Semi-infinite Medium Research Articles

Guiding near-source elastic waves in a semi-infinite medium.

In this work, we propose elastic metamaterials with phase discontinuities to steer the propagation of near-source bulk waves in a semi-infinite elastic medium. Our design exploits an array of embedded subwavelength resonators with tailored masses to attain a complete phase shift spanning [Formula: see text]. This phase control allows for diverse wave functionalities, such as directional refraction and energy focusing. Through the use of dispersion diagrams and the generalized Snell's law, along with a multiple scattering formulation, we analytically demonstrate the effectiveness of our design in achieving the desired wavefront manipulation. The proposed design has the potential to advance the field of guiding elastic waves using metamaterials and find practical applications in areas such as isolating ground-borne vibrations in densely urbanized regions and energy harvesting. This article is part of the theme issue 'Current developments in elastic and acoustic metamaterials science (Part 1)'.

Exact Solutions for Radiative Transfer with Partial Frequency Redistribution

The construction of exact solutions for radiative transfer in a plane-parallel medium has been addressed by Hemsch and Ferziger in 1972 for a partial frequency redistribution model of the formation of spectral lines consisting in a linear combination of frequency coherent and fully incoherent scattering. The method of solution is based on an eigenfunction expansion of the radiation field, leading to two singular integral equations with Cauchy or Cauchy-type kernels, that have to be solved one after the other. We reconsider this problem, using as starting point the integral formulation of the radiative transfer equation, where the terms involving the coupling between the two scattering mechanisms are clearly displayed, as well as the primary source of photons. With an inverse Laplace transform, we recover the singular integral equations previously established and, with Hilbert transforms as in the previous work, recast them as boundary value problems in the complex plane. Their solutions are presented in detail for an infinite and a semi-infinite medium. The coupling terms are carefully analyzed and consistency with either the coherent or the incoherent limit is systematically checked. We recover the important result of the previous work that an exact solution exists for an infinite medium, whereas for a semi-infinite medium, which requires the introduction of half-space auxiliary functions, the solution is given by a Fredholm integral equation to be solved numerically. The solutions of the singular integral equations are used to construct explicit expressions providing the radiation field for an arbitrary primary source and for the Green’s function. An explicit expression is given for the radiation field emerging from a semi-infinite medium.

Comparative analysis of double and single porosity effects on SH-wave induced vibrations in periodic porous lattices

The manuscript presents a novel study comparing the impact of single and double porosity on horizontally polarized shear wave (SH-wave) propagation in corrugated elastic void materials. The considered mathematical model comprises two cases; the first one depicts SH-wave propagation through a void porous layer having creased boundaries and resting over a heterogeneous anisotropic fluid-saturated fractured porous half-space, whereas according to the second case, heterogeneous anisotropic fluid-saturated porous semi-infinite medium without fractures has been considered. In both cases, rigidity and density of the half-space vary quadratically with depth. The separable variable method is used to attain the complex frequency equation for each case that leads to two different dispersion relations associated with two distinct wave fronts. The first wave front depends on the void parameters, whereas the second wave front defines the propagation of SH-waves in an elastic layer without void pores. In each case, the complex dispersion relation has been separated into two equations that illustrate the dispersion and attenuation properties of SH-waves. Using the variations in the inhomogeneity, position, fluctuation, flatness, total porosity, and anisotropy parameters, case I and case II have been compared in each graphical execution. In addition, the surface response of shear stress and displacement have been implemented graphically.

Scattering of SH wave in an elastic half-space by a semi-elliptical crater with surface elasticity

Cratering occurs inevitably on the surface of materials at all scales due to comprehensive factors. The presence of craters may have a significant influence on the mechanical performance of the surface of an elastic body. In this paper, we study the SH wave propagation in an elastic semi-infinite medium whose surface has a semi-elliptical crater. We explore the scattering of the SH wave by the crater when surface elasticity is incorporated in the model of deformation to account for possible small-scale effects of the crater. We obtain a series solution for the scattered wave function by employing the Mathieu functions in the corresponding elliptical coordinate system. We discuss in some numerical examples the dynamic stress concentration around the crater relative to the wave frequency and few geometric parameters.

