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Pole skipping is a recently discovered subtle effect in the thermal energy density retarded two point function at a special point in the complex $(\omega,p)$ planes. We propose that pole skipping is determined by the stress tensor contribution to many-body chaos, and the special point is at $(\omega,p)_\text{p.s.}= i \lambda^{(T)}(1,1/u_B^{(T)})$, where $\lambda^{(T)}=2\pi/\beta$ and $u_B^{(T)}$ are the stress tensor contributions to the Lyapunov exponent and the butterfly velocity respectively. While this proposal is consistent with previous studies conducted for maximally chaotic theories, where the stress tensor dominates chaos, it clarifies that one cannot use pole skipping to extract the Lyapunov exponent of a theory, which obeys $\lambda\leq \lambda^{(T)}$. On the other hand, in a large class of strongly coupled but non-maximally chaotic theories $u_B^{(T)}$ is the true butterfly velocity and we conjecture that $u_B\leq u_B^{(T)}$ is a universal bound. While it remains a challenge to explain pole skipping in a general framework, we provide a stringent test of our proposal in the large-$q$ limit of the SYK chain, where we determine $\lambda,\, u_B,$ and the energy density two point function in closed form for all values of the coupling, interpolating between the free and maximally chaotic limits. Since such an explicit expression for a thermal correlator is one of a kind, we take the opportunity to analyze many of its properties: the coupling dependence of the diffusion constant, the dispersion relations of poles, and the convergence properties of all order hydrodynamics., Comment: 39 pages, 13 figures

The linear stability analysis has been analyzed for an electrified liquid sheet confined between two symmetric dielectric movable ambient gas. The present study aims to estimate the planer jet systems response when applying a tangential electric field. The dispersion relations for varicose and sinuous modes have been formulated. The electrical Euler number and some flow parameters have been tested for their influence on the electrified liquid sheet instability. In other words, the effects of electric field on the dominant wave number, and instability range (cutoff wave number) are discussed. The results clarify that the tangential electric field has a stabilizing effect for sinuous mode over the whole range of gas velocity ratio. On the other hand, for varicose mode, the electric field slightly improves the system stability to a certain limit, but it has a great role in stability for higher values of gas velocity ratio.

An analytical study on the propagation characteristics of shear wave in a spherical layered structure comprising of an isotropic sandy material medium covered by a concentric spherical transversely isotropic sandy material layer of finite thickness has been carried out in the present work. The source of excitation for wave propagation is considered in the outer layer and the analytical treatment with contour integration and Fourier-Legendre Expansion theorem is utilized to accomplish closed form of dispersion relation. Further, Debye Asymptotic Expansion analysis is used to deal the complication caused by the inclusion of Bessel's function, Hankel's function and their derivatives in the solution procedure. With help of Debye Asymptotic Expansion analysis, the obtained dispersion relation converts in the form of elementary functions and found to be in well-agreement with pre-established and classical results through the particular cases. Numerical computations are done to illustrate the influence of the affecting parameters (anisotropy, sandiness and radii ratio) on dispersion curves graphically. Moreover, to expose the impact of anisotropy, numerical data of three distinct transversely isotropic materials (Magnesium, Zinc and Beryl) are taken into the account to establish the comparative study with isotropic material which serves the major highlights of the present study.

Accurate Rayleigh scattering and absorption cross sections of atmospheric gases are essential for understanding the propagation of electromagnetic radiation in planetary atmospheres. Accurate extinction cross sections are also essential for calibrating high-finesse optical cavities and differential optical absorption spectroscopy and for accurate remote sensing. In this study, we measured the scattering and absorption cross sections of carbon dioxide, nitrous oxide, sulfur hexafluoride, oxygen, and methane in the continuous wavelength range of 307–725 nm using broadband cavity-enhanced spectroscopy (BBCES). The experimentally derived Rayleigh scattering cross sections for CO2, N2O, SF6, O2, and CH4 agree with refractive index-based calculations, with a difference of (0.4 ± 1.2) %, (−0.6 ± 1.1) %, (0.9 ± 1.4) %, (2.8 ± 1.2) %, and (0.9 ± 2.2) %, respectively. The O2–O2 collision-induced absorption and absorption by methane are obtained with high precision at the 0.8 nm resolution of our BBCES instrument in the 307–725 nm wavelength range. New dispersion relations for N2O, SF6, and CH4 were derived using data in the UV–vis wavelength range. This study provides dispersion relations for refractive indices, n-based Rayleigh scattering cross sections, and absorption cross sections based on more continuous and more extended wavelength ranges than available in the current literature.

