Quasi-longitudinal Waves Research Articles

Estimation of the coupling loss factors of structural junctions with in-plane waves by means of the inverse statistical energy analysis problem

The Coupling Loss Factors (CLF) to describe the vibration transmission at junctions in the context of Statistical Energy Analysis (SEA) are determined. Out-of-plane energy (bending waves) and in-plane energy (treating transverse shear waves and quasi longitudinal waves as a single subsystem) as well as their interactions are considered. The CLF values are obtained from the inverse SEA problem where the energies and input powers are computed numerically by means of a Spectral Finite Element model. A set of L, T and X-junctions with variable dimensions and material properties typical from heavy building structures is considered. The Transmission Loss (TL) of the junctions is computed from the CLF values. Afterwards the TL results are approximated by means least squares leading to simplified expressions for each transmission path. The TL SEA results are compared with the TL predicted by means of a semi-infinite wave-based model.

Analysis of Two Colliding Thin Spherical Shells

In the present paper, the collision of two elastic spherical shells is investigated using the wave theory of impact. The model developed here suggests that after the moment of impact quasi-longitudinal and quasi-transverse shock waves are generated, which then propagate along the spherical shells. The solution behind the wave fronts is constructed with the help of the theory of discontinuities. Since the local bearing of the materials of the colliding elastic shells is taken into account, then the solution in the contact domain is found via the Hertz contact theory.

Reflection of plane waves on the stress-free and rigid boundary surfaces of pre-stressed piezoelectric-orthotropic substrate: A comparative approach

This study delves with the reflection phenomenon of plane-waves due to incidence of quasi-longitudinal (QL) wave at a semi-infinite structure comprising of pre-stressed piezoelectric-orthotropic substrate (POS) with stress-free (Case I) and rigid (Case II) bounding surfaces. The compact-form expressions of reflection-coefficients (RCs) of reflected QL wave, quasi-transverse (QT) wave and evanescent-electroacoustic (EA) wave for different cases of pre-stressed POS are deduced. To unravel effects of higher anisotropy in substrate, comparative analysis in pre-stressed transversely isotopic piezoelectric substrate (TIPS) medium with stress-free (Case III) and rigid (Case IV) boundaries is carried out. PVDF and ALN materials are considered for numerical computations. For validation, energy-partition and conditions of grazing incidence for all the undertaken cases are analyzed.

Three-dimensional nonreciprocal transmission in a layered nonlinear elastic wave metamaterial

In this investigation, three-dimensional nonreciprocal transmission is realized by the combination of structural asymmetry and material nonlinearity. Analytical solutions of the coupled quasi-longitudinal (qP) and quasi-shear (qS1 and qS2) waves are derived. According to Bloch’s law and the stiffness matrix method, we obtain the band gaps and the transmission coefficients of the fundamental and double frequency waves. We also discuss the effects of the incident azimuthal angle on the propagation behaviors in nonlinear phononic crystals and elastic wave metamaterials. This present work is expected to help design practical devices of nonreciprocal transmission with vector characteristics.

Effects of initial stresses on reflection phenomenon of plane waves at the free surface of a rotating piezothermoelastic fiber-reinforced composite half-space

In this paper the impact of initial stresses on reflection characteristics of plane waves at the free surface of a rotating piezothermoelastic fiber-reinforced composite (PTFRC) half-space has been investigated. Three theories of thermoelasticity viz. Green-Lindsay, Lord-Shulman and Classical dynamical coupled theories have been used to explore the problem. The expressions of the amplitude ratios of four types of reflected waves viz. quasi-longitudinal (qP), quasi-transverse (qSV), thermal (T-mode) and electro acoustic (EA) waves have been obtained with the aid of suitable boundary conditions which exhibited their reliance on the incident angle, the properties of the rotating media and varying magnitudes of initial stresses and the impacts of the aforesaid parameters have been extensively studied and demonstrated graphically. Also, the energy ratio of the reflected waves has been derived for different values of initial stresses and it has been showed that the total energy flux at the surface attains value unity.

