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Shear horizontal surface acoustic waves (SAW) propagation in an ultra-thin functionally graded piezoelectric (FGP) layer bonded to a homogeneous substrate is analytically formulated in the mathematical framework of surface/interface elasticity theory. It is assumed that the FGP over-layer is made of a hexagonal 6-mm crystal with a single axis of rotational symmetry coinciding with the axis of polarization. The mechanical and piezoelectric properties of the layer are assumed to vary linearly with thickness. The half-space is made of a transversely isotropic material. It should be mentioned that this model is of great interest in investigating SAW devices at high frequencies. Accounting for the surface/interface effects, the pertinent electromechanical dispersion relation for both cases of the electrically open and short surfaces is derived analytically using Wentzel–Kramers–Brillouin singular perturbation theory. The effects of the surface/interface parameters and the gradient parameter which indicate the variation of the piezoelectric properties of the FGP layer on the dispersion relation, the electromechanical coupling factor, displacement field, electric potential function, electric displacement field, and stress field are studied numerically, and the results are compared with those obtained from classical theory.

The propagating characteristics of the water and muddy waves are one of the concerns of theoretical studies because of the following facts: (1) With the development of the economy in the estuaries, local residents and government paid increasing attention to the ecological environment of the estuarine beaches The propagating characteristics of the water and muddy waves are closely related to the ecological environment, the concern of the government and the residents. (2) The propagating characteristics of the mud are the bottleneck of the study of the coastal and beach dynamics because of the complicated constitutive relation of the mud. The different constitutive models may lead to different explanations of its mechanism. Hence, the proper selection of the model is one of the keys to reveal the true kinematic properties of the mud in the estuaries. This paper first establishes a power law model of the constitutive relation of the mud. Based on this model, a gravity wave theory is proposed. According to the mechanism of the mud wave transportation, the coupled mud-water wave field can be divided into two layers. The upper layer is described as the viscous Newtonian fluid, whereas the lower high-concentration mud layer is described as the power law fluid. Next, the equations of the proposed model are discretized and the calculations are made by using the difference method. Then, the propagating characteristics are discussed, and the dispersion relations of the water and mud waves are analyzed in detail.

Generalized Kramers–Kronig (K–K) type dispersion relations and sum rules are derived in the static limit for the moments of the degenerate four wave mixing susceptibility. The degenerate nonlinear susceptibility is different from a typical use of the conventional K–K dispersion relations, which assume absence of complex poles of a function in the upper half of complex frequency plane, whereas degenerate susceptibility has simultaneous poles in both half planes. In the derivation of the generalized K–K relations the poles and their order are taken into account by utilization of the theorem of residues. The conventional K–K relations can be used to estimate the real and imaginary parts of the second and higher powers of the susceptibility as the effect of the poles is reduced due to a faster convergence of the dispersion relations. The present theory is directly applicable to higher order susceptibilities and can be used in testing of theoretical models describing the degenerate four wave mixing susceptibility in nonlinear optical and terahertz spectroscopy.

This paper reports dynamical effects in one-dimensional locally resonant piezoelectric metastructures leveraged by nonlinear electrical attachments featuring either combined quadratic and quartic, or essentially quartic potentials. The nonlinear electromechanical unit cell is built upon a linear host oscillator coupled to a nonlinear electrical circuit via piezoelectricity. Semi-analytical harmonic balance (HB)-based dispersion relations are derived to predict the location and edges of the nonlinear attenuation band. Numerical responses show that weakly and moderately nonlinear piezoelectric metastructures (NPMSs) promote a class of nonlinear attenuation band where a bandgap and a wave supratransmission band coexist, while also imparting nonlinear attenuation at the resonances around the underlying linear bandgap. Besides, strongly nonlinear regimes are shown to elicit broadband chaotic attenuation. Negative capacitance (NC)-based essentially cubic piezoelectric attachments are found to expand the aforementioned effects over a broader bandwidth. Excellent agreement is found between the predictions of the HB-based dispersion relations and the nonlinear transmissibility functions of undamped and weakly damped NPMSs at weakly and moderately nonlinear regimes, even in the presence of NC circuits. This research is expected to pave the way toward fully tunable smart periodic metastructures for vibration control via nonlinear piezoelectric attachments.

The narrow band gap width caused by local resonance is the shackle for the further development of conventional metamaterials. In this paper, a novel type of hierarchical metamaterial containing parallel multiple resonators is presented, whose ultra-broadband vibration suppression property originates from a rational gradient design. Characterization of dispersion relations and bandgap characteristics of the one-dimensional outward-hierarchical lattice system reveals the variation of bandgap boundaries with relevant parameters. Thus, two gradient design approaches are proposed to overlap and merge the bandgaps. These designs are verified on the analogous continuum models by finite element simulations, and the ultra-wide wave/vibration attenuation band is consistent with the prediction of band gap diagram. Moreover, the range of the total band gap is determined by the boundary parameters of the gradient design, and its integrity is affected by the setting of gradient interval. In particular, the relationship between the gradient interval setting and the generation of overlapping gaps is analyzed. The gradient outward-hierarchical metamaterial is demonstrated to be an effective solution for elastic wave bandgap engineering, which can also be considered as a complement to existing inward-hierarchical configurations.

This paper addresses the wave propagation problem within the context of the theory of materials with triple porosity under local thermal non-equilibrium. The time-harmonic wave solution is studied and the dispersion relation is explicitly established as an eighth-degree algebraic equation. It is shown that there are two shear waves undamped in time, that are non-dispersive and unaltered by the presence of the pore system or by thermal effects. Also, there are seven longitudinal wave solutions that are dispersive and damped in time: one longitudinal quasi-elastic wave, three longitudinal quasi-pore standing waves and three other longitudinal quasi-thermal standing waves. An illustrative simulation on a material like Berea sandstone shows that the effects of coupling triple porosity with local thermal imbalance and mechanical deformations leads to an increase in the propagation speed of the longitudinal quasi-elastic wave accompanied by an increase in its rate of decrease over time, as well as to a relaxation of the decrease in time for the quasi-pore and quasi-thermal standing waves.

