Elementary Waves Research Articles

Analytical modelling for predicting the sound field of planar acoustic metasurface

An analytical model is built to predict the acoustic fields of acoustic metasurfaces. The acoustic fields are investigated for a Gaussian sound beam incident on the acoustic metasurfaces. The Gaussian sound beam is decomposed into a set of discrete elementary plane waves. The diffraction caused by the acoustic metasurfaces can be obtained using this analytical model, which is validated with the numerical simulations for the different incident angles of the Gaussian sound beam. This model overcomes the limitation of the method based on the generalised Snell's law which can only predict the direction of a specific diffracted order. Actually, this analytical model can be also used to predict the sound fields of acoustic metasurfaces under any incident sound if its Fourier transforms exist. This conclusion is demonstrated by studying the sound field for a point sound source incident on the acoustic metasurface. The acoustic admittances of acoustic metasurfaces are required in the calculation of the analytical model. Therefore, a numerical method for obtaining the effective acoustic admittances is proposed for the structurally complex metasurfaces without the analytical expressions of material properties, such as equivalent density and sound speed.

Interaction of weak shocks in drift-flux model of compressible two-phase flows

We consider the Riemann problem for an isothermal no-slip compressible gas-liquid drift-flux model of multi-phase flows governing two mass conservation equations and one mixture momentum equation. We establish a global result for the existence and uniqueness of a solution to the Riemann problem. We obtain one parameter family of elementary wave curves explicitly. Further, we derive a necessary and sufficient condition to ensure the existence of a shock wave or a simple wave for a 1-family and a 3-family of characteristics on the initial data in the solution of the Riemann problem. Finally, we prove the result of Von Neumann’s concerning the overtaking of two weak shocks of same family.

Spectrum of Langmuir fluctuations trapped in a cavity of a collisional plasma

The Langmuir fluctuations trapped and enhanced inside a cavity of the electron concentration in a plasma are considered. Stationary fluctuation spectrum is calculated with regard to finite collisional damping of the Langmuir waves and nonequilibrium fluctuation source. It is shown that the fluctuation spectrum obtained exhibits a frequency modulation due to the reflection and superposition of the elementary Langmuir waves in the irregularity. The modulation depth and period as a function of the irregularity size and the electron–ion collision frequency is calculated. These results may help to find from experimental observations such important plasma parameters as the irregularly size and lifetime, as well as the electron collision frequency.

Interaction of the elementary waves of isentropic flow in a variable cross-section duct

Interaction of the elementary waves of isentropic flow in a variable cross-section duct

Global Solution of a Nonlinear Conservation Law with Weak Discontinuous Flux in the Half Space

This paper is concerned with the initial-boundary value problem of a nonlinear conservation law in the half space R+= {x |x > 0} where a>0 , u(x,t) is an unknown function of x ∈ R+ and t>0 , u ± , um are three given constants satisfying um=u+≠u- or um=u-≠u+ , and the flux function f is a given continuous function with a weak discontinuous point ud. The main purpose of our present manuscript is devoted to studying the structure of the global weak entropy solution for the above initial-boundary value problem under the condition of f '-(ud) > f '+(ud). By the characteristic method and the truncation method, we construct the global weak entropy solution of this initial-boundary value problem, and investigate the interaction of elementary waves with the boundary and the boundary behavior of the weak entropy solution.

The Boyd Conjecture

The Boyd Conjecture is that the amplitude of an elementary ray from the Theory of Elementary Waves (TEW) is thephysical analog of a probability amplitude from Quantum Mechanics (QM). Boyd learned TEW from his cousin Lewis E.Little, who had said that quantum math is the roadmap to the world of elementary waves, but we didn’t know how to readthe map. He suggested that Boyd, with a degree in mathematics, should work on deciphering the map. In 2014 Boydbegan thinking that amplitudes are the core of QM, and elementary rays the core of TEW: perhaps they were equivalent.Both decisively influence particle behavior without conveying any energy. This essay discusses the history, strengths,weakness and implications of this conjecture.

