Governing Wave Propagation Research Articles

Optimal resonator damping for wave propagation control in mechanical metamaterials

A novel approach to the optimal damping of linearly damped resonators embedded in metamaterial systems is proposed with the aim of minimizing the metamaterials’ response when the external excitation frequency lies within one of the bandgaps. The equation governing wave propagation in the metamaterial system is obtained via a Galerkin projection combined with the quasi-periodicity ansatz of the Floquet–Bloch theorem. It is shown that an optimality criterion can be obtained for the resonator damping for any excitation frequency by extending the Den Hartog theory of fixed points in the frequency response functions. A numerical example of a honeycomb metamaterial is discussed to show how the proposed method works in a practical application. A full numerical optimization is carried out to study the quality factor of the metamaterials’ response with respect to the resonator damping ratio while proving its effectiveness.

DH-GAN: a physics-driven untrained generative adversarial network for holographic imaging.

Digital holography is a 3D imaging technique by emitting a laser beam with a plane wavefront to an object and measuring the intensity of the diffracted waveform, called holograms. The object's 3D shape can be obtained by numerical analysis of the captured holograms and recovering the incurred phase. Recently, deep learning (DL) methods have been used for more accurate holographic processing. However, most supervised methods require large datasets to train the model, which is rarely available in most DH applications due to the scarcity of samples or privacy concerns. A few one-shot DL-based recovery methods exist with no reliance on large datasets of paired images. Still, most of these methods often neglect the underlying physics law that governs wave propagation. These methods offer a black-box operation, which is not explainable, generalizable, and transferrable to other samples and applications. In this work, we propose a new DL architecture based on generative adversarial networks that uses a discriminative network for realizing a semantic measure for reconstruction quality while using a generative network as a function approximator to model the inverse of hologram formation. We impose smoothness on the background part of the recovered image using a progressive masking module powered by simulated annealing to enhance the reconstruction quality. The proposed method exhibits high transferability to similar samples, which facilitates its fast deployment in time-sensitive applications without the need for retraining the network from scratch. The results show a considerable improvement to competitor methods in reconstruction quality (about 5 dB PSNR gain) and robustness to noise (about 50% reduction in PSNR vs noise increase rate).

Measurement of the electric permittivity using Bleustein–Gulyaev wave sensor

We present a novel compact electric permittivity sensor that exploits Bleustein–Gulyaev waves propagating along the surface of shear-poled piezoelectrics. We formulate the dynamic nonlinear electromechanical partial differential equations of motion governing wave propagation under electromagnetically quasistatic conditions. The permittivity of the medium-under-test was found to influence the sensor eigenvalues, enabling the implementation of a frequency-shift permittivity sensor. Solution of the equations of motion demonstrates resonance of the first and third modes when excited using an interdigitated transducer. We fabricated two sensor prototypes on shear-poled PZT4 and LiNbO3 substrates and used a Vector Network Analyzer to observe the shift in their fundamental natural frequency in the presence of various media-under-test. S 11 measurements show deterministic and repeatable shifts in the resonant frequency of the first mode of the LiNbO3 sensor measured at MHz for ethanol and MHz for deionized water where the bare surface frequency was initially at MHz.

Vibration analysis of laminated planar frame structures

A wave-based model is developed to analyze the free vibrations of symmetrically and asymmetrically laminated cross-ply planar frame structures. The model includes the effects of shear deformation, rotary inertia and the elastic coupling due to structural anisotropy. The frame is modeled as an assemblage of beam segments, whereby each segment acts as a one-dimensional waveguide. The system matrices governing wave propagation, reflection and transmission along the connected segments are derived. An advantage of the proposed model over numerical methods, such as the finite element method, is that exact solutions for the equations of motion are used to derive the system matrices of the frame. The matrices are subsequently assimilated to give a complete description of the wave train present during free vibration. The approach, which is well suited for implementation in conventional computing programs, can be used to determine the exact natural frequencies and mode shapes of the frame. By using the proposed model, the free vibration characteristics of inclined laminated portal frames are analyzed. The effects of stacking sequence, frame angle, relative frame length, orthotropicity ratios and boundary conditions on the exact natural frequencies, and in certain cases the mode shapes, are evaluated and discussed. The numerical results presented in this work are in close agreement with finite element (FE) simulations and where possible, published results.