Construction of Short-Time Heat Conduction Solutions in One-Dimensional Finite Rectangular Bodies

Abstract The concept of both penetration and deviation times for rectangular coordinates along with the principle of superposition for linear problems allows short-time solutions to be constructed for a one-dimensional (1D) rectangular finite body from the well-known solutions of a semi-infinite medium. Some adequate physical considerations due to thermal symmetries with respect to the middle plane of a slab to simulate homogeneous boundary conditions of the first and second kinds are also needed. These solutions can be applied at the level of accuracy desired (one part in 10A, with A = 2, 3, …, 15) with respect to the maximum temperature variation (that always occurs at the active surface and at the time of evaluation) in place of the exact analytical solution to the problem of interest consisting of an infinite number of terms and, hence, unapplicable.

Goos–Hänchen shift for coupled vibrational modes in a semiconductor structure

We present a theoretical investigation of the Goös–Hanchen shift (GHS) experienced by acoustic and optical vibrational modes reflected and transmitted from the surfaces of a semiconductor thin film sandwiched between two semi-infinite media. Our study focuses on the impact of the incident angle on the GHS, considering the coupling between longitudinal and transverse modes. For acoustic vibrations, our findings reveal that the GHS can reach magnitudes up to seven times larger than the thickness of the thin film and up to 20 times larger than the incident wavelength. Besides, it is shown that this significant amplification of the GHS highlights the strong influence of the incident angle and the frequency of the modes involved. In the case of optical vibrations, we observe even more pronounced GHS values, exceeding 30 times the incident wavelength. This demonstrates the potential of GHS in acoustical systems, which opens up possibilities for applications in the design of acoustic devices.

Elucidating the coherent and incoherent reflection of plane-waves in finite thickness media: a practical pedagogical application

The propagation of light through a thin film interfaced between two semi-infinite media serves as a compelling illustration for elucidating electromagnetic wave interactions with matter at the undergraduate level. Despite its pedagogical significance and diverse technological applications, this model often receives inadequate attention in foundational electromagnetism literature, limiting early student exposure to this emblematic concept. In this pedagogical initiative, we present a comprehensive analysis of electromagnetic wave propagation through a dielectric medium positioned between semi-infinite media. We examine interference phenomena arising from back-and-forth reflected waves within the dielectric, focusing on the coherent and incoherent reflection regimes as limiting cases. Employing rigorous analytical treatment, we delineate transmittance and reflectance profiles, offering students a lucid understanding of how the refractive index’s real and imaginary components compete and manifest under specific conditions. This analytical approach enhances students’ comprehension of electromagnetic wave behavior within diverse mediums. Furthermore, we extend this theoretical foundation to practical applications, emphasizing renewable energy contexts. By calculating absorptance, we estimate the maximum photo-generated current and power conversion efficiency of a prototype solar cell, establishing a tangible link between theoretical knowledge and real-world solar energy utilization.

Cluster modeling of the interaction of statyonary SH-waves with a system of curvilinear cracks in a half-space

A method is proposed for solving problems of mathematical physics for semi-infinite media containing systems of curvilinear cuts. Based on singular integral equations (SIE), a unified approach to solving the problem has been developed. Numerical implementation was carried out using parallelisation. A graph of the duration of the initial cluster hour is given for calculating an array of searched functions in the context of a parabolic form as a function of the number of processes for one variant of implementation. It is shown that the entire algorithm scales well and has an efficient number of processes. The expected result was obtained, because the structure of the calculation process in models based on SIE is mostly unchanged and built on well-defined procedures. It turned out that 150–200 processes are effective. An accuracy of 10–12 was achieved with the number of collocation points of the contour of each section N = 300, because the algorithm for numerical solution of the problem uses interpolation by Chebyshev polynomials in accordance with the fact that the unknown function has a key feature at the ends of the section, which causes a higher speed of convergence of the algorithm. The study of the question of the further increase in the accuracy of the result was not conducted. The corresponding dynamic boundary value problems for a restrained and force-free half-plane are studied. The influence of the curvature of defects, their interaction and proximity of the boundary on the magnitude and nature of the change in the dynamic stress intensity coefficients was studied. To check the reliability of the algorithm, two tests were carried out: removing three inhomogeneities at a distance of 106 of their length from each other and saturating the system by increasing the number of geometrically equal reflectors to 13–15. When removed, the characteristics of each reflector tend to the characteristics of a single one, and when saturated, they tend to the results of the corresponding periodic problem. The agreement between the results showed good reliability of the algorithm. The proposed method can be used to assess the influence of various mechanical or geometric factors on the strength of bodies with defects.

