Homogeneous Half-space Research Articles

Study of different theories of thermoelasticity under the propagation of Rayleigh waves in thermoelastic medium

The present research is on the propagation of Rayleigh waves in a homogenous thermoelastic solid half-space by considering the compact form of six different theories of thermoelasticity. The medium is subjected to an insulated boundary surface that is free from normal stress, tangential stress, and a temperature gradient normal to the surface. After developing a mathematical model, a dispersion equation is obtained with irrational terms. To apply the algebraic method, this equation must be converted into a rational polynomial equation. From this, only those roots are filtered out, which has satisfied both of the above equations for the propagation of waves decaying with depth. With the help of these roots, different characteristics are computed numerically, like phase velocity, attenuation coefficient, and path of particles. Various particular cases are compared graphically by using phase velocity and attenuation coefficient. The elliptic path of surface particles in Rayleigh wave propagation is also presented for the different theories using physical constants of copper material for different depths and thermal conductivity.

THEPORE: A software package for modeling THErmo-PORo-elastic displacements

THEPORE (THErmo-POro-Elastic solutions) is an open source software to perform forward and inverse modeling of ground displacements induced by thermo-poro-elastic sources. The software, implemented in MATLAB, offers a library of analytical and semi-analytical solutions to compute ground displacements induced by thermo-poro-elastic deformation sources of different geometries, embedded in an elastic, homogeneous and isotropic half-space. The solutions have been verified against finite-element simulations. THEPORE includes also an inversion procedure of the deformation data to constrain the source parameters that better fit the observed signals.The software's functionality is showcased by inverting the GPS deformation data recorded on Vulcano Island at the onset of the 2021 unrest, in order to estimate the position and volume change of the source responsible for the observed deformations. The results encourage to consider THEPORE as a practical tool suitable for a fast preliminary estimation of the deformation source during a volcanic crisis.

Comment on: "Anomalous reflection from a two-layered marine sediment" [J. Acoust. Soc. Am. 155, 1285-1296 (2024)] (L).

Buckingham [(2024). J. Acoust. Soc. Am. 155, 1285-1296] analyzed the dependence of the reflection coefficient on the grazing angle in two-layer marine sediment model. The upper layer in his model consists of a fine-grained material (mud), while seawater and the basement below the mud layer are treated as homogeneous halfspaces. Buckingham's analyses revealed several narrow spikes in this dependence that appeared only in the presence of a sound velocity gradient in the mud layer, a phenomenon he called acoustic glint. His derivation was accomplished for certain specific dependencies of the sound velocity on the depth. Surprisingly, the authors appear to reach the conclusion that for a slightly different vertical sound speed profile in the mud layer the spikes are no longer present in the dependence of the reflection coefficient on the grazing angle. More precisely, the same problem is examined in this letter for the case of an n2-linear layer (often called Airy medium). Acoustic glint effect is therefore very sensitive to the exact parametrization of the mud layer.

The Problem of Heat and Mass Transfer of a Half-Space with an Air Flow with Periodic Changes in Air Temperature

A system of differential equations, boundary and initial conditions for calculating temperature and moisture content fields in a homogeneous halfspace, the boundary of which is blown by an air flow, is formulated. The heat exchange of the boundary with the air medium occurs according to the Newtonian law, and mass transfer — according to the Dalton evaporation law. In the initial state, the air and the material have the same temperature, and water vapor, both near the surface of the material and outside the boundary layer, is in a saturation state, so there is no exchange of heat and moisture between the air and the material. At some point, the air temperature begins to produce small harmonic fluctuations near its initial value. For a mathematical model of heat and mass transfer, in which the movement of moisture to the surface is considered to be caused only by a drop in moisture content, i.e. the phenomenon of thermal diffusion is neglected, an asymptotic time view of the temperature and moisture content fields is obtained. The moisture content field has the form of a damped harmonic wave, and the temperature field is represented by the superposition of two waves of the same type, which have the same frequency, but different attenuation coefficients and phase velocities. The calculation of the basic wave parameters for a material with clay characteristics is carried out, and the results obtained are compared with experimental data. The constructed solution is a generalization of the Fourier formulas known in the literature, which are valid only in a situation when the material does not contain moisture, and according to the harmonic law, not the air temperature changes, but the surface temperature. The results obtained in the article will be used in geocryology as a theoretical tool in the study of seasonal fluctuations in the thermal and physical state of the soil, which is an important task in planning economic activities in the field of the distribution of frozen rocks.