The correlation between electrical properties and surface plasmon resonance (SPR) wavelength was discussed using ITO grating coupler. ITO films were prepared on SiO $$_{2}$$ /Si substrate by RF sputtering with thermal annealing, and the grating was directly fabricated on the sample surface by focused ion beam. Electron density and mobility were evaluated by fitting optical reflection spectra according to Drude model, and then SPR wavelength was estimated from the dispersion relation of surface plasmon. Characteristic SPR dip was observed in polarized reflection spectrum, however, measured wavelength was different from theoretically estimated value. The discrepancy was explained by introducing the assumption based on effective medium theory, in which the grating is assumed as a thin film with effective refractive index.

The surface plasmon polaritons (SPPs) have been demonstrated with significant advantages in nano-photonic devices due to their ability to control and manipulate light. In this paper, the transmission characteristics of light and field properties of SPPs on the surface of different shapes of gratings in Kretschmann–Raether configurations are studied. With increase of frequency, the strength of SPPs and the light-SPP conversion efficiency decrease, and the conversion efficiency of grating is lower than that of the smooth gold film. The minimum reflection spectrum is obtained through connecting the minimum reflection coefficients corresponding to different incident angles. The valley points of the minimum reflection spectrums are close in gratings with different width while quite different in gratings with different groove depths, which means that the depth of the grooves in grating can regulate the maximum transmission frequency of light. The effective plasma frequency of metal grating is proposed to replace the plasma frequency of the gold film, which shows that the groove depth is negatively relevant to effective plasma frequency of the gold grating and the frequency of SPPs while the groove width does not affect them. Synthesizing the simulation results, we found that the effective plasma frequency of gold grating and dispersion relations of SPPs on the surface of grating change with the propagation trajectory of SPPs. The physical mechanism of this phenomenon is explained through an analogy of the surface wave on grating to slow wave in slow-wave-structure of vacuum electronic devices.

In this paper, a pre-compressed one-dimensional granular crystal model is studied. The bright analytic single and multiple solitary wave solutions in more general forms than those obtained the KdV system in the previous studies are derived by using the homogeneous balance principle and Hirota’s bilinear method. The difference between the present solutions and those from the KdV system are investigated both analytically and numerically. By analyzing the dispersion relation and the collision process of solitary waves, we find that there are two types of double-solitary waves in the pre-compressed granular crystal model. The geometric and numerical analysis of dynamic behaviors of the solutions is presented with emphasis on the relation between the double-solitary waves and elastic collision between single-solitary waves. We find that the opposite collision between single-solitary waves may be stable and thus generate a stable double-solitary wave. It is concluded that the collision is a special stable double-solitary wave solution. We further propose a possible way to determine the stability of multiple solitary waves qualitatively. The results of this paper provide a theoretical basis for finding stable multiple solitary wave solutions.

The study presents an analysis of free induction decay (FID) in the diluted magnetic composite using the model of spin waves in continuous media. It takes into account the dipole–dipole interaction of the bulk lattice atoms and between them and ferromagnetic dopant ions, and also the indirect exchange interaction of the dopants. A dispersion relation of the magnetic waves is evaluated from the magnetization evolution equation. There are two spectrum branches of spin waves. Their basic frequencies are determined by the gyromagnetic ratio and interaction parameters of the dopant ions and bulk lattice atoms, and by the dopant concentration. The FID asymptotic behaviour is evaluated.

The present investigation deals with the study of higher-order nonlinearity and dispersion to electron-acoustic waves in an unmagnetized and collisionless plasma system incorporating immobile ions, cool adiabatic electrons and nonthermal-Tsallis distributed hot electrons. Employing proper stretched coordinates and Reductive Perturbation Method (RPM), dispersion relation and Korteweg de-Vries (KdV) equation have been derived to the lower-order potential. An inhomogeneous KdV-type equation emerges to the next higher order that accounts for fifth-order dispersion. This study focuses on revealing the dependency of the behaviour of dressed electron-acoustic waves on nonthermal parameter (β), hot-to-cool electron density ratio (α), hot-to-cool electron temperature ratio (θ), Mach number ( $${{\Delta} M}$$ ) and nonextensive parameter (q). Only negative potential structures are reported for which the numerical results are interpreted in the form of two and three-dimensional profiles. The numerical results reflect the decrease in the amplitude of dressed solitary structures, thus obliging simulation of electron-acoustic solitary behaviour in the auroral regions and magnetosphere of Earth.

To gain a complete view on the unique spatial attenuation phenomenon arising from the periodicity and the material viscoelasticity of the two-dimensional phononic crystals with finite out-of-plane extension, this paper presents a theoretical study of the viscoelastic phononic crystal slabs to investigate the wave transmission characteristics in terms of the dispersion relations for the propagating, evanescent, and complex modes. A numerical approach based on the theory of three-dimensional elasticity is proposed to eliminate the restrictions of plane analysis and to obtain the complex band diagrams using the wave expansion functions combined with the finite discretization technique. This formulation allows one to perform the simulations associated with the nonlinear frequency-dependent viscoelastic property which is accurately described by a generalized Maxwell model in this work. Numerical results are presented to show the validity of the established model and to reveal the dissipation mechanisms inside and outside the band gaps as well as to investigate the effects induced by the material viscoelasticity on the attenuation diagrams. Furthermore, the influences of the out-of-plane extension, the geometrical size and the shape of the etcfhed holes are detailed discussed to obtain thorough understandings of such a finite solid viscoelastic periodic lattice which is indispensable to practical applications.