Research On Soil Displacement Prediction Method Caused By High-pressure Jet Grouting Pile Group Construction Under Adjacent Pile Response

This paper analyzes the soil displacement effect during high-pressure jet grouting pile construction by simplifying the construction process as a series of pressure-controlled spherical cavity expansions in a semi-infinite soil medium. Based on the elastic-plastic solution of pressure-controlled spherical cavity expansion theory, a calculation method is proposed to estimate the soil displacement caused by high-pressure jet grouting pile construction. Non-linear contact theory between the pile and soil is introduced, and the finite difference method is employed to determine the internal forces and deformations of the pile. The proposed method is applied to field construction cases, and its validity is verified by comparing the results with on-site monitoring data. The research findings reveal that the soil displacement effect decreases with depth and primarily occurs in shallow soil layers during the construction of individual high-pressure jet grouting piles. The lateral displacement at the ground surface increases first and then decreases with the increase in horizontal distance, while the ground uplift exhibits an exponential decay trend.

Transient thermoelastic response in a semi-infinite medium subjected to a moving heat source: an implementation of the Moore–Gibson–Thompson model with higher-order memory-dependent derivatives

Transient thermoelastic response in a semi-infinite medium subjected to a moving heat source: an implementation of the Moore–Gibson–Thompson model with higher-order memory-dependent derivatives

Influence of the Schottky Junction on the Propagation Characteristics of Shear Horizontal Waves in a Piezoelectric Semiconductor Semi-Infinite Medium

In this paper, a theoretical model of the propagation of a shear horizontal wave in a piezoelectric semiconductor semi-infinite medium is established using the optimized spectral method. First, the basic equations of the piezoelectric semiconductor semi-infinite medium are derived with the consideration of biased electric fields. Then, considering the propagation of a shear horizontal wave in the piezoelectric semiconductor semi-infinite medium, two equivalent mathematical models are established. In the first mathematical model, the Schottky junction is theoretically treated as an electrically imperfect interface, and an interface characteristic length is utilized to describe the interface effect of the Schottky junction. To legitimately confirm the interface characteristic length, a second mathematical model is established, in which the Schottky junction is theoretically treated as an electrical gradient layer. Finally, the dispersion and attenuation curves of shear horizontal waves are numerically calculated using these two mathematical models to discuss the influence of the Schottky junction on the dispersion and attenuation characteristics of shear horizontal waves. Utilizing the equivalence of these two mathematical models and the above numerical results, the numerical value of the interface characteristic length is reliably legitimately confirmed; this value is independent of the thickness of the upper metal layer, the doping concentration of the lower n-type piezoelectric semiconductor substrate, and biasing electric fields. Only the biasing electric field parallel to the Schottky junction can provide an evident influence on the attenuation characteristics of shear horizontal waves and enhance the interface effect of the Schottky junction. Since the second mathematical model is also a validation of our previous mathematical model established through the state transfer equation method, some numerical results calculated using these two mathematical models are compared with those obtained using the previous method to verify the correctness and superiority of the research work presented in this paper. Since these two mathematical models can better calculate the dispersion and attenuation curves of high-frequency waves in micro- and nano-scale piezoelectric semiconductor materials, the establishment of mathematical models and the revelation of physical mechanisms are fundamental to the analysis and optimization of micro-scale resonators, energy harvesters, and amplifications.