The extensive arch-shaped features, solar prominences, are observed subtending high into the solar corona. Considering the prominences to be unbound, homogeneous and isothermal, invoking the background flow, in presence of the Newtonian radiation and turbulent viscosity, to investigate the spatial damping of compressional MHD waves, we derived a dispersion relation and solved it numerically for both the Kraichnan and Kolmogorov turbulences for a time span of 10 - 5 s to 10 5 s. For Kraichnan and Kolmogorov turbulences, we investigated the effect of background flow as well as variable physical parameters (e.g. electron density, magnetic field and angle between propagation vector and magnetic field) on the damping length of fast and slow modes. Our findings reveal that the background flow does not show any Doppler shift in the damping length of Fast as well as Slow modes for shorter periods. However, for the longer periods, the fast modes show significantly smaller Doppler shift but interestingly, we noticed that the slow modes exhibit a substantial Doppler shift.

This paper is dedicated to the mathematical modeling of torsional surface waves in a heterogeneous monoclinic crystal cladded with a void type porous layer and gravitating viscous-sandy-elastic semi infinite substratum subjected to imperfect corrugated boundary surface. The model is solved analytically by using an implicit separable variable technique together with a Whittaker’s function and Bessel’s functions. This study focused on the determination of dispersion equation constituents from displacement components of the media by using suitable boundary conditions. The isotropic limit tests expose strong symphony together with the isotropic results attainable in literature. Detailed analysis through numerical simulations and graphical illustration of the dispersion equation includes discussion on the various rheological parameters such as shape of the irregular surfaces, Biot’s gravity parameter, sandy parameter, heterogeneity parameter, dissipation factor etc., on the dispersion curves of torsional wave.

We suggest a novel approach to predict and clarify the transformation of an arbitrarily polarized wave into two circularly polarized ones, based on the peculiar features of their transmission and reflection from a stack with regularly arranged gyrotropic layers. In spite of the fact that in the Faraday geometry gyrotropic media possess four electromagnetic eigenmodes, we have managed to modify the problem to be analytically resolved within the conventional transfer-matrix method. As a result, we have obtained and analyzed the dispersion relation, as well as the transmission and reflection coefficients, which are applicable for any elliptic polarization of a wave irradiating a gyrotropic superlattice with finite number of unit cells. It is shown that there exists a wide range of wave frequency and magneto-optical parameter (proportional to external dc magnetic field) inside of which the wave of given circular polarization is perfectly transmitted, while the wave with the opposite circular polarization is completely reflected. Thus, under certain conditions, the incident plane wave (taken as an appropriate example) splits up into the transmitted and reflected circular waves with mutually inverse rotations. We indicate conditions when a relatively small variation of the magnetization can reverse the polarizations mentioned above.

Distributions different from those predicted by equilibrium statistical mechanics are commonplace in a number of physical situations, such as plasmas and self-gravitating systems. The best strategy for probing these distributions and unavailing their origins consists in combining theoretical knowledge with experiments, involving both direct and indirect measurements, as those associated with dispersion relations. This paper addresses, in a quite general context, the signature of nonequilibrium distributions in dispersion relations. We consider the very general scenario of distributions corresponding to a superposition of equilibrium distributions, that are well-suited for systems exhibiting only local equilibrium, and discuss the general context of systems obeying the combination of the Schrödinger and Poisson equations, while allowing the Planck’s constant to smoothly go to zero, yielding the classical kinetic regime. Examples of media where this approach is applicable are plasmas, gravitational systems, and optical molasses. We analyse in more depth the case of classical dispersion relations for a pair plasma. We also discuss a possible experimental setup, based on spectroscopic methods, to directly observe these classes of distributions.

We propose a three-dimensional mathematical model to describe dynamical processes of membrane fission. The model is based on a phase field equation that includes the Gaussian curvature contribution to the bending energy. With the addition of the Gaussian curvature energy term numerical simulations agree with the predictions that tubular shapes can break down into multiple vesicles. A dispersion relation obtained with linear analysis predicts the wavelength of the instability and the number of formed vesicles. Finally, a membrane shape diagram is obtained for the different Gaussian and bending modulus, showing different shape regimes.