Investigation on the effect of porosity on wave propagation in FGM plates resting on elastic foundations via a quasi-3D HSDT

In this paper, wave propagation in functionally graded (FG) porous plates resting on Winkler-Pasternak foundation is studied using a quasi-3D shear deformation theory. The proposed theory has a new displacement field that includes indeterminate integral terms and contains fewer unknown variables taking into account the effect of transverse shear and thickness stretching. The parameters of the elastic foundation are introduced in the present formulation following the mathematical model of Pasternak. In addition, the effect of porosity is studied. The material of the FG plate is inhomogeneous and the material properties are assumed to vary continuously in the thickness direction according to a power law of the volume fraction. The equations governing wave propagation in the plates resting on an elastic foundation are derived using Hamilton's principle. Then, the dispersion relationship between frequency and wave number is solved analytically. A comprehensive numerical result is accomplished to evaluate the effects of the volume fraction index, the porosity, thickness ratio (h/a), and the wave number on the wave propagation in functionally graded porous plates are discussed in detail.

Virtual elements for sound propagation in complex poroelastic media

We develop a novel Virtual Element Method (VEM) to resolve the mixed Biot displacement-pressure formulation governing wave propagation in porous media. Within this setting, the weak form of the governing equations is discretized using implicitly defined canonical basis functions and the resulting integral forms are computed using appropriate polynomial projections. The projection operator accounting for the solid, fluid, and coupling phases of the problem are presented. Different boundary, interface and excitation conditions are accounted for. The convergence behaviour, accuracy, and efficiency of the method is examined through a set of illustrative examples. A node insertion strategy is proposed to resolve non-conforming interfaces that naturally arise in multilayered systems. Finally the power of the VEM is exploited to examine the acoustic response of composite materials with periodic and non-periodic inclusions of complex geometries.

Qualitative analysis and wave propagation of the nonlinear model for low-pass electrical transmission lines

This paper studies a nonlinear partial differential equation governing wave propagation in nonlinear low-pass electrical transmission. A travelling wave transformation is used to convert this equation into a 2D dynamical system which is proved to be a Hamiltonian system. Based on qualitative theory of the dynamical systems, a comprehensive qualitative study of bifurcation and phase portrait for this system is carried. We construct some new traveling wave solutions and they are graphically clarified.

Fractional low-pass electrical transmission line model: Dynamic behaviors of exact solutions with the impact of fractionality and free parameters

In this study, we focus on extracting soliton solutions to the differential equation of fractional order governing wave propagation in low-pass electrical transmission lines by the (G'/G2)-expansion method. By a simple linear fractional transformation, we convert the model equation to an ordinary differential equation. Thereupon, through the use of the (G'/G2)-expansion method, various types of solitary wave solutions to the equation of interest are achieved. All the received solutions are verified using Maple software. To show the dynamical behavior of some of the acquired solutions representing singular-periodic, singular, anti-kink, periodic, and bright soliton solutions, their two and three-dimensional profiles are displayed by selecting suitable values of the solutions’ free parameters. Impacts of the fractionality and free parameters on the dynamical behavior of the attained soliton solutions are presented graphically and discussed elaborately. A comparison of our explored solutions with those obtained in the literature is also made. The comparison, wherever available, shows no noticeable difference between our attained results and the published ones. The results that came out through this investigation expose the productivity and powerfulness of the adoptive method for excerpting a diversity of wave solutions to nonlinear evolution equations arising in the fields of mathematics, engineering, and physics.

