Inhomogeneous Wave Research Articles

On the viscoelastic-electromagnetic-gravitational analogy

The analogy between electromagnetism and gravitation was achieved by linearizing the tensorial gravitational equations of general relativity and converting them into a vector form corresponding to Maxwell’s electromagnetic equations. On this basis, we use the equivalence with viscoelasticity and propose a theory of gravitational waves. We add a damping term to the differential equations, which is equivalent to Ohm’s law in electromagnetism and Maxwell’s viscosity in viscoelasticity, to describe the attenuation of the waves. The differential equations in viscoelasticity are those of cross-plane shear waves, commonly referred to as SH waves. A plane-wave analysis gives the phase velocity, the energy velocity, the quality factor and the attenuation factor of the field as well as the energy balance. To obtain these properties, we use the analogy with viscoelasticity; the properties of electromagnetic and gravitational waves are similar to those of shear waves. The presence of attenuation means that the transient field is generally a composition of inhomogeneous (non-uniform) plane waves, where the propagation and attenuation vectors do not point in the same direction and the phase velocity vector and the energy flux (energy velocity) are not collinear. The polarization of cross-plane field is linear and perpendicular to the propagation-attenuation plane, while the polarization of the field within the plane is elliptical. Transient wave fields in the space–time domain are analyzed with the Green function (in homogeneous media) and with a grid method (in heterogeneous media) based on the Fourier pseudospectral method for calculating the spatial derivatives and a Runge–Kutta scheme of order 4 for the time stepping. In the examples, wave propagation at the Sun–Earth and Earth–Moon distances using quadrupole sources is considered in comparison to viscoelastic waves. The Green and grid solutions are compared to test the latter algorithm. Finally, an example of propagation in heterogeneous media is presented.

Reflection, transmission, and AVO response of inhomogeneous plane waves in thermoporoelastic media with two-temperature equations of heat conduction

We develop a modified fluid-saturated thermoporoelastic model by introducing two temperature equations to account for the temperature differences between the solid skeleton and the pore filling. The modified two-temperature generalized thermoporoelastic (TTG) equation is an extension of the classical single-temperature Lord-Shulman (LS), Green-Lindsay, and generalized LS theories. It predicts four compressional waves and one shear wave based on the analysis of inhomogeneous plane waves. We study the exact reflection and transmission (R/T) coefficients at the interface separating two thermoporoelastic half-spaces and develop an amplitude-variation-with-offset (AVO) approximation. Comparison with the Biot poroelastic case of the water/oil contact shows that the TTG model reproduces the exact R/T results. The AVO response of oil, gas, and real CO2 geosequestration reservoirs illustrates the practical applicability of our model and provides the theoretical basis for the exploration of high-temperature resources.

The propagation of inhomogeneous waves in an orthotropic viscoelastic medium

Abstract The present article deals with the propagation of inhomogeneous waves in an orthotropic viscoelastic medium. For chosen directions of propagation and a real finite inhomogeneity parameter, a complex slowness vector is specified to define the propagation of an inhomogeneous incident wave. Then, the reflection and transmission of plane waves at a plane interface between two orthotropic viscoelastic half-spaces are discussed. In this incidence, horizontal slowness determines the slowness vectors for all reflected and transmitted waves. For each reflected and transmitted wave, the corresponding slowness vector is resolved to define its phase direction, phase velocity and attenuation angle. Appropriate boundary conditions on this wave field determine the amplitude ratios for reflected and transmitted waves relative to the incident wave. The numerical examples are provided to show the effect of the inhomogeneity of the incident wave on the propagation characteristic as well as the reflection and transmission coefficients. The existence of homogeneous, inhomogeneous incident waves also is investigated.

