Classical Wave Research Articles

Magnonic active ring co-processor

In this work, we consider the possibility of building a magnonic co-processor for special task data processing. Its principle of operation is based on the natural property of an active ring circuit to self-adjust to the resonant frequency. The co-processor comprises a multi-path active ring circuit where the magnetic part is a mesh of magnonic waveguides. Each waveguide acts as a phase shifter and a frequency filter at the same time. Being connected to the external electric part, the system naturally searches for the path which matches the phase of the electric part. This property can be utilized for solving a variety of mathematical problems including prime factorization, bridges of the Konigsberg problem, traveling salesman, etc. We also present experimental data on the proof-of-the-concept experiment demonstrating the spin wave signal re-routing inside a magnonic matrix depending on the position of the electric phase shifter. The magnetic part is a 3 × 3 matrix of waveguides made of single-crystal yttrium iron garnet Y3Fe2(FeO4)3 films. The results demonstrate a prominent change in the output power at different ports depending on the position of the electric phase shifter. The described magnonic co-processor is robust, deterministic, and operates at room temperature. The ability to exploit the unique physical properties inherent in spin waves and classical wave superposition may be translated into a huge functional throughput that may exceed 1060 operations per meter squared per second for 50×50 magnetic mesh. Physical limits and constraints are also discussed.

The viscoacoustic Green's function for the Helmholtz equation in a velocity gradient interface model

SUMMARY In viscoacoustically stratified media where the density is constant, 3-D wave propagation in the frequency–radial wavenumber domain is governed by the Helmholtz equation. In the case that the model is a velocity gradient interface where the squared velocity in depth is represented by a smooth Heaviside function of the Fermi–Dirac distribution type, the Helmholtz equation for a point source at arbitrary location is shown to have analytical solution for the Green's function. The velocity depth profile, which is a modification of the Epstein profile which has been thoroughly studied in different branches of physics, is described by four parameters: the velocities at minus and plus infinity, the reference depth of the gradient interface, and its smoothness. The Helmholtz equation is first solved in a model where the point source is absent. The solution to the source-free equation has four unknown constants that must be determined. The radiation conditions at minus and plus infinity and two conditions on the Green's function at the source depth allow the constants to be found. The Green's function solution can be represented in two mathematically equivalent algebraic forms involving ordinary hypergeometric functions. The first form allows a numerically stable implementation over all wavenumber components. The second form allows a physical, intuitive interpretation and is expressed mathematically as the sum of two terms. Each of the terms contains the product of constant-velocity reference phase shift functions and hypergeometric functions which take the role to adjust the amplitude and phase shift calculated by the reference phase shift functions to account for the depth varying velocity profile. Inverse Fourier transforms take the Green's function from the frequency–wavenumber domain to the frequency–space domain or time–space domain. The Green's function solution is valid for any sharpness of the interface. Selected numerical results are presented for the 1-D and 2-D Helmholtz equation to demonstrate the influence of the velocity gradient zone on the wavefield. The 1-D solution in an acoustic model is compared in time domain to the classical finite-difference wave propagation solution. For the purpose of interpretation of seismograms, we model for comparison the wavefield response in a model of two half-spaces in welded contact. For brevity, the latter model is referenced as the HS model.