Nonlinear propagation of magnetoacoustic waves in magnetized, collisionless, warm bi-ion plasmas with positive and negative ions of different masses and charge states are investigated. The two-fluid model is used here. Using reductive perturbation technique, the Korteweg-de Vries equation is derived as nonlinear dispersion relation, and its solution for the magnetoacoustic solitons propagating in the direction perpendicular to the external magnetic field is obtained. Using experimental values for the plasma parameters, magnetoacoustic soliton structures for K-fullerene plasma are illustrated along with some other useful alkali-fullerene plasmas.

Using the linearized Vlasov–Maxwell model and the polarization tensor components for weakly magnetized ( $$\left| \omega -{\mathbf {k}}\cdot {\mathbf {v}}\right| >\varOmega $$ ) electron plasma the dispersion relations of parallel and perpendicular propagating electromagnetic modes are evaluated under high frequency/long wavelength limit (i.e., $$\omega >{\mathbf{k }}\cdot {\mathbf{v }}$$ ). The propagation characteristics of R–L and ordinary waves are discussed in the presence of isotropic Fermi–Dirac distribution function in a relativistic regime. The Polylog functions that arise due to arbitrary/partial degeneracy are examined in various degeneracy and relativistic limits. The graphical results are also extracted in the presence of quantum non-degenerate, degenerate and fully degenerate regimes due to the variation in relativistic parameter $$(\frac{T}{ m_{0}c^{2}})$$ , Lorentz factor ( $$\gamma $$ ) and degeneracy parameter $$(\frac{\mu }{T})$$ in weakly and strongly relativistic limits. Furthermore, the derived results are found to be in complete agreement with the previous investigations.

We use the Hamiltonian structure of equatorial wave-current interactions in the case of non-zero constant vorticity in the near-surface homogeneous layer, studying the dispersion relation for linear long waves outrunning the currents (downstream waves). We also investigate the behavior of the lifespan of the solution of the evolution equation for nonlinear long internal waves with respect to vorticity values.

In this paper, propagation of surface waves in nonlocal transversely isotropic liquid-saturated porous solid half-space has been investigated. The model is proposed for two different situations of which one is for excluding fluid nonlocal effect, and the other is for including fluid nonlocal effect. The existence of a new wave that arises due to the presence of nonlocality parameter in the medium as well as some critical circular frequencies is derived. Dispersion equation for the propagation of Rayleigh-type surface waves and their conditions of existence at the free surface of transversely isotropic liquid-saturated porous solids has been obtained. The boundary may be opened surface pores or sealed surface pores. Dependence of the velocities of surface waves, Rayleigh wave type on the direction of propagation and frequency has been illustrated. These numerical results report the phenomena such as anomalous negative dispersion of wave velocities with frequency.

In this paper we investigate the influence of rotational viscosity on the plane shear-rotational wave propagation in the linear isotropic reduced Cosserat medium. In such a continuum body points possess independent rotational and translational degrees of freedom, but the material does not resist to the gradient of microrotation. Stress tensor is asymmetric, but the couple stress is zero. For the elastic reduced Cosserat medium a band gap is present, where no running waves exist, and the logarithmic decrement is infinitely large. It was shown before that introducing viscoelasticity we may qualitatively change the dispersion behaviour. This work investigates the case of rotational viscosity, related to the difference between micro- and macrorotation. We show that in this case running decaying waves appear in the band gap domain, and their logarithmic decrement becomes finite. The dispersion relation for the real part of the wave number has a decreasing part. This means that the group velocity of these waves is opposed to their phase velocity, the behaviour characterizing double negative acoustic metamaterials.

With the incremental thermal field theory by Lee and Yong, the Mindlin plate equations for the analysis of thickness-shear vibrations of SC-cut quartz crystal with the consideration of thermal effect have been obtained. By assuming straight-crested waves in a rectangular plate, the dispersion relations, frequency spectra, mode shapes, and frequency–temperature relations of the thickness-shear mode of SC-cut quartz crystal plates are calculated with plate configurations. The computational results from SC-cut quartz crystal plates are compared with those from AT-cut, which shows that SC-cut resonators have better frequency–temperature relation as a validation. For practical applications, such results will be useful in the precision design and improvement of SC-cut quartz crystal resonator structures.

Head-on collisional phenomena of ion acoustic (IA) waves in electron–positron–ion magnetized plasmas are investigated through linear and nonlinear analyses. The plasma system is considered collisionless and comprising cold ions, kappa distributed cold electrons, and positrons. Dispersion relation is derived with the aid of the linear perturbation method. The two sided nonlinear Korteweg-de Vries (KdV) equations are derived employing extended Poincare-Lighthill-Kuo (ePLK) method. The effects of wave frequency on linear group velocity with species density ratio are studied. Further, the production of ion acoustic solitons in terms of ion gyrofrequency ( $${\omega }_{\mathrm{g}}$$ ), superthermality parameters for electrons ( $${\kappa }_{\mathrm{e}}$$ ), and positrons ( $${\kappa }_{\mathrm{p}}$$ ) is investigated. Additionally, the effects of obliqueness (angle between magnetic field and wave propagation vector) on nonlinearity coefficient and phase shifts with density ratios (positron to electron and ion to electron) are demonstrated.

Dispersion analysis of Lamb waves propagating in anisotropic functionally graded plates with transverse inhomogeneity reveals that the high frequency asymptotes associated with the interfacial Stoneley waves disappear, while in stratified plates the corresponding high frequency asymptotes exist. The analysis utilizes a variant of the Cauchy sextic formalism coupled with the exponential fundamental matrices for constructing explicit dispersion equation.