Numerical study on transient harbor oscillations induced by successive solitary waves

Tsunamis are traveling waves which are characterized by long wavelengths and large amplitudes close to the shore. Due to the transformation of tsunamis, undular bores have been frequently observed in the coastal zone and can be viewed as a sequence of solitary waves with different wave heights and different separation distances among them. In this article, transient harbor oscillations induced by incident successive solitary waves are first investigated. The transient oscillations are simulated by a fully nonlinear Boussinesq model, FUNWAVE-TVD. The incident successive solitary waves include double solitary waves and triple solitary waves. This paper mainly focuses on the effects of different waveform parameters of the incident successive solitary waves on the relative wave energy distribution inside the harbor. These wave parameters include the incident wave height, the relative separation distance between adjacent crests, and the number of elementary solitary waves in the incident wave train. The relative separation distance between adjacent crests is defined as the ratio of the distance between adjacent crests in the incident wave train to the effective wavelength of the single solitary wave. Maximum oscillations inside the harbor excited by various incident waves are also discussed. For comparison, the transient oscillation excited by the single solitary wave is also considered. The harbor used in this paper is assumed to be long and narrow and has constant depth; the free surface movement inside the harbor is essentially one-dimensional. This study reveals that, for the given harbor and for the variation ranges of all the waveform parameters of the incident successive solitary waves studied in this paper, the larger incident wave heights and the smaller number of elementary solitary waves in the incident tsunami lead to a more uniform relative wave energy distribution inside the harbor. For the successive solitary waves, the larger relative separation distance between adjacent crests can cause more obvious fluctuations of the relative wave energy distribution over different resonant modes. When the wave height of the elementary solitary wave in the successive solitary waves equals to that of the single solitary wave and the relative separation distance between adjacent crests is equal to or greater than 0.6, the maximum oscillation inside the harbor induced by the successive solitary waves is almost identical to that excited by the single solitary wave.

Topological magnon bound states in periodically modulated Heisenberg XXZ chains

Strongly interacting topological states in multi-particle quantum systems pose great challenges to both theory and experiment. Recently, bound states of elementary spin waves (magnons) in quantum magnets have been experimentally observed in quantum Heisenberg chains comprising ultracold Bose atoms in optical lattices. Here, we explore a strongly interacting topological state called topological magnon bound-state in the quantum Heisenberg chain under cotranslational symmetry. We find that the cotranslational symmetry is the key to the definition of a topological invariant for multi-particle quantum states, which enables us to characterize the topological features of multi-magnon excitations. We calculate energy spectra, density distributions, correlations and Chern numbers of the two-magnon bound-states and show the existence of topological protected edge bound-states. Our study not only opens a new prospect to pursue topological magnon bound-states, but also gives insights into the characterization and understanding of strongly interacting topological states.

Wave Interactions and Stability of Riemann Solutions of the Aw–Rascle Model for Generalized Chaplygin Gas

In this paper, we mainly study the interactions of the elementary waves of the Aw–Rascle model for the generalized Chaplygin gas. With the characteristic analysis method, the solutions are obtained constructively when the initial data are three piecewise constant states. Moreover, by studying the limits of the perturbed Riemann solutions as the perturbed parameter $\varepsilon$ tends to zero, we find that the Riemann solutions are stable for such perturbations on the initial data.

Two dimensional Riemann problem for a 2 × 2 system of hyperbolic conservation laws involving three constant states

Zhang and Zheng (1990) conjectured on the structure of a solution for a two-dimensional Riemann problem for Euler equation. To resolve this illuminating conjecture, many researchers have studied the simplified 2 × 2 systems. In this paper, 3-pieces Riemann problem for two-dimensional 2 × 2 hyperbolic system is considered without the restriction that each jump of the initial data projects one planar elementary wave. We classify twelve topologically distinct solutions and construct analytical and numerical solutions. The computed numerical solutions clearly confirm the constructed analytic solutions.