Dispersive landslide

Considering the non-hydrostatic mass flow model (Pudasaini, 2022 [1]), here, I derive a novel dispersive wave equation for landslide. The new dispersive wave for landslide recovers the classical dispersive water waves as a special case. I show that the frequency dispersion relation for landslide is inherently different than the classical frequency dispersion for water waves. The wave frequency with dispersion increases non-linearly as a function of the wave number. For dispersive landslide, the wave frequency without dispersion appears to heavily overestimate the dispersive wave frequency for higher wave number. Due to the dispersion term emerging from the non-hydrostatic contribution for landslide, the phase velocity becomes a function of the wave number. This gives rise to the group velocity that is significantly different from the phase velocity, characterizing the dispersive mass flow. The dispersive phase velocity and group velocity decrease non-linearly with the wave number. Yet, the group velocity is substantially lower than the phase velocity. I analytically derive a dispersion number as the ratio between the phase velocity and the group velocity, which measures the deviation of the group velocity from the phase velocity, provides a dynamic scaling between them and summarizes the overall effect of dispersion in the mass flow. The dispersion number for landslide increases rapidly with the wave number, which is in contrast to the dispersion in water waves. With the definition of the effective dispersive lateral stress, I prove the existence of an anti-restoring force in landslide. I reveal the fact that due to the anti-restoring force, landslides are more dispersive than the piano strings. So, the wave dispersion in landslide is fundamentally different than the wave dispersion in the piano string. My model constitutes a foundation for the wave phenomenon in dispersive mass flows.

Sharp Resolvent Estimate for the Damped-Wave Baouendi–Grushin Operator and Applications

In this article we study the semiclassical resolvent estimate for the non-selfadjoint Baouendi-Grushin operator on the two-dimensional torus $\mathbb{T}^2=\mathbb{R}^2/(2\pi\mathbb{Z})^2$ with H\"older dampings. The operator is subelliptic degenerating along the vertical direction at $x=0$. We exhibit three different situations: (i) the damping region verifies the geometric control condition with respect to both the non-degenerate Hamiltonian flow and the vertical subelliptic flow; (ii) the undamped region contains a horizontal strip; (iii) the undamped part is a line. In all of these situations, we obtain sharp resolvent estimates. Consequently, we prove the optimal energy decay rate for the associated damped waved equations. For (i) and (iii), our results are in sharp contrast to the Laplace resolvent since the optimal bound is governed by the quasimodes in the subelliptic regime. While for (ii), the optimality is governed by the quasimodes in the elliptic regime, and the optimal energy decay rate is the same as for the classical damped wave equation on $\mathbb{T}^2$. Our analysis contains the study of adapted two-microlocal semiclassical measures, construction of quasimodes and refined Birkhoff normal-form reductions in different regions of the phase-space. Of independent interest, we also obtain the propagation theorem for semiclassical measures of quasimodes microlocalized in the subelliptic regime.

About theory of hybride TWTO and an amplifire with a complex permittivity

The purpose of this work is to construct a nonlinear theory of a hybrid between travelling wave tube (TWT) and an amplifier with a complex permittivity. Methods. The following model is considered: an ion-compensated one-dimensional electron beam penetrates the input travelling wave tube section, then flies into a medium with a complex permittivity, and then enters the output travelling wave tube section. A linear theory of this hybrid is constructed, and its results are compared with the results of the well-known linear theory of travelling wave tube. A nonlinear theory of this hybrid was constructed by a modified wave method, and the results were compared with the nonlinear travelling wave tube theory obtained by the classical Ovcharov–Solntsev’s wave method. In addition, to test the limits of applicability of the obtained results, a stationary nonlinear theory of the hybrid was constructed, obtained using the large particle method. The results of this theory were also compared with the stationary nonlinear travelling wave tube theory constructed using the large particle method. Results and conclusion. Based on the results of the developed theories, it is shown that, under certain parameters, the linear theory and nonlinear theories (both by the modified Ovcharov–Solntsev’s wave method and by the large particle method) make it possible to obtain comparable results both in the case of a classical travelling wave tube and for the hybrid under study. It is shown that under certain parameters, due to the resistive instability, the bunching of electrons can be noticeably improved and, as a result, the gain of the hybrid can exceed the gain in a classical travelling wave tube with the same parameters and the same total length of the device in the linear mode of operation. In the nonlinear mode of operation, the specified hybrid, with optimal environmental parameters, can have significantly higher values of output power and efficiency than travelling wave tube with the same value of the space charge parameter and the Pierce parameter.

