovered.

A theoretical model for the study of thermal fracture of functionally graded thermal barrier coatings

The problem of fracture of functionally graded coatings (FGCs) on a homogeneous substrate (a semi-infinite medium) is investigated under the influence of thermal and/or mechanical loads (e.g. a heat flux, residual thermal stresses caused by cooling-heating, tension). These loads reflect the most important cases, which arise during the exploitation of FGC structures. The FGC contains pre-existing systems of cracks, such as edge, internal and interface cracks. The mathematical description of the model is based on singular integral equations. The properties of the FGC are continuous functions of the thickness coordinate. Furthermore, the non-homogeneity of the functionally graded material is revealed in the form of corresponding inhomogeneous traction distributions on the surfaces of cracks. This method is approximate and used with the assumption, that the gradation of material properties of the FGC with the depth of the layer is not abrupt. The influence of residual stresses caused by temperature changes on ΔT (cooling) is investigated in detail. Different crack patterns (which are reported in experiments and available in the literature) are studied by carrying out numerical experiments with respect to stress intensity factors, fracture angles (a deviation of cracks from the initial direction of propagation) and critical loads, when the propagation of cracks starts. The proposed model in combination with a detailed parametric analysis can help to optimize FGCs in order to improve the fracture resistance of FGC/homogeneous systems.

(Invited) A One Dimensional Crevice Experiment for Determining the Critical Factors Contributing to Crevice Corrosion Stability and Repassivation

From its inception more than 40+ years ago, the one-dimensional pencil electrode experiment (1-D pit) has allowed investigators to investigate factors on pitting corrosion such as ohmic drop, concentration of species as a function of pit depth and the stability of salt films.[1,2] In fact, in recent work, Srinivasan and Kelly have developed a 1-D transport model of repassivation for pencil electrodes made of SS 316L.[3] Unfortunately, for the phenomena of crevice corrosion, the understanding of these critical factors (that no doubt control crevice propagation and repassivation) are far less understood. The primary reason for this has been the inability to simultaneously measure the location of the active crevice front (e.g. its depth) and the area of the front (which fixes the current density). These two parameters are easily measured in the 1-D pit experiment. In this presentation, we will introduce a method for measuring both the depth of the active front in crevice corrosion and the active area. The method is similar to that which Pickering introduced years ago,[4] we record video of a propagating crevice in the optical microscope. In this technique the crevice is formed by a set of washers, a piece of acrylic that acts as a crevice former and the metal specimen, nickel alloy 625. The potential of the assembly is controlled potentiostatically using a traditional three electrode set up and the current recorded as afunciton of time. A stereo microscope equipped with a digital camera is used to record images of the initiation and propagation of crevice corrosion. Image processing software is used to quantify the area of the active front. A typical result is presented in Figure 1 which shows the curent density of the active crevice front as a function of crevice depth. As seen in this figure, the crevice propagates from the deepest regions of the crevice towards the crevice mouth. this initial propagation direction is a result of an increase in the concentration of metal salts ahead of the active front and a corresponding decrease in pH. It can also be seen in this figure that at distances greater than 1 cm from the mouth, the crevice is able to propagate at current densities less than 1 mA/cm2. Once the active front reaches a distance of 1cm, an exponential increase in curent density is reorded. This increase is attributed to the IR* mechanism.[4] With the knowledge of the current density and distance from the crevice mouth we are able to use a simplification of the Nernst-Plank equation to calculate the concenrtation of species in solution. As might be anticipated from Figure 1, the concetration of metal salts increases with increasing current density. As such, concentration near the mouth is greater than at the tip, the saturation limit being reached at a distance of approximately 0.4cm from the mouth. Acknowledgments Work on this project was made possible by a grant from the US DoD TCC #1000000149, program directors Gregory A. Shoales, USAFA, and Daniel Dunmire, Corrosion Policy Oversight. References JW Tester, HS Isaacs, JES, 122, 1438, 1975.RC Newman, HS Isaacs, JES, 130, 1621, 1983.J Srinivasan, RG Kelly, JES, 163, C768, 2016.K Cho, HW Pickering, JES, 138, L56, 1991. Figure 1 Current density as a function of distance inside an alloy 625 crevice. Figure 1