Evanescent waves in two-dimensional fluid-saturated porous metamaterials with a transversely isotropic matrix

Wave propagation in a two-dimensional periodic fluid-saturated porous metamaterial (FSPM) is investigated. The constitutive relation considered for fluid-saturated porous materials is based on Biot’s homogeneization theory. Such media generally support two shear and two longitudinal elastic waves. Anisotropic wave propagation results both from anisotropy of the solid matrix and from the periodic structure of the metamaterial. Special attention is devoted to the dispersion and attenuation of evanescent Bloch waves in a FSPM whose matrix is transversely isotropic in which case in-plane elastic waves are a superposition of one shear wave and of two longitudinal waves. Bloch waves, complex band structures, and transmission properties are obtained numerically using finite element analysis. The case of homogeneous fluid-saturated porous media is considered since numerical simulations can be compared directly with analytic results in this case. The effects of material anisotropy and of fluid viscosity on wave propagation in two-dimensional FSPM are then discussed. It is found that wave-number band gaps appear in the complex band structure of the lossless FSPM due to the interference of waves with different but nonorthogonal polarizations. Wave-number band gaps are connected continuously by complex-frequency bands, hence, defining stop bands in the time domain. No complete frequency band gaps are found in two-dimensional lossless FSPM, in contrast to the one-dimensional case [Y.-F. Wang et al. Phys. Rev. B 99, 134304 (2019)] due to the presence of the additional quasishear wave accompanying the two quasi-longitudinal waves. Both the complex band structure and the transmission properties are affected by the anisotropy of the solid matrix. Wide transmission dips come up when viscosity is introduced as a result of the strong attenuation and the coupling of all wave polarizations. Concurrently, wave-number band gaps are washed out by viscosity. This theoretical paper has relevance to practical applications of fluid-saturated porous metamaterials, e.g., in concrete structures and geological soils.

Anisotropy of acousto-optic figure of merit for the case of a Bragg-diffracted wave propagating along an optic axis in optically biaxial crystals: Tl3AsS4 crystals.

In the present work, we analyze acousto-optic (AO) interactions under the condition when a diffracted optical wave propagates along one of the optic axes in optically biaxial crystals. We find that the number of AO interaction types is reduced from nine, which is typical for the most general case, to six, since some of the isotropic and anisotropic diffraction types manifest themselves simultaneously. We reveal that, in the case of the interactions of quasi-ordinary and quasi-extraordinary optical waves, the dependences of effective elasto-optic coefficients and AO figure of merit on the tracing angle of an incident optical beam around the optic axis are symmetric with respect to the plane that is perpendicular to the plane of optic axes and contains the principal axis Z. We determine the geometry of AO interactions at which the highest AO figure of merit can be expected. Finally, we show that, in the case of propagation of diffracted light along one of the optic axes in the biaxial Tl3AsS4 crystals, the highest AO figures of merit, M2=1290×10-15s3/kg and M2=1305×10-15s3/kg, can be reached when the quasi-longitudinal acoustic wave interacts with the quasi-ordinary or quasi-extraordinary optical waves in the plane of optic axes.

Линии тока и вектор плотности потока энергии при возбуждении колебаний пьезоэлектрическим преобразователем в слоистом фононном кристалле

The use of periodic composite structures has a great potential for improving sensors/actuators and self-adjusting optics, as well as for active vibration suppression and energy storage systems. At the same time, the employment of dielectric elastomers offers even greater possibilities since they provide tools for the active control of wave energy fluxes. The mathematical formulation of the problem considers more general state equations for piezoelectric bodies. Namely, the absence of symmetry in elastic and piezoelectric tensors is taken into account. The present study investigates the features of elastic wave excitation by a surface piezoelectric transducer (generally made of dielectric elastomer) in multi-layered periodic composites or so-called phononic crystals. A hybrid approach is used to solve the boundary value problem, which involves application of the spectral element method and the boundary integral equation method. Employing the energy streamlines and the power density vector, wave phenomena associated with the interaction of a piezoelectric transducer with a layered phononic crystal at frequencies belonging to four frequency ranges (band gaps, pass bands and the frequency ranges, in which only a quasi-transverse or quasi-longitudinal waves propagate without attenuation) are analyzed.

