Time-averaged Energy Density Research Articles

High-frequency vibration analysis of laminated composite plates using energy flow and shear deformation theories

Energy Flow Analysis (EFA) is an efficient approach for characterizing high-frequency vibrations through time-averaged energy density distributions. This study employs EFA to investigate vibroacoustic behavior in laminated composite plates subjected to high-frequency excitation. Governing equations of motion are derived using Classical Plate Theory (CPT), First Order Shear Deformation Theory (FSDT), and Higher Order Shear Deformation Theories (HSDT). Wave propagation parameters like wave number and group velocity are obtained from each theory and compared to assess accuracy. Energy density and intensity formulations are then developed based on classical solutions of the governing equations. EFA analysis indicates that HSDT provides more accurate predictions of wave parameters, especially at very high frequencies where it accounts better for shear deformation effects. Validation against classical energy density solutions shows acceptable accuracy with less than 0.5dB difference in the far-field. Comparisons further demonstrate the superiority of HSDT over FSDT for thicker plates in both classical and EFA analyses. This research establishes the effectiveness of EFA for high-frequency vibration analysis of composite laminates. The main novelty of this research lies in the integration of HSDT into the application of EFA for composite laminates, providing a more precise consideration of shear deformation effects at high frequencies. Specifically, HSDT enhances modeling of critical shear deformation effects at elevated frequencies. EFA delivers computationally efficient solutions while maintaining acceptable accuracy, improving characterization and design of composite structures under high-frequency loading.

Derivation of expression of time-averaged stored energy density of electromagnetic waves

Derivation of expression of time-averaged stored energy density of electromagnetic waves

Born approximation of acoustic radiation force and torque

The Born approximation provides a simple method for calculating acoustic radiation force and torque on compressible objects of arbitrary shape, which may be inhomogeneous. The specific acoustic impedance throughout the object must not differ significantly from that of the background fluid. Also, the incident field must not be too similar to a progressive plane wave, because the approximation is associated with radiation forces due to time-averaged energy density gradients in the incident field. Comparisons with a complete theory for radiation force and torque exerted by standing plane waves on compressible spheroids, based on expansions of the incident and scattered fields in spheroidal wave functions, indicate that the Born approximation is accurate for impedance contrasts up to at least 30% and for spheroids with sizes on the order of a wavelength. This presentation will review the derivation, validation, and applications of the Born approximation. Closed-form solutions for spheres and finite cylinders are presented and for selected distributions of material inhomogeneity. Applications include acoustofluidic separation of biological targets, and acoustic forces on nonspherical objects near interfaces. The approximation also serves as a useful benchmark for other methods of calculation. [T.S.J. was supported by an ARL:UT McKinney Fellowship in Acoustics.]

Energy Flow Analysis of High-Frequency Flexural Vibration of Wedge Beam Structures

Wedge members with variable thickness are widely used in ship structures, aerospace structures, and building structures. Considering their application scenarios, their high-frequency vibration characteristics have important research value. Energy finite element analysis (EFEA) is a powerful tool to predict high-frequency vibration response. EFEA is essentially applying finite element analysis to solve the energy density governing equation. However, the equation for wedge beam is missing. Energy flow analysis aims to obtain energy density governing equation. The energy flow model of wedge beam with variable thickness is established in this paper. Firstly, according to the geometric acoustic approximation method, the general displacement solution of the bending deformation equation for a wedge beam is deduced. Furthermore, the wave dispersion relation of bending deformation of wedge beam is derived. Then, the relationship between the time-averaged vibration energy density and energy flow of the wedge beam is also derived. Also, the governing equation taking energy density as a variable of the wedge beam structure is derived through the energy balance relationship in the microelement body. The governing equation is numerically solved, and the energy density distribution on the wedge beam structure is calculated. Finally, the energy density distribution of uniform beam and wedge beam under the same excitation is analyzed and summarized; the influence of different excitation frequencies and power exponent of thickness change on the performance of wedge beam structure is also summarized. Geometrically, the wedge beam satisfies the one-dimensional acoustic black hole structure. The calculation results show that the energy density on the beam structure increases with the decrease of thickness and reaches its maximum near the tip. The wedge acoustic black hole beam structure has a good energy absorption effect with a frequency between 250 Hz and 2000 Hz. With the further increase of excitation frequency, the energy absorption effect of the wedge beam structure has dropped significantly.