Elementary wave interactions in blood flow through artery

In this paper, we consider the Riemann problem and interaction of elementary waves for the quasilinear hyperbolic system of conservation laws that arises in blood flow through arteries. We study the properties of solution involving shocks and rarefaction waves and establish the existence and uniqueness conditions. We show that the Riemann problem is solvable for arbitrary initial data under certain condition and construct the condition for no-feasible solution. Finally, we present numerical examples with different initial data and discuss all possible interactions of elementary waves.

Global stability of combination of viscous contact wave with rarefaction wave for compressible Navier–Stokes equations with temperature-dependent viscosity

We study the nonlinear stability of a composite wave pattern, which is a combination of a viscous contact wave with a rarefaction wave, to the Cauchy problem of one-dimensional compressible Navier–Stokes equations for a viscous and heat conducting ideal polytropic gas with large initial perturbation when the transport coefficients depend on both temperature and density. Our main idea is to use the “smallness mechanism” induced by the structures of the equations under consideration and the smallness of the strengths of the two elementary waves to control the possible growth of the solutions caused by the nonlinearities of the equations, the interactions between the solutions themselves and the wave pattern, and the interactions of waves between different families. The main ingredient in the analysis is to derive the uniform positive lower and upper bounds on the specific volume and the temperature.

On the Shallow Water Equations

AbstractWe studied the shallow water equations of nonlinear conservation laws. First we studied the parametrisation of nonlinear elementary waves and hence we present the solution to the Riemann problem. We also prove the uniqueness of the Riemann solution. The Riemann invariants are formulated. Moreover we give an interesting application of the Riemann invariants. We present the shallow water system in a diagonal form, which admits the existence of a global smooth solution for these equations. The other application is to introduce new conservation laws for the shallow water equations.

The Riemann problem for a one-dimensional nonlinear wave system with different gamma laws

The Riemann problem for a one-dimensional nonlinear wave system with different gamma laws is considered. By the properties of wave curves, we observe that this system does not contain the composite wave compared to the barotropic models of gas dynamics with different pressure laws. Under some initial value data, the Riemann solution is constructed. Using the interaction of the elementary waves, we consider the generalized Riemann problem and discover that the Riemann solution is stable for such perturbation of the initial data.

Theoretical Analyses and Design of Circular Array to Generate Orbital Angular Momentum

Uniform circular array (UCA) is a pioneering approach to generate radio twisted beams carrying orbital angular momentum (OAM). However, faulty antenna elements in a UCA could render the array into an imperfect UCA (IUCA), consequently creating distortion for OAM-based communication or radar systems. To analyze the effects of an IUCA on generating OAM, elementary wave function and wave transformation are derived to obtain field equations of a general circular array. Further derivation of IUCA shows that the field of an IUCA is a superposition of OAM modes, which is caused by polluted orthogonalities of OAM modes. To achieve pure OAM modes, the orthogonal matching pursuit algorithm, the convex optimization tool, and the analytical solution are employed to design a feeding network to retrieve the orthogonal property at the cost of strong gain loss. Then, the reduced constraints method is proposed for the IUCA to achieve high-purity and low-loss OAM mode. Finally, four circular arrays with different shapes are explored for limited installation space to generate twisted beams.

Non‐selfsimilar global solutions to a two‐dimensional system of conservation laws

This paper considers the two‐dimensional Riemann problem for a system of conservation laws that models the polymer flooding in an oil reservoir. The initial data are two different constant states separated by a smooth curve. By virtue of a nonlinear coordinate transformation, this problem is converted into another simple one. We then analyze rigorously the expressions of elementary waves. Based on these preparations, we obtain respectively four kinds of non‐selfsimilar global solutions and their corresponding criteria. It is shown that the intermediate state between two elementary waves is no longer a constant state and that the expression of the rarefaction wave is obtained by constructing an inverse function. These are distinctive features of the non‐selfsimilar global solutions. Copyright © 2017 John Wiley & Sons, Ltd.