Scattering of waves at the interface of two distinct anisotropic media

The present paper aims to study the reflection/transmission of two-dimensional plane elastic waves (quasi-longitudinal displacement waves) at the interface between a self-reinforced half-space and an orthotropic micropolar elastic solid half-space. Three kinds of coupled elastic waves are reflected in the orthotropic micropolar half-space, and two types of waves are transmitted in the self-reinforced medium. The phase velocities obtained in closed form depend upon the angle of propagation and material parameters. The boundary conditions at the interface of the two media are solved, and reflection/transmission coefficients are obtained using Cramer’s rule. Numerical examples are given, and dependence of the reflection/transmission coefficients on the incident angle has been shown graphically. Also, the comparisons between anisotropic and isotropic cases have been marked distinctly. The obtained results may be utilised for the interpretation and analysis of geophysical data.

Propagation of Rayleigh waves at the boundary of an orthotropic elastic solid: Influence of initial stress and gravity

Propagation of harmonic plane waves is considered in an orthotropic elastic medium in the presence of initial stress and gravity. Roots of a quadratic equation define the propagation of one quasi-longitudinal wave and one quasi-transverse wave in a symmetry plane in this medium. These two waves are coupled in the identical phase to define the propagation of Rayleigh waves at the boundary of the medium. Two conditions at the stress-free boundary translate into a complex frequency equation, which explains the dispersive behavior of this Rayleigh wave. For the presence of radical terms, this complex equation is rationalized into a real algebraic equation. Only one root of this algebraic equation satisfies the mother frequency equation and hence represents the propagation of dispersive Rayleigh waves at the boundary of the orthotropic solid. The influence of initial stress and gravity on velocity and polarization of Rayleigh waves is observed through a numerical example.

Acoustic and double elastic shock waves in single-crystal graphene

Double elastic shock waves are rarely observed in two-dimensional (2D) materials and normally unexpected for elastically isotropic 2D crystals such as graphene. With large-scale molecular dynamics simulations, we show that in single-crystal graphene shock-loaded along nonzigzag and nonarmchair directions, double elastic shock waves (quasilongitudinal and quasitransverse) can emerge. Quantitative acoustic wave equation analysis reveals that shock-induced symmetry reduction in lattice, as well as in elastic stiffness tensor, gives rise to the normally unexpected quasitransverse wave following the quasilongitudinal wave.

Reflection of photothermoelastic waves in a semiconductor material with different relaxations

In this article, reflection of thermoelastic waves in semiconducting medium is studied. The selected elastic medium is a half-space with an exponentially decaying heat source. Firstly, the governing equations are formulated with the help of generalized thermoelastic model initiated by Green and Lindsay. Interface of the solid is exposed to a heat flow having an exponentially decaying pulse. Harmonic wave solution is adopted to find the solution. It is found that, for the selected half-space model, four types of coupled waves are generated after reflection, namely quasi-longitudinal wave $${\text{P}}_{\text{q}}$$ , quasi-transverse wave $${\text{SV}}_{\text{q}}$$ , quasi-thermal wave $${\text{T}}_{\text{q}}$$ , and quasi-plasma wave $${\text{PL}}_{\text{q}}$$ . Different characteristics of waves subjected to angular frequency of incident wave are calculated numerically for the medium. Effect of the incident angle on amplitudes of reflected waves has also been calculated in terms of amplitude ratios for a semiconductor like material. Comparison of results obtained in the context three different models of thermoelastic theories is also presented, and the findings are highlighted.

Reflection phenomena of waves through rotating elastic medium with micro-temperature effect

Abstract In this article, we analyzed the effect of variable thermal conductivity on reflected elastic waves. The waves are propagating through a thermoelastic medium rotating with some angular frequency. The concept of micro-temperature is also been considered, in which microelements of the medium contain a high temperature. A heat conduction phenomenon is encountered by dual phase-lag heat conduction model. P (or SV)-type wave is incident on the medium with some specific angle of incidence. After reflection from the surface incident, P-wave is converted into quasi longitudinal and quasi transverse waves and propagates back into the medium. Helmholtz’s potential function along with the harmonic wave solution is used to obtain the solution of the model. Analytically, we calculated the amplitude ratios and attenuation factor for each reflected wave against the angle of incidence. The obtained results are also represented graphically for different values of rotational frequency and variable thermal conductivity for a particular material.

Lamb Waves and Adaptive Beamforming for Aberration Correction in Medical Ultrasound Imaging.