We analyze the chirality of dispersion relations and propagation lengths of two fundamental modes (m=0 and 1) in single silver nanowire coated by chiral TDBC. The systems of both Ag nanowires only coated by left TDBC (ALM) and right TDBC (ARM) are considered, respectively. Finite-difference time-domain method is used to do the mode analysis. The results show that the dispersion relations curves of both ALM and ARM are similar to magnetic hysteresis loops. There is a dip in the propagation length of ALM or ARM versus the incident frequency at the resonant frequency of left TDBC or right TDBC. The propagation length is reduced shapely in chiral TDBC due to the dissipation of TDBC. The difference of the propagation length $$|\Delta L|$$ becomes larger with the incident frequency far away from the resonant frequency of chiral TDBC for these two modes. The maximum difference of the real part of the effective index between ALM and ARM ( $$\Delta n'$$ ) is the linear function of m and the radius of Ag nanowire. With the radius decreasing from 100nm to 10nm, the propagation length for m=0 decreases and the propagation length for m=1 increases. Also, the maximum difference of the propagation length for m=1 increases. The maximum normalized difference between Ag nanowires coated by left TDBC and right TDBC in both the dispersion relation for $$m=0$$ (0.024) and the propagation length for $$m=1$$ (0.16) are quite larger than the one of only chiral TDBC (1/3000). According to our analysis, chiral molecules can be distinguished in either the dispersion relation or the propagation length, and our proposed structure has potential applications in chiral molecules detecting.

Artificial materials, known as phononic crystals (PCs) and acoustics/elastic metamaterials (AMs/EMs), have been proved to have band-gaps within which no flexural wave can propagate through, which have received great attentions. Among AMs/EMs, locally resonant (LR) beams are frequently encountered in the field of mechanical, civil and aeronautical engineering. However, the analysis of finite LR beams lacks of an efficient and exact approach. The primary objective of this study is to employ and develop an exact and efficient wave-based vibration analysis approach to study a finite Timoshenko LR beam carrying periodic separated force and moment type resonators. The proposed wave-based analysis approach is a powerful method that provides the exact solution in vibration analysis for structural waves that propagate along uniform structural elements and reflect and transmit at structural discontinuities. With the approach, the modeling and analysis process can be well simplified, which improves the efficiency in the design and analysis of finite LR beams. The frequency response results with three sets of resonators obtained through wave-based analysis approach show great agreements with both simulation and experimental results. Moreover, the measured results of the resonator with higher natural frequency show the promising advantage in achieving broader band-gap in low-frequency range, because the propagation attenuation in the experiment causes the absence of the narrow pass-bands which exists in the theoretical results. This paper proves that the wave-based vibration analysis approach was an exact and efficient method in solving complex vibration problem in finite periodic LR structures. In addition, the rich dispersion relation and wide band-gap property of the separated force and moment type resonators are demonstrated with the theoretical, simulation, and experimental results.

The current paper is concerned with the nonlinear stability analysis of rotating magnetic fluid columns. The rotation sources are a mixture of both uniform and oscillating behavior. The motivation behind tackling this topic is the increasing interest in atmospheric and oceanic motions. The system consists of two magnetic phase fluid that fills two infinite vertical cylinders. An azimuthal uniform magnetic field is penetrated on the system. The governing equations of motion, in terms of the Coriolis force and reduced pressure, along with Maxwell’s equation in the quasi-static approximations are considered. Consequently, the disturbance of the interface has an azimuthal behavior. The fluids are fully saturated in porous media. In light of the implication of the nonlinear boundary conditions, the solutions of the linearized equations of motion resulted in a nonlinear characteristic dispersion equation. Utilizing the homotopy perturbation technique, this equation is analyzed. A modification of the latter equation is made to seem like a nonlinear Klein–Gordon equation. The stability criteria are realized in linear as well as nonlinear approaches. A set of diagrams is graphed to illustrate the effects of several non-dimensional numbers on the stability profile in resonance as well as non-resonance cases.

A simplified second strain gradient Euler–Bernoulli beam theory with two non-classical elastic coefficients in addition to the classical constants is presented. The governing equation and the associated classical and non-classical boundary conditions are derived with the aid of variational principles. The simplified second strain gradient theory is governed by an eighth-order differential equation with displacement, slope, curvature and triple derivative of displacement as degrees of freedom. This theory can be reduced to the first strain gradient and classical Euler–Bernoulli beam theories. Analytical solutions for static behaviour, free vibration and stability analyses are presented for different boundary conditions and length scale parameters. Using the numerical Laplace transform, a spectral element is developed for dynamic analysis of a cantilever beam subjected to a Gaussian pulse. Further, spectrum and dispersion relations are derived to study wave propagation characteristics. The gradient effects on the structural response are assessed and compared with the corresponding first strain gradient and classical beam theories. Observations show that the second strain gradient theory exhibiting stiffer behaviour in comparison to the first strain gradient and classical theories. The beam deflection decreases whereas frequencies and buckling load increase for increasing values of the gradient coefficient in comparison to the first strain gradient and classical theories. The forced response for a finite beam reveals a decrease in the amplitude and a shift to smaller time values with an increase in the value of length scale parameter. Additionally, the second strain gradient beam shows a dispersive behaviour, and for a given frequency the wavenumber decreases and the phase speed increases with an increase in the length scale parameter as compared to the first strain gradient beam theory.

This paper presents a theoretical study on the propagation of electromagnetic surface waves supported by the metamaterial-filled bi-layered graphene structure. The effective surface conductivity approach based upon the Kubo’s formalism is used to model the atomically thick graphene layer, while the analytical modeling of the metamaterials used the framework of causality principle-based Kramers-Kroing relations. With respect to the index of refraction, the two configurations of metamaterials have been considered—i.e., double positive (DPS) and double negative (DNG). The impedance boundary conditions (IBCs) have been employed at the interface for the computation of characteristics equations of odd and even surface modes under DPS and DNG configurations. The numerical results regarding the dispersion relations, effective mode index, and phase speed for the odd and even modes of electromagnetic surface waves supported by each configuration of structure have been presented. Further, the influence of chemical potential and thickness of metamaterial filling in parallel graphene layers on the wave propagation characteristics has been analyzed. To check the accuracy of the numerical results, special conditions were employed, which converge to the published results. The proposed work may have potential applications in plasmonic based logic designs, renewable energy devices, THz surface wave guides, and near-field communication devices.

