Plane Wave Propagation Research Articles

On the representation of solutions and the propagation of plane waves under strain gradient theory with dual-phase-lag thermoelasticity

The present work concerns the size-dependent thermoelasticity with the dual-phase-lag model proposed by Chandrasekharaiah and Tzou. In the first part, we study the Galerkin-type representation of the solution of the field equations. Then, we find the fundamental solutions for the steady vibrations of the field equations. In the second part, we analyze the effect of the strain gradient length parameter on the propagation of plane harmonic waves with assigned frequency in an infinite space. We derive the exact dispersion relation solutions for the longitudinal plane waves. Several characterizations of the wave fields, like phase velocity, specific loss, and penetration depth, are obtained. Numerical results are presented to show the strain gradient length parameter effect on the wave field. The validity of numerical findings of different characterization works has been deduced, and a comparison is made with some earlier work. Some limiting cases are also noted and discussed.

Polyvinylidene fluoride transducer shape optimization for the characterization of anisotropic materials.

In the context of ultrasonic determination of mechanical properties, it is common to use oblique incident waves to characterize fluid-immersed anisotropic samples. The lateral displacement of the ultrasonic field owing to leaky guided wave phenomena poses a challenge for data inversion because beam spreading is rarely well represented by plane wave models. In this study, a finite beam model based on the angular spectrum method was developed to estimate the influence of the transducer shape and position on the transmitted signals. Additionally, anisotropic solids were considered so that the beam skewing effect was contemplated. A small-emitter large-receiver configuration was chosen, and the ideal shape and position of the receiving transducer were obtained through a meta-heuristic optimization approach with the goal of achieving a measurement system that sufficiently resembles plane wave propagation. A polyvinylidene fluoride receiver was fabricated based on the findings and tested in three cases: a single-crystal silicon wafer, a lightly anisotropic stainless-steel plate, and a highly anisotropic composite plate. Good agreement was found between the measurements and the plane wave model.

Effect of tube dimensions on resonant frequency of stepped acoustic resonator

An investigation of the effects of tube dimensions on resonant frequencies of stepped acoustic resonators is conducted. A linear acoustic theory is employed to model the plane wave propagation in the stepped resonator. The linear eigenvalue problem obtained by the method of separation of variables is solved to derive a frequency equation. Then, an experimental apparatus is established, in which the tube dimensions can be changed to adjust frequency. The variation pattern of resonant frequency with different lengths and diameters of the sub-tubes is analyzed theoretically and experimentally. The results suggest that the overall length determines the approximate scope of the resonant frequencies while the exact values of resonant frequencies and the relationship between resonant frequencies of each mode are determined by the ratio of sub-tube lengths and the ratio of sub-tube diameters. The calculated results are compared with the measured data. The comparative research declares that the derived frequency equation is an effective tool to calculate the resonant frequency of stepped resonator with adjustable size used in wide-band microphone calibration.

Constitutive modelling and wave propagation through a class of anisotropic elastic metamaterials with local rotation

Elastic metamaterials are typically periodic materials possessing unit cells endowed with engineered architecture much smaller than the typical phenomenological length scale. The development of continuum models capable of accurately representing the effects of this aforementioned architecture is extremely challenging, and hence a sparsely developed area. This paper develops a novel 2-D continuum model capable of capturing the dynamic behaviour of a class of anisotropic elastic metamaterials with local rotational elements in the long wavelength limit. A constitutive relation incorporating these local rotational effects is proposed, and ratified using a representative discrete model using linear Hookean springs and identical rigid disks. The new continuum model is used to generate a dispersion relation for harmonic plane waves propagating in an arbitrary direction, which is subsequently compared to the dispersion behaviour of the original discrete model. The general behaviour of this continuum when subjected to 2-D planar harmonic wave propagation in the anisotropic medium is then analysed, with specific attention given to the effect of material anisotropy and wave propagation direction. This work is the first of its kind to create a new continuum model of a class of anisotropic elastic metamaterials with local rotational effects.