Mode transition in a standing-wave thermoacoustic engine: A numerical study

This study investigates the mode transition phenomenon in a standing-wave thermoacoustic engine (TAE) by means of computational fluid dynamics (CFD). Firstly, the steady-state responses of the TAE at selected temperature ratios are examined via continuous wavelet transform. The bifurcation diagram and spectral map indicate that, as the temperature ratio increases, the TAE experiences a series of bifurcations, through which first-mode periodic oscillations, quasiperiodic oscillations and second-mode periodic oscillations occur. Secondly, the TAE performances in the initial decay/build-up, nonlinear saturation and steady states are studied. The onset of the first and/or second acoustic mode is identified via dynamic mode decomposition. The oscillation frequencies and growth/attenuation rates from CFD agree well with those from the reduced-order network model. Nonlinear mode competition takes place during saturation in which the growth of one acoustic mode is affected or even totally inhibited by the growth of the other. At steady state, periodic oscillations exhibit a closed loop in the phase space whilst quasiperiodic oscillations generate a torus. The time-averaged acoustic energy density, acoustic intensity and efficiency increase with increasing temperature ratio. Finally, parametric studies are conducted to investigate the effects of the gap between stack plates and stack position on mode transition. It is found that the TAE will exhibit second-mode oscillations if the stack is near the closed end or the gap is small. Results in this study indicate that mode transition could become a novel approach to match the TAE with external loads for higher electric power outputs.

Examining axion-like particles with superconducting radio-frequency cavity

We address production of massive axion-like particles by two electromagnetic modes inside a superconducting radio-frequency (SRF) cylindrical cavity. We discuss in detail the choice of pump modes and cavity design. We numerically compute time-averaged energy density of produced axion field for various cavity modes and wide range of axion masses. This allows us to estimate optimal conditions for axion production within a cavity. In addition, we consider photon regeneration process initiated by produced axion field in a screened radio-frequency cavity and derive constraints in parameter space (gaγγ, ma) for different choice of pump modes.

Effective Electrodynamics Theory for the Hyperbolic Metamaterial Consisting of Metal–Dielectric Layers

In this work, we study the dynamical behaviors of the electromagnetic fields and material responses in the hyperbolic metamaterial consisting of periodically arranged metallic and dielectric layers. The thickness of each unit cell is assumed to be much smaller than the wavelength of the electromagnetic waves, so the effective medium concept can be applied. When electromagnetic (EM) fields are present, the responses of the medium in the directions parallel to and perpendicular to the layers are similar to those of Drude and Lorentz media, respectively. We derive the time-dependent energy density of the EM fields and the power loss in the effective medium based on Poynting theorem and the dynamical equations of the polarization field. The time-averaged energy density for harmonic fields was obtained by averaging the energy density in one period, and it reduces to the standard result for the lossless dispersive medium when we turn off the loss. A numerical example is given to reveal the general characteristics of the direction-dependent energy storage capacity of the medium. We also show that the Lagrangian density of the system can be constructed. The Euler–Lagrange equations yield the correct dynamical equations of the electromagnetic fields and the polarization field in the medium. The canonical momentum conjugates to every dynamical field can be derived from the Lagrangian density via differentiation or variation with respect to that field. We apply Legendre transformation to this system and find that the resultant Hamiltonian density is identical to the energy density up to an irrelevant divergence term. This coincidence implies the correctness of the energy density formula we obtained before. We also give a brief discussion about the Hamiltonian dynamics description of the system. The Lagrangian description and Hamiltonian formulation presented in this paper can be further developed for studying the elementary excitations or quasiparticles in other hyperbolic metamaterials.