Dispersion of a passive tracer by water waves is of significant importance for many scientific and technological problems including bio-diversity of marine life, ecological impact of anthropogenic incidents, planning of rescue operation and global oceanic transport. Formally, turbulent dispersion of a passive tracer by surface waves is a Brownian motion caused by a prescribed noise, viz., random fluctuations of the wave field. From this perspective, it is similar to the conventional dispersion by tracer particles by turbulent flows initially described in the seminal work of Richardson, Taylor and Obukhov. The additional challenges of this problem are imposed by the complexity of the underlining wave field—different dispersion relations and correlation structure, directionality and its spread, wave breaking—and this complexity necessitates further theoretical and experimental research. The aim of the present study is experimental validation of scaling relations for the mean drift and mean variance of tracer particles predicted by the wave turbulence theory. We report results of a set of targeted experiments in a large three-dimensional wave tank where the positions of the tracer particles—modelled as surface drifters—were tracked down with optical cameras. The experimental data are analysed and discussed in the light of the weak turbulence theory.

The size-dependent effect arising in micro/nano-structures has attracted considerable attention in academic and industrial communities. The theories developed to characterize such effect mainly include the nonlocal elasticity theory, the strain gradient theory, the modified coupled stress theory and the more recent nonlocal strain gradient elasticity etc. Meanwhile, the thermal-induced deformation or stress in micro/nano-structures is increasingly becoming a vital issue in their designs and applications. Nevertheless, theoretical investigations to predict the thermoelastic performances of micro/nano-structures are not so common, especially in the case of nonlocal strain gradient elasticity incorporating thermoelastic coupling effect. In present work, investigation of thermoelastic wave propagation in micro-beam is conducted in the context of the nonlocal strain gradient elasticity and the G-N theory, taking the thermoelastic coupling effect into account. The governing equations are formulated based on the Euler–Bernoulli beam model and the G-N theory. By assuming the wave-type solutions, the equations are solved and the dispersion relation between frequency and wave number and the relation between phase velocity and wave number are determined respectively. In calculation, the above two relations are fully investigated and comparisons on them under different theories are provided accordingly. Some new findings are presented and discussed in detail. It is hoped that the present work may provide some guidelines in designing and optimizing micro-structures.

An exact approach is used to investigate the propagation of a Love-type wave in a low-velocity piezoelectric-viscoelastic material (PV) stratum bonded to a functionally graded transversely isotropic viscoelastic (FGTIV) material substrate. The dispersion relations are obtained for electrically open (EO) and electrically short (ES) conditions on the upper free surface of the considered structure by using Valeev’s method of infinite determinants. The expression of group velocity of a Love-type wave in a low-velocity layered structure is also obtained in compact form. The obtained closed-form dispersion relations of a Love-type wave are matched with the standard result of a Love-type wave when reduced to the isotropic case. In order to observe the concealed characteristics of a Love-type wave, a detailed study is carried out graphically, to forefront the effect of various influencing parameters, viz. piezoelectric coupling parameter and magnifying gradient (MG) parameters. The profound impact of mass loading on the upper free surface of the considered layered structure is also studied.

By employing Mindlin’s Form II gradient elasticity theory (strain gradient elasticity theory), we investigate the SH surface wave propagating in a strain-gradient layered half-space. The dispersion equation of the SH surface wave is derived analytically. For the general case where both the surface layer and the half-space are strain-gradient elastic materials, the dispersion equation involves 10 material constants. The dispersion equation of the general case can degenerate to that of several special cases by dropping some material parameters. The existence regions of the dispersion curves are discussed in detail. The effects of strain-gradient elastic constants on the dispersion curves are examined. It is seen that the features of SH surface waves in strain-gradient elastic materials are much richer than those in classical elastic materials. Ministry of Education (MOE) The authors would like to thank the financial support from Singapore Ministry of Education Academic Research Fund Tier 1 (RG185/18). Jianmin Long also acknowledges the support from the National Natural Science Foundation of China (11702081) and the Fundamental Research Funds for the Central Universities, Hohai University (2019B08714).

The propagation of dust-acoustic waves in a dusty plasma bounded in finite symmetrical cylindrical geometry has been theoretically investigated. By using the reductive perturbation method, we obtain a quasi-nonlinear Schrodinger equation. It is shown that there is a dark envelope soliton under certain conditions, while its amplitude will decrease with the time. The dependence of both the dispersion relation and group velocity of envelope solitary waves on the radius of cylinder has been given. Moreover, we define the duration time of dark envelope solitons and give the theoretical ones. It is found that the duration time increases as the radius of cylinder increases, while it decreases as the viscosity coefficient of dusty plasmas increases.

Nowadays, the viscoelastic carbon nanotubes (CNTs) have been used as one of the most promising candidates for nanodevices (or nanoelectronics) and super-strong reinforcement fibers in polymer nanocomposites. In recent years there have been many papers that considered the effects of material length scales in the investigations of the material constants related on the structural parameters on the wave propagation in the viscoelastic CNTs. By using different approaches, for examples, Eringen’s differential nonlocal model, modified couple stress theory, and strain gradient theories, the stiffness softening or strengthening effect can be well predicted. However, the influence of the rapid stiffness enhancing effect on the wave characteristics of the viscoelastic CNTs is still not reported. To deal with such problem, this work aims to investigate the transverse wave propagation in viscoelastic single-walled carbon nanotubes (SWCNTs) adhered by surface material based on the nonlocal second-order strain gradient elasticity theory. The characteristic equation of wave motion of the viscoelastic SWCNTs with surface effect is systematically formulated to achieve the analytical dispersion relation between the wave frequency (or phase velocity) and the wave number. In the end, some most important conclusions are concluded.