Influence of metal on the far-exploding surface on fragment deformation behavior under contact explosion

Metals exhibit diverse failure behavior under impact loading. In the context of fragment warheads, preformed fragments also undergo fracture and crushing behaviors when subjected to explosive loading, potentially diminishing the terminal effect and damage capability of the warhead. To address this issue, metal disks of varying impedance were applied to the far-exploding surface of the fragments, and their influence on fragment deformation behavior was examined. The experimental results revealed that when metal disks were attached to the far-exploding surface of the fragments, their fracture behavior changed, and the recovered fragments remained intact axially. Additionally, the axial length of the recovered fragments decreased as the impedance of the metal disk on the far-exploding surface increased. To elucidate the underlying mechanism of this experimental phenomenon, the variation in fragment pressure during the propagation process was calculated by employing theories of planar detonation waves and shock wave propagation in the study. The results indicate that when the impedance of the metal disks on the far-exploding surface is higher than that of the fragments, it leads to an increase in internal pressure and the formation of a compression zone within the fragments, thereby preventing fragment fracture. Conversely, lower impedance results in the formation of a tensile effect within the fragments. The theoretical and experimental results were consistent. Finally, based on the dimensional analysis, the dimensionless models were established to predict fragment deformation and internal pressure values influenced by the metal disk on the far-exploding surface.

Signal detection

Acoustic emission (AE) occurring in concrete fracture process is a combination of concrete fracture at local material level and the propagation of induced elastic waves at structural level. Simulation of the fracture-induced AE in concrete needs a model that is able to address the fracture behavior and the wave propagation at the same time. However, to date, no available modelling tools can simulate the whole fracture-induced AE process. Among others, the lattice models have the potential to achieve the simulation of the whole fracture-induced AE process. However, the lattice models are currently restricted to statistical analysis of AE events in fracture process. This paper investigates the feasibility and limitation of lattice model for the simulation of elastic wave propagation process. Several simulation cases are performed considering the propagation of both plane waves and point-source waves. The simulation results, in terms of wave velocity and geometric spreading loss of wave amplitudes, agree well with the analytical solutions. This study represents the first step for the simulation of complete fracture-induced AE waves in concrete.

Characterization of Elastic Constants of Langatate Single Crystals with 32 Symmetry Using Ultrasonic Pulse‐Echo Technique

AbstractIn this study, the propagation of plane waves in the lanthanum gallium tantalate (langatate, LGT) single crystals is investigated. Moreover, the flight time of different waves in the LGT rectangular parallelepiped sample is measured using the ultrasonic pulse‐echo (UPE) technique, and the elastic constants of the LGT sample are determined. The experimental results clearly show echoes corresponding to the longitudinal and transverse waves along the x‐axis. The waves along the z‐axis have a similar property. However, the waves along the y‐axis are more complex than those along the x‐ and z‐axes. The echoes corresponding to the quasi‐longitudinal waves along the y‐axis are clear, but those corresponding to the transverse and quasi‐transverse waves along the y‐axis are not. The elastic constant can be accurately determined if the wave echoes corresponding to this constant propagate without distinct distortion and are clear; otherwise, it may be impossible to accurately determine the constant using UPE. All elastic constants except of the LGT single crystals can be determined using UPE from one sample. This study uses UPE to provide a reference for the characterization of elastic constants of piezoelectric crystals with 32 symmetry from one sample.

Self-similar flow behind a shock wave in a gas under the effect of viscosity, heat conduction, and variable ambient density

Self-similar solutions have been investigated to describe the propagation of planar shock waves in a non-ideal gas generated by a piston under viscous stress and heat flux. The equation of state for non-ideal gas incorporates the correction in pressure and volume of the gas. The piston position and ambient density vary exponentially with time. Newton’s law of viscosity is used for the viscous stress and Fourier’s law of heat conduction is taken for heat flux. The viscosity coefficient is taken as constant whereas the thermal conductivity coefficient varies with temperature and density following the power law. The shock jump conditions have been derived for the viscous non-ideal gas using integral form of conservation laws. The shock Reynolds number Re s has been introduced to study the effect of viscosity on shock propagation in non-ideal gas. It is found that similarity solution exists only in an ideal gas under the condition that the ambient density exponent is equal to twice the shock position exponent. This study shows that shock Reynolds number Re s and heat conduction parameter Γ c can be used to control the variation of the flow variables and piston position significantly. The shock strength decreases with increase in the value of shock Reynolds number Re s but is independent of the heat conduction parameter Γ c . The pressure, density, and adiabatic compressibility have significant deviations from high to low viscous flow of ideal gas but the velocity and heat flux undergo negligible change. The results do not support the claim of negligible effect of viscosity in earlier studies and establish the impact of viscosity and heat flux on shock propagation in an ideal gas.