Розглядається задача про поширення квазілембових хвиль у попередньо деформованому нестисливому пружному шарі, що взаємодіє з півпростором в'язкої стисливої рідини. Дослідження проводиться на основі тривимірних лінеаризованих рівнянь теорії пружності скінченних деформацій для нестисливого пружного шару і тривимірних лінеаризованих рівнянь Нав’є—Стокса для півпростору в'язкої стисливої рідини. Застосовуються постановка задачі та підхід, засновані на використанні представлень загальних розв’язків лінеаризованих рівнянь для пружного тіла і рідини. Отримано дисперсійне рівняння, яке описує поширення нормальних хвиль у гідропружній системі. Побудовано дисперсійні криві квазілембових хвиль у широкому діапазоні частот. Проаналізовано вплив скінченних початкових деформацій у нестисливому пружному шарі, а також півпростору в'язкої рідини на фазові швидкості, коефіцієнти загасання, дисперсію квазілембових мод і поверхневу нестійкість гідропружного хвилеводу. Числові результати представлені у вигляді графіків і дано їх аналіз. The problem of the propagation of quasi-Lamb waves in a pre-deformed incompressible elastic layer that interacts with the half-space of an viscous compressible fluid is considered. The study is conducted on the basis of the three-dimensional linearized equations of elasticity theory of finite deformations for the incompressible elastic layer and on the basis of the three-dimensional linearized Navier–Stokes equations for the half-space of a viscous compressible fluid. The problem formulation and the approach, which are based on the utilization of representations of the general solutions of the linearized equations for an elastic solid and a fluid, are applied. Next, applying the Fourier method, we arrive at three eigenvalue problems for the equations of motion of the elastic body and the fluid. Solving them, we find the eigenfunctions. Substituting the general solutions into the boundary conditions, we obtain a homogeneous system of linear algebraic equations for the arbitrary constants. From the condition for the existence of a nontrivial solution, we derive the dispersion equation. It describes the propagation of normal waves in the hydroelastic system. The dispersion curves for quasi-Lamb waves over a wide range of frequencies are constructed. The effect of the finite initial deformations in an elastic layer, the thickness of the elastic layer, and the half-space of viscous compressible fluid on the phase velocities, attenuation coefficients, and dispersion of quasi-Lamb modes are analyzed. It follows from the graphical material presented above that, in the case of compression with 0.54, i.e., with a 46–percent reduction in the length of the highly elastic incompressible body, the phase velocities of the surface waves (Stoneley waves and Rayleigh waves) vanish. This indicates that the surface instability develops at 0.54 for a highly elastic incompressible non-Hookean body initially in a plane stress-strain state. We should point out that these figures agree with results obtained earlier in the theory of stability and correspond to the critical value of the con-traction parameter. In the case of highly elastic incompressible bodies, the linearized wave theory makes it possible to study not only general and several specific wave processes, but also the conditions under which the surface instability begins in elastic bodies and hydroelastic systems. It also follows from the graphs that the viscous fluid slightly affects the surface instability of hydroelastic systems. The numerical results are presented in the form of graphs, and their analysis is given.

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.

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.

Wavefield decomposition can be used to extract effective information in reverse time migration and full-waveform inversion. The wavefield decomposition methods based on the Hilbert transform (HTWD) and the Poynting vector (PVWD) are the most commonly used. The HTWD needs to save the wavefields at all time steps or introduce additional numerical simulation, which increases the computational cost. PVWD cannot handle multiwave arrivals, and its performance is poor in complex situations. We have developed an efficient wavefield decomposition method based on the Hilbert transform (EHTWD). EHTWD constructs two wavefields to replace the original wavefield and the wavefield after the Hilbert transform. The first wavefield is obtained by using the dispersion relation to modify the frequency components. The other wavefield is obtained by time difference approximation. Therefore, there is a 90° phase change between the two wavefields. In EHTWD, we only need two wavefields at different moments, which avoids the need for additional numerical simulation. EHTWD is also suitable for wavefield decomposition in arbitrary directions. Compared to HTWD, the computational complexity can be greatly reduced with the decrease of the number of imaging time slices. The numerical examples of wavefield decomposition demonstrate that our method can realize wavefield decomposition in any direction. The examples of imaging decomposition and real data also indicate that EHTWD suppresses imaging noise effectively.

The motion of domain walls in chiral magnets with symmetry broken relative to spatial inversion is investigated theoretically. It was found experimentally that such walls, which are known as the Dzyaloshinskii domain walls [22], exhibit unique dynamic properties. For chiral ferromagnets and antiferromagnets, nonlinear regimes of the motion of such a wall with the velocity up to its critical value have been investigated. The dynamics of walls in antiferromagnets can be described analytically, while for ferromagnets, numerical analysis must be used. Simple approximate expressions derived for ferromagnets describe dynamic characteristics of walls to a high degree of accuracy in the entire domain of their existence. The dispersion relations (i.e., the dependences of the energy of a wall on its linear momentum) are determined. For ferromagnets with purely uniaxial anisotropy, the critical velocity of a wall is determined exclusively by the chiral interaction and vanishes in the absence of this interaction. For antiferromagnetic, the role of the chiral interaction is not so obvious: like in the standard models for antiferromagnets, the structure of a moving wall can be obtained with the help of the Lorentz transformation with a chosen speed equal to the velocity of magnons in the linear part of the spectrum; however, the dispersion relation is more complicated and cannot be described by Lorentz-invariant expressions.

Effect of plasmon–magnon interaction on the propagation of a hybrid wave along the graphene surface in the graphene–antiferromagnetic insulator system is studied. Three antiferromagnetic materials NiO, FeF2, and MgF2 with different antiferromagnetic resonance frequencies in the terahertz range and different damping rates of magnons are considered. It is found that the plasmon–magnon interaction is manifested in a change in the dispersion relation near the antiferromagnetic resonance frequency, and the efficiency of this interaction depends on the position of the Fermi level in graphene and the damping of magnons.

