Energy Flux Vector Research Articles

Electromagnetic energy flux vector for a dispersive linear medium

The electromagnetic energy flux vector in a dispersive linear medium is derived from energy conservation and microscopic quantum electrodynamics and is found to be of the Umov form as the product of an electromagnetic energy density and a velocity vector.

Harmonic waves in thermoviscoelastic solids

The constitutive properties and time-harmonic wave propagation in thermoviscoelastic solids are investigated through the energy flux. First the dissipative character of the model is characterized by requiring that the divergence of the energy flux vector be non-positive. The attention is then restricted to planarly stratified solids and the governing equations are shown to satisfy a first-order system of the Stroh-form. The corresponding scalar energy flux is examined and shown to satisfy a decay property which, thanks to a previous result, provides existence and uniqueness for the reflection–transmission process generated by a layer. The required definiteness property of an appropriate matrix is shown to hold in particular cases.

Finite-Amplitude Damped Inhomogeneous Waves in a Deformed Blatz-Ko Material

In this paper special Blatz-Ko nonlinear elastic materials are considered, which are characterized by a constitutive constant and a constitutive function. We deal with the propagation of finite-amplitude inhomogeneous plane waves in such materials subjected to an arbitrary static homogeneous deformation. Linearly polarized transverse “damped” inhomogeneous plane wave solutions are explicitly obtained. Such waves are attenuated (or amplified) both in space and time (time-harmonic inhomogeneous plane waves obtained previously by Destrade appear as a special case). The properties of the energy flux vector and energy density associated with these wave solutions are investigated. With an appropriate concept of mean, it is seen that the “mean” energy-flux vector and the “mean” energy density satisfy two relations which are independent of the constitutive constant and constitutive function of the model, and of the homogeneous static deformation of the material. These relations are the same as those obtained by Hayes in the general context of time-harmonic inhomogeneous plane waves in linear systems. However, here, the theory is nonlinear, and the finite-amplitude waves are not time-harmonic.

Diffraction of a plane electromagnetic wave by a slot in a conducting screen of arbitrary thickness

A 2D theoretical model of the diffraction of a plane electromagnetic wave by a slot in a perfectly conducting screen is constructed based on the partial domain method. The Tikhonov regularization is used to solve a system of algebraic equations for the slot-mode amplitudes. This makes it possible to extend the domain of applicability of the theory to conducting screens of arbitrary thickness and to significantly increase the accuracy of solution in the cases when the slot width and the screen thickness are comparable to the wavelength of the diffracted radiation. The absence of a continuous passage to the limit of an infinitesimal screen thickness from the case of an arbitrarily small finite thickness is demonstrated. The boundary conditions for the energy-flux vector are considered. A concept of the energy potential that is convenient for the computer calculations of the energy-flux lines of 2D diffraction fields is introduced.

Moving oscillating loads in 2D anisotropic elastic medium: Plane waves and fundamental solutions

The paper considers a concentrated point force moving with constant velocity and oscillating with constant frequency in an unbounded homogeneous anisotropic elastic 2D medium. Such problems come from the problems of a source that acts along the line in the corresponding three-dimensional anisotropic medium. Fundamental solutions for two-dimensional problems with moving oscillating sources lay the foundation for constructing solutions for more complicated problems and applying the boundary integral equation method. The properties of plane waves, their phase, slowness and ray or group velocity curves are determined in a moving coordinate system. The use of Fourier integral transform techniques and the properties of plane waves enables to obtain an explicit representation for elastodynamic Green’s tensor for all types of the source motion as a sum of integrals over a finite interval. The quasistatic and dynamic components of the Green’s tensor are obtained. The stationary phase method is employed to derive an asymptotic approximation of the far wave field. Simple formulae for the Poynting energy flux vectors for moving and stationary observers are also presented. It is noted that in far zones the wave fields are subdivided into separate cylindrical waves under kinematics and energy. It is shown that motion brings some difference in the far field properties, exemplified by the modification of the wave propagation zones and the change in their number, emergence of fast and slow waves under trans- and superseismic motion and etc.