3D Seismic-Wave Modeling with a Topographic Fluid–Solid Interface at the Sea Bottom by the Curvilinear-Grid Finite-Difference Method

ABSTRACTThe curvilinear-grid finite-difference method (FDM), which uses curvilinear coordinates to discretize the nonplanar interface geometry, is extended to simulate acoustic and seismic-wave propagation across the fluid–solid interface at the sea bottom. The coupled acoustic velocity-pressure and elastic velocity-stress formulation that governs wave propagation in seawater and solid earth is expressed in curvilinear coordinates. The formulation is solved on a collocated grid by alternative applications of forward and backward MacCormack finite difference within a fourth-order Runge–Kutta temporal integral scheme. The shape of a fluid–solid interface is discretized by a curvilinear grid to enable a good fit with the topographic interface. This good fit can obtain a higher numerical accuracy than the staircase approximation in the conventional FDM. The challenge is to correctly implement the fluid–solid interface condition, which involves the continuity of tractions and the normal component of the particle velocity, and the discontinuity (slipping) of the tangent component of the particle velocity. The fluid–solid interface condition is derived for curvilinear coordinates and explicitly implemented by a domain-decomposition technique, which splits a grid point on the fluid–solid interface into one grid point for the fluid wavefield and another one for the solid wavefield. Although the conventional FDM that uses effective media parameters near the fluid–solid interface to implicitly approach the boundary condition conflicts with the fluid–solid interface condition. We verify the curvilinear-grid FDM by conducting numerical simulations on several different models and compare the proposed numerical solutions with independent solutions that are calculated by the Luco-Apsel-Chen generalized reflection/transmission method and spectral-element method. Besides, the effects of a nonplanar fluid–solid interface and fluid layer on wavefield propagation are also investigated in a realistic seafloor bottom model. The proposed algorithm is a promising tool for wavefield propagation in heterogeneous media with a nonplanar fluid–solid interface.

Solitary pulses of a conformable nonlinear differential equation governing wave propagation in low-pass electrical transmission line

In this work, we investigate soliton solutions for the conformable nonlinear differential equation governing wave-propagation in low-pass electrical transmission lines. Adopting two integration techniques, we construct dark and bright solitary waves, jacobian elliptic function solutions and trigonometric solutions. The obtained results are relevant and will probably help to carry data and codify them in telecommunication.

The double auxiliary equations method and its application to space-time fractional nonlinear equations

This paper reflects the execution of a reliable technique which we proposed as a new method called the double auxiliary equations method for constructing new traveling wave solutions of nonlinear fractional differential equation. The proposed scheme has been successfully applied on two very important evolution equations, the space-time fractional differential equation governing wave propagation in low-pass electrical transmission lines equation and the time fractional Burger’s equation. The obtained results show that the proposed method is more powerful, promising and convenient for solving nonlinear fractional differential equations (NFPDEs). To our knowledge, the solutions obtained by the proposed method have not been reported in former literature.

Physically significant nonlocal nonlinear Schrödinger equation and its soliton solutions

A new integrable nonlocal nonlinear Schroedinger (NLS) equation with clear physical motivations is proposed. This equation is obtained from a special reduction of the Manakov system, and it describes Manakov solutions whose two components are related by a parity symmetry. Since the Manakov system governs wave propagation in a wide variety of physical systems, this new nonlocal equation has clear physical meanings. Solitons and multi-solitons in this nonlocal equation are also investigated in the framework of Riemann-Hilbert formulations. Surprisingly, symmetry relations of discrete scattering data for this equation are found to be very complicated, where constraints between eigenvectors in the scattering data depend on the number and locations of the underlying discrete eigenvalues in a very complex manner. As a consequence, general $N$-solitons are difficult to obtain in the Riemann-Hilbert framework. However, one- and two-solitons are derived, and their dynamics investigated. It is found that two-solitons are generally not a nonlinear superposition of one-solitons, and they exhibit interesting dynamics such as meandering and sudden position shifts. As a generalization, other integrable and physically meaningful nonlocal equations are also proposed, which include NLS equations of reverse-time and reverse-space-time types as well as nonlocal Manakov equations of reverse-space, reverse-time and reverse-space-time types.