Rayleigh-type waves are investigated in a layered model composed of thermoelastic medium with double porosity structure. Using appropriate boundary conditions, the dispersion relation is obtained, ...

A van der Waals bonded moire bilayer formed by sequential growth of TiSe2 and TiTe2 monolayers exhibits emergent electronic structure as evidenced by angle-resolved photoemission band mapping. The two monolayers adopt the same lattice orientation but incommensurate lattice constants. Despite the lack of translational symmetry, sharp dispersive bands are observed. The dispersion relations appear distinct from those for the component monolayers alone. Theoretical calculations illustrate the formation of composite bands by coherent electronic coupling despite the weak interlayer bonding, which leads to band renormalization and energy shifts.

In this present study, weighted total acceleration for Kinematic Free Surface Boundary Condition (KFSBC) and in momentum Euler equation was formulated. Furthermore, by using both aforementioned equations, the nonlinear water wave dispersion equation was then formulated. The wavelength obtained from dispersion equation is determined by weighting coefficient. The weighting coefficient value was determined by using the maximum wave height and critical wave steepness criteria which have been obtained from the previous studies.

Two-dimensional van der Waals (vdW) crystals can sustain various types of polaritons with strong electromagnetic confinements, making them highly attractive for nanoscale photonic and optoelectronic applications. While extensive experimental and numerical studies have been devoted to the polaritons of the vdW crystals, analytical models are sparse. Particularly, applying the model to describe polariton behaviors that are visualized by state of the art near-field optical microscopy requires further investigations. In this study, we develop an analytical waveguide model to describe polariton propagations in vdW crystals. The dispersion contours, dispersion relations, and localized electromagnetic field distributions of polariton waveguide modes are derived. The model is verified by real-space optical nano-imaging and numerical simulation of phonon polaritons in α-MoO3, which is a vdW biaxial crystal. Although we focus on α-MoO3, the proposed model is valid for other polaritonic crystals within the vdW family given the corresponding dielectric substitutions. Our model therefore provides an analytical rationale for describing and understanding the localized electromagnetic fields in vdW crystals that are associated with polaritons.

A slender thread of elastic hydrogel is susceptible to a surface instability that is reminiscent of the classical Rayleigh-Plateau instability of liquid jets. The final, highly nonlinear states that are observed in experiments arise from a competition between capillarity and large elastic deformations. Combining a slender analysis and fully three-dimensional numerical simulations, we present the phase map of all possible morphologies for an unstable neo-Hookean cylinder subjected to capillary forces. Interestingly, for softer cylinders we find the coexistence of two distinct configurations, namely, cylinders-on-a-string and beads-on-a-string. It is shown that for a given set of parameters, the final pattern is selected via a dynamical evolution. To capture this, we compute the dispersion relation and determine the characteristic wavelength of the dynamically selected profiles. The validity of the "slender" results is confirmed via simulations and these results are consistent with experiments on elastic and viscoelastic threads., Comment: 14 pages, 6 figures

At present, mechanics of damage media, which studies both the stress–strain state of media and the damage accumulation in materials, is being actively developed. In this study, a self-consistent problem, including the dynamic equation of the theory of elasticity and the kinetic equation of damage accumulation in a material, is formulated for an isotropic elastic half-space with damage in the material. It is assumed that damage is distributed uniformly over the medium. The surface-wave propagation along the free boundary of damaged half-space has been investigated. The wave propagates horizontally and decays in the vertical direction. All processes are assumed to be homogeneous along the third axis. It is shown that a self-consistent system with boundary conditions expressing the absence of stress at the half-space boundary is reduced to a complex dispersion equation in this case. In the limiting case (damage-free material), the obtained dispersion equation is reduced to the classical dispersion equation for a Rayleigh wave in the polynomial form (the surface wave propagates along the half-space boundary without dispersion and attenuation). If damage is present in the medium, the surface wave attenuates along the propagation direction, and low-frequency disturbances have frequency-dependent dissipation and dispersion. It is shown that dispersion has abnormal character. It is established that, in the high-frequency region, the phase and group velocities increase and decrease, respectively, with an increase in the damage coefficient. At low frequencies, both velocities increase with a decrease in the damage coefficient.

This study considers to solve the complex dispersion equation for propagation of Rayleigh waves in a dissipative porous composite. This porous medium is a viscoelastic porous solid frame saturated ...

Considering the modified Lorentz dispersion relation, combined with the Dirac equation and Rarita–Schwinger equation of fermions in stationary axisymmetric Sen black hole space–time, the fermion tunneling radiation of the black hole is modified accurately, and meaningful physical quantities such as the modified fermion tunneling rate, event horizon temperature, and entropy of the black hole are obtained. The discussion of the conclusions shows that the effect of the Lorentz dispersion relation and Lorentz violation theory on particle dynamics must be considered in curved space–time during the study of quantum theory and Hawking tunneling radiation.