Fundamental solutions in antiplane elastodynamic problem for anisotropic medium under moving oscillating source

In present article we consider the problems of concentrated point force which is moving with constant velocity and oscillating with cyclic frequency in unbounded homogeneous anisotropic elastic two-dimensional medium. The properties of plane waves and their phase, slowness and ray or group velocity curves for 2D problem in moving coordinate system are described. By using the Fourier integral transform techniques and established the properties of the plane waves, the explicit representation of the elastodynamic Green's tensor is obtained for all types of source motion as a sum of the integrals over the finite interval. The dynamic components of the Green's tensor are extracted. The stationary phase method is applied to derive an asymptotic approximation of the far wave field. The simple formulae for Poynting energy flux vectors for moving and fixed observers are presented too. It is noted that in the far zones the cylindrical waves are separated under kinematics and energy. It is shown that the motion bring some differences in the far field properties. They are modification of the wave propagation zones and their number, fast and slow waves appearance under trans- and superseismic motion and so on.

Experimental test of the compatibility of the definitions of the electromagnetic energy density and the Poynting vector

It is shown that the generally accepted definition of the Poynting vector and the energy flux vector defined by means of the energy density of the electromagnetic field (Umov vector) lead to the prediction of the different results touching electromagnetic energy flux. The experiment shows that within the framework of the mentioned generally accepted definitions the Poynting vector adequately describes the electromagnetic energy flux unlike the Umov vector. Therefore one can conclude that a generally accepted definitions of the electromagnetic energy density and the Poynting vector, in general, are not always compatible.

Vector-phase characteristics of acoustic fields: Statistical analysis and measurement algorithms

Statistical characteristics of an acoustic field that are based on the data of simultaneous measurements of pressure and acoustic velocity vector are investigated. Conditions of the formation of vector-phase characteristics of acoustic field are formulated in relation to the dispersion properties of the medium. Crosscorrelation functions of the components of the vector field are presented. Expressions for the characteristic functionals of vector-phase relationships in acoustic fields and, in particular, for the acoustic energy flux are derived with the use of functional methods. Algorithms of space-time processing of the energy flux vector and the optimum measurement algorithm for a Gaussian vector-phase field are considered. The signal-to-noise ratio is determined as the quality index of vector reception algorithms, and its relation to the corresponding parameter of scalar pressure field measurements is revealed. Indices of relative efficiency of vector algorithms are determined depending on the dispersion characteristics of the medium (the flux algorithm) and the dimension of the input vector of observations (the optimum algorithm).

Tidal currents, energy flux and bottom boundary layer thickness in the Clyde Sea and North Channel of the Irish Sea

A high-resolution three-dimensional model of the Clyde Sea and the adjacent North Channel of the Irish Sea is used to compute the major diurnal and semidiurnal tides in the region, the associated energy fluxes and thickness of the bottom boundary layer. Initially, the accuracy of the model is assessed by performing a detailed comparison of computed tidal elevations and currents in the region, against an extensive database that exists for the M2, S2, N2, K1 and O1 tides. Subsequently, the model is used to compute the tidal energy flux vectors in the region. These show that the major energy flux is confined to the North Channel region, with little energy flux into the Clyde Sea. Comparison with the observed energy flux in the North Channel shows that its across-channel distribution and its magnitude are particularly sensitive to the phase difference between elevation and current. Consequently, small changes in the computed values of these parameters due to slight changes of the order of the uncertainty in the open-boundary values to the model, can significantly influence the computed energy flux. The thickness of the bottom boundary layer in the region is computed using a number of formulations. Depending upon the definition adopted, the empirical coefficient C used to determine its thickness varies over the range 0.1 to 0.3, in good agreement with values found in the literature. In the North Channel, the boundary layer thickness occupies the whole water depth, and hence tidal turbulence produced at the sea bed keeps the region well mixed. In the Clyde Sea, the boundary layer thickness is a small fraction of the depth, and hence the region stratifies.

Separation of Variables Applied to the Wave Equation in Laterally Inhomogeneous Media

—The new eigenfunction expansion formula based upon the method of separation of variables is derived. The method gives the exact transport equation and the generalized eikonal equation without the need of asymptotic series expansions. The generalized eikonal equation extends the classical eikonal equation to a rapidly varying medium. It governs the wave propagation with dispersive phase velocity depending upon the amplitude. Both equations are separated by making use of conventional phase-ray coordinates. The solution is expressed in terms of the elementary Sturm-Liouville eigenfunctions or modes for the vibrating string. The generalized eikonal equation is solved by employing the modified Hamiltonian formalism. This results in frequency-dependent phase-ray trajectories associated with the energy flux vector field.