Experimental Study of Alfvén Wave Reflection from an Alfvén-speed Gradient Relevant to the Solar Coronal Holes

We report the first experimental detection of a reflected Alfvén wave from an Alfvén-speed gradient under conditions similar to those in coronal holes. The experiments were conducted in the Large Plasma Device at the University of California, Los Angeles. We present the experimentally measured dependence of the coefficient of reflection versus the wave inhomogeneity parameter, i.e., the ratio of the wavelength of the incident wave to the length scale of the gradient. Two-fluid simulations using the Gkeyll code qualitatively agree with and support the experimental findings. Our experimental results support models of wave heating that rely on wave reflection at low heights from a smooth Alfvén-speed gradient to drive turbulence.

Математическое моделирование волнового поля методом конечных разностей

The system of differential equations of hydrodynamics is reduced to the inhomogeneous wave equation, for which a discrete model is constructed. The computational domain is a rectangle covered with a two-dimensional uniform calculation grid. Discrete analogues of the wave equation for Dirichlet and Neumann boundary conditions were developed. The paper describes the choice of calculation grids. Analytical expressions describing the approximation error in spatial coordinate directions for the second derivative operator based on the second and fourth order of accuracy schemes were obtained and, in this range of error, the sizes of grids were estimated. The computational experiments demonstrated that, to keep the calculation error in the specified limit 0.1- 1%, it is necessary to perform calculations on grids with the number of nodes per half wavelength in the range from 9 to 30 when using the second-order of accuracy scheme, and from 4 to 6 when using the fourth-order of accuracy scheme. It is important to choose the values of the time step and the weight parameter. At the weight parameter σ =1/12, the relative error is significantly less than at σ =0 (corresponding to an explicit scheme) and at σ =1/4 (corresponding to a symmetrical setting of coefficients in a weight scheme). The optimal values of the weight parameter were calculated in the context of maintaining the propagation frequency of oscillatory processes. The analysis of the discrete model gave a condition for the stability of the difference scheme, and an expression describing the approximation error in a time variable, depending on time steps and spatial coordinate directions, was determined. It was established that the approximation of initial conditions makes a smaller contribution to the total error than the approximation of the equation for subsequent time layers. Based on the proposed algorithms and approaches, a software package was designed to simulate the propagation of vibrations in a two-dimensional computational domain. A number of computational experiments were carried out to study, for example, the propagation of acoustic waves from antennas with different directional characteristics and the scattering of waves on the obstacles of different types. To determine the farfields of an acoustic antenna, it is proposed to make the calculation window movable and to find its location in space. This significantly reduces the calculation time for the propagation of sound waves over long distances.

Revisiting optical absorption power density of inhomogeneous plane waves

We review the analytical expressions for the complex Poynting’s vector in the case of arbitrary plane-waves in a lossy isotropic medium. We demonstrate how these expressions can be used to recover the optical absorption power density Q, considering the divergence of the time-averaged Poynting vector. This quantity, proportional to the imaginary part of the dielectric function and the field intensity, i.e. Q∝|E|2ɛ”, is usually established for harmonic fields, using the Poynting’s identity. The derivation from the complex Poynting vector expression is more direct for TE-polarized homogeneous waves, but the derivation encompasses the other cases like inhomogeneous TM plane waves. As an application, the optical absorption profile Q(x) within 1D multilayers is detailed using matrix transfer method for both TE and TM plane waves, including the evanescent case.

Inhomogeneous Wave Propagation in Triple-Porosity Medium

Inhomogeneous Wave Propagation in Triple-Porosity Medium

Double wavefront reversal during six-wave interaction on Kerr nonlinearity in a waveguide with infinitely conducting surfaces