High-Speed Train-Induced Vibration of Bridge–Soft Soil Systems: Observation and MTF-Based ANSYS Simulation

In this paper, a multi-transmitting formula (MTF) was integrated into ANSYS software through secondary development, enabling dynamic finite element simulation of wave propagation in infinite domains. The numerical reliability and accuracy of the MTF were verified through a plane wave problem involving a homogeneous elastic half-space, as well as 3D scattering and source problems in a three-layered soil site. Additionally, a comparative analysis of various artificial boundaries was conducted to highlight the advantages of the MTF. Field observations of environmental vibrations caused by high-speed railway operations revealed localized amplification of vibrations along the depth direction at the Kunshan segment of the Beijing–Shanghai high-speed railway. Based on these observations, a series of numerical analyses were conducted using the customized ANSYS integrated with the MTF to investigate the underlying causes and mechanisms of this phenomenon, as well as the spatial variation characteristics of foundation vibrations induced by bridge vibrations during high-speed train operations. This study reveals the mechanism by which the combined effect of bridge piles and soft soil layers influences the depth variation in peak ground accelerations during site vibrations. It also demonstrates that the presence of bridge piers and pile foundations effectively reduces vibration intensity in the vicinity of the railway, playing a crucial role in mitigating vibrations induced by high-speed train operations.

Analytical solution to 2D dynamic soil-tunnel-building interaction under seismic SH-wave excitation considering the influence of the stiffness of tunnel and building's foundation

This investigation examines the structure-soil-structure interaction (SSSI) problem between a tunnel and nearby aboveground building, considering the impact of the stiffness of tunnel and building's foundation. A two-dimensional (2D) model is utilized, which includes a flexible circular tunnel and a three-story building anchored by a flexible semi-circular foundation embedded in a homogeneous half-space. The response of the system to incident plane SH-wave excitation is solved analytically using the wave function expansion method. The analysis focuses on how the system response is affected by the stiffness of the tunnel and the building's foundation. Tunnel stiffness plays a leading role in tunnel response analysis. The model with a rigid tunnel and rigid foundation can offer a reasonable system response overall, but it fails to accurately represent tunnel deformation and stress distribution. Depending on the stiffness of the building's foundation, the maximum value of the inter-story displacement of the first floor can be overestimated by around 70 %, with even greater overestimations for higher floors. The results help understand observed seismic behaviors and provide insight into the effects of foundation and tunnel stiffness on the tunnel-building interaction system.

Semi-analytical method for calculating ground vibrations from a tunnel in a homogeneous half-space with an irregular surface

This paper proposes a novel two-dimensional (2D) semi-analytical method for calculating ground vibrations from a tunnel situated in a homogeneous half-space with an irregular surface. The circular tunnel is conceptualized as an elastic solid, while the soil is modelled as an elastic, isotropic, and homogeneous half-space with an irregular surface. A virtual horizontal interface is introduced to divide the soil domain into an irregular region with an arbitrary-shaped surface and a half-space with a circular hollow. The wavefield scattered by the irregular surface is simulated by the boundary integral equation. Through the application of the discrete wavenumber approach and expansion of wavefield on the irregular surface, the boundary integral equation is transformed into a set of simultaneous matrix equations containing unknown expansion coefficients. By utilizing the transformation between cylindrical and plane waves, the boundary conditions on soil-tunnel interface are satisfied, yielding a solution for a harmonic point load acting on the tunnel in a half-space with an irregular surface. The proposed method can simulate topographies with arbitrary shapes and only requires the discretization of the irregular part of the ground surface, which provides a highly efficient tool to investigate the propagation characteristics of train-induced vibrations in complex topographies. The proposed method is verified through comparisons with the existing analytical methods and the finite element method. The dynamic responses of a tunnel embedded in a half-space with a Gaussian-shaped surface are meticulously investigated. A case study involving a site for the Shenzhen metro line 5 is performed. The numerical results demonstrate that the presence of irregular topography alters the distribution of ground vibrations, leading to significant differences in responses within the irregular region and certain disparities in the regular region. The effect of the location, size, and undulating direction of the irregular topography on ground vibrations is frequency-dependent.