Characteristics of Radiated Sound Field from a Duct with Cavities at Open End

The sound characteristics of the radiated waves from the open end of the duct with cavities have been studied numerically and experimentally for various depths of the cavities in connection with the sound reduction for the wide ranged frequencies. The sound pressure level (SPL), time history of the sound pressure and acoustic energy flux vector, i.e., instantaneous intensity, etc., are computed by a finite difference method. The present study shows that the lager depth of the nearest cavity from the duct yields the best configuration for the reduction of radiated power in the wide ranged frequencies. The present result also shows that the instantaneous intensity makes clear the interaction of the duct and each cavity. It is confirmed that the sound wave at the open end of the resonating cavity and the one of the duct are nearly equal in magnitude but reverse in phase. The computed results are in good agreement with experimental data.

In the Chapman-Enskog Expansion¶the Burnett Coefficients Satisfy¶the Universal Relation ω 3 +ω 4 + θ 3 = 0

The Chapman–Enskog expansion when applied to a gas of spherical molecules yields formal expressions for the stress deviator P and energy-flux vector q, P=μP(1)+μ2P(2)+…, q=μq(1)+μ2q(2)+…. The Burnett terms P(2), q(2) depend on 11 coefficients ωi, 1≦i≦6, θ&i, 1≦i≦ 5. This paper shows that ω3+ω4+θ3= 0.

Dynamics of rarefied granular gases.

This paper presents quite general bidimensional gas-dynamic equations--derived from kinetic theory-which include the fourth cumulant kappa(r,t) as a dynamic field. The dynamics describes a low-density system of inelastic hard spheres (disks) with normal restitution coefficient r. Two illustrative examples are given and the role of kappa in them is discussed. Our general gas-dynamic equations would deal with 9 hydrodynamic fields (which corresponds to 14 in three-dimension). These fields are the standard hydrodynamic fields plus the components p(ij) of the traceless part of the pressure tensor, the energy flux vector Q and the fourth cumulant kappa. The present formulation requires no constitutive equations. The two examples are: the well-known homogeneous cooling state and a system, with and without gravity, steadily heated by two parallel walls. In the first case, the dynamics yield a description of the homogeneous cooling state consistent with known results adding extra details mainly about the transient time behavior. The steadily heated system kept in a static state gives rise to quite simple but nontrivial equations. In the case with gravity, it is shown that when kappa is included as a dynamic field, the formalism leads to a non-Fourier law already to first order in dissipation. Setting gravity g=0 a perturbative solution is shown and favorably compared with observations obtained from molecular dynamics (MD). In both cases, with and without gravity, kappa is not homogeneous. An analytic extension suggests a divergent situation for a small negative value of q, which originates in the unavoidable extension of the formalism to exothermic collisions associated with a restitution coefficient larger than one. This divergent behavior is observed in MD.

Theoretical investigation of the influence of a diaphragm on the flat surface of a solid immersion lens on beam parameters

The optical property of a solid immersion lens with a flat sur- face, coated with an opaque thin film with a rectangular hole is investi- gated on the basis of a three-dimensional vector mathematical model. The laser beam is focused on the center of the hole. It is shown that the side lobe in the energy flux vector distribution in the focal plane of a solid immersion lens with a large numerical aperture (.1.5) significantly de- creased the optical efficiency of the lens. The results of digital simulation show that an opaque thin film with a hole, equal or larger than the width of the main lobe, significantly increases the transmission coefficient be- cause it closes the inverse electromagnetic energy flux in the side lobe. The hole has an unimportant influence on the energy flux vector distri- bution in the main light spot when its dimensions are equal or larger than the dimension of the central lobe of the laser beam. The distribution of electrical and magnetic intensity is also investigated. © 2001 Society of

Flux expressions and NEMD perturbations for models of semi-flexible molecules

We derive energy and momentum flux expressions, for systems composed of a general class of semi-flexible molecules, in the Ciccotti-Ferrario-Ryckaert linear constraint formalism. According to this formalism, the whole set of Cartesian coordinates is divided into basic (independent) and secondary (dependent) subsets. It is found that energy and momentum flux vectors have a simple and general expression using both basic and secondary coordinates. In the case of non-equilibrium molecular dynamics, we give general and simple heat and shear flow algorithms, deriving the dissipative fluxes in the space of all Cartesian coordinates. In comparison with previous derivations for some models of flexible molecules, the present approach has more general applicability. Moreover, it leads to more efficient equilibrium and non-equilibrium molecular dynamics calculations of transport coefficients, and requires minimal programming effort.