An investigation of equatorial near-inertial wave dynamics under complete Coriolis parameters is performed in this paper. Starting from the basic model equations of oceanic motions, a Korteweg de Vries equation is derived to simulate the evolution of equatorial nonlinear near-inertial waves by using methods of scaling analysis and perturbation expansions under the equatorial beta plane approximation. Theoretical dynamic analysis is finished based on the obtained Korteweg de Vries equation, and the results show that the horizontal component of Coriolis parameters is of great importance to the propagation of equatorial nonlinear near-inertial solitary waves by modifying its dispersion relation and by interacting with the basic background flow.

Metamaterials are artificially structured materials that enable wave attenuation in band gaps. However, opening an ultralow-frequency band gap is still a challenge, since it is hard to realize near-zero stiffness in a traditional way. In this paper, a one-dimensional tunable quasi-zero-stiffness (QZS) metamaterial is engineered for ultralow-frequency (about a few tens Hertz) wave attenuation. Design optimization on the configuration of this new metamaterial is conducted to achieve quasi-zero stiffness. The dispersion relation is derived theoretically based on a lumped diatomic chain model, and then the band structure is revealed. The characteristics of longitudinal wave propagation in the metamaterial are studied by both numerical analyses and FE simulations, which are also validated by experimental tests. The results indicate that the stiffness is deformation-related, and the band gap can be tuned substantially by just changing the pre-compression. Therefore, the quasi-zero stiffness and then the ultralow-frequency band gap can be fulfilled by pre-compressing the metamaterial properly.

The problem of the propagation of acoustic waves in a layer of a compressible viscous fluid that interacts with an elastic half-space is solved using the three-dimensional linear equations of classical elasticity theory for the solid and the three-dimensional linearized Navier–Stokes equations for the compressible viscous fluid. The problem statement and approach based on the application of the general solutions of linear equations for elastic bodies and linearized equations for the fluid are employed. The dispersion equation describing the propagation of quasi-Lamb waves in the hydroelastic system is derived. Dispersion curves for normal waves in a wide frequency range are plotted. The effect of the thickness of the liquid layer on the phase velocities and attenuation coefficients of acoustic waves is analyzed. It is shown that the influence of fluid viscosity on the parameters of the wave process is associated with the properties of wave localization. Using the developed approach and obtained results, the limits of the applicability of the models of wave processes based on the model of an ideal fluid are established. The numerical results are presented in the form of graphs and analyzed.

A second-order plate theory for a lithium niobate piezoelectric plate with a ferroelectric inversion layer is established. The theory describes coupled extensional, thickness-stretch and symmetric thickness-shear motions of the plate. The two-dimensional theory obtained is validated by comparing the dispersion relations of the relevant waves with the three-dimensional exact theory. For long waves with small wave numbers, the dispersion curves obtained from the plate theory and the three-dimensional theory have the same cutoff frequencies and curvatures. Therefore, the plate theory is useful in the design of devices operating with these waves. A piezoelectric gyroscope based on symmetric thickness-shear modes is analyzed as an example.

Parametrically excited standing waves are observed on a cylindrical fluid filament. This is the cylindrical analog of the Faraday instability in a flat surface or spherical droplet. Using Floquet theory, a linear stability analysis is carried out on a viscous cylindrical fluid surface, which is subjected to a time-periodic radial acceleration. Viscosity of the fluid has a significant impact on the critical forcing amplitude as well as the dispersion relation of the non-axisymmetric patterns. The effect of viscosity on the threshold of the pattern with azimuthal wavenumber $$m=1$$ shows a different dependence from $$m>1$$ . It is also observed that the effect of viscosity is greater on the threshold with higher m.

Honeycomb phononic crystal can obtain wider band gaps in the low frequency based on local resonance theory. Its band structure can be adjustable if we change the height of the cores, which means different kinds of honeycomb phononic crystal can be selected on the basis of different damping demands. Meanwhile, the point defects and line defects affect the localized modes of sound waves and propagation characteristics, the dispersion relations and the displacement fields of the eigenmodes are calculated in the defected systems, as well as the propagation behaviors in the frequency ranges of the band structure, which are also discussed in detail. We constructed the model based on the periodic boundary condition and calculated the band structure according to Bloch theory, and also performed a series of simulation through the COMSOL software, showing that honeycomb has excellent features in reducing noise and vibration, which has a far-reaching influence in designing the new type of acoustic wave devices.

Love-type wave propagation through inhomogeneous dual porous layer has been investigated in this present article. The heterogeneous fluid-saturated dual porous stratum is bounded by a non-homogeneous viscous liquid layer and an isotropic half-space. Impacts of viscosity, inhomogeneity, matrix porosity along with fracture porosity have been calculated in detail. Navier–Stokes equation has been used to acquire the velocity component in heterogeneous viscous liquid layer. Separable variable method has been performed to convert partial differential equations into ordinary differential equations. Elimination of arbitrary constants from boundary conditions leads to complex dispersion relation of Love-type wave propagation. The complex equation consists of Whittaker functions and their derivatives which are expanded up to second term by approximating large parameters. Dispersion and attenuation equations of Love-type wave have been decoupled for implementing several graphs which illustrate reverberations of heterogeneity parameter, porosity, volume fraction of fractures, density on dispersive and damping nature of Love-type wave. Fundamental mode and higher modes of Love-type wave are observed through graphical execution. Effects of inhomogeneity parameters are also portrayed through surface plotting. Correlation of liquid layer and fractured porous layer in crustal region has been established both analytically and graphically which is also validated by applying particular conditions. Heterogeneity parameter, volume fraction of fractures, porosity, density have major impact on dispersion and attenuation of Love-type wave propagating in dual porous medium. This solid–liquid collaborative study unlocks a different area of future research.