SeismicQof inhomogeneous plane waves in porous media

Seismic waves in an attenuative porous medium are generally inhomogeneous waves, which have different directions of propagation and attenuation. The dissipation factors (1/ Q) of inhomogeneous waves are strongly dependent on the degree of wave inhomogeneity and cannot be expressed correctly with the usual 1/ Q expressions valid only for homogeneous waves. We have used the differing definitions of 1/ Q for inhomogeneous waves (i.e., the ratio of the time-averaged dissipated energy density to the time-averaged strain energy density or time-averaged total energy density) and the complex form of the energy balance equations of poroviscoelastic media to derive concise and explicit expressions for the dissipation factors. They are given as simple functions of the material parameters and the wave inhomogeneity parameter for inhomogeneous SV-waves and fast and slow P-waves. The isotropic, poroviscoelastic medium under consideration is upscaled from effective Biot theory for a double-porosity solid, which is the most general theory to describe wave propagation in a reservoir. We find that, if the inhomogeneity parameter is infinite (i.e., the inhomogeneity angle is 90°) for all three Biot waves, then the dissipation factors only depend on the ratio of the imaginary to the real part of the complex shear modulus. Our explicit expressions for the dissipation factors of poroviscoelastic materials also are reduced to obtain their counterparts for viscoelastic media as a special case. The inhomogeneous waves in an example poroviscoelastic material are used to demonstrate that the 1/ Q values of the three Biot waves strongly depend on the inhomogeneity parameters and furthermore the different definitions may cause significant differences of 1/ Q values. We find that the dissipation factor of fast P-waves may decrease with the increasing degree of inhomogeneity, which contradicts previously published results.

Stored energy density of electromagnetic wave in dispersive media

In this work, we shall demonstrate that usual expression of time-averaged stored energy density (TASED) for disperisve medium is invalid, which is mainly attributed to the fact that there exist mathematical errors in derivation of this expression. Noting connection between TASED and time-averaged Poynting vector (TAPV) and the case that TAPV of a pulse is the sum of TAPVs of its components, similarly, it is verified that TASED of a pulse in disperisve medium can be taken as the sum of TASEDs of all its components. This work may be helpful to gain a better understand of stored energy of electromagnetic wave in material media.

Three-dimensional direct numerical simulation of helicon discharge

With the detailed consideration of electrochemical reactions and collision relations, a direct numerical simulation model of helicon plasma discharge with three-dimensional fluid-dynamic equations is proposed in the present work. It can improve the precision of results and widen the model applicability by discarding the small perturbation theory in previous helicon models which are partially analytical in essence. Under the assumption of weak ionization, the Maxwell equations coupled with the plasma parameters are directly solved in the whole computational domain. Thus the energy deposited from electromagnetic wave to plasma can be then easily calculated. The values of plasma parameters which include electron density, mean electron energy and heavy species density are obtained by solving a set of drift-diffusion equations. Meanwhile, seven kinds of chemical reactions in the plasma and two kinds of surface reactions on the wall are taken into account. All of the partial differential equations are solved by the finite element solver of COMSOL MultiphysicsTM with the full coupled method.#br#The results of numerical cases employing argon as the working medium show that there exists a sharp density jump from a low to high value as the radiofrequency power is raised. The density jump phenomenon is in accordance with the experimental results of Toki (Toki K, Shinohara S, Tanikawa T, Shamrai K P 2006 Thin Solid Films 506-507 597) and Chen (Chen F F 2007 Plasma Sources Sci. Technol. 16 593). The electron temperature decreases with an increase of the gas pressure, which is similar to Toki's (Toki K, Shinohara S, Tanikawa T, Shamrai K P 2006 Thin Solid Films 506-507 597) measurement by a RF compensation probe. In comparison with the classical sheath theory, the simulation result demonstrates that the distribution of parameters such as particle number density, the Deby length, electric potential and electron temperature can be solved exactly. In addition, the phenomenon of low-field density peak in helicon discharge was studied in the work. Previous research by Chen (Chen F F 2003 Phys. Plasmas 10 2586) suggests that this peak is caused by constructive interference from the reflected wave. The effect of length of the discharge chamber on the relation of electron density and background magnetic field is investigated numerically. The results validate the mechanism of wave interference reflected by endplates of the discharge chamber. Furthermore, the time-averaged magnetic energy density has more than one peak on the axial direction. Comparing the distribution of the magnetic energy density to that of the dimensionless amplitude of the helicon wave and the TG wave in the one-dimensional undamped condition, it found that the length of peak to peak of the helicon wave is just as twice as that of the magnetic energy density, which indicates that the substance of wave interference is involved in the standing wave generated by the helicon wave and its reflected wave from endplates.