Identification of Mechanical Models for Golf Ball Materials Using a Viscoelastic Split Hopkinson Pressure Bar

A viscoelastic (polymeric) split Hopkinson pressure bar (SHPB) was used as a means of determining the dynamic characteristics of low-impedance or soft materials. The present viscoelastic SHPB consists of polymethyl methacrylate (PMMA) bars to account for the impedance mismatch between test specimens and metallic bars. Wave propagation analysis of strain pulses in a PMMA bar was executed in the frequency domain to identify its mechanical model using elementary one-dimensional wave theory. The SHPB made of PMMA bars was applied to evaluate the high strain rate compressive properties of core and cover materials for a two-piece golf ball within nearly 0.10 strain. Complex compliances of polybutadiene rubber (core) and ionomer resin (cover) specimens, which are the ratios of the strain to stress in the frequency domain, were determined to identify the respective mechanical models. To validate the accuracy of the mechanical models for golf ball materials, finite element investigations on axial collisions between the golf ball and a long elastic bar were conducted for comparison with measured contact force histories. It is demonstrated that three-element solid models can describe the dynamic behavior of both the core and cover materials within a given frequency range. The limitations of the models are also discussed.

Paul Dirac’s view of the Theory of Elementary Waves

Is science open to a new idea? Thomas Kuhn says paradigm shifts sound like gibberish to scientificleaders, and are rejected for that reason. The Theory of Elementary Waves (TEW) is such an idea:quantum particles follow waves moving in the opposite direction. Time always goes forwards. Wefocus on Paul Dirac’s 1930 book The Principles of Quantum Mechanics, applied to TEW. We keepDirac notation and quantum math but replace the picture of how nature is organized. Waveinterference and probabilistic effects occur prior to particle emission. Wave function collapse occursat emission & there is no further interference. We have launched a successful program of teachingthis form of physics in the format of YouTube music videos of five minutes duration. Some of ourvideos have been watched 40,000 times: within YouTube search for “Jeffrey H Boyd” to watch theseamusing videos including one in which Yoda (from Star Wars) solves what Richard Feynman calledthe “Fundamental Mystery of Quantum Mechanics.”

Construction of Global Weak Entropy Solution of Initial-Boundary Value Problem for Scalar Conservation Laws with Weak Discontinuous Flux

This paper is concerned with the initial-boundary value problem of scalar conservation laws with weak discontinuous flux, whose initial data are a function with two pieces of constant and whose boundary data are a constant function. Under the condition that the flux function has a finite number of weak discontinuous points, by using the structure of weak entropy solution of the corresponding initial value problem and the boundary entropy condition developed by Bardos-Leroux-Nedelec, we give a construction method to the global weak entropy solution for this initial-boundary value problem, and by investigating the interaction of elementary waves and the boundary, we clarify the geometric structure and the behavior of boundary for the weak entropy solution.

The Barotropic Instability of the Oceanic Jet Currents

In this article we present the results of the laboratory experiments with shift currents in the homogenous shallow water generated by differentially rotating elements of the vessel bottom: central axisymmetric disk and outer coupled rings. Based on the experimental results, we explain the generation of vortex rings due to the development of barothropic or shift instability of currents. We show that the surface of the differentially rotating liquid in the horizontal rate shift zone creates the conditions for generating chains of vortices rotating clockwise. When changing the values of rate shift and the rotation rate of the whole system, the formation of modes of instability of vortex structures is observed, the conditions for which were quantified on the laboratory current stability diagram in Rossby and Ekman numbers. To interpret the experimental data we considered the solutions of the equations for quasi-two-dimensional geophysical currents in the form of elementary waves of the current function disturbance. We estimated the parameters of the perturbations development at the meanders of the different parts of Gulf Stream using the calculation of the increment of experimental curves for neutral stability. The evaluation results provide a basis for the development of realistic approaches to understanding the processes of generation and evolution of synoptic vortices in the meanders of intense oceanic jet currents.