Surface acoustic waves in laterally periodic superlattices

The existence of surface acoustic waves (SAWs) is studied in laterally periodic superlattices, modelled as an anisotropic elastic half-space with an arbitrary periodic variation of its material properties along the stratification direction (call it X) parallel to the surface. Unlike a homogeneous half-space, such a structure allows for more than one (dispersive) SAW. Specifically, it is shown that any superlattice with a generic shape of periodicity profile admits at most three SAW dispersion branches ω(Kx), i.e., at most three different SAW frequencies at any fixed Bloch wavenumber Kx. Moreover, the total number of SAWs at fixed Kx in a pair of superlattices with periodicity profiles obtained from one another by the inversion of the axis X cannot exceed three either. At least one SAW branch must exist in one of these two superlattices unless the bulk-wave threshold is the so-called exceptional (i.e., admits surface skimming wave). The SAW branch is unique in the particular case of a superlattice invariant to the inversion X→−X. The above general results are illustrated by the perturbation theory derivations for the weakly modulated superlattices. Explicit leading-order formulas are obtained for the quasi-Rayleigh wave branch evolving from the Rayleigh wave in each of the mutually ”inverse” superlattices and for the quasibulk wave branch evolving from the exceptional bulk-wave threshold in one of the superlattices.

Rayleigh wave propagation in centrosymmetric materials with micro-stiffness, flexoelectric and micro-inertia effects

A theoretical investigation of Rayleigh waves propagation in polarized media has been carried out using a reformulated flexoelectric theory for isotropic dielectrics with micro-inertia effect. Within this non-classical theory, the internal energy density is the functional of the strain tensor, dilatation gradient, deviatoric part of stretch gradient and rotation gradient tensors, polarization vector, and polarization gradient. The obtained system of governing equations additionally contains three material length-scale parameters to account the micro-stiffness effect, one material constant to capture the micro-inertia effect, two flexoelectric constants to describe the flexoelectric effect and three length scale parameters related to the polarization gradient. To solve the coupled governing equations, the method of Lamé-type potentials for mechanical displacement and electric polarization vectors is used. The influences of various factors such as micro-stiffness, flexoelectricity, electric quadrupoles and micro-inertia effects on the phase velocity of the Rayleigh waves in a homogeneous isotropic half-space are studied. It is found that above effects become significant with the increase of the wavenumber. This study can be important for the investigation of high frequency surface acoustic waves in dielectric materials.

Lateral Dynamics of a Pile in a Homogeneous Half-Space with Vlasov–Leont′ev Foundation Model and Expressions for a Rigid Disk

Lateral Dynamics of a Pile in a Homogeneous Half-Space with Vlasov–Leont′ev Foundation Model and Expressions for a Rigid Disk

Anomalous reflection from a two-layered marine sediment.

This paper concerns the theory of acoustic reflection from a two-layered marine sediment, the upper layer of which consists of a fine-grained material (mud). The seawater above and basement below the layer are treated as homogeneous half-spaces. Within the mud layer, the density is taken to be constant, and three sound speed profiles are considered: uniform, linear, and inverse-square. The reflection coefficient exhibits a background component that is similar in all three cases, exhibiting only a weak sensitivity to the gradient of the profile, the frequency, and the depth of the layer. Additionally, the two profiles with a non-zero gradient, linear and inverse-square, exhibit a sequence across grazing angle of narrow spikes of total reflection. The angular distribution of this acoustic glint is highly sensitive to the frequency and depth of the layer, and mildly so to the gradient. As the gradient approaches zero, the glint vanishes and the reflection coefficient reduces identically to the form of a uniform sound speed profile. If it were detectable, the angular distribution of the glint, observed at several frequencies, could constitute a unique, sensitive set of "fingerprints," allowing the depth and sound speed gradient of the mud layer to be inferred.

Pulsed Electromagnetic Excitation of a Narrow Slot Between Two Dielectric Halfspaces

The transient electromagnetic (EM) excitation of a narrow slot in a perfectly electrically conducting (PEC) screen that separates two homogeneous dielectric halfspaces, a simplified model of a typical feeding structure of leaky lens antennas, is analyzed numerically in the time domain (TD). The problem is formulated using the TD reciprocity theorem of the time-convolution type and subsequently solved with the aid of the Cagniard-DeHoop method of moments (CdH-MoM). Numerical results are validated using a general-purpose EM-field solver.