Spectral singularities in the TE and TM modes of a [formula omitted]-symmetric slab system: Optimal conditions for realizing a CPA–laser

Among the interesting outcomes of the study of the physical applications of spectral singularities in PT-symmetric optical systems is the discovery of CPA–lasers. These are devices that act both as a threshold laser and a coherent perfect absorber (CPA) for the same values of their physical parameters. Unlike a homogeneous slab that is made to act as a CPA, a slab CPA–laser would absorb the incident waves coming from the left and right of the device provided that they have appropriate intensity and phase contrasts. We provide a comprehensive study of one of the simplest experimentally accessible examples of a CPA–laser, namely a PT-symmetric optical slab system consisting of a balanced pair of adjacent or separated gain and loss components. In particular, we give a closed form expression describing the spectral singularities of the system which correspond to its CPA–laser configurations. We determine the intensity and phase contrasts for the TE and TM waves that are emitted (absorbed) whenever the slab acts as a laser (CPA). We also investigate the behavior of the time-averaged energy density and Poynting vector for these waves. This is necessary for determining the optimal values of the physical parameters of the system that make it act as a CPA–laser. These turn out to correspond to situations where the separation distance s between the gain and loss layers is an odd multiple of a characteristic length scale s0. A curious by-product of our study is that, except for the cases where s is an even integer multiple of s0, there is a critical angle of polarization beyond which the energy of the waves emitted from the lossy layer can be larger than the energy of those emitted from the gain layer.

Energy flow, energy density of Timoshenko beam and wave mode incoherence

Time-averaged energy flow and energy density are of significance in vibration analysis. The wave decomposition method is more fruitful and global in physical sense than the state variables depicted point by point. By wave approach, the Timoshenko beam vibration field is decomposed into two distinct modes: travelling and evanescent waves. Consequently, the power and energy functions defined on these waves’ amplitude and phase need to be established. However, such formulas on Timoshenko beam are hardly found in literatures. Furthermore, the incoherence between these two modes is of theoretical and practical significance. This characteristic guarantees that the resultant power or energy of a superposed wave field is equal to the sum of the power or energy that each wave mode would generate individually. Unlike Euler–Bernoulli beam, such incoherence in the Timoshenko beam case has not been theoretically proved so far. Initially, the power and energy formulas based on wave approach and the corresponding incoherence proof are achieved by present work, both in theoretical and numerical ways. Fortunately, the theoretical and numerical results show that the travelling and evanescent wave modes are incoherent with each other both on power and energy functions. Notably, the energy function is unconventional and self-defined in order to obtain the incoherence. Some remarkable power transmission characteristics of the evanescent wave are also illustrated meanwhile.

Simulation of Low-Intensity Ultrasound Propagating in a Beagle Dog Dentoalveolar Structure to Investigate the Relations between Ultrasonic Parameters and Cementum Regeneration

The therapeutic effect of low-intensity pulsed ultrasound on orthodontically induced inflammatory root resorption is believed to be brought about through mechanical signals induced by the low-intensity pulsed ultrasound. However, the stimulatory mechanism triggering dental cell response has not been clearly identified yet. The aim of this study was to evaluate possible relations between the amounts of new cementum regeneration and ultrasonic parameters such as pressure amplitude and time-averaged energy density. We used the finite-element method to simulate the previously published experiment on ultrasonic wave propagation in the dentoalveolar structure of beagle dogs. Qualitative relations between the thickness of the regenerated cementum in the experiment and the ultrasonic parameters were observed. Our results indicated that the areas of the root surface with greater ultrasonic pressure were associated with larger amounts of cementum regeneration. However, the establishment of reliable quantitative correlations between ultrasound parameters and cementum regeneration requires more experimental data and simulations.

Vector analyses of linearly and circularly polarized Bessel beams using Hertz vector potentials

Using the transverse Hertz vector potentials, vector analyses of linearly and circularly polarized Bessel beams of arbitrary orders are presented in this paper. Expressions for the electric and magnetic fields of vector Bessel beams in free space that are rigorous solutions to the vector Helmholtz equation are derived. Their respective time averaged energy density and Poynting vector are also obtained, in order to exhibit their non-diffracting properties. Polarization patterns and magnitude profiles with different parameters are displayed. Particular emphasis is placed on the cases where the ratio of wave number over its transverse component k/kt approximately equals to one and largely exceeds it, which corresponding to the nonparaxial and paraxial condition, respectively. These results allow us to recognize that the vector Bessel beams exhibit new and important features, compared with the scalar fields.

Efficient Analysis of the Electromagnetic Energy in Dispersive Composite Materials

This paper presents an analysis for electromagnetic energy in dispersive composite media based on classical electrodynamics. An investigation of the relation between reactance rate and electromagnetic energy derived from the Fosters theorem is conducted. The material energy of this kind is discussed in the frequency band of the left handed property. It is illustrated that the time average electromagnetic energy density is still positive even after an effect of dissipation is cancelled in the left handed band, which is reasonable in physics.