Structural signatures of igneous sheet intrusion propagation

The geometry and distribution of planar igneous bodies (i.e. sheet intrusions), such as dykes, sills, and inclined sheets, has long been used to determine emplacement mechanics, define melt source locations, and reconstruct palaeostress conditions to shed light on various tectonic and magmatic processes. Since the 1970's we have recognised that sheet intrusions do not necessarily display a continuous, planar geometry, but commonly consist of segments. The morphology of these segments and their connectors is controlled by, and provide insights into, the behaviour of the host rock during emplacement. For example, tensile brittle fracturing leads to the formation of intrusive steps or bridge structures between adjacent segments. In contrast, brittle shear faulting, cataclastic and ductile flow processes, as well as heat-induced viscous flow or fluidization, promotes magma finger development. Textural indicators of magma flow (e.g., rock fabrics) reveal that segments are aligned parallel to the initial sheet propagation direction. Recognising and mapping segment long axes thus allows melt source location hypotheses, derived from sheet distribution and orientation, to be robustly tested. Despite the information that can be obtained from these structural signatures of sheet intrusion propagation, they are largely overlooked by the structural and volcanological communities. To highlight their utility, we briefly review the formation of sheet intrusion segments, discuss how they inform interpretations of magma emplacement, and outline future research directions.

Zitterbewegung of exciton-polaritons

Macroscopic wave packets of spin-polarized exciton-polaritons in two-dimensional microcavities experience the zitterbewegung, the effect manifested by the appearance of the oscillatory motion of polaritons in the direction normal to the initial propagation direction. The oscillating trajectories of exciton-polaritons are adjustable by the control parameters: the splitting of the longitudinal and transverse exciton-polariton modes, the wave vector and the width of the resonant cw pump. Our theoretical analysis supported by the numerical calculations allowed to optimize values of the control parameters suitable for a direct experimental observation of the zitterbewegung effect.

Internal wave attractors in three-dimensional geometries: trapping by oblique reflection

We study experimentally the propagation of internal waves in two different three-dimensional (3D) geometries, with a special emphasis on the refractive focusing due to the 3D reflection of obliquely incident internal waves on a slope. Both studies are initiated by ray tracing calculations to determine the appropriate experimental parameters. First, we consider a 3D geometry, the classical set-up to get simple, two-dimensional (2D) parallelogram-shaped attractors in which waves are forced in a direction perpendicular to a sloping bottom. Here, however, the forcing is of reduced extent in the along-slope, transverse direction. We show how the refractive focusing mechanism explains the formation of attractors over the whole width of the tank, even away from the forcing region. Direct numerical simulations confirm the dynamics, emphasize the role of boundary conditions and reveal the phase shifting in the transverse direction. Second, we consider a long and narrow tank having an inclined bottom, to simply reproduce a canal. In this case, the energy is injected in a direction parallel to the slope. Interestingly, the wave energy ends up forming 2D internal wave attractors in planes that are transverse to the initial propagation direction. This focusing mechanism prevents indefinite transmission of most of the internal wave energy along the canal.

Modeling of thermomechanical fracture of functionally graded materials with respect to multiple crack interaction

Different aspects of thermomechanical fracture of functionally graded materials (FGMs) are considered. Among them are the crack interaction problems in a functionally graded coating on a homogeneous substrate (FGM/H). The interaction between systems of edge cracks is investigated, as well as, how this mutual interaction influences the fracture process and the formation of crack patterns. The problem is formulated with respect to singular integral equations which are referred to the boundary equation methods. The FGM properties are modeled by exponential functions. The main fracture characteristics are calculated, namely, the stress intensity factors, the angles of deviation of the cracks from their initial propagation direction and the critical stresses when the crack starts to propagate. The last two characteristics are calculated using an appropriate fracture criterion. The problem contains different parameters, such as the geometry (location and orientation of cracks, their lengths, and the width of the FGM layer) and material parameters, i.e. the inhomogeneity parameters of elastic and thermal coefficients of the functionally graded material. The influence of these parameters on the thermo-mechanical fracture of FGM/H is investigated. As examples the following real material combinations are discussed: TiC/SiC, Al2O3/MoSi2, MoSi2/SiC, ZrO2/nickel and ZrO2/steel.