The comparison between effects of heterogeneous and homogeneous half-spaces underlying homogeneous layer on solitary Love waves

Abstract A comparative analysis is performed on the effects of heterogeneity in both linear and non-linear material characteristics of half-space on the propagation of bright and dark solitary Love waves in a non-linear layered half-space. The layer is assumed to be homogeneous, non-linear and elastic, while the half-space is vertically heterogeneous. The problem is formulated for two types of elastic materials, incompressible and generalized neo-Hookean materials, and the differences caused by the two materials in the formulation are revealed. For the media composed of generalized neo-Hookean materials, a non-linear Schrödinger equation describing the self interaction of Love waves is obtained using multiple scales method. The differences in the effects of linear and non-linear material properties of the half-space on both the existence and non-linear evolution of bright and dark solitary Love waves are compared graphically.

Axial stiffness of floating pile foundations and rigid footings with Vlasov-Leont′ev foundation model and expressions for a rigid disk

In the manuscript, a hybrid formulation to evaluate the axial dynamic and static head stiffness of pile foundations and rigid footings floating in a homogeneous half-space is demonstrated. In the formulation, the soil adjacent to the pile is modeled with the Vlasov-Leont′ev foundation model, and the underlying soil beneath the pile is modeled by a spring-dashpot system. The spring-dashpot system is described with expressions obtained from the solution of a vertically vibrating rigid massless disk on the surface of a homogeneous half-space. The expressions are obtained from the literature reported by several authors. The results obtained from the proposed formulation and the expressions are compared with rigorous formulations in the literature for pile foundations and rigid footings. It is found, the formulation demonstrated produces accurate dynamic and static head stiffness for pile foundations and rigid footings. Besides, it captures the essential mechanics of a foundation floating in a homogeneous half-space. Based on an investigative study, it is found, the expression proposed by Veletsos and Verbic accurately models and simulates the homogeneous half-space underneath the pile foundations and rigid footings.

IMPROVEMENT OF THE NON-DESTRUCTIVE METHOD FOR DETERMINING THE MECHANICAL CHARACTERISTICS OF ELEMENTS OF MULTILAYER STRUCTURES ON THE EXAMPLE OF PAVEMENTS

The article is devoted to solving the urgent problem of improving the method of nondestructive testing of the condition of non-rigid road clothes. In addition to the previously developed approach to determining the mechanical parameters of materials of structural layers of nonrigid road coverings based on solving the inverse coefficient problem of restoring operational elastic modules providing specified displacement fields, an approach was developed that allows determining the parameters of energy dissipation in the structure of multilayer road pavement based on the correction of dynamic hysteresis loops recorded in the field and calculated using mathematical model of dynamic VAT. The article presents the results of numerical simulation of dynamic hysteresis loops for two different variants of multilayer structures. The possibilities of correcting the shape of hysteresis loops and their area by varying the tangents of the energy loss angles in layers of a multilayer medium and the elastic modulus of a homogeneous half-space are shown. The complex correction of the elastic modulus values of the structural layers of road clothes and the calculated loops of dynamic hysteresis made it possible to fully take into account the processes of energy attenuation at a distance from the point of application of the load. During the correction, it was found that the values of the elastic modulus of the road layers and the tangents of the loss angles in them have a complex effect on the areas of dynamic hysteresis loops and the nature of energy attenuation at a distance from the point of application of the load. At the same time, the elastic modulus of the underlying half-space is not limited in thickness to the greatest extent on the area of dynamic hysteresis loops (which led to an increase in the elastic modulus of the underlying half-space from 120 and 150 MPa for road construction variants with a reinforced and non-reinforced base to 170 and 160 MPa, respectively), and a decrease in the dissipated energy at a distance from the point of application This is primarily due to the tangent of the angle of energy loss in the half-space. The obtained values of the tangents of the loss angles are obviously related to the patterns of energy dissipation at the boundaries of the contacting layers of the pavement, and also take into account all possible anomalies and delaminations in the calculated structures. Within the framework of this article, calculated hysteresis loops on the surface of the pavement were obtained for the first time, the deformation energy was calculated based on their areas, and the possibility of their comparison with those registered experimentally was proved.

