Analytical Solution In Time Domain Research Articles

Laser-induced ultrasound in multiple thin layers-An analytical solution.

Laser-induced ultrasound is based on the thermo-elastic conversion of absorbed short light pulses to pressure pulses. In the work presented here, laser-induced ultrasound in a planar structure of interconnected layers with variations in optical, thermal, and mechanical properties is studied. Layered structures can be used for generating wideband ultrasonic pulses specific to a chosen application. An analytical time-domain solution is derived for the resulting pressure transmitted from the layered structure. The solution is derived for an arbitrary number of layers with an arbitrary optical absorption profile. Free space Green's functions with image sources are used to derive the solution. A solution employing the Beer-Lambert law is also proposed. The simplification with reflections only at the boundaries is in agreement with previous published results. The spectral properties of the generated pulse are derived, where the effects of optical absorption coefficients and layer thicknesses are shown. The analytical solution is compared to one-dimensional (1D) simulations and a three-dimensional (3D) simulation, realised as a two-dimensional (2D) axially symmetric case, using the matlab toolbox k-Wave. The 3D simulation on-axis pressure agrees well with the 1D analytical solution when the diameter of the laser beam is larger by approximately 1 order of magnitude than the thickness of the planar layered structure.

Time domain analysis of impulse electromagnetic field at the interface of two media

Background. Ultrashort pulses of electromagnetic field are widely used in construction, archeology and demining, etc., by constructing effective georadars of the underlying surface, but theoretical study of physical processes of irradiation of medium is had a limit range of simplified model problems, usually in frequency domain. Therefore, the solutions of the problems of penetration of a pulsed wave with arbitrary time dependence into material medium are of special importance for understanding the possibilities and limitations of georadar’s study. Objectives. To obtain the analytical solution in time domain of the problems of reflection and propagation of a impulse electromagnetic wave through the interface of two media, which is the first model approximation to the description of physical processes that occur during operation of pulsed ultrawideband radar. Materials and methods. The problem of irradiation by nonstationary electric field of a lossless medium with a given permittivity is solved analytically by application of evolutionary approach. It consists in solving of Cauchy’s problem for the second-order partial differential equation Klein–Gordon type with respect to evolutionary coefficients. The components of the electromagnetic field in free space are found by integration by spectral parameters and summation by angular modes with appropriate combinations of basis functions. Results. Cauchy’s problems for differential equations that describe the behavior of reflected and refracted waves are solved. The electrical transverse components of the reflected and refracted waves as a function of time on the longitudinal axis were found for the case of irradiation with the step-like time dependence. Graphs of dependence of electric components on time and coordinates are plotted and analyzed. Conclusion. The phenomenon of an electromagnetic missiles in the medium that was irradiated by a pulsed electromagnetic wave of ultrashort duration was demonstrated for the first time. The obtained results can be generalized for the case of an arbitrary impulse by the Duhamel’s integral method. In addition, the electric field for observation point that do not lie on the longitudinal axis can be considered. An even more interesting continuation of the researches in terms of energy analysis is the study of the behavior of longitudinal electric and transverse magnetic components.

Time domain analysis and localization of a non-local PML for dispersive wave equations

In this work we design and analyze new perfectly matched layers (PML) for a dispersive waves equation: the Klein Gordon equation. We show that because of the dispersion, classical PMLs do not guarantee the convergence to zero of the error, which hampers the precision in long time simulation. We propose to consider a non-local PML for which we can obtain explicit uniform estimates for the reflected analytical solution in time domain, given by an integral representation formula. This uniform estimates ensure the convergence of the error to zero at fixed time t and guarantee the accuracy of the layer. For the implementation of the new PML, we propose a localization technique that we validate numerically.

Convolutive modeling of cyclic voltammetry, AC-voltammetry, sine wave voltammetry and impedance spectroscopy with interfacial CPE behaviour and uncompensated ohmic resistances: A Unified Theory

The constant phase element (CPE) is a well-established circuit component for the frequency domain simulation of electrochemical reactions at disperse and heterogeneous electrodes. Computing the time-dependent current response of voltammetric experiments with interfacial CPE behaviour is, however, not a straightforward task, owing to the lack of analytical time domain solutions of the convoluted capacitive current. In this paper, a universal framework for the convolutive treatment of cyclic voltammetry (CV), alternating current cyclic voltammetry (ACCV), (large) sine wave voltammetry (LSWV) and electrochemical impedance spectroscopy (EIS) in presence of interfacial CPEs and uncompensated Ohmic resistances is presented. A combination of numerical and analytical inverse Laplace transformation techniques allows the accurate assessment of the non-ideal capacitive behaviour, which can be used in turn for classical convolution algorithms. The convolutive approach bears the advantage that different electrode geometries with any kind of spatial boundary conditions as well as first order homogeneous chemical reactions, coupled to the charge transfer step, can be implemented readily. This offers an exceptionally large degree of flexibility in the simulations for the electrochemical system at hand.