Dispersion Analysis of Plane Wave Propagation in Lattice-Based Mechanical Metamaterial for Vibration Suppression

Phononic crystals based on lattice structures provide important wave dispersion characteristics as band structures, showing excellent compatibility with additive manufacturing. Although the lattice structures have shown the potential for vibration suppression, a design guideline to control the frequency range of the bandgap has not been well established. This paper studies the dispersion characteristics of plane wave propagation in lattice-based mechanical metamaterials to realize effective vibration suppression for potential aerospace applications. Triangular and hexagonal periodic lattice structures are mainly studied in this paper. The influence of different geometric parameters on the bandgap characteristics is investigated. A finite element approach with Floquet–Bloch’s principles is implemented to effectively evaluate the dispersion characteristics of waves in lattice structures, which is validated numerically and experimentally with a 3D-printed lattice plate. Based on numerical studies with the developed analysis framework, the influences of the geometric parameters of lattice plate structures on dispersion characteristics can mainly be categorized into three patterns: change in specific branches related to in-plane or out-of-plane vibrations, upward/downward shift in frequency range, and drastic change in dispersion characteristics. The results obtained from the study provide insight into the design of band structures to realize vibration suppression at specific frequencies for engineering applications.

Examining basic theorems and plane waves in the context of thermoelastic diffusion using a multi-phase-lag model with temperature dependence

This study employs a novel mathematical formulation of temperature-dependent thermoelastic diffusion with multi-phase delays (TDMT). The TDMT model, which is assumed, is obtained by converting the fundamental rules of Fourier’s and Fick’s laws into higher order time derivatives of the heat flow vector, the temperature gradient, diffusing mass flux, and the chemical potential gradient. Fundamental theorems such as energy, uniqueness, reciprocity theorems, and variational criterion are established using the fundamental equations for the assumed model TDMT. The reciprocity theorem is applied to a particular scenario by using instantaneous concentrated body forces, heat sources, chemical potential sources, and moving heat sources as examples. It is noted that both the variational criterion and these theorems depend on the field variables’ susceptibility as well as the fluctuations in the higher order temporal derivatives’ parameters. Additionally, the assumed model’s two-dimensional (2D) plane wave propagation is described. Three longitudinal waves are identified: the longitudinal (P) wave, the thermal (T) wave, the chemical potential (CP) wave, and the linked transverse (SV) wave. Wave properties (phase velocity and attenuation coefficient) are calculated numerically and shown visually. A few special examples are also investigated and compared with the established findings. The framework for additional research into basic issues in thermoelastic continua under various physical field variables is provided by this work. Numerous applications in material science, geomechanics, soil dynamics, and the electronic sector can be made of the current findings.

Propagation of plane waves at the initially stressed surface of an orthotropic nonlocal rotating half space under dual-phase-lag model

PurposeThe purpose of the current manuscript is to investigate the reflection of plane waves in a rotating, two-dimensional homogeneous, initially stressed, nonlocal orthotropic thermoelastic solid half-space based on dual-phase-lag model.Design/methodology/approachThe reflection phenomenon has been utilized to study the effects of initial stress, rotation and nonlocal parameter on the amplitude ratios. During the reflection phenomenon three coupled waves, namely quasi displacement primary wave (qP), quasi thermal wave (qT) and quasi displacement secondary wave (qSV) have been observed in the medium, propagating with distinct velocities. After imposing the suitable boundary conditions, amplitude and energy ratios of the reflected waves are obtained in explicit form.FindingsWith the support of MATLAB programming, the amplitude ratios and energy ratios are plotted graphically to display the effects of rotation, initial stress and nonlocal parameters. Moreover, the impact of anisotropy and phase lags is also observed on the reflection coefficients of the propagating waves.Originality/valueIn the current work, we have considered rotation and nonlocality parameters in an initially stressed orthotropic thermoelastic half-space, which is lacking in the published literature in this field. The introduction of these parameters in a nonlocal orthotropic thermoelastic medium provides a more realistic model for these studies. The present work is valuable for the analysis of orthotropic thermoelastic problems involving rotation, initial stress and nonlocality parameters.