A new class of small initial data which may shift the critical power and lifespan estimates for the classical damped wave equations

In this manuscript, lifespan of solutions to semilinear classical damped wave equations is investigated when the sum of initial position and speed is $ 0 $ pointwisely. The results in the existing literature are considered mainly in the case when the sum is positive. Here, it is shown that lifespan is extended with appropriate initial data with modified ODE approach. The result implies that the blowup mechanics for semilinear damped wave equations cannot be directly described by using the scaling property and the corresponding weak form approach. The results in this work leave different directions for further study of the lifespan behavior of the solutions to semilinear damped wave equations.

The <i>n</i>th-order self-resonance law of single wave train for surface capillary-gravity waves in deep water

Wave-wave resonance mechanism plays a fundamental and prominent role in the process of energy transfer and distribution, whether it is in microscopic or macroscopic matter. For the most extensive and intuitive ocean surface wave motion on earth, it is bound to be even more so. Can we extract the general wave-wave resonance law from it, especially the most special and brief resonance law for single wave train? To this end, according to a set of classical methods proposed by Phillips for initiating modern water wave dynamics with the specific 4-wave resonance conditions, and starting from the basic governing equations of ocean deep-water surface capillary-gravity waves, the first-order differential equation, and the second-, third- and fourth-order integral differential ones, which are becoming more and more complex but tend to be complete, of the Fourier components of free surface displacement are respectively given by the Fourier-Stieltjes transformation and perturbation method. Under a set of symbol system, which is self-created, self-evident and concise, these equations are solved in turn to obtain the first-order free surface displacement of single wave train, the Fourier coefficients of the second-, third- and fourth-order non-resonant and resonant free surface displacements, and the second-, third- and fourth-order resonant conditions, thus leading to the general nth-order self-resonance law of single wave train. This completely reveals the rich connotation of single wave resonance dynamics of ocean surface capillary-gravity waves, effectively expands the application range of the classical single wave resonance solutions given by Phillips for ocean surface gravity waves, lays the foundation for depicting single and multiple resonance interaction mechanisms of double and multi-wave trains of ocean surface waves, and so provides a typical example for the exploration of single-wave resonance law in all wave fields.

Propagation of shear waves in viscoelastic layered structure

• Shear wave through orthotropic material on functionally graded viscoelastic substrate. • Analytical dispersion relation. • Influence of material parameters on dispersion. The presented analytical study deals with the propagation of shear waves (S waves) in the layered structure containing an orthotropic material layer bonded with an heterogeneous viscoelastic half-space. Separate governing differential equations are applied while obtaining the analytical solution of transverse displacement in the respective media. The dispersion relation (complex form separated into real and imaginary parts) is finally derived in the closed form for the considered system employing suitable boundary conditions and the derived displacements. The obtained results are also shown to match with the classical Love waves equation (for isotropic media case). The phase- and damping velocity terms are plotted against the wavenumber (all quantities in dimensionless form) for the different values of material parameters. The effect of heterogeneity parameter, viscoelastic parameter, and the layer thickness on the wave propagation is graphically presented that can be used while exploring the nature of S wave propagation through similar composite structures.

How Classical, Quantum-Mechanical, and Relativistic Wave and Field Equations Are Uniformly Generated by Velocity-Field Divergence Equations

How Classical, Quantum-Mechanical, and Relativistic Wave and Field Equations Are Uniformly Generated by Velocity-Field Divergence Equations

Manipulation of electromagnetic waves induced by pseudomagnetic fields in two dimensional photonic crystals