The optical properties of macroscopic materials are described, basing only on classical arguments. Starting from Maxwell’s equations, the propagation of electromagnetic waves in isotropic, linear, and homogeneous materials is treated. To this aim, complex macroscopic quantities as the dielectric function, the optical conductivity, and the refractive index are introduced, together with the dispersion relations that they obey. The reflection and transmission properties at the interface between different materials are analyzed, and the appropriate functions are defined for both unpolarized and polarized light waves, with the addition of the ellipsometric functions that are particularly interesting from the application point of view.

" 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."

The problem of the field of internal gravity waves in a stratified medium of finite depth is considered for model distributions of background shear flows. The constant distribution of the buoyancy frequency and various linear dependences of background shear flow on the depth are used to solve the problem analytically. The dispersion relations expressed in terms of the modified Bessel function of imaginary index are obtained. In satisfying the Miles–Howard stability condition in the case of large Richardson numbers, the Debye asymptotics of the modified Bessel function of imaginary index are used to construct the analytical solutions. The properties of the dispersion relation are studied and the basic analytical characteristics of the dispersion curves are investigated as functions of the parameters of background shear flows. The phase patterns of the excited fields are calculated numerically for various wave generation models.

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.

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.

Equivalent staggered-grid (ESG) as a new family of schemes has been utilized in seismic modeling, imaging, and inversion. Traditionally, the Taylor series expansion is often applied to calculate finite-difference (FD) coefficients on spatial derivatives, but the simulation results suffer serious numerical dispersion on a large frequency zone. We develop an optimized equivalent staggered-grid (OESG) FD method that can simultaneously suppress temporal and spatial dispersion for solving the second-order system of the 3D elastic wave equation. On the one hand, we consider the coupling relations between wave speeds and spatial derivatives in the elastic wave equation and give three sets of FD coefficients with respect to the P-wave, S-wave, and converted-wave (C-wave) terms. On the other hand, a novel plane wave solution for the 3D elastic wave equation is derived from the matrix decomposition method to construct the time–space dispersion relations. FD coefficients of the OESG method can be acquired by solving the new dispersion equations based on the Newton iteration method. Finally, we construct a new objective function to analyze P-wave, S-wave, and C-wave dispersion concerning frequencies. The dispersion analyses show that the presented method produces less modeling errors than the traditional ESG method. The synthetic examples demonstrate the effectiveness and superiority of the presented method.

This paper is concerned with the Rayleigh wave dispersion in an incompressible functionally graded orthotropic half-space loaded by a thin fluid-saturated aeolotropic porous layer under initial stress. Both the layer and half-space have subjected to the incompressible in nature. The particle motion of the Rayleigh type wave is elliptically polarized in the plane, which described by the normal to the surface and the focal point along with wave generation. The dispersion of waves refers typically to frequency dispersion, which means different wavelengths travel at a different velocity of phase. To deal with the analytical solution of displacement components of Rayleigh type waves in a layer over a half-space, we have taken the assistance of different methods like exponential, characteristic polynomial and undetermined coefficients. The dispersion relation has been derived based upon suitable boundary conditions. The finite difference scheme has been introduced to calculate the phase velocity and group velocity of the Rayleigh type waves. We also have derived the stability condition of the finite difference scheme (FDS) for the phase and group velocities. If a wave equation has to travel in the time domain, it is necessary to achieve both accuracy and stability requirements. In such cases, FDS is preferred because of its power, accuracy, reliability, rapidity, and flexibility. The effect of various parameters involved in the model like non-homogeneity, porosity, and internal pre-stress on the propagation of Rayleigh type waves have been studied in detail. Graphical representations for the effects of various parameters on the dispersion equation have been represented. Numerical results demonstrated the accuracy and versatility of the group and phase velocity depending on the stability ratio of the FDS.

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.