Erosion of a California Undercurrent eddy by bottom topography

AbstractSubsurface eddies are ubiquitous features in eastern boundary current systems. These phenomena tend to modulate the across‐shore distribution of heat and biogeochemical tracers. A California Undercurrent eddy was detected by shipboard observations in October 2009 off the northern Baja California continental slope. The spatiotemporal variation in the California Undercurrent eddy is investigated by using a mesoscale‐resolving hindcast ocean simulation. A poleward coastal current that is driven by an upwelling‐wind relaxation event and the coastline geometry instigated the separation of the California Undercurrent from the slope, forming a meander‐like structure, which evolved as a mesoscale eddy‐like structure. The latter structure evolved as a subsurface eddy with a warm anomaly core, a distinctive feature of eddies that form from the California Undercurrent. During the initial stage, the subsurface eddy presented a cone‐shape form, with the maximum amplitude of the relative vorticity in the upper section. The inviscid effect of the irregular bottom topography altered both the initial direction of propagation and the initial eddy shape: the propagation direction of the eddy changed from north‐south to southwest, and the shape changed from a cone shape to a subsurface lens shape. The strong eddy‐topography interactions triggered a sign change in the heat flux direction from the environment toward the eddy and vice versa through the horizontally divergent component of the velocity field, which accelerated the eddy's decay. This study shows the relevance of the synoptic wind stress events and the irregularity of the bottom topography on the mesoscale eddy activity in the southern portion of the California Current.

On spacetime structure and electrodynamics

Electrodynamics is the most tested fundamental physical theory. Relativity arose from the completion of Maxwell–Lorentz electrodynamics. Introducing the metric [Formula: see text] as gravitational potential in 1913, versed in general (coordinate-)covariant formalism in 1914 and shortly after the completion of general relativity, Einstein put the Maxwell equations in general covariant form with only the constitutive relation between the excitation and the field dependent on and connected by the metric in 1916. Further clarification and developments by Weyl in 1918, Murnaghan in 1921, Kottler in 1922 and Cartan in 1923 together with the corresponding developments in electrodynamics of continuous media by Bateman in 1910, Tamm in 1924, Laue in 1952 and Post in 1962 established the premetric formalism of electrodynamics. Since almost all phenomena electrodynamics deal with have energy scales much lower than the Higgs mass energy and intermediate boson energy, electrodynamics of continuous media should be applicable and the constitutive relation of spacetime/vacuum should be local and linear. What is the key characteristic of the spacetime/vacuum? It is the Weak Equivalence Principle I (WEP I) for photons/wave packets of light which states that the spacetime trajectory of light in a gravitational field depends only on its initial position and direction of propagation, and does not depend on its frequency (energy) and polarization, i.e. nonbirefringence of light propagation in spacetime/vacuum. With this principle it is proved by the author in 1981 in the weak field limit, and by Lammerzahl and Hehl in 2004 together with Favaro and Bergamin in 2011 without assuming the weak-field condition that the constitutive tensor must be of the core metric form with only two additional degrees of freedom — the pseudoscalar (Abelian axion or EM axion) degree of freedom and the scalar (dilaton) degree of freedom (i.e. metric with axion and dilaton). In this paper, we review this connection and the ultrahigh precision empirical tests of nonbirefringence together with present status of tests of cosmic Abelian axion and dilaton. If the stronger version of WEP is assumed, i.e. WEP II for photons (wave packets of light) which states in addition to WEP I also that the polarization state of the light would not change (e.g. no polarization rotation for linear polarized light) and no amplification/attenuation of light, then no Abelian (EM) axion and no dilaton, and we have a pure metric theory.