Many interesting phenomena, such as quantization of Landau levels and quantum Hall effect, can occur in an electronic system under a strong magnetic field. However, photons do not carry charge, and they do not have many properties induced by external magnetic fields, either. Recently, the pseudomagnetic field, an artificial synthetic gauge field, has attracted intense research interest in classical wave systems, in which the propagation of the wave can be manipulated like in a real magnetic field. The photonic crystal is an optical structure composed of periodic material distributions and provides a good platform for studying the control of electromagnetic waves. In this work, we construct a uniform pseudomagnetic field by introducing uniaxial linear gradient deformation of metallic rods in a two-dimensional photonic crystal. The strong pseudomagnetic field leads to the quantization of photonic Landau levels in photonic crystal. The sublattice polarization of <i>n</i> = 0 Landau level is also demonstrated in our simulations. Unlike the real magnetic field, the pseudomagnetic fields of photonic crystal is opposite in two inequivalent energy valleys, and the time-reversal symmetry of the system is not broken. Our designed gradient photonic crystals support the transport of edge state in the gap between <i>n</i> = 0 and <i>n</i> = ±1 Landau levels. The edge state can propagate unidirectionally when it is excited by a chiral source. When a gaussian beam impinges on the photonic crystal, the propagating paths of two splitting beams can be controlled, which gives rise to the bend of two beams. Two photonic crystals with opposite pseudomagnetic fields are assembled together, and the interesting phenomenon of “snake-state” can be obtained. Our proposal opens the way for designing information processing devices by manipulating electromagnetic waves.

IDENTIFICATION OF GEOMECHANICAL AND GEOTECHNICAL STRUCTURES USING THE VIBROACOUSTIC METHOD FOR AUTOMATED STABILITY MONITORING SYSTEMS OF GROUND BUILDINGS AND MINE WORKINGS

Purpose. Identification of geomechanical and geotechnical structures by vibroacoustic method to improve reliability and automation level of ground buildings and mine working stability monitoring. Methods. Classical wave theory, experimental studies in laboratory and field conditions. Results. The most important feature of acoustic wave propagation in fractured media is the complexity of the wave pattern, since at each boundary between adjacent layers four new waves are formed – two reflected (longitudinal and transverse) and two refracted, each of them, when falling on another boundary, will cause a similar process. In the near-surface layer, classical acoustic waveguides are formed – sections of rock limited in one or two directions by voids, cracks or other media. As a result, the divergence of waves to the sides is eliminated or reduced, so their propagation along the layers occurs with less attenuation than in an unbounded homogeneous or inhomogeneous medium. To control elements of geomechanical and geotechnical structures, normal waves (modes) propagated in such acoustic waveguides without changing their structure should be used. When upper limit of the frequency range for recording acoustic signal of the response of a plane-parallel or block structure to the applied shock is lowered, the range of possible sounding bases expands. Within this range, the first mode of normal resonant oscillations is registered, and its limits shift towards an increase of the absolute value. The identification of geomechanical and geotechnical structures for systems of automated vibroacoustic control of stability of the mine workings was carried out. Originality. It is established that system control is ensured by identifying local geomechanical objects as the systems with lumped parameters, and changes in the load of the system elements should be controlled by analyzing the compliance between parameters of the system’s response to the exciting stimulation and the quasi-stationary criterion. In contrast to the known methods, automated control should be carried out with taking into account the stages of the oscillatory process development. Practical implications. The research results contribute to improvement of reliability and automation of the ground buildings and mine working stability monitoring. Keywords: vibroacoustic methods, identification of geomechanical structures, ground buildings and mine workings stability, automation of mine workings stability monitoring.

An even entire function of order one is a special solution for a classical wave equation in one-dimensional space

In this article we consider an even entire function of order one as the solution of the classical wave equation in one-dimensional space. We suggest a conjecture that this function has only purely real zeros in the entire complex plane. This is given as a new prospective on a nice connection with number theory and wave equation.