For a number of materials with cubic lattice structure the dispersion relations of the Debye bosons (sound waves) and of the acoustic phonons along [ζ 0 0] direction have been analyzed quantitatively. When all phonon modes are excited, that is, for temperatures of larger than the Debye temperature the dispersion of the mass-less Debye bosons exhibits a pronounced non-linearity, which is explained by interactions with the phonon background. For the exponent x in the dispersion relation ~qx of the Debye bosons, the rational values of x=1/4, 1/3, 1/2, 2/3 and 3/4 could be established firmly. The discrete values of x show that there are distinct modes of interaction with the phonons only. It is furthermore shown that for many materials the dispersion of the acoustic phonons along [ζ 0 0] direction follows a perfect sine function of wave vector, which is known to be the dispersion of the linear atomic chain. This dispersion is unlikely to be the intrinsic behavior of three-dimensional solids. It is argued that the sine-function is induced by the Debye boson-phonon interaction. Quantitative analyses of the temperature dependence of the heat capacity show that the heat capacity can be described accurately over a large temperature range by the expression cp=c0-B‧T-ε. The constants c0 and B are material specific and define the absolute value of the heat capacity. However, for the exponent ε the same rational value occurs for materials with different chemical compositions and lattice structures. The temperature dependence of the heat capacity therefore exhibits universality. This universality must be considered as a non-intrinsic dynamic property of the atomistic phonon system, arising from the Debye boson-phonon interaction. The discrete modes of boson-phonon interaction are essential for the observed universality classes of the heat capacity. Safely identified values for ε are ε=1, 5/4 and 4/3. The fit values for c0 are generally larger than the theoretical Dulong-Petit value. Universal exponents are identified also in the temperature dependence of the coefficient of the linear thermal expansion, α(T). Since the universality in α(T) holds for the same thermal energies (temperatures) as for the ~qx functions in the dispersion of the Debye bosons it can be concluded that the Debye bosons also determine the temperature dependence of α(T). Our results show that the dynamics of the atomic lattice is modified for all temperatures by the Debye bosons. Atomistic models restricting on inter-atomic interactions therefore are neither sufficient to explain the phonon dispersion relations nor the detailed temperature dependence of the heat capacity.

We study the characteristic velocities of the extended thermodynamics of molecular rotational- and vibrational- relaxation processes for a van der Waals gas. The lower and upper bounds of the characteristic velocity are, respectively, estimated by the one of rarefied gases and the one evaluated on the spinodal curve. We also study the dispersion relation of linear waves, in particular, the phase velocity, attenuation factor and attenuation per wavelength in a low frequency region.

In transversely isotropic (TI) media, accurate and efficient pure P-wave extrapolation is the basis of seismic imaging and inversion algorithms. For pure P-wave equations in TI media, combining the finite-difference (FD) method with the pseudo-spectral (PS) method is an effective solution. However, if the mixed-domain pure P-wave equation contains multiple wavenumbers, using the hybrid approach will involve multiple Fourier transforms, which will result in high computational costs. Referring to the weak anisotropy approximation, we propose a new approximate P-wave phase velocity expression. We then use an optimization strategy to reduce the number of wavenumber terms in the corresponding dispersion relations and derive the mixed-domain pure P-wave equations in 2D and 3D tilted transversely isotropic (TTI) media. Through numerical experiments in 2D and 3D TTI models, we verify the feasibility and efficiency of the proposed mixed-domain P-wave equations.

Topological Dirac semimetals made of two-dimensional transition-metal dichalcogenides (TMDCs) have attracted enormous interest for use in electronic and optoelectronic devices because of their electron transport properties. As van der Waals materials with a strong interlayer interaction, these semimetals are expected to support layer-dependent plasmonic polaritons yet to be revealed experimentally. Here, we demonstrate the apparent retardation and attenuation of mid-infrared (MIR) plasmonic waves in type-II Dirac semimetal platinum tellurium (PtTe2) nanoribbons and nanoflakes by near-field nanoimaging. The attenuated dispersion relations for the plasmonic modes in the PtTe2 nanoribbons (15-25 nm thick) extracted from the near-field standing-wave patterns are applied for the fitting of PtTe2 permittivity in the MIR regime, indicating that both free carriers and Dirac fermions are involved in MIR light-matter interaction in PtTe2. The annihilation of plasmonic modes in the ultrathin (

This paper considers the thermoelastic theory with two temperatures that involves higher gradients of thermal and mechanical effects. The wave propagation question is addressed within the class of waves of assigned wavelength. Considering harmonic in time wave solutions, it is found that the transverse waves are undamped in time and non-dispersive, and they are not altered by the thermal effects. Conversely, the longitudinal waves are dispersive and damped in time; the dispersion relation is established like a cubic equation and the effects of conductive temperature are explicitly presented. Rayleigh surface waves are also studied and an explicit secular equation is derived by using wave solutions damped in time. Illustrative examples are numerically analyzed and graphically depicted. The results achieved are meaningful because they are able to bring information about the propagation of waves with assigned length and, moreover, they are in agreement with the results regarding the wave speed of travelling discontinuities. Also the structure of the wave solutions provides information upon asymptotic stability.