Phase aberration in transcranial ultrasound imaging (TUI) caused by the human skull leads to an inaccurate image reconstruction. In this article, we present a novel method for estimating the speed of sound and an adaptive beamforming technique for phase aberration correction in a flat polyvinylchloride (PVC) slab as a model for the human skull. First, the speed of sound of the PVC slab is found by extracting the overlapping quasi-longitudinal wave velocities of symmetrical Lamb waves in the frequency-wavenumber domain. Then, the thickness of the plate is determined by the echoes from its front and back side. Next, an adaptive beamforming method is developed, utilizing the measured sound speed map of the imaging medium. Finally, to minimize reverberation artifacts caused by strong scatterers (i.e., needles), a dual probe setup is proposed. In this setup, we image the medium from two opposite directions, and the final image can be the minimum intensity projection of the inherently co-registered images of the opposed probes. Our results confirm that the Lamb wave method estimates the longitudinal speed of the slab with an error of 3.5% and is independent of its shear wave speed. Benefiting from the acquired sound speed map, our adaptive beamformer reduces (in real time) a mislocation error of 3.1, caused by an 8 mm slab, to 0.1 mm. Finally, the dual probe configuration shows 7 dB improvement in removing reverberation artifacts of the needle, at the cost of only 2.4-dB contrast loss. The proposed image formation method can be used, e.g., to monitor deep brain stimulation procedures and localization of the electrode(s) deep inside the brain from two temporal bones on the sides of the human skull.

Degenerate Structure of Transformation Twins and Estimation of Dislocation Density in Martensite Crystals

In the dynamic theory of martensitic transformations, the wave mechanism of controlling martensite crystal growth is determined by the superposition of wave beams of quasi-longitudinal (or longitudinal) waves carrying the “tensile–compression” deformation in the orthogonal directions. The wave beam formation is considered to be a result of the formation of excited (vibrational) states. The existence of transformation twins is interpreted as a result of a matched propagation with respect to long-wave (l waves) and short-wave (s waves) shifts. The matching condition is analyzed for the γ–α martensitic transformation in iron-base alloys. It is shown for the first time that the transition to a degenerate twin structure with the allowance for the medium discreteness enables one to estimate the dislocation density in crystals with habit {557}, which agrees with that observed experimentally.

Analysis of wave scattering in 3-layer piezo composite structure [Pb[Zr x Ti1- x ]O3-ALN-Pb[Zr x Ti1- x ]O3

An analytical model has been considered to study the scattering phenomenon of the quasi-plane wave in functionally graded piezoelectric material (FGPM) layer sandwiched between two similar piezoelectric (PE) media. In more contrast, the effect of material gradient factor on amplitude ratios of various reflected and transmitted waves has been investigated. The composite structure consists of an FGPM layer bonded between two similar transversely isotropic PE half-spaces. It has been shown that incident quasi-longitudinal wave (qP-mode) at lower interface generates reflected and transmitted waves namely qP and quasi-shear vertical waves (qSV-mode). These transmitted waves further give rise to reflected and transmitted waves at the upper interface. At the interfaces of the FGPM layer, the stresses, displacements and electric potential functions are continuous. Due to the scattering of waves, 12 independent amplitude ratios of displacement components are obtained. Amplitude ratios of various reflected and transmitted waves (for the electrically short case) have been obtained by solving a set of linear algebraic equations (using Crammer’s rule). The impact of gradient factor has been examined on amplitude ratios against the angle of incidence and presented graphically. The plot of the slowness section is also demonstrated through graph. In addition, deduced results are compared with the earlier studies and found to be in agreement with the findings.

Stability analysis of the mixed variable method and its application in wave reflection and transmission in multilayered anisotropic structures

This paper focuses on the reflection and transmission (R/T) problem of elastic waves in multilayered anisotropic structures. Stability analysis of the mixed variable method (MVM) for computing the R/T coefficients of elastic waves in multilayered anisotropic structures is presented. For this purpose, a detailed comparison of the MVM with the other two widely used methods, namely the transfer matrix method (TMM) and the stiffness matrix method (SMM), is made. Although the TMM, the SMM and the MVM are mathematically equivalent, they are quite different in numerical stability. The theoretical analysis shows that the MVM is unconditionally stable for arbitrary wavenumber–thickness products, whereas the TMM and the SMM may become unstable for large or small wavenumber–thickness products, respectively. This conclusion is numerically verified by various examples. Finally, the R/T coefficients of elastic waves in generally anisotropic multilayered structures bounded by two semi-infinite spaces are calculated using the MVM for a quasi-longitudinal or quasi-transverse wave incidence, and the effects of incident angles and wavenumber–thickness products on the R/T coefficients are discussed in detail through an example.