Reconstruction of Co-Seismic and Post-Seismic Processes for the February 6, 2023 Earthquake in Turkey from Data of Satellite SAR Interferometry

Using different methods for processing SAR images from the Sentinel-1A satellite, the displacement fields were determined in the region of the East Anatolian Fault Zone (EAFZ) and the Sürgü-Çardak faults, as well as a small fault on the continuation of the East Hatay fault zone, which rupture initiated a series of catastrophic earthquakes in Turkey on February 6, 2023. DInSAR and offset methods were applied. The most detailed data on the displacements were obtained by the offset method using images from the descending orbit. When constructing the model from the available SAR data, the data with the maximum signal-to-noise ratio were selected. For the northern part of the region, above 37.4° N, the range displacements obtained by the offset method from a descending orbit were used. South of parallel 37.0° N, we used azimuth displacements from the same descending orbit. The model of the seismic rupture was constructed on the basis of solution of (Pollitz, 1996) of the problem of deformations at the surface of a layered spherical Earth caused by along dip and strike displacements on a rectangular fault located inside the planet. Pollitz (1996) demonstrated that ignoring the radial layering of the planet leads to errors up to 20%, with the largest errors occurring in the presence of a large strike-slip component. Ignoring sphericity also introduces an error when using the solution in the framework of the idealization of an elastic homogeneous half-space with a flat free surface (Okada, 1985) which was used when constructing USGS and (Barbot et al., 2023) models. In our model the surfaces of seismic rupture are approximated by 19 rectangular elements along the strike, divided into three levels along the dip. Another element approximated a rupture along the extension of the East Hatay Fault Zone. As in the models of other authors (USGS; Barbot et al., 2023), in our model in the southern part of the EAFZ, the displacements increase from south to north, and are mainly concentrated in the upper part of the Earth’s crust to a depth of 10 km. At the southern end of this rupture, displacements in our model with an amplitude of up to 2 m are obtained at the lower levels of the model, and at its upper level, the displacements were only 0.11 m, and in this area on February 20, 2023 an earthquake of magnitude 6.3 occurred with a hypocenter depth of 11.5 km. The main displacements on the EAFZ are determined on its central segment. Here, the displacements go to a greater depth, their value reaches 10.2 m. On the Sürgü-Çardak fault, significant displacements occurred down to a depth of 20 km; displacements exceeded 10.2 m. In our model, at the northeast end of the seismic rupture along the EAFZ, a displacement area of 6.8 m overlaps with the southwest end of the seismic rupture model of the Doğanyol-Sivrice earthquake of January 24, 2020 with Mw 6.7, published on the USGS website. Therefore, our model does not confirm the hypothesis of the presence of a seismic gap here, which, according to (Barbot et al., 2023), is a zone of a possible nearest earthquake.

Analysis of three-dimensional cracks horizontally placed in transversely isotropic FGMs under interior compressive stresses

This paper develops the dual boundary element method (DBEM) using a new fundamental singular solution. This new solution is the elastic fields of transversely isotropic multilayered materials of infinite extent under point concentrated loads. The discretization technique is utilized to approximate arbitrary variations of functionally graded materials (FGMs) with transverse isotropy in a graded direction. The proposed DBEM is used to analyze the three-dimensional crack problems in FGM halfspaces. The penny-shaped and elliptical cracks are horizontally placed on the interface between the FGM layer and homogeneous halfspace and are subjected to uniform compressive stresses on crack surfaces. The stress intensity factors are provided to understand the fracture behaviors of transversely isotropic FGMs. An influential factor is defined to describe the influence of the anisotropy of the FGMs on stress intensity factors. Results show that the anisotropy and heterogeneity of materials exert an obvious influence on the fracture properties of crack problems.