Constitutive equations for viscoelastic surface diffusion: Effects of deformation and thermal history

In this paper we derive constitutive equations for surface diffusion in a non-isothermal fluid–fluid interface, consisting of a mixture of linear viscoelastic components with fading memory. The constitutive equation contains three main contributions to the total diffusive flux: thermal diffusion, diffusion resulting from the thermal history of the components, and mechanical diffusion. The mechanical diffusion term incorporates the effects of ordinary diffusion, forced diffusion, and diffusion arising from surface tension gradients. It also incorporates diffusion caused by inertial effects, diffusion resulting from hydrostatic normal stresses, diffusion arising from hydrodynamic viscous stresses, and contributions arising from the deformation history of the individual components. The history terms in the constitutive equation are all expressed in terms of memory integrals. In addition to an equation for the surface mass flux vector, the analysis also provides constitutive equations for the surface thermal energy flux vector, and the surface stress tensor. All these constitutive equations are derived using the jump entropy inequality.

Energy flux for damped inhomogeneous plane waves in viscoelastic fluids

The propagation of inhomogeneous plane waves in the context of the linearized theory of incompressible viscoelastic fluids is considered. The angular frequency and the slowness vector are both assumed to be complex. As in incompressible purely viscous fluids, two kinds of waves may propagate: A “zero pressure wave” for which the increment in pressure due to the wave is zero, and a “universal wave” which is independent of the viscoelastic relaxation modulus. The balance of energy is written using a decomposition of the stress power into a reversible component and a dissipative component proposed namely by P.W. Buchen [J. R. Astr. Soc. 23 (1971) 531–542]. For the inhomogeneous waves, a “weighted mean” energy flux vector, “weighted mean” energy density and “weighted mean” energy dissipation are introduced. It is shown that they satisfy two modulus independent relations. These generalize to the case of viscoelasticity relations previously obtained in other contexts.

Up-to-date state and outlook for the intensity measurement method in underwater acoustics

The acoustic intensity measurement method consisting of four components of the acoustic field [acoustic pressure p(t) and three orthogonal components of the particle velocity vector V(t){Vx(t),Vy(t),Vz(t)} simultaneous measurements] did not occupy a suitable place in underwater acoustics until now. Simultaneous measurement of four components of the acoustic field enables one to construct a point receiving system possessing direct characteristics in a wide frequency range (for example, from 10 to 100 Hz). By now unconventional results of investigations of underwater acoustic noises, reverberation, point source capacity, seismic noises, and underwater navigation were obtained. The intensity measurement method has given one a chance to research the acoustic field characteristics that have never been revealed during measurements of only acoustic pressure, i.e., characteristics connected with energy flux density vector. For example, a compensation of opposite energy fluxes of ambient noise and tone or signal similar to noise; horizontal energy flux of dynamic noise connected with propagational direction and the extent of wind near-surface roughness; splitting of ambient noise total field energy density into an isotropic component connected with the isotropic (diffuse) part of the field and an anisotropic one connected with the anisotropic (coherent) part of the field related to the energy transfer in the ocean. The isotropic component (possessing zero averaged flux) existence in the field of ambient noise allows for decreasing the threshold noise level for receiving systems measuring acoustic intensity, thus enables one to research a thin structure of the acoustic field of the noise and signal. Measurements of only acoustic pressure cannot give one such information. The acoustic intensity measurement method development will allow for constructing the most up-to-date acoustic devices for oceanologic processes dynamic investigation on the bases of ambient noise and sounding signal vector characteristic measurements.