Exact results for the Schrödinger equation with moving localized potential

We have provided an alternative method to solve the time-dependent Schrödinger equation for a moving delta function potential with a time-varying strength. We have employed the Galilean transformation to get the equation corresponding to a static delta potential with a time-varying strength. We apply the Feynman's operator technique along with the Green's function method to solve the equation. One of the observation is if a particle initially in the non eigenstate, can posses eigen state after encountered by a attractive potential well in the same initial location. By applying our proposed method, we have derived the analytical time domain solution for a constant and time inversely decaying strength of a uniformly moving well. We observed that the particle is unlikely to be found in the moving well for this decaying strength at large times unlike the previously observed in the case of constant strength.

Modelling laser ultrasound waveforms: The effect of varying pulse duration and material properties.

Optical generation of ultrasound using nanosecond duration laser pulses has generated great interest both in industrial and biomedical applications. The availability of portable laser devices using semiconductor technology and optical fibres, as well as numerous source material types based on nanocomposites, has proliferated the applications of laser ultrasound. The nanocomposites can be deposited on the tip of optical fibres as well as planar hard and soft backing materials using various fabrication techniques, making devices suitable for a variety of applications. The ability to choose the acoustic material properties and the laser pulse duration gives considerable control over the ultrasound output. Here, an analytical time-domain solution is derived for the acoustic pressure waveform generated by a planar optical ultrasound source consisting of an optically absorbing layer on a backing. It is shown that by varying the optical attenuation coefficient, the thickness of the absorbing layer, the acoustic properties of the materials, and the laser pulse duration, a wide variety of pulse shapes and trains can be generated. It is shown that a source with a reflecting backing can generate pulses with higher amplitude than a source with an acoustically-matched backing in the same circumstances when stress-confinement has not been satisfied.

Analytical solution of diffusion probability for a flat potential with a localized sink

We give a method for finding the exact analytical solution in time domain for the problem of a particle undergoing diffusive motion on a flat potential in the presence of a new localized sink. The diffusive motion is described by using the Smoluchowski equation. This analytical method with this new localized sink is very general and can be used to deal with other potentials. This model is well suited for understanding the problem of electronic relaxation of a molecule immersed in polar solvent using reaction–diffusion approach.

АЛГОРИТМ ПОСТРОЕНИЯ РАЦИОНАЛЬНЫХ АППРОКСИМАЦИЙ ФУНКЦИЙ В ИЗОБРАЖЕНИЯХ ЛАПЛАСА С ПРИМЕНЕНИЕМ К ДИНАМИЧЕСКИМ ЗАДАЧАМ ПОРОУПРУГОСТИ

In this paper, the Vector Fitting method is used for fitting functions in Laplace domain by pole relocating. The used version of the method is described for one function and for vector functions. An algorithm based on the Vector Fitting method for obtaining strictly proper rational approximations of vector functions is presented. Algorithm produces approximations with increasing orders until the specified convergence condition is satisfied. Specific recommendations for selecting algorithm parameters are given. The proposed algorithm allows to adaptively select frequency samples that are used for obtaining rational approximations, without any previously known information about the nature of functions in Laplace domain. Time-domain solutions are obtained by analytical inversion of the rational approximations. The developed algorithm is tested in detail on the problem of wave propagation in one-dimensional poroelastic column with finite length. The model of linear isotropic fully saturated poroelasticity proposed by Biot is employed. For the general case, analytical solutions of the problem are presented in Laplace domain for displacements, stresses, pore pressures and flux. For the limiting value of the permeability of a poroelastic material, the corresponding analytical solutions are also given in time domain. Rational approximations of solutions in Laplace domain obtained with the proposed algorithm are presented in details for two values of the permeability. The convergence of the algorithm is investigated depending on the number of complex frequencies. It was found that with an increase in the order of approximation, the relative error changes similarly for all considered functions. It is shown that when parameters of the algorithm are chosen within the recommended ranges, the solutions in time domain are stable and the amplitude of the oscillations caused by truncation of the high frequency parts of the solutions remains moderate over the entire specified time interval. For the limiting value of the permeability, high accuracy of the obtained approximate results in comparison with analytical solutions in time domain is clearly demonstrated.