Free electron emission in vacuum assisted by photonic time crystals

The Cerenkov radiation and Smith–Purcell (SP) effect state that free electron emission occurs exclusively in dielectrics when the velocity of the particles exceeds the speed of light in the medium or in the vicinity of periodic gratings close to each other within a vacuum. We demonstrate that free electrons in a vacuum can also emit highly directional monochromatic waves when they are in close proximity to a medium that is periodically modulated temporally, suggesting the existence of the temporal SP effect. The momentum band gaps of time-varying media, such as photonic time crystals (PTCs), create new pathways for the injection of external energy, allowing the frequency, intensity, and spatial distribution of electromagnetic fields to be controlled. Moreover, the PTC substrate enables the conversion of localized evanescent fields into amplified, highly directional propagating plane waves that are only sensitive to the velocity of particles and the modulation frequency, which allows us to observe and utilize Cerenkov-like radiation in free space. Our work presents significant opportunities for the utilization of time-varying structures in various fields, including particle identification, ultraweak signal detection, and improved radiation source design.

Response of stiffness and viscosity on the energy ratios at piezo-visco-thermo-elastic medium

This article presents a mathematical framework that characterizes a transversely isotropic piezo-visco-thermo-elastic medium within the context of the dual-phase lags heat transfer law (PVID) applied to an elastic medium (ES). Specifically, the study investigates the propagation of plane waves within the elastic medium and their interaction with the imperfect interface of the ES/PVID media. This interaction results in two waves reflecting back into the elastic medium and four waves propagating through the piezo-visco-thermo-elastic medium. The research explores the distribution of energy between the reflected and transmitted waves by analyzing amplitude ratios at the boundary interfaces, considering factors such as phase delays, viscosity effects, and wave frequency. The study illustrates the influence of boundary stiffness and viscosity parameters on these energy ratios through graphical representations. The study's findings are consistent with the principles of the energy balance law, and the research also delves into specific cases of interest. Overall, this investigation provides insights into wave behavior within complex media and offers potential applications across various fields.

Complex slowness vector for generalised propagation of harmonic plane waves at the boundaries of real materials

Complex slowness vector for generalised propagation of harmonic plane waves at the boundaries of real materials

Method of characteristics for transient, planar, cylindrical flows

Cylindrical radiation of individual pulses is studied and strong differences from spherical radiation are highlighted. Prior to this study, the class of available analytical solutions for cylindrical radiation was rather limited. It included (i) a well-known solution for a steady-state, time-harmonic line source and (ii) the solution for a delta-pulse line source. The first of these has no analytical counterpart for individual pulses. By means of the second (actually the Green's function) a more general class of line source solutions can be formulated, but each one is in the form of an integral that has to be evaluated numerically and requires tedious preparation of a singular integrand, making it difficult to achieve perfect accuracy. In the present paper, use is made instead of a particular family of analytically-exact (linear) solutions that has been developed recently. It is used to assess the accuracy of solutions obtained numerically by the method of characteristics (MoC), which is then applied to more general sources than those amenable to analytical treatment. The MoC analysis is specially formulated to reduce the influence of terms involving the reciprocal of the radius (such terms are numerically unmanageable close to the origin). It is found that the qualitative behaviour of waves radiating from the surface of a cylinder of finite radius can depend strongly on the duration of the initiating disturbance and that this can be interpreted in a convolution-like manner. After validation, the numerical method is coupled with a conventional MoC analysis of planar wave propagation and is used to simulate the reflection and radiation of an initially planar wavefront arriving at a flanged duct exit. The numerical representation is one-dimensional in both domains – (x,t) inside the duct and (r,t) outside it. Although its use implies identical behaviour in all radial directions in the external domain, the solution is found to be in close agreement with circumferential-average values from a CFD solution that simulates azimuthal variations as well as radial variations. Whilst being less comprehensive than the CFD simulation, the MoC analysis has the important practical benefit of making highly efficient use of human and computational resources. By comparing the MoC and CFD solutions, it is found that the nominally-cylindrical radiation from the duct approximates closely to radiation from a line source a small distance inside the duct (not at the exit plane). An approximate location of the effective source is quantified.