" Three-layer slab waveguide with a graded-index film and a nonlinear substrate is investigated. The nonlinear substrate is considered of Kerr-type and the film index is assumed to change linearly as we move from the film-clad to the film- substrate interfaces. The solutions of the fields in the guiding film contains Airy functions which can be written as Bessel functions of order 1/3. The dispersion relation is derived in terms of three well-known normalized parameters V, a and b and the dispersion properties are plotted and studied. Many interesting features are obtained such as the nonexistence of cut-off thickness due to the nonlinear substrate and the value of the normalized guide index does not exceed unity which means they correspond to guided modes." " Three-layer slab waveguide with a graded-index film and a nonlinear substrate is investigated. The nonlinear substrate is considered of Kerr-type and the film index is assumed to change linearly as we move from the film-clad to the film- substrate interfaces. The solutions of the fields in the guiding film contains Airy functions which can be written as Bessel functions of order 1/3. The dispersion relation is derived in terms of three well-known normalized parameters V, a and b and the dispersion properties are plotted and studied. Many interesting features are obtained such as the nonexistence of cut-off thickness due to the nonlinear substrate and the value of the normalized guide index does not exceed unity which means they correspond to guided modes."

This article investigates the interaction between a surface gravity wave that propagates over an elastic plate based on linear viscoelastic foundation. The plate is considered to be thin and infinite and is modeled based on the Euler–Bernoulli beam theory. Static and dynamic boundary conditions are applied to the Laplace equation of the fluid domain. The dispersion relation of the wave–plate system is derived and ratio of surface wave amplitude and plate deflection is proposed. Considering dimensionless dispersion relation, two modes of propagating wave are attained. Problem is analyzed for two cases of presence and absence of viscous damping coefficient in the foundation of the elastic plate. It is shown that flexural rigidity of the submerged plate has considerable effect on wave decay and plate vibration. It is illustrated that shallowness has noticeable effect on the wave propagation frequency and a critical shallowness demarcates damped or overdamped excitation of the elastic plate based on the viscoelastic foundation. Moreover, effects of flexural rigidity of the plate, foundation stiffness coefficient, and foundation viscous coefficient on phase and group velocities of wave are discussed in the present study.

In this paper, by applying the correction of Lorentz dispersion relation, the fermion tunneling radiation of the stationary axisymmetric Kerr-TAUB-NUT black hole in curved space-time is studied. Based on the modified Lorentz dispersion relation, the revised Rarita-Schwinger equation has been obtained. Then, we get the dynamic equation of fermions via semi-classical theory, which is the modified Hamilton-Jacobi equation. So at the event horizon the tunneling radiation in the space-time of stationary axisymmetric Kerr-TAUB-NUT black hole has been effectively investigated. At last, the modified tunneling radiation and Hawking temperature have been obtained in the same space-time.

The dispersion relation and growth rate γ of the parallel propagating Weibel mode in an unmagnetized, anisotropic non-thermal plasma are investigated using Cairns distribution function. The real and imaginary frequencies, considering Cairns distribution function in the limits of ξ >> 1 and ξ > 1), the real frequency of the wave under consideration is obtained, while in the lower limit (ξ <

The problem of the propagation of linear Rossby waves in horizontally inhomogeneous non-zonal flows is studied. The explicit solution within the geometric optics (WKBJ) approximation is found to be identical to the exact Cauchy problem solution for the case of a constant horizontal velocity shear.The effect of the short-wave transformation of Rossby waves near the so-called critical layer is detailed for the arbitrary direction of non-zonal flow. In the general case, this transformation can occur in two ways: (1) as an adhering, a monotonic approaching of wave packets to the critical layer for an infinitely long time. The sign of the intrinsic frequency of the packet remains the same all the time; (2) as an adhering with overshooting when the wave packet, first, crosses its critical layer at finite wavenumber. The wave changes the sign of the intrinsic frequency when overshooting the critical layer and then keeps the sign when it is adhering to this layer asymptotically similarly to the previous scenario. The latter regime does not exist for zonal flows, that degenerates the short-wave dynamics of Rossby waves in this special case. On the contrary, the anisotropy of the dispersion relation permits both positive and negative frequencies in a non-zonal flow. It allows for the effective use of the concept of waves of negative energy for the analysis of the stability of large-scale currents.

Highly lossy nature of metals has severely limited the scope of practical applications of plasmonics. The conventional approach to circumvent this limitation has been to search for new materials with more favorable dielectric properties (e.g., reduced loss), or to incorporate gain media to overcome the inherent loss. In this study, however, we turn our attention to the source and show that the wealth of new SPP modes with simultaneous complex frequencies and complex wave vectors that are otherwise unreachable can be excited by imposing temporal decay on the excitation. Therefore, to understand the possible implications of these new modes and how to be able to tune them for specific applications, we propose a framework of pseudo-monochromatic modes that are generated by introducing exponential decays into otherwise monochromatic sources. Within this framework, the dispersion relation of complex SPPs is re-evaluated and cast to be a surface rather than a curve, depicting all possible ω − k pairs (both complex in general) that are supported by the given geometry. To demonstrate the potentials of the complex modes and the use of the framework to study them selectively, we have chosen two important, and somewhat limiting, features of SPPs to investigate; resolution in plasmonic lenses and propagation length in SPP waveguides. While the former is mainly used to validate the proposed method and the framework on the recent improvement of resolution in plasmonic superlenses, the latter provides a novel approach to extend the propagation length of the SPP modes in planar waveguides significantly. Since the improvement in propagation length due to the introduction of temporal decay to the excitation is rather counter-intuitive, the dispersion-based theoretical predictions (the proposed approach) have been validated via the FDTD simulations of Maxwell’s equations in the same geometry without any a priori assumptions on the frequency or the wave vector.