Shock compression/release of magnesium single crystals along a low-symmetry orientation: Role of basal slip

To gain insights into the relative contributions of different plastic deformation mechanisms, particularly basal slip, for shocked hexagonal close-packed (hcp) metals, magnesium (Mg) single crystals were subjected to shock compression and release along a low-symmetry (LS) orientation to 1.9 and 4.8 GPa elastic impact stresses. LS-axis is a “nonspecific” direction resulting in propagation of quasilongitudinal and quasishear waves. Wave profiles, measured using laser interferometry, show a small elastic wave followed by two plastic waves in compression; release wave profiles exhibited a structured response for the higher stress and a smooth response for the lower stress. The LS-axis wave profiles are significantly different than profiles published previously for c- and a-axes, demonstrating that Mg single crystals exhibit strong anisotropy under shock compression/release. Numerical simulations, using a time-dependent anisotropic modeling framework, show that shock wave loading along the LS-axis involves the simultaneous operation of multiple deformation mechanisms. Shock compression along LS-axis is dominated by basal slip while prismatic slip and pyramidal I {101¯1}⟨112¯3⟩ slip play a smaller role; coupling between longitudinal and shear deformations was observed. The unloading response is dominated by basal slip with some contribution from prismatic slip; pyramidal I slip is not activated. The present results, unlike results obtained for c- and a-axes, show that the deformation mechanism observed under quasistatic loading conditions along LS-axis is not sufficient to determine the shock response along this orientation. Although requiring numerical simulations for wave analysis, shock propagation along a LS-orientation provides new insights into the plastic deformation response of hcp metal single crystals.

Impact of the Space Charge Distribution in the Model Source of Quasi-Electrostatic Whistler-Mode Waves on the Effective Length of a Short Receiving Antenna

In our previous paper [1] it was shown, using the theory of antennas in plasmas, that the effective length leff of a receiving dipole antenna can be much larger than its geometric length in case quasielectrostatic chorus emissions propagating close to the resonance cone are received. In order to simplify calculations of leff, it was proposed to use a model (“effective”) source of such emissions because taking into account all of the real source properties (that are determined by the nonlinear processes of interaction between waves and charged particles in a wide region of space) would lead to unreasonably cumbersome calculations. The present paper analyzes how the effective length of the spacecraft-borne receiving antenna depends on the parameters of the space charge distribution in the model source of quasi-electrostatic chorus emissions. It is found that the length leff decreases as a power function (with exponent −1/2) with increasing distance (along the group velocity resonance cone) between the model source and the receiving antenna. This relationship is correct up to the distance at which the effective length becomes of the order of the geometric length. At longer distances, the radiation field loses its resonance nature because of the excitation of an electromagnetic (quasi-longitudinal) wave. It is shown that under conditions of the Earth’s magnetosphere, the approximation we used can remain valid up to distances of the order of the geomagnetic field line length, which confirms the importance of the discussed effect for correct interpretation of the electric wave measurement data in the whistler-mode frequency range. It is also shown that the length leff changes by less than 10% when the characteristic scale of spectrum of the charge distribution along the model source varies as Δ ∼ (0.1–1.0) kobs, where the wave number kobs corresponds to the observed spectral maximum of radiation at a given frequency.

Diffraction Resonances of Acoustic Stresses in the Crystal Layer

The six-beam scattering of elastic waves by a crystal film is investigated. Cases of symmetric diffraction of waves inside the crystal layer are analyzed. In this scattering geometry, the projections of the wave vectors on the normal to the film surface are pairwise equal to each other up to the sign. Under certain conditions, quasi-shear waves modulate the structure in the path of quasi-longitudinal waves. If the period of the modulated structure is close to the length of a quasi-longitudinal wave, the amplitude of oscillations of the latter is amplified. The explanation of this effect is similar to the description of parametric resonance.This effect is illustrated by examples of calculations of the scattering spectra of elastic waves by cubic and tetragonal crystals in a three-layer model. Calculations are based on the transfer matrix method.