Construction of solutions of a two‐dimensional Riemann problem for a thin film model of a perfectly soluble antisurfactant solution

This article is concerned with formulation of three‐dimensional thin film model for an antisurfactant solution and hence constructing unique global solution for a two‐dimensional Riemann problem for the corresponding reduced hyperbolic form. We develop six geometrically different structures of the solution using generalized characteristic analysis method while relaxing the restriction that only one planar elementary wave is developed at the interface of each initial discontinuity. We analyze the interactions of classical and nonclassical waves in detail to construct the global solution of the corresponding 2‐D Riemann problem. Further, we provide the expressions for strength, location, and propagation speed of delta shock wave at each interaction point. Moreover, we compare these solutions with the solutions of a one‐dimensional rotated initial value problem and prove that our solutions are globally unique.

Global bifurcation structure and some properties of steady periodic water waves with vorticity

This paper studies the classical water wave problem with vorticity described by the Euler equations with a free surface under the influence of gravity over a flat bottom. Based on fundamental work [5], we first obtain two continuous bifurcation curves which meet the laminar flow only one time by using the modified analytic bifurcation theorem. They are symmetric waves whose profiles are monotone between each crest and trough. Moreover, we show a connection between the concavity and convexity of wave profile and the monotonicity of the vertical velocity component v along the free surface. As an important application, we make up the missing major aspect on the behavior of v as mentioned in [4, Sect 4.4] for small amplitude waves. In addition, for favorable vorticity, we prove that the vertical displacement of water waves decreases with depth.

Second Chern crystals with inherently non-trivial topology.

Chern insulators have been generalized to many classical wave systems and thereby lead to many potential applications such as robust waveguides, quantum computation and high-performance lasers. However, the band structure of a material can be either topologically trivial or non-trivial, depending on how the crystal structure is designed. Here, we propose a second Chern crystal in a four-dimensional parameter space by introducing two extra synthetic translation dimensions. Since the topology of the bulk bands in the synthetic translation space is intrinsically non-trivial, our proposed four-dimensional crystal is guaranteed to be topologically non-trivial regardless of the crystal's detailed configuration. We derive the topologically protected modes on the lower dimensional boundaries of such a crystal via dimension reduction. Remarkably, we observe the one-dimensional gapless dislocation modes and confirm their robustness in experiments. Our findings provide novel perspectives on topologically non-trivial crystals and may inspire designs of classical wave devices.

Tunable Rectification of Diffusion-Wave Fields by Spatiotemporal Metamaterials.

The diffusion process is the basis of many branches of science and engineering, and generally obeys reciprocity between two ports of a linear time-invariant medium. Recent research on classical wave dynamics has explored the spatiotemporal modulation to exhibit preferred directions in photons and plasmons. Here we report a distinct rectification effect on diffusion-wave fields by modulating the conductivity and observe nonreciprocal transport of charges. We experimentally create a spatiotemporal diffusion metamaterial, in which a mode transition to zero frequency is realized. A direct current component thereby emerges, showcasing a biased effect on the charge diffusion when the incident fundamental frequency is a multiple of the system modulation frequency. These results may find applications spanning a plethora of diffusive fields in general.

Theory of holey twistsonic media

Rotating two overlapping lattices relative to each other produces the well known moiré interference patterns and has surprisingly led to strongly correlated superconductivity in twisted bilayer graphene. This seminal effect that is associated with electrons occupying flat dispersion bands has stimulated a surge of activities in classical wave physics such as acoustics to explore equivalent scenarios. Here, we mimic twisted bilayer physics by employing a rigorous sound wave expansion technique to conduct band engineering in holey bilayer plates, i.e., twistsonic media. Our numerical findings show how one flexibly is able to design moiré sound interference characteristics that alone are controlled by the twist angle and the interlayer air separation. More specifically, our numerical approach provides a significant advantage in both computational speed and storage size in comparison with widely used commercial finite-element-method solvers. We foresee that our findings should stimulate further studies in terms of band engineering and exotic topological twisted phases.