Purpose Quantitatively detecting surface defects in a circular annulus with high levels of accuracy and efficiency has been paid more attention by researchers. The purpose of this study is to investigate the theoretical dispersion equations for circumferential guided waves and then develop an efficient technique for accurate reconstruction of defects in pipes. Design/methodology/approach The methodology applied to determine defects in pipelines includes four steps. First, the theoretical work is carried out by developing the appropriate dispersion equations for circumferential guided waves in a pipe. In this phase, formulations of strain-displacement relations are derived in a general equidistant surface coordinate. Following that, a semi-analytical finite element method (SAFEM) is applied to solve the dispersion equations. Then, the scattered fields in a circular annulus are calculated using the developed hybrid finite element method and simulation results are in accord with the law of conservation of energy. Finally, the quantitative detection of Fourier transform (QDFT) approach is further enhanced to efficiently reconstruct the defects in the circular annuli, which have been widely used for engineering applications. Findings Results obtained from four numerical examples of flaw detection problems demonstrate the correctness of the developed QDFT approach in terms of accuracy and efficiency. Reconstruction of circumferential surface defects using the extended QDFT method can be performed without involving the analytical formulations. Therefore, the streamlined process of inspecting surface defects is well established and this leads to the reduced time in practical engineering tests. Originality/value In this paper, the general dispersion equations for circumferential ultrasonic guided waves have been derived using an equidistant surface coordinate and solved by the SAFEM technique to discover the relationship between wavenumber of a wave and its frequency. To reconstruct defects with high levels of accuracy and efficiency, the QDFT approach has been further enhanced to inspect defects in the annular structure.

The present paper deals with the propagation of surface waves in an isotropic and homogeneous nonlocal generalized thermoelastic solid half-space with voids. The dispersion relations for Rayleigh-t...

We study the semiclassical approximation of the lower bands of the Schrodinger operator with a periodic two-dimensional potential with a trigonal symmetry and consider the cases where the potential has one or two wells in the elementary cell. We obtain the exponentially small asymptotic behavior of the band width and find the dispersion relations. We investigate the form of the Bloch functions. Solving this problem is the first step in studying the more complicated (and more physically interesting) problem of tunnel effects in rotating dimers.

A theoretical approach is taken into consideration to investigate the propagation behaviour of shear acoustic waves in a piezoelectric cylindrical layered structure composed of a piezoelectric material cylinder imperfectly bonded to a concentric functionally graded piezoelectric material (FGPM) cylindrical layer of finite width. The functional gradient in the FGPM cylindrical layer is considered to vary continuously along the radial direction (function of radial coordinate), and the imperfection of the interface of the cylindrical structure is taken into account which may practically exist due to some mechanical and/or electrical damage. By means of mathematical transformation, the governing electromechanical coupled field differential equations are reduced to Bessel’s equations. An analytical treatment has been employed to determine the dispersion relations of propagating shear acoustic waves for both electrically short and electrically open conditions, which are further validated by reducing the obtained results to the pre-established standard results and classical Love wave equation as a special case of the problem. The effects of functional gradient parameter, radii ratio, wave number, order of Bessel’s function appearing in the dispersion relations and mechanical/electrical imperfection parameters associated with the imperfect bonding of a piezoelectric material cylinder and FGPM layer on the phase velocity of shear acoustic waves have been reported through numerical simulation and graphical demonstration. For the sake of numerical computation, the data of PZT-5H for the FGPM cylindrical layer and AlN for a piezoelectric material cylinder have been considered.

The problem of constructing solutions that describe internal gravity waves in a stratified medium with finite depth and multidirectional shear flows has been considered. The model constant distribution of the Brunt–Vaisala frequency and the linear dependence of shear flows on depth have also been considered, and an analytical solution of the problem, which is expressed through the modified Bessel functions with an imaginary index, has been obtained. Using the Debye asymptotics of the modified Bessel function, analytical expressions have been obtained for dispersion relations and the phase characteristics of wave fields have been studied. The dependences of the wave characteristics of excited fields on the main parameters of the used models of stratification, flows, and generation modes have also been studied.

—The problem of constructing the solutions that describe harmonic internal gravity waves in a semi-infinite stratified medium with shear flow is considered. The model constant distribution of the Brunt–Vaisala frequency and the linear dependence of shear flow as functions of the depth are used and an analytical solution of the problem expressed in terms of the modified Bessel functions of the imaginary index is obtained. The analytical expressions for the dispersion relations are obtained using the Debye asymptotics of the modified Bessel function of the imaginary index and the phase characteristics of the wave fields are investigated. The wave characteristics of the excited fields are investigated as functions of the main parameters of the used stratification models, flows, and generation regimes.

A wide-angle Fourier migrator is proposed to more accurately image complex media with strong lateral velocity contrasts. The broadband wave-equation migrator is developed based on the pseudo-Pade approximation, where the Pade coefficients are independent of spatial coordinates, leading to a pure Fourier transform-based matching solution for one-way wavefield extrapolation. We use genetic algorithms to estimate the constant Pade coefficients more accurately than is feasible with conventional least-squares methods. Because of the global features of pure Fourier migrators, we present an angle-partitioning optimization scheme with dip focusing to improve the performance of the Fourier migrator for super-wide-angle waves and strong velocity contrasts. The wavefield gradient is used to calculate propagation angles during dual-domain wavefield extrapolation. Particular attention is paid to the first-order optimized pseudo-Pade Fourier (OPF1) migrator, which significantly improves the split-step Fourier (SSF) method for strong lateral variations at the cost of one additional Fourier transform in each step. Wavefield extrapolation based on the OPF1 method actually constitutes linear interpolation in the wavenumber domain between two split-step terms. We benchmark the OPF1 migrator with other typical migrators based on the exact dispersion equation. Numerical experiments with impulse responses, the SEG/EAGE salt model and 3D field data demonstrate the excellent performance and efficiency of seismic imaging with the OPF1 migrator.