Subordination in a Class of Generalized Time-Fractional Diffusion-Wave Equations

Abstract Motivated by recently proposed generalizations of the diffusion-wave equation with the Caputo time fractional derivative of order α ∈ (1, 2), in the present survey paper a class of generalized time-fractional diffusion-wave equations is introduced. Its definition is based on the subordination principle for Volterra integral equations and involves the notion of complete Bernstein function. Various members of this class are surveyed, including the distributed-order time-fractional diffusion-wave equation and equations governing wave propagation in viscoelastic media with completely monotone relaxation moduli.

Polynomial approach for modeling a piezoelectric disc resonator partially covered with electrodes

The frequency spectrum of a partially metallized piezoelectric disc resonator was studied using Legendre polynomials. The formulation, based on three-dimensional equations of linear elasticity, takes into account the high contrast between the electroded and non-electroded regions. The mechanical displacement components and the electrical potential were expanded in a double series of orthonormal functions and were introduced into the equations governing wave propagation in piezoelectric media. The boundary and continuity conditions were automatically incorporated into the equations of motion by assuming position-dependent physical material constants or delta-functions. The incorporation of electrical sources is illustrated. Structure symmetry was used to reduce the number of unknowns. The vibration characteristics of the piezoelectric discs were analyzed using a three-dimensional modelling approach with modal and harmonic analyses. The numerical results are presented as resonance and anti-resonance frequencies, electric input admittance, electromechanical coupling coefficient and field profiles of fully and partially metallized PIC151 and PZT5A resonator discs. In order to validate our model, the results obtained were compared with those published previously and those obtained using an analytical approach.

Information Transmission Using the Nonlinear Fourier Transform, Part I: Mathematical Tools

The nonlinear Fourier transform (NFT), a powerful tool in soliton theory and exactly solvable models, is a method for solving integrable partial differential equations governing wave propagation in certain nonlinear media. The NFT decorrelates signal degrees-of-freedom in such models, in much the same way that the Fourier transform does for linear systems. In this three-part series of papers, this observation is exploited for data transmission over integrable channels such as optical fibers, where pulse propagation is governed by the nonlinear Schr\"odinger equation. In this transmission scheme, which can be viewed as a nonlinear analogue of orthogonal frequency-division multiplexing commonly used in linear channels, information is encoded in the nonlinear frequencies and their spectral amplitudes. Unlike most other fiber-optic transmission schemes, this technique deals with both dispersion and nonlinearity directly and unconditionally without the need for dispersion or nonlinearity compensation methods. This first paper explains the mathematical tools that underlie the method.

A coarse space for heterogeneous Helmholtz problems based on the Dirichlet-to-Neumann operator

The Helmholtz equation governing wave propagation and scattering phenomena is difficult to solve numerically. Its discretization with piecewise linear finite elements results in typically large linear systems of equations. The inherently parallel domain decomposition methods constitute hence a promising class of preconditioners. An essential element of these methods is a good coarse space. Here, the Helmholtz equation presents a particular challenge, as even slight deviations from the optimal choice can be devastating.In this paper, we present a coarse space that is based on local eigenproblems involving the Dirichlet-to-Neumann operator. Our construction is completely automatic, ensuring good convergence rates without the need for parameter tuning. Moreover, it naturally respects local variations in the wave number and is hence suited also for heterogeneous Helmholtz problems. The resulting method is parallel by design and its efficiency is demonstrated on 2D homogeneous and heterogeneous numerical examples.