Evaluation model of circular tunnel lining subjected to dislocation earthquake point source in semi-infinite media

Earthquakes are generated by fault dislocations, propagates outward in the form of seismic waves in the stratum, and cause the mechanical response of the tunnel lining. In this paper, an evaluation model of circular tunnel lining under the action of dislocation earthquake point source in semi-infinite space is proposed, which describes the effect of the earthquake point source on the tunnels. First, the tunnel-source coordinate system is defined, the Green function-based equation for the stress field due to dislocation source in a semi-infinite medium is derived, and the source time function is introduced. Then, based on the seismic stress state at the tunnel location, the stress response of the tunnel-ground system is derived, and a miniature dynamic finite element model is developed to verify the evaluation model. Compared with existing related research, this evaluation model consists of the dislocation earthquake source and the local tunnel system, instead of only focusing on local tunnel systems, which can take into account the spatial location of the earthquake source and the tunnel, the magnitude, and the type of earthquake. In addition, the parameters such as source fault type, epicenter distance, magnitude, earthquake rise time, and tunnel azimuth are also discussed. The evaluation model can analyze the response of the tunnel during an earthquake at a larger scale.

The Vertical Behaviors of Dissimilar Pile Composite Foundations over Inclined Bedrock

Pile composite foundations (PCFs) have been commonly applied in reinforcement engineering to adjust the vertical stiffness of foundations, due to the displacement control design criteria for foundations. PCFs that have dissimilar pile lengths, located over inclined bedrock, have shown significantly different vertical behaviors from PCFs with equal pile lengths, placed over a semi-infinite medium. However, the vertical behaviors of dissimilar PCFs over inclined bedrock cannot be predicted with the current theoretical methods, although they have been widely adopted in engineering. An analytical method is proposed in this investigation to analyze the vertical bearing characteristics of dissimilar PCFs over inclined bedrock. A pile–soil system is decomposed into fictitious piles and extended soil, and then a control equation to determine the axial force along the fictitious piles is established, stemming from the compatibility conditions between them. The vertical behaviors of dissimilar PCFs can be obtained by solving the control equation with iterative procedures, and the equation is verified by two field load tests of single piles from the Honghe bridge and a numerical case. Then, the settlement and load transfer behaviors of 3 × 1 dissimilar PCFs and their influence factors are analyzed, and the results are as follows. (1) Obvious differences can be observed concerning the axial force distribution, settlement w, and load-sharing ratio (LSR) of each pile element for different pile–soil stiffness ratios (Ep/Es). (2) The LSR of pile 1 increases from 0.074 to 0.253 for the rigid pile and from 0.062 to 0.161 for the flexible pile condition when the cushion stiffness Kc changes from 1 × 104 kN/m to 3 × 108 kN/m. The non-dimensional vertical stiffness of the foundation, N0/wdEs, increases from 10.21 to 28.95 for the rigid pile condition and increases from 8.69 to 14.44 for the flexible pile condition, when Kc increases from 1 × 104 kN/m to 4 × 105 kN/m. (3) The neutral layer depth of the pile zn, the average settlement w, and the differential settlement wd of each element head decrease with Kc, and no negative friction zone exists (zn = 0 m) for all the pile elements when Kc> 2 × 105 kN/m. (4) The N0/wdEs decreases with the distance between the pile bottom and the inclined bedrock Δ. For the rigid and flexible pile conditions, the N0/wdEs is 22.16 and 13.48 for Δ = 1 m, and 13.13 and 10.10 for Δ = 8 m. The wd reaches 16.7 mm and 4.0 mm for Δ = 1 m and Δ = 8 m, respectively. (5) The N0/wdEs increases almost linearly with an increase in l/d for the rigid pile condition, and it gradually decreases for the flexible pile condition. The developed model can improve the design and analysis of PCFs located over inclined bedrock under vertical loading.