Electromagnetic energy momentum in dispersive media

The standard derivations of electromagnetic energy and momentum in media take Maxwell's equations as the starting point. It is well known that for dispersive media this approach does not directly yield exact expressions for the energy and momentum densities. Although Maxwell's equations fully describe electromagnetic fields, the general approach to conserved quantities in field theory is not based on the field equations, but rather on the action. Here an action principle for macroscopic electromagnetism in dispersive, lossless media is used to derive the exact conserved energy-momentum tensor. The time-averaged energy density reduces to Brillouin's simple formula when the fields are monochromatic. The time-averaged momentum density for monochromatic fields corresponds to the familiar Minkowski expression $\mathbf{D}\ifmmode\times\else\texttimes\fi{}\mathbf{B}$, but for general fields in dispersive media the momentum density does not have the Minkowski value. The results are unaffected by the debate over momentum balance in light-matter interactions.

Radiation from an accelerated quark via AdS/CFT correspondence

In this paper we investigate radiation by an accelerated quark in a strongly coupled gauge theory via anti―de Sitter/conformal field theory (AdS/CFT) correspondence. According to the AdS/CFT dictionary, we can read off the energy density or energy flux of the radiation from the asymptotic gravitational field in AdS bulk sourced by an accelerated string trailing behind the quark. In the case of an oscillating quark with frequency Ω, we show that the time averaged energy density is asymptotically isotropic, and it falls off as (g 2 YM N) 1/2 Ω 4 /R 2 with distance R from the source. In a toy model of a scattered quark by an external field, we simply estimate the Poynting vector by the bremsstrahlung radiation and show that the energy flux is anisotropic outgoing radiation. Based on these investigations, we discuss the properties of strongly coupled gauge theory radiation in comparison with electromagnetic radiation.

Disk illumination by black hole superradiance of electromagnetic perturbations

Using the Kerr-Schild formalism to solve the Einstein-Maxwell equations, we study energy transport due to time-dependent electromagnetic perturbations around a Kerr black hole, which may work as a mechanism to illuminate a disk located on the equatorial plane. For such a disk-hole system it is found that the energy extraction from the hole can occur under the well-known superradiance condition for wave frequency, even though the energy absorption into the hole should be rather dominant near the polar region of the horizon. We estimate the efficiency of the superradiant amplification of the disk illumination. Further we calculate the time-averaged energy density distribution to show explicitly the existence of a negative energy region near the horizon and to discuss the possible generation of a hot spot on the disk.

Lateral resolution improvement in two-photon excitation microscopy by aperture engineering

A technique for resolution improvement in two-photon excitation (2PE) fluorescence microscopy based on radially-symmetric annular binary filter (consist of central circular aperture and a concentric peripheral annulus) is proposed. Resolution improvement is achieved by engineering the aperture of the objective lens in a way so as to enhance high spatial frequencies. The structure of the electromagnetic field in the regions of focus and nearby regions are determined. The central lobe of the time-averaged electric energy density is considerably reduced for both linearly- and circularly-polarized illuminated light. An impressive combined comparative percentage improvement of 40% and 53.71% both at low ( α = 30) and high ( α = 60) aperture angle is obtained for linearly-polarized light. Proposed aperture engineering technique complements conventional, confocal, two-photon fluorescence microscopy, and may facilitate working at low-to-medium magnifications and large free-working distances.

Hamiltonian Structure for Dispersive and Dissipative Dynamical Systems

We develop a Hamiltonian theory of a time dispersive and dissipative inhomogeneous medium, as described by a linear response equation respecting causality and power dissipation. The proposed Hamiltonian couples the given system to auxiliary fields, in the universal form of a so-called canonical heat bath. After integrating out the heat bath the original dissipative evolution is exactly reproduced. Furthermore, we show that the dynamics associated to a minimal Hamiltonian are essentially unique, up to a natural class of isomorphisms. Using this formalism, we obtain closed form expressions for the energy density, energy flux, momentum density, and stress tensor involving the auxiliary fields, from which we derive an approximate, ``Brillouin-type,'' formula for the time averaged energy density and stress tensor associated to an almost mono-chromatic wave.