Tuning of magnetosplamon coupling between graphene scatterers for the optimal design of adjustable metasurfaces

The resonance characteristics of magnetically-biased graphene micro-scatterers are thoroughly investigated in the present work using both eigenvalue and full-wave solvers. Initially, the graphene surface conductivity is presented in a tensor form due to the application of a magnetostatic bias field, which is perpendicular to the material’s surface. Then, the simple case of a graphene disk scatterer is examined, and a properly modified eigenvalue formulation is utilized to extract the plasmonic fundamental frequencies. The validity of the modal analysis is verified via a full-wave analysis that involves a plane-wave propagation and the extraction of the subsequent absorption cross-section utilizing the Finite-Difference Time-Domain method. Additionally, the dependence of a single disk scatterer resonances with the magnetostatic bias is evaluated, highlighting that as the bias field is increased, every edge mode degenerates into two sub-modes with an augmented difference between the resonant frequencies. Finally, the plasmonic coupling between adjacent scatterers is studied considering a periodic arrangement, similar to a metasurface, indicating the additional coupling modes as well as the adjustability of the properties with multiple degrees of freedom.

Transmission-mode geometric-phase signatures of circular Bragg phenomenon

A dielectric structurally chiral medium (DSCM) exhibits the circular Bragg phenomenon, whereby circular-polarization-state selective reflection occurs in a specific spectral regime that depends on the direction of propagation of the incident plane wave. Theoretical analysis shows that the geometric-phase spectrum of the plane wave transmitted through a DSCM slab contains a signature of the circular Bragg phenomenon, provided that the incident plane wave is not right-circularly polarized, regardless of the structural handedness of the DSCM. As the number of structural periods in the DSCM slab increases with the structural period fixed, the spectrum of the transmission-mode geometric phase evolves but without an easily discernible pattern. A reversal of structural handedness affects but does not lead to a simple change in the transmission-mode geometric phase.

Reflection of coupled elastic waves in nonlocal rotating viscoelastic media

In this paper, propagation of homogeneous harmonic plane waves has been studied in an infinite homogeneous isotropic nonlocal viscoelastic rotating medium. Due to the rotational effects, two coupled plane waves, namely a quasi longitudinal wave and a quasi transverse wave are found in the medium under study. The dispersion equations is derived analytically and represented numerically. The problem of reflection of the quasi longitudinal wave at a stress-free solid half-space of the considered medium is also investigated. Various analytical results for the phase speed, attenuation coefficients, reflection coefficients and their corresponding energy ratios are depicted graphically for a particular model of interest to highlight the effect of nonlocal as well as rotational parameter.

Aeroacoustic two-port characterisation technique for ventilation valves

This paper describes a measurement method to characterise the low-frequency aeroacoustic properties of ventilation extraction and supply valves based on an active two-port formulation for network modelling purposes. The method uses a more compact controlled environment compared to commonly used semi-(anechoic) or reverberations rooms, and relies on a radiation model that assumes the valve radiates similarly to plane waves from an open duct. Using a loudspeaker excitation inside the set-up, a passive two-port model representing the reflection and transmission properties of the valve is obtained. This passive two-port model is then extended to an active two-port model including the source properties by characterising a reflection-free source vector. The resulting properties do not include the influence of the radiation impedance of the environment and can be used to estimate the reflection coefficient and the sound power radiated from the valve when the valve is backed by a different environment. Predictions of the one-port reflection coefficient and the predicted radiated airborne sound power, obtained from the experimentally characterised two-port properties of a ventilation valve, are compared to measurements using established methods in a semi-anechoic environment, showing good agreement in the targeted frequency range where plane wave propagation can be assumed.

Spherical shock waveform reconstruction by heterodyne interferometry.

The indirect measurement of shock waveforms by acousto-optic sensing requires a method to reconstruct the field from the projected data. Under the assumption of spherical symmetry, one approach is to reconstruct the field by the Abel inversion integral transform. When the acousto-optic sensing modality measures the change in optical phase difference time derivative, as for a heterodyne Mach-Zehnder interferometer, e.g., a laser Doppler vibrometer, the reconstructed field is the fluctuating refractive index time derivative. A technique is derived that reconstructs the fluctuating index directly by assuming plane wave propagation local to a probe beam. With synthetic data, this approach is compared to the Abel inversion integral transform and then applied to experimental data of laser-induced shockwaves. Time waveforms are reconstructed with greater accuracy except for the tail of the waveform that maps spatially to positions near a virtual origin. Furthermore, direct reconstruction of the fluctuating index field eliminates the required time integration and results in more accurate shock waveform peak values, rise times, and positive phase duration.