A three-dimensional magnetoelastic valley Hall insulator with tunable elastic wave route and frequency

Topological insulators (TIs) are a new type of quantum state materials. Due to their novel physical properties, such as topological protection defect immunity to edge states, TIs have become the focus of attention in condensed matter and material physics. At present, the research on TIs has been gradually extended to classical wave fields such as electromagnetic waves, acoustic waves, and elastic waves, and has aroused extensive research interest. However, for elastic wave systems, most TIs cannot actively control topological interface states due to the limitation of fixed structure, which hinders their application in practical situations. Here, we propose a kind of tunable three-dimensional (3D) valley Hall insulator composed of magnetoelastic materials. First, the topological phase transition can be induced by the asymmetric geometry. Then, the working frequency of topological interface states can be changed by using static magnetic fields. Second, topological phase transformation can also be induced by independently tuning the distribution of static magnetic fields or pre-stress in each unit. Based on this, reconfigurable propagation routes of interface states with arbitrary shapes can be realized by tuning the distribution of static magnetic fields or pre-stress in each unit. Finally, considering the sandwich structure composed of different magnetic fields or pre-stress distribution modes, the waveguide with tunable width and route is designed by coupling edge and bulk states, which is convenient for application and better energy transfer. This study provides a reference for the design of a tunable intelligent elastic waveguide.

Realization of Photonic Topological Insulators at Terahertz Frequencies Characterized by Time-Domain Spectroscopy

Topological systems are inherently robust to disorder and continuous perturbations, resulting in dissipation-free edge transport of electrons in quantum solids, or reflectionless guiding of photons and phonons in classical wave systems characterized by topological invariants. Realization of robust, lossless topological insulators in the technologically important terahertz (THz) frequency range has drawn immense attention. In this paper, we propose a Kagome lattice structure that supports topological edge states with valley-dependent transport. Optical topological insulators are subsequently fabricated, in which mirror symmetry is broken and degenerate states are lifted at $K({K}^{\ensuremath{'}})$ valleys in the band structure. Theoretical and numerical analyses show that four types of edge states can be obtained. The performance of the fabricated topological waveguides is characterized with THz time-domain spectroscopy (TDS). Nearly identical THz wave transmission in the 0.437--0.453-THz range is verified with a straight-line topological waveguide and Z-shaped waveguides with multiple sharp turning corners, which quantitatively illustrate the strong backscattering suppression effects from the nontrivial topology edge states of the structures. Particularly, the THz TDS measurements provide important extra time-domain information about the topological waveguiding mode, which is generally unavailable from previous topological insulator investigations solely replying on power spectrum measurements.

A Multi‐Layer Linear Rossby Wave Dispersion Relation for Vertical Tilt of Mesoscale Eddies

AbstractVertical tilt of mesoscale eddies has been frequently observed in world oceans. The observations of composite eddies, individual eddies from an oceanic reanalysis product, and idealized numerical experiments of mesoscale eddies all show that the eddy vertical tilt is gradually increased with time in eddy mature stage, implying that the eddy translation speed changes with depth. However, eddy vertical tilt cannot be explained by the classic linear Rossby wave dispersion relation, since the classic zonal translation speed () of linear Rossby waves is depth‐independent from the quasi‐geostrophic potential vorticity (QGPV) equation. To deduce a depth‐dependent linear Rossby wave dispersion relation, we solved the QGPV equation by assuming that the stratified water column in oceans can be separated into a number of fine layers with constant stratification in each layer, whereas the stratification changes among layers. A multi‐layer linear Rossby wave (MLRW) dispersion relation in stratified oceans is then deduced. The MLRW dispersion relation shows that changes with depth. In each layer, is proportional to stratification and layer thickness but inversely proportional to Earth's rotation. The estimated structure from the MLRW dispersion relation is consistent with the structure of eddies in numerical experiments and the observed composite eddies. Noticeably, there are some discrepancies between the theoretical estimates and those from observations and numerical simulations, primarily due to the sensitivity of the theoretical estimates to layer thickness. The study attempts to provide an approach to describe and predict mesoscale eddy vertical tilt.