Time-domain instability mechanism for artificial boundary condition of semi-infinite medium described by discrete rational approximation

The use of discrete rational approximation functions to represent foundation dynamic impedance plays a crucial role in the time-domain analysis of soil-structure interaction regarding artificial boundary. In modeling, the rational approximation functions need to be transformed into time-domain recursive models for incorporation into time-history analysis. However, this method also suffers from uncertainties in parameter identification and instability in time-domain analysis when the time-domain recursive models of soil are coupled with upper structure. The stability issue in artificial boundary coupled time-domain computation has not been effectively addressed. This paper establishes the closed-loop transfer function of the coupled system and analyzes the causes of instability in the coupled system based on gain margin analysis. Different numerical simulations demonstrate that the major cause of artificial boundary system instability is the description errors of discrete rational approximation functions beyond the fitting frequency range.

Corrosion at the Steel–Medium Interface

Corrosion on the interface between a metal alloy, such as steel, and a wet, permeable non-metallic medium is of considerable practical interest. Examples include the interface between steel and water, the atmosphere or concrete, as for steel reinforcement bars; between metal and soil, as for buried cast iron or steel pipes; deposits of some type, as in under-deposit corrosion; and the interface with insulation, protective coatings, or macro- or micro-biological agents. In all cases, corrosion initiation depends on the characteristics of the interfacial zone, both of the metal and the medium, and the spatial variability. For (near-)homogeneous semi-infinite media with good interfacial contact, the pitting, crevices and general corrosion of the metal will be largely controlled by the metal (micro-)characteristics, including its inclusions, imperfections and surface roughness. In other cases, these may be overshadowed by the macro-characteristics of the medium and the degree of interfacial contact, possibly with severe resulting corrosion. Where the build-up of corrosion products can occur at the interface, they will dominate longer-term corrosion and govern the long-term corrosion rate. For media of finite thickness, diffusion issues and material deterioration may also be involved. The practical implications are outlined. It is argued that with the presence of a suitable medium, it is possible to achieve negligible long-term corrosion but only if certain practical actions are taken.

Solution of the fractional diffusion equation by using Caputo-Fabrizio derivative: application to intrinsic arsenic diffusion in germanium

In this work, we focused on solving the space-time fractional diffusion equation with an application on the intrinsic arsenic diffusion in germanium. At first we have treated the differential equation in a semi-infinite medium by using Caputo-Fabrizio fractional derivative. We have introduced the Laplace transform to solve this type of equations. Secondly, Based on the obtained solution, we have simulated an profile of arsenic diffusion in germanium under intrinsic conditions. Accurate simulations have been achieved showing that the fractional derivative orders affect on the estimation of the diffusion coefficient, where increasing the time fractional derivative order α reduces the value of the diffusion coefficient, while increasing the space fractional derivative order β increases the value of the diffusion coefficient.

Influence of Homo- and Hetero-Junctions on the Propagation Characteristics of Radially Propagated Cylindrical Surface Acoustic Waves in a Piezoelectric Semiconductor Semi-Infinite Medium

This paper theoretically investigates the influence of homo- and hetero-junctions on the propagation characteristics of radially propagated cylindrical surface acoustic waves in a piezoelectric semiconductor semi-infinite medium. First, the basic equations of the piezoelectric semiconductor semi-infinite medium are mathematically derived. Then, based on these basic equations and the transfer matrix method, two equivalent mathematical models are established concerning the propagation of radially propagated cylindrical surface acoustic waves in this piezoelectric semiconductor semi-infinite medium. Based on the surface and interface effect theory, the homo- or hetero-junction is theoretically treated as a two-dimensional electrically imperfect interface in the first mathematical model. To legitimately confirm the interface characteristic lengths that appear in the electrically imperfect interface conditions, the homo- or hetero-junction is equivalently treated as a functional gradient thin layer in the second mathematical model. Finally, based on these two mathematical models, the dispersion and attenuation curves of radially propagated cylindrical surface acoustic waves are numerically calculated to discuss the influence of the homo- and hetero-junctions on the dispersion and attenuation characteristics of radially propagated cylindrical surface acoustic waves. The interface characteristic lengths are legitimately confirmed through the comparison of dispersion and attenuation curves calculated using the two equivalent mathematical models. As piezoelectric semiconductor energy harvesters usually work under elastic deformation, the establishment of mathematical models and the revelation of physical mechanisms are both fundamental to the analysis and optimization of micro-scale surface acoustic wave resonators, energy harvesters, and acoustic wave amplification based on the propagation of surface acoustic waves.