Background. Generation of a wave with a double reversed wavefront in multimode waveguides increases the efficiency of six-wave radiation converters and expands the possibilities of its use in adaptive optics problems and the conversion of complex spatially inhomogeneous waves. Aim. The quality of double wavefront reversal during six-wave interaction in a waveguide with infinitely conducting surfaces with Kerr nonlinearity is analyzed for the ratio of the wave numbers of the pump waves equal to 2 and 0,5, and the condition that one of the pump waves excites the zero mode of the waveguide, and the amplitude distribution of the other pump wave excites the edges of the waveguide are described by a Gaussian function. Methods. The influence of pump wave parameters on the half-width and contrast of the amplitude modulus of the object wave was studied using numerical methods. A wave from a point source located on the front face of the waveguide was used as a signal wave. Results. The dependences of the half-width and contrast of the amplitude modulus of the object wave on the ratio between the width of the waveguide and the width of the Gaussian pump wave are obtained. Conclusion. It is shown that the maximum change in the characteristics of a wave with a double reversed wavefront is observed when the width of the Gaussian pump waves changes in the range from 0,3 to 2 half-widths of the waveguide.

Near-field scattering by the method of locally subsonic waves

A technique is developed for determining the sound field scattered by a compact body when it is close enough to an acoustic source to be in its near field. Our approach is based on the fact that large regions of many near fields may be well approximated at each point in space by a subsonic plane wave (also called an inhomogeneous plane wave, or an evanescent wave). Such a wave is defined by the property that in one direction it propagates with subsonic phase speed, while in a perpendicular direction it has exponential amplitude variation. Hence by defining a canonical problem, compact scattering of a subsonic plane wave, and solving it, we are able to give a unified analytical treatment of many near-field scattering problems. Our approach draws on the formulae of Rayleigh scattering (as applied to an incident field with complex wavenumber) and the asymptotic theory of the wave equation. For an arbitrary three-dimensional multipole, we determine in full detail how its subsonic wave structure depends on the spherical harmonic parameters ( m , n ) , and show that our approach has a very large region of validity.

Deformation of inhomogeneous vector optical rogue waves in the variable coefficients coupled cubic–quintic nonlinear Schrödinger equations with self-steepening

Deformation of inhomogeneous vector optical rogue waves in the variable coefficients coupled cubic–quintic nonlinear Schrödinger equations with self-steepening

Акустический пограничный слой твёрдой абсолютно теплопроводной поверхности

The paper presents the analysis results of formation theoretical descriptions of an acoustic boundary layer near solid absolutely thermally conductive surface, obtained by G. Kirchhoff and L.D. Landau. In both cases, the acoustic boundary layer is formed by inhomogeneous viscous and thermal waves in the wall layer of a liquid medium in contact with the surface of a solid body, from which a plane traveling sound wave is reflected. Based on the analysis, conclusions can be drawn: the analyzed problem solutions are physically sound, independent and complementary to each other. During the formation of an acoustic boundary layer, viscous and thermal waves are excited synchronously in pairs. Inside the acoustic boundary layer, each pair of inhomogeneous waves propagates towards each other. Inhomogeneous waves originate on parallel surfaces that limit the volume of the acoustic boundary layer. The analysis of the process of transformation of heat waves into additional one-dimensional inhomogeneous waves, the appearance of which in the boundary layer was predicted by G. Kirchhoff. It is shown that when interacting with the surface of the body of a traveling sound wave in the sound frequency range, these waves do not affect the formation of the boundary layer. The expressions allowing for a numerical estimation of the heat dissipation power density in the boundary layer are refined. A formula has been obtained that allows us to determine the proportion of the energy of the sound wave that is absorbed in the acoustic boundary layer. In practice, the results obtained in the article can be used, for example, in aeroacoustics to assess the dissipative properties of solid surfaces.

Potentials and fields of a charge set suddenly from rest into uniform motion

The fact that electromagnetic effects propagate at the speed of light suggests how the Lorenz-gauge scalar and vector potentials of a uniformly moving point charge must be modified when the charge was initially at rest and then set suddenly into uniform motion. The modified potentials are shown to satisfy the requisite inhomogeneous wave equations. The gauge function of the transformation of these potentials to the Coulomb gauge is calculated in closed form. It is validated by confirming that the Coulomb-gauge vector potential that is calculated using it yields together with the Coulomb-gauge scalar potential the same electric and magnetic fields as those calculated with the Lorenz-gauge potentials.