The Maxwell and adjoint systems for complex media: Physical significance and applications

Useful information on the solutions to the Maxwell system of first order linear differential equations may be obtained with the aid of two distinct, but not independent, sets of adjoint equations, the formally adjoint system which reduces to the complex conjugate equations in loss free media, and the Lorentz adjoint system which reduces to the time reversed equations in loss free media. The formally adjoint system is used to define a generalized energy flux vector, to decompose wavefields into constituent modes and to define their amplitudes. The Lorentz adjoint system is used to construct time reversed, “complementary” media, and time reversed wavefields and sources that are Lorentz reciprocal to the given wavefields and sources. In some problems, such as the determination of reciprocal scattering matrices or reciprocal modal Green’s functions, it is necessary to employ both adjoint systems. 1. Maxwell, formally adjoint and complex conjugate systems The Maxwell system of equations in the frequency domain may be written compactly as L · e ≡ [ jωK(r, ω) + D ] · e(r, ω) = −j(r, ω), (1) L is the Maxwell operator; eT = [E,H] and jT = [Je,Jm] are the 6-element electromagnetic field and current vectors; K(r, ω) is the constitutive tensor and D the differential operator: [ D B ] = [ e ξ η μ ] · [ E H ] ≡ K · e, D = [ 0 −∇× I ∇× I 0 ] = D. (2) The formal structure (1) of the Maxwell system applies also to compressible magnetoplasmas, which support acoustic field variables v and p, the respective macroscopic plasma velocity and pressure. The wavefield then has (at least) 10 scalar components, eT ≡ [E,H,v, p], and K and D are 10 × 10 matrices [1–3]. Chiral and other spatially dispersive media have a slightly different structure, 1383-5416/98/$8.00  1998 – IOS Press. All rights reserved 136 C. Altman and K. Suchy / The Maxwell and adjoint systems for complex media containing differential operators (such as curl terms) in the constitutive tensor. Thus for chiral media the proposed constitutive relations [4,5] are D = e[E + β∇×E ], B = μ[H + β∇×H ], but in the Fourier (ω, k) domain (for plane wave solutions for instance) the system has the same formal structure as the others, with the chiral system described by a biisotropic constitutive tensor K(ω, k) [6, Section 1.2c]. The system of equations formally adjoint to the Maxwell system is obtained transposing all matrix operators and reversing the signs of all differential operators [7,8], L · e ≡ [ jωKT −DT ] · e = −j (3) yielding a set of formally adjoint fields eT = [E,H ]. This system is nonphysical: KT could in principle represent a physical constitutive tensor but the symmetric differential operator DT = D has the opposite sign to that in the Maxwell system. The given and formally adjoint systems yield a mathematical relationship called the Lagrange identity [7], e · L · e− e · L · e = ∇ ·w. (4) w is called the bilinear concomitant [8]. Substitution of (1), (2) and (3) in (4) yields w = E ×H +E ×H. (5) The complex conjugate of (1) in a source free region, j = 0, yields −L∗ · e∗ = [ jωK∗ −D ] · e∗ = 0. (6) If the medium is dissipationless K is hermitian, K∗ = KT, and the complex conjugate system (6) is identical to the formally adjoint system (3). Then e∗ = e, and w (5) reduces to the Poynting vector. Note that since we are working in the frequency domain all wave fields, including the complex conjugate wave fields, have the same exp(jωt) Fourier transform time dependence. 2. Modal biorthogonality and modal amplitudes in a waveguide For propagation in a longitudinally homogeneous waveguide the gradient operator ∇ is split into longitudinal (parallel to the waveguide axis, z) and transverse parts, ∇ = ẑ ∂ ∂z +∇t, ∇t = x ∂ ∂x + ŷ ∂ ∂y . (7) If there are no currents in the guide (j = j = 0) then, with (1) and (3) in (4), the generalized energy flux density w (5), has zero divergence, ∇ · w = 0 even in dissipative media. Integrating over the transverse (x, y) plane we obtain, with Gauss, − ∂ ∂z ∫

Energy flux associated with a plane crack along an (hyper)elastic bimaterial interface

A numerical procedure, incorporated with the finite element solutions, is developed to evaluate the energy flux vector for a crack located along the interface of 2-D hyperelastic bimaterial solids. The formulation is considered with finite strains for use with both linear and nonlinear material behavior. The formulation is verified to be path-independent in a modified sense and so the near-tip region, where singular mechanical behavior dominates, is always included. Special attention is hence addressed on appropriate modeling of the singular behavior. The numerical results show good accuracy without using any particular singular finite elements.