Multilayered elastic medium reinforced with interfacial thin film: A theoretical model for geogrid reinforced HIR asphalt pavement

Aiming at modeling the reinforcement behavior of geogrid reinforced HIR asphalt pavement structure accurately and efficiently, a mathematical formulation using the Generalized Kelvin Solution (GKS) based method for multi-layer three-dimensional elastic system with the consideration of elastic membrane without thickness is put forward in this thesis to simulate the geogrid reinforcement for HIR pavement. Solutions in transform and physical domain, solutions for z = 0, and specific solutions of surface displacements are determined. Four special cases including homogeneous half space with buried and surface thin film reinforcement, Boussinesq's solution, unreinforced multilayered half space are introduced to validate the solution. Moreover, the mechanical effect of geogrid on the deformation, stress state and fatigue life of the pavement structure is investigated using the proposed model and solution. The geogrid stiffness has significant effect on the vertical and radial strain. After the reinforcement, both the radial stress and hoop stress near the interface reduce to a very low level, the stress concentration is effectively alleviated. Meanwhile, geogrid reinforcement plays a positive role on enhancing the fatigue damage resistance of pavement stricture Our proposed method provides a novel way to model reinforcement mechanism of the geogrid on pavement structure, which is helpful to engineering practice.

Reconstruction of Co-Seismic and Post-Seismic Processes for the February 6, 2023 Earthquake in Turkey from Data of Satellite SAR Interferometry

Abstract—Using different methods for processing SAR images from the Sentinel-1A satellite, the displacement fields were determined in the region of the East Anatolian Fault Zone (EAFZ) and the Sürgü-Çardak faults, as well as a small fault on the continuation of the East Hatay fault zone, which rupture initiated a series of catastrophic earthquakes in Turkey on February 06, 2023. DInSAR and offset methods were applied. The most detailed data on the displacements were obtained by the offset method using images from the descending orbit. When constructing the model from the available SAR data, the data with the maximum signal-to-noise ratio were selected. For the northern part of the region, above 37.4° N, the range displacements obtained by the offset method from a descending orbit were used. South of parallel 37.0° N, we used azimuth displacements from the same descending orbit. The model of the seismic rupture was constructed on the basis of solution of (Pollitz, 1996) of the problem of deformations at the surface of a layered spherical Earth caused by along dip and strike displacements on a rectangular cut located inside the planet. Pollitz (1996) demonstrated that ignoring the spherical layering of the planet leads to errors up to 20%, with the largest errors occurring in the presence of a large strike-slip component. Ignoring sphericity also introduces an error when using the solution in the framework of the idealization of an elastic homogeneous half-space with a flat free boundary (Okada, 1985) which was used when constructing USGS and (Barbot et al., 2023) models. In our model the surfaces of seismic rupture are approximated by 19 rectangular elements along the strike, divided into three levels along the dip. Another element approximated a rupture along the extension of the East Hatay Fault Zone. As in the models of other authors (USGS; Barbot et al., 2023), in our model in the southern part of the EAFZ, the displacements increase from south to north, and are mainly concentrated in the upper part of the Earth’s crust to a depth of 10 km. At the southern end of this rupture, displacements in our model with an amplitude of up to 2 m are obtained at the lower levels of the model, and at its upper level, the displacements were only 0.11 m, and in this area on February 20, 2023 an earthquake of magnitude 6.3 occurred with a hypocentre depth of 11.5 km. The main displacements on the EAFZ are determined on its central segment. Here, the displacements go to a greater depth, their value reaches 10.2 m. On the Sürgü-Çardak fault, significant displacements occurred down to a depth of 20 km; displacements exceeded 9.8 m. In our model, at the northeast end of the seismic rupture along the EAFZ, a displacement area of 6.8 m overlaps with the southwest end of the seismic rupture model of the Doğanyol-Sivrice earthquake of January 24, 2020 with Mw 6.7, published on the USGS website. Therefore, our model does not confirm the hypothesis of the presence of a seismic gap here, which, according to (Barbot et al., 2023), is a zone of a possible nearest earthquake.

Transient scattering of a Rayleigh wave by a cluster of subwavelength resonators—Towards asymptotic modeling of seismic surface metabarriers

A seismic metabarrier (intended for surface waves mitigation effect) is modeled as a cluster of single-degree-of-freedom resonator units deposited on the surface of an isotropic homogeneous elastic half-space. It is assumed that each resonator has a frictionless flat rigid base of diameter much smaller than the wavelength of an incoming surface wave. The slow-motions asymptotic method is applied for constructing the first-order asymptotic model of multiple time-dependent scattering of a pulsed Rayleigh wave with respect to the vertical displacements of the resonators (including their rigid bases and inertial elements) and the normal contact forces (time-dependent integral characteristics of the contact reactions beneath the resonator bases). Both the stationary and transient multiple scattering scenarios are considered. The variation of the amplitude reduction factor due to the model parameters variation is studied in detail.