Estimation of the stress intensity factor of an interface crack by the scattered SH-far field

The diffraction of the plane elastic SH-wave from a semi-infinite interface crack on junction of two elastic semi-infinite media is studied. The crack is modelled by the mathematical cut with no stress on its faces. The displacement and the stress fields are continuous outside of the crack. The wave diffraction problem is reduced to the solution of the mixed boundary value problem for Helmholtz equation. We search the solution which satisfies the Neumann boundary condition on the crack faces and the continuity condition for the stress and displacement fields outside of the crack. The radiation condition at the infinity and the Meixners condition at the crack tip must be also satisfied. Using the Fourier integral transform, the mixed boundary value problem is reduced to the Wiener-Hopf functional equation which is valid in the given strip of regularity in the complex plane. The method of factorization and decomposition, as well as the Liouvilles theorem, are used to solve this equation. Its kernel function is factorized and represented as a product of two split functions that are regular in the overlapping half-planes. These ones allow for the simple poles outside of the regularity regions. The solution of the Wiener-Hopf equation is presented in analytical form. The scattered displacement field is found for an arbitrary frequency and sounding angle by applying an inverse Fourier integral transform to the solution. The asympŹtotic formula for the stress intensity factor (SIF) at the crack tip is obtained. The correlation between complex amplitude of the SH-wave far scattered displacement field and the SIF caused by this field is obtained for an arbitrary radiation angle, frequency and medium parameters. This allows us to express the SIF through the amplitude of the far field if the radiation frequency, sounding angle as well as the physical characteristics of materials are known. It is shown that in the plane that is normal to the tip of the crack the ratio of SIF for the given junction and two fixed values of the sounding angle or two sounding frequencies is proportional to the ratio of the scattered fields. It is found that under above mentioned condition the proportionality rate does not depend on material properties. The obtained relations can be applied for estimation of the SIF with changing frequency and sounding angle.

Analytical solution for the single scattered radiance of two-layered turbid media in the spatial frequency domain, Part 1: Scalar radiative transfer equation

In this paper analytical solutions to the radiative transport equation are derived for the angular resolved single scattered radiance in a semi-infinite turbid medium having depth dependent optical properties. Explicit formulas are given for a two-layered turbid medium. A difference in the refractive index between the surrounding non-scattering medium and the two-layered medium is considered. The result displays a linear relationship between the phase functions of both layers and the radiance. In order to verify the results, the Monte Carlo method is used to compare calculations for varying configurations of the medium and phase functions. For all cases, the analytical solution and the Monte Carlo simulations show excellent agreement with one another.

An Analytical Approach to Contaminant Transport with Spatially and Temporally Dependent Dispersion in a Heterogeneous Porous Medium

This study presents an analytical solution to the one-dimensional advection-dispersion equation (ADE) for a semi-infinite heterogeneous aquifer system with space and time-dependent groundwater velocity and dispersion coefficient. The dispersion coefficient is assumed to be proportional to the groundwater flow velocity. In addition, retardation factor, first-order decay and zero-order production terms are also considered. Contaminants and porous media are assumed to be chemically inert. Initially, it is assumed that some uniformly distributed solutes are already present in the aquifer domain. The input point source is considered uniformly continuous and increasing nature in a semi-infinite porous medium. The solutions are obtained analytically using the Laplace Integral Transform Technique (LITT). The nature of the concentration profile of the resulting solution for different parameters in different time domains is illustrated graphically.