Modeling of edge cracks interaction

From experimental and theoretical investigations it is known that cracks are sensitive to geometry, e.g., to the inclination angle to the load. A small deviation of a crack from the normal direction to a tensile load causes mixed mode conditions near the crack tip which lead to deviation of the crack from its initial propagation direction. Besides, the presence of other cracks, inhomogeneities, surfaces and their interaction causes additional deformations and stresses which also have influence on the initiation of the crack propagation and on the direction of this propagation. The aim of this paper is to show the effects of the interaction of edge cracks on further crack formation. The main fracture characteristics, such as, stress intensity factors, fracture angles and critical loads are provided in this study. A series of illustrative examples is presented for different geometries of arbitrarily inclined edge cracks

Experimental Field Tests and Finite Element Analyses for Rock Cracking Using the Expansion of Vermiculite Materials

In the previous research, laboratory tests were performed in order to measure the expansion of vermiculite upon heating and to convert it into expansion pressure. Based on these test results, this study mainly focuses on experimental field tests conducted to verify that expansion pressure obtained by heating vermiculite materials is enough to break massive and hard granite rock with an intention to excavate the tunnel. Hexahedral granite specimens with a circular hole perforated in the center were constructed for the experimental tests. The circular holes were filled with vermiculite plus thermal conduction and then heated using the cartridge heater. As a result, all of hexahedral granite specimens had cracks in the surface after 700-second thermal heating and were finally spilt into two pieces completely. The specimen of larger size only requires more heating time and expansion pressure. The material properties of granite rocks, which were obtained from the experimental tests, were utilized to produce finite element models used for numerical analyses. The analysis results show good agreement with the experimental results in terms of initial cracking, propagation direction, and expansion pressure.

Time-dependent fracture formation around a borehole in a clay shale

Opalinus Clay, a Mesozoic clay shale, has been chosen as host rock formation for the disposal of radioactive waste in Switzerland. For this study, borehole damage zones were utilized as a proxy for an excavation damage zone that forms around a circular, mechanically excavated tunnel in intact Opalinus Clay. Pilot boreholes were resin-impregnated and over-cored at different times after drilling, to provide insight into the time-dependent formation of fractures on both the micro- and the macro-scale. Observed fractures were characterized in terms of failure mode, their relation to the rock anisotropy, and the in-situ stress tensor.The analyses show that fractures that form in the short term initiate as shear fractures at the pilot-borehole wall and propagate parallel to bedding. Typically, a dominant shear fracture tangential to the pilot borehole wall was observed. Upon propagation of these shear fractures, wing cracks, horsetail splays and second-order shears form sub-parallel and sub-perpendicular to bedding planes, forming a complex fracture network, which extends a quarter pilot-borehole diameter into the rock mass. In the longer term, tangential shear fractures tend to propagate in a direction opposite to the initial propagation direction. In addition, new bedding-parallel fractures deeper in the rock develop, leading to the formation of thin slabs, buckling of the slabs when unsupported and eventually progression of the buckling zone deeper into the rock mass. Buckling is associated with the formation of extensional fractures normal to bedding in the center and lateral to the buckling zone. The zone of buckled rock slabs was found to have an extension of more than one borehole diameter at the time of preservation with resin.In the short term, the axis connecting the maximum failure depth on opposing sides of the borehole is parallel to the minimum stress direction in a plane normal to the borehole axis. In the long term, this axis rotates significantly towards the maximum stress direction, primarily as a consequence of tangential shear fracture propagation, slab formation and buckling. Dissipation of excess pore pressures may be the key process underpinning longer-term fracture propagation and formation.

Propagation of P- and S-waves in initially stressed gravitating half space

The present paper contributes in studying the phase velocities of P- and S-waves in a half space subjected to a compressive initial stress and gravity field. The density and acceleration due to gravity vary quadratically along the depth. The dispersion equation is derived in a closed form. It is shown that the phase velocities depend not only on the initial stress, gravity, and direction of propagation but also on the inhomogeneity parameter associated with the density and acceleration due to gravity. Various particular cases are obtained, and the results match with the classical results. Numerical investigations on the phase velocities of P- and S-waves against the wave number are made for various sets of values of the material parameters, and the results are illustrated graphically. The graphical user interface model is developed to generalize the effect.