The origin of traveling waves in an emperor penguin huddle

Emperor penguins breed during the Antarctic winter and have to endure temperatures as low as −50 °C and wind speeds of up to 200 km h−1. To conserve energy, they form densely packed huddles with a triangular lattice structure. Video recordings from previous studies revealed coordinated movements in regular wave-like patterns within these huddles. It is thought that these waves are triggered by individual penguins that locally disturb the huddle structure, and that the traveling wave serves to remove the lattice defects and restore order. The mechanisms that govern wave propagation are currently unknown, however. Moreover, it is unknown if the waves are always triggered by the same penguin in a huddle. Here, we present a model in which the observed wave patterns emerge from simple rules involving only the interactions between directly neighboring individuals, similar to the interaction rules found in other jammed systems, e.g. between cars in a traffic jam. Our model predicts that a traveling wave can be triggered by a forward step of any individual penguin located within a densely packed huddle. This prediction is confirmed by optical flow velocimetry of the video recordings of emperor penguins in their natural habitat.

Mechanical Properties of Viscoelastic Media by Local Frequency Estimation of Divergence-Free Wave Fields

Magnetic resonance elastography (MRE) is an imaging modality with which mechanical properties can be noninvasively measured in living tissue. Magnetic resonance elastography relies on the fact that the elastic shear modulus determines the phase velocity and, hence the wavelength, of shear waves which are visualized by motion-sensitive MR imaging. Local frequency estimation (LFE) has been used to extract the local wavenumber from displacement wave fields recorded by MRE. LFE -based inversion is attractive because it allows material parameters to be estimated without explicitly invoking the equations governing wave propagation, thus obviating the need to numerically compute the Laplacian. Nevertheless, studies using LFE have not explicitly addressed three important issues: (1) tissue viscoelasticity; (2) the effects of longitudinal waves and rigid body motion on estimates of shear modulus; and (3) mechanical anisotropy. In the current study we extend the LFE technique to (1) estimate the (complex) viscoelastic shear modulus in lossy media; (2) eliminate the effects of longitudinal waves and rigid body motion; and (3) determine two distinct shear moduli in anisotropic media. The extended LFE approach is demonstrated by analyzing experimental data from a previously-characterized, isotropic, viscoelastic, gelatin phantom and simulated data from a computer model of anisotropic (transversely isotropic) soft material.

Construction, in vitro and in vivo evaluation of an in-house conductance meter for measurement of skin hydration

Different probes are used in dermato-cosmetic research to measure the electrical properties of the skin. The principle governing the choice of the geometry and material of the measuring probe is not well defined in the literature and some device's measuring principles are not accessible for the scientific community. The purpose of this work was to develop a simple inexpensive conductance meter for the objective in vivo evaluation of skin hydration.The conductance meter probe was designed using the basic equation governing wave propagation along Transverse Electromagnetic transmission lines. It consisted of two concentric copper circular electrodes printed on FR4 dielectric material.The performance of the probe was validated by evaluating its measurement depth, its ability to monitor in vitro water sorption–desorption and in vivo skin hydration effect in comparison to that of the Corneometer CM 825. The measurement depth of the probe, 15μm, was comparable to that of CM 825. The in vitro readings of the probe correlated strongly with the amount of water adsorbed on filter paper. Skin hydration after application of a moisturizer was monitored effectively by the new probe with good correlation to the results of CM 825.In conclusion, a simple probe for evaluating skin hydration was made from off-the-shelf materials and its performance was validated in comparison to a commercially available probe.

Analyzing of the effect of the dusty fluid, anisotropy and layered of the wall on the wave propagation

This study considers the propagation of time harmonic waves in, prestressed, anisotropic elastic tubes filled with viscous fluid containing dusty particles. The fluid is assumed to be incompressible and Newtonian. The tube material is considered to be incompressible, anisotropic, and elastic. The tube is subjected to a static inner pressure Pi and an axial stretch λ. Utilizing the theory of “Superposing small deformations on large initial static deformations”, differential equations governing wave propagation inside the tube are obtained in terms of cylindrical coordinates. Analytical solutions for the equations of motion for the dust and the fluid are obtained, and expressed numerically. The dispersion relation is obtained as a function of the stretch, the thickness ratio and the parameters for dusty particles.