Preliminary Implications of Viscoelastic Ray Theory for Anelastic Seismic Tomography Models

ABSTRACT The recent developments in general viscoelastic ray theory provide a rigorous mathematical framework for anelastic seismic tomography. They provide closed-form solutions of forward ray-tracing and simple inverse problems for anelastic horizontal and spherical layered media with material gradients. They provide ray-tracing computation algorithms valid for all angles of incidence that account for changes in wave speed, attenuation, and trajectory of anelastic P and S body waves induced by anelastic boundaries. They account for theoretical predictions that seismic waves refract as inhomogeneous waves across anelastic boundaries for all angles of incidence, which in turn accounts for energy carried by plane waves along seismic boundaries at head wave critical angles and wide-angle refracted (WAR) ray paths that are not predicted by elastic models. Exact viscoelastic ray-tracing numerical results for various models provide examples that illustrate the effects of anelastic boundaries on the travel times and amplitudes of seismic waves. They show the effects are strongly dependent on angle of incidence. For near-critical and wide angles of incidence the anelastic effects on travel times and amplitudes can be large and are not explained by elastic ray theory, but the effects on travel times can be relatively small and difficult to distinguish from those for elastic media for pre-near-critical angles of incidence. The results for some models indicate that reflected anelastic WAR waves may be observable at the surface and possibly account for some prominent seismic arrivals not explained by elasticity. These preliminary results suggest that the application of exact viscoelastic ray-tracing computation algorithms to exploration and teleseismic data sets can reveal new insights regarding the properties and distribution of anelastic materials in the Earth.

Experimental reproduction of inhomogeneous fjord waves

Waves in coastal areas and fjords can present inhomogeneities that affect the responses of very large floating structures. However, spatial inhomogeneities of waves are usually not included in model testing. In this paper, the feasibility of generating inhomogeneous wave conditions in an Ocean basin is investigated. For that purpose, a fast linear numerical model of a basin of constant depth is applied, and an optimization method is presented, that allows to find the control signal for the multiflap wavemaker. Comparisons with experiments in SINTEF’s Ocean basin demonstrate the validity of the method for generating simple monochromatic synthetic waves, as well as short-crested irregular waves representing realistic conditions in Sulafjorden.

Insights on electromagnetic field distribution due to a vertical electric dipole source inside an infinite steel casing

Energized vertical steel casings have been developed to excite and monitor changes in the electrical properties of resistivity anomalies associated with hydrocarbon production and [Formula: see text] injection, among other applications. Accordingly, the spatial distribution of the fields expected in the medium surrounding a casing source has received considerable attention. However, investigations of the in-borehole distribution of the fields, currents, and charges have been lacking. Such an analysis is the objective of this work. Inside the borehole, the electric field is dominantly vertical, owing to the azimuthal symmetry of the charge density induced at the inner and outer casing boundaries. At the inner surface, charges accumulate dominantly in the vicinity of the source. At the outer surface, charges distribute along the extent of the casing, driven by the vertical channeling and radial leakage of the current flow. Inductive effects arise as the source oscillates in time, effectively resulting in a vanishing flux of azimuthal magnetic field and switching of the sign of the charge density along the vertical extent of the pipe. The alternating charge distribution yields fields propagating inward and outward within the metallic medium, describing inhomogeneous Zenneck surface waves that can interfere as they propagate and superimpose up and down the borehole. In the surrounding medium, the azimuthal magnetic field is driven by the current flowing vertically along the borehole, whereas a dominantly radial electric field is driven by the charge distributed on the outer casing surface.