In actual line operation, the critical velocity is one of the key physical quantities of rail design owing to its great influence on the riding comfort and safety of vehicles due to the wheel-rail contact loss caused by the abrupt change of rail foundation rigidity, rail wear, or abruptness irregularities on rail. In this study, the short floating slab track (SFST) structure is regarded as a double-layer system. The Euler beam and the rigid body model are adopted for the rail and the floating slab, respectively, and the dispersion equation and the theoretical critical velocity of the rail structure under ideal conditions are deduced. Besides, this study considers the implementation of the SFST in the vehicle-structure coupling system. The alterable element method is introduced for accurately simulating the change of the wheel-rail contact state and coding a vehicle-structure dynamic analysis program (VSDAP) to calculate the critical velocity of rail structures from the dynamic response of vehicles and rail structures. The principle of its design at the beginning of the design is given on the basis of the theoretical value of the critical velocity and the simulation of the dynamic response, which can provide reference for practical engineering design.

In this study, the effect of the induced electromagnetic fields on the linear instability of a viscoelastic nanofluid liquid layer is discussed. Two physical cases are considered for the channel flow. The first one is considered for the free–free boundaries and the second for the rigid–rigid boundaries. The instability is investigated analytically due to the normal mode analysis and numerically according to the relation between the energy density function and time. The Routh–Hurwitz criteria are applied. The dispersion relation is obtained as a sixth-degree equation of the growth rate. Also, the stationary, as well as oscillatory states of the thermal Rayleigh number, are obtained. The numerical solution is compared with the exact solution for a special choice of the parameters and the results show higher match between the results. The main important result of the study concludes that the Brownian motion of the nanoparticles destabilizes the nanofluid layer. This means that the pure fluid layer is more stable than the nanofluid layer. Also, the results confirm that the flow between the rigid–rigid boundaries is more stable than the free–free boundaries. Finally, due to the Lorentz force, the induced magnetic field destabilizes the motion due to the temperature rise.

The dispersion relation and the nonlinear behaviour of dispersive Alfven waves (DAWs) in a plasma, which is composed by hot electrons, cold positrons and ions, are studied from the multi-component fluid theory. It can be indicated from the linear dispersion relation that the coupling of positron acoustic waves with shear Alfven waves results from the effect of the finite ion gyroradius. In our mode, the pressure and the inertia of the positron acoustic wave are, respectively, provided by the hot electrons and the cold positrons, which are different from the ones in a plasma consisting of two-group positrons with different temperatures. Furthermore, the existence and properties of DAW solitons are analysed from the Sagdeev pseudopotential equation, which indicates the existence of super-Alfvenic or sub-Alfvenic solitary DAWs, as well as the amplitude and width of DAW solitons depend on the positron concentration and the electron temperature. When the effects of cold positrons can’t be neglected, there may coexist the compressive and rarefactive DAW solitons. When the concentration of positron is negligibly small, the properties of DAW solitons in our model are similar to the ones of kinetic Alfven wave solitons in a plasma composed by hot electrons and cold ions.

It is a challenge to choose the appropriate kernel function in nonlocal problems. We tackle this challenge from the aspect of nonlocal wave propagation and study the dispersion relation at the analytical level. The kernel function enters the formulation as an input. Any effort to narrow down this function family is valuable. Dispersion relations of the nonlocal governing operators are identified. Using a Taylor expansion, a selection criterion is devised to determine the kernel function that provides the best approximation to the classical (linear) dispersion relation. The criterion is based on selecting the smallest coefficient in magnitude of the dominant term in the Taylor expansion after the constant term. The governing operators are constructed using functional calculus, which provides the explicit expression of the eigenvalues of the operators. The ability to express eigenvalues explicitly allows us to obtain dispersion relations at the analytical level, thereby isolating the effect of discretization on the dispersion relation. With the presence of expressions of eigenvalues of the governing operator, the analysis is clear and accessible. The choices made to obtain the best approximation to the classical dispersion relation become completely transparent. We find that the truncated Gaussian family is the most effective compared to power and rational function families.

In this study, the wave propagation properties of lattice metamaterials with Koch fractal structures are investigated in terms of band structures and directional wave propagation. The analytical models of lattice metamaterials are established using the finite element method, and the dispersion relation is solved using the Bloch’s theorem. The band structures of the lattice metamaterials with different numbers of iterations are studied, and the group velocities at a selected frequency are calculated to analyze the directional wave propagation characteristics. Furthermore, dynamic responses of the finite structures are calculated using commercial finite element software to verify the band gaps and directional wave propagation behaviors in the lattice metamaterials. The results show that multiple and low band gaps are present in the lattice materials with various geometric parameters of the Koch fractal, and the position of the lowest band gap decreases as the number of iterations increases. The results indicate the potential applications of lattice metamaterials with Koch fractals for vibration isolation and multi-functional design.

A comparative analysis of steady flow excited by an oscillating core in a rotating spherical cavity with a liquid is carried out in different cases: when the core is free and performs differential rotation, and when the core differential rotation is absent. In the cavity reference frame the core, whose density is less than the density of the liquid, oscillates near the cavity center under the action of transverse to the rotation axis external force field. In both cases, the oscillations lead to the appearance of almost two-dimensional axisymmetric azimuthal steady flow, with several inflection points in the velocity profile. An increase in the amplitude of core oscillations leads to a loss of stability of the axisymmetric flow. In the supercritical region, there is a series of threshold transitions associated with various instability modes. The instability manifests itself in the development of an azimuthal periodic system of vortices elongated parallel to the rotation axis. The modes differ in an azimuthal wavenumber, the location of the vortices (distance from the axis of rotation), and the azimuthal drift rate of the vortex system relative to the cavity. It is shown that the core differential rotation modifies the azimuthal velocity profile, resulting in a change in the instability thresholds. Due to an additional azimuthal flow, the drift velocity of the same type vortices in the two cases is different. The effect of the core differential rotation on the dispersion relations for various instability modes has been investigated.