Using a Gurevich-Krylov solution that describes the propagation of nonlinear magnetoacoustic waves in a cold plasma, we construct solutions of various other nonlinear systems. These include, for example, Madelung fluid, reaction diffusion, Broer-Kaup, Boussinesq, and Hamilton-Jacobi-Bellman systems. We also construct dilaton field solutions for a Jackiw-Teitelboim black hole with a negative cosmological constant. The black hole metric corresponds to a cold plasma metric by way of a change of variables, and the plasma dilatons and cosmological constant also have an expression in terms of parameters occurring in the Gurevich-Krylov solution. A dispersion relation, moreover, links the magnetoacoustic system and a resonance nonlinear Schrodinger equation.

This study is devoted to investigate the propagation of Rayleigh-type waves in transversely isotropic nonlocal piezoelastic half-space. When the stress-free boundary is maintained at charge-free condition, the dispersion equation for the propagation of Rayleigh waves at the free surface of transversely isotropic piezoelastic solids has been obtained. Based on the obtained dispersion equation, the effect of the nonlocality on the speed of Rayleigh wave is numerically considered. The dependence of velocities of plane waves in transversely isotropic nonlocal piezoelastic medium on the direction of propagation as well as non-dimensional frequency parameter has been also illustrated.

The derivation of mean-field limits for quantum systems at zero temperature has attracted many researchers in the last decades. Recent developments are the consideration of pair correlations in the effective description, which lead to a much more precise description of both spectral properties and the dynamics of the Bose gas in the weak coupling limit. While mean-field results typically lead to convergence for the reduced density matrix only, one obtains norm convergence when considering the pair correlations proposed by Bogoliubov in his seminal 1947 paper. In this article, we consider an interacting Bose gas in the case where both the volume and the density of the gas tend to infinity simultaneously. We assume that the coupling constant is such that the self-interaction of the fluctuations is of leading order, which leads to a finite (nonzero) speed of sound in the gas. In our first main result, we show that the difference between the N-body and the Bogoliubov description is small in $$L^2$$ as the density of the gas tends to infinity and the volume does not grow too fast. This describes the dynamics of delocalized excitations of the order of the volume. In our second main result, we consider an interacting Bose gas near the ground state with a macroscopic localized excitation of order of the density. We prove that the microscopic dynamics of the excitation coming from the N-body Schrodinger equation converges to an effective dynamics which is free evolution with the Bogoliubov dispersion relation. The main technical novelty are estimates for all moments of the number of particles outside the condensate for large volume, and in particular control of the tails of their distribution.

The article focuses on the interaction of Rossby waves in the ocean with zonal jet flows. A new approach is proposed to show that nonlinearity in the long-wave approximation exactly compensates the Doppler shift. A new dispersion relation for the Rossby waves interacting with the jets is deduced from the nonlinear theory. The conclusion is verified using satellite altimetry data of the Antarctic Circumpolar Current (ACC). For the ACC area, we compare empirical velocities obtained from the altimetry data with theoretical phase velocities of Rossby waves calculated from nonlinear dispersion relation using the equivalent beta effect. The comparison shows that the new dispersion relation based on the nonlinear approach is capable of describing both the westward and the eastward propagation of mesoscale eddies in the field of sea level anomalies that can be identified as manifestation of Rossby waves in the ocean.

We present an analysis of propagation of coupled electromagnetic TE-TE waves in a nonlinear plane waveguide located between two half-spaces with constant permittivities in which the permittivity of...

In the present work, we have performed numerical simulations of carry-along (sound drift) effect in ultrasonic flow-meter with various steering angles using an accurate, dispersion relation preserving ( DRP ) scheme. Validation has been performed with the experimental results by Kang et al. (2017) for the case of wave propagation along and opposite to the direction of the flowing medium. Linearized Euler equations have been solved for the simulation of the ultrasonic acoustic field. Various velocity profiles associated with uniform, laminar and turbulent are applied in acoustic field simulations while keeping the same average mean flow velocity. Results highlight that the final steered angle does not depend on the velocity profile but depends on the average mean velocity of the flow. However, flow measurement will be affected by the velocity profile since the propagating time depends upon the velocity profile. Present simulations can be used to design the ultrasonic flow-meter with a phased array transducer with steering & focusing capabilities to achieve high signal to noise ratio ( SNR ) .