A hybrid finite volume - spectral element method for aeroacoustic problems

We propose a hybrid Finite Volume (FV) - Spectral Element Method (SEM) for modeling aeroacoustic phenomena based on the Lighthill's acoustic analogy. First the fluid solution is computed employing a FV method. Then, the sound source term is projected onto the acoustic grid and the inhomogeneous Lighthill's wave equation is solved employing the SEM. The novel projection method computes offline the intersections between the acoustic and the fluid grids in order to preserve the accuracy. The proposed intersection algorithm is shown to be robust, scalable and able to efficiently compute the geometric intersection of arbitrary polyhedral elements. We then analyze the properties of the projection error, showing that if the fluid grid is fine enough we are able to exploit the accuracy of the acoustic solver and we numerically assess the obtained theoretical estimates. Finally, we address two relevant aeroacoustic benchmarks, namely the corotating vortex pair and the noise induced by a laminar flow around a squared cylinder, to demonstrate in practice the effectiveness of the projection method when dealing with high order solvers. The flow computations are performed with OpenFOAM [50], an open-source finite volume library, while the inhomogeneous Lighthill's wave equation is solved with SPEED [34], an open-source spectral element library.

On the eigenvalues and eigendisplacement of the critical mode in horizontally layered media

Wave propagation in horizontally layered media is a classical problem in seismic-wave theory. In semi-infinite space, a nondispersive Rayleigh wave mode exists, and the eigendisplacement decays exponentially with depth. In a layered model with increasing layer velocity, the phase velocity of the Rayleigh wave varies between the S-wave velocity of the bottom half-space and that of the classical Rayleigh wave propagated in a supposed half-space formed by the parameters of the top layer. If the phase velocity is the same as the P- or S-wave velocity of the layer, which is called the critical mode or critical phase velocity of surface waves, the general solution of the wave equation is not a homogeneous (expressed by trigonometric functions) or inhomogeneous (expressed by exponential functions) plane wave, but one whose amplitude changes linearly with depth (expressed by a linear function). Theories based on a general solution containing only trigonometric or exponential functions do not apply to the critical mode, owing to the singularity at the critical phase velocity. In this study, based on the classical framework of generalized reflection and transmission coefficients, the propagation of surface waves in horizontally layered media was studied by introducing a solution for the linear function at the critical phase velocity. Therefore, the eigenvalues and eigenfunctions of the critical mode can be calculated by solving a singular problem. The eigendisplacement characteristics associated with the critical phase velocity were investigated for different layered models. In contrast to the normal mode, the eigendisplacement associated with the critical phase velocity exhibits different characteristics. If the phase velocity is equal to the S-wave velocity in the bottom half-space, the eigendisplacement remains constant with increasing depth.

Dynamic response of a sea-crossing cable-stayed suspension bridge under simultaneous wind and wave loadings induced by a landfall typhoon

Sea-crossing bridges with long spans are sensitive to wind and wave environments. The estimation of the dynamic response under simultaneous wind and wave loadings is hence important for the safety of bridges, especially during the landfall of typhoons when extreme wind and wave actions usually occur. In this paper, a framework combining the simulation of typhoon wind and wave fields, calculation of aerodynamic and hydrodynamic loadings, and structural analysis was adopted to assess the dynamic response of a sea-crossing bridge. A cable-stayed suspension bridge located in the typhoon-affected area was taken as an example. The inhomogeneity of wave loading, the time-varying dynamic response of bridges during the landfall process, and the effect of critical wave load calculation on extreme response were presented and discussed. Results demonstrated that wave loadings at different pies show strong inhomogeneity during the landfall process. The typhoon-induced simultaneous wind and wave loadings could affect not only the amplitude of the power spectrum density but also the number of dominant modes of the response. In addition, the error brought by excluding the wave inhomogeneity and the time lag of wind and wave in estimating the extreme response was acceptable when the maximum wind loadings were adopted.

Inhomogeneous wave propagation in porothermoelastic medium with dual-phase-lag heat conduction

Inhomogeneous wave propagation in porothermoelastic medium with dual-phase-lag heat conduction

A priori estimates for derivative solutions of one-dimensional inhomogeneous wave equations with an integral load in the main part

A priori estimates for derivative solutions of one-dimensional inhomogeneous wave equations with an integral load in the main part