Semi‐analytical solutions of two‐dimensional plane strain consolidation in unsaturated soils subjected to the lateral semipermeable drainage boundary

SummaryThe study presents semi‐analytical solutions of two‐dimensional plane strain consolidation problem in unsaturated soils incorporating the lateral semipermeable drainage boundary by adopting Fourier sine series and Laplace transform. The two‐dimensional plane strain consolidation equations in the form of two‐order partial differential equations with three variables are firstly converted to two‐order partial differential equations with two variables, which are similar to those of one‐dimensional consolidation problem. The four‐order ordinary differential equations about excess pore‐air and excess pore‐water pressures are got by applying Laplace transform and the substitution method. Then, the solutions of excess pore pressures and settlement are achieved in the Laplace transform domain. Afterwards, on the basis of Crump's method, the inverse Laplace transform is conducted to obtain the analytical solutions in time domain. The comparison is conducted to verify the exactness of the obtained solutions, and the two‐dimensional plane strain consolidation property with the lateral semipermeable drainage boundary is illustrated and discussed. Parametric studies are demonstrated for the excess pore pressures and normalized settlement with the change of the boundary parameters, air‐water and lateral‐vertical permeability coefficients, and the distance and depth. It can be found that the lateral semipermeable drainage boundary impedes the consolidation rate obviously, and when different investigated parameters are adopted, the consolidation property is similar to each other under the later permeable and semipermeable drainage boundary conditions.

Semi-analytical solutions to two-dimensional plane strain consolidation for unsaturated soil

This paper presents semi-analytical solutions to two-dimensional plane strain consolidation of unsaturated soils under different initial and boundary conditions. By applying the finite sine and Laplace transforms, the partial differential equations are converted to the ordinary differential equations. The semi-analytical solutions are obtained by using the finite sine inversion transform. Crump’s method is adopted to perform the inverse Laplace transform to obtain analytical solutions in time domain. The present solutions are more general and can be degenerated into conventional solutions to one-dimensional consolidation of unsaturated soils. Several examples are provided to investigate two-dimensional plane strain consolidation behavior of unsaturated soils.

Vibration of Nonuniform Beams Under Moving Point Loads: An Approximate Analytical Solution in Time Domain

The forced-free responses of nonuniform beams under moving point loads are analyzed in this paper. Simple approximate analytical formulae for the forced responses of undamped nonuniform beams, derived using the fundamental mode by the Rayleigh–Ritz (R–R) method, are presented. The responses of both simply supported and clamped–clamped beams are analyzed. The responses are also determined by the finite element method (FEM) in which nonuniform elements are used for fast convergence. It is found that the present method yields results that are very close to those obtained by the FEM. As this method does not require time integration, it is faster and computationally more efficient. Though the single-mode analysis of forced vibration of uniform beams under moving loads has been done by several researchers, its application to nonuniform beams has not been reported.

Non-uniform Euler-Bernoulli beams under a single moving oscillator: An approximate analytical solution in time domain

Dynamic responses of simply supported non-uniform beams traversed by a moving oscillator are analysed in this paper. An approximate analytical method based on Rayleigh-Ritz (R-R) formulation is developed. The fundamental approximate mode obtained from R-R method is used in the present formulation to determine the responses of the beam and the oscillator. Effects of surface irregularities on the displacement and acceleration responses of the beam and the vehicle are also analysed. The results are compared with those obtained using Finite element method (FEM). A numerical example is provided to illustrate the validity of the present method which shows that the proposed method is simple, computationally more efficient compared to FEM and gives fairly good results. Though the single-mode approach used in the present paper is a classical one and numerous studies on the responses of uniform beams under moving loads have been reported in the past, its application to non-uniform beams (for which there does not exist any closed form expression for mode shapes) under a moving load, especially a moving oscillator, is presented for the first time.

Efficient evaluation of edge diffraction integrals using the numerical method of steepest descent

For the problem of edge diffraction from an edge of finite length a frequency-domain solution, obtained from an analytical time-domain solution, has been presented by Svensson et al. [Acta. Acust. Acust. 95, 568-572]. This formulation takes the form of a Fourier-type integral whose evaluation is expensive in the high frequency range. This paper demonstrates that by using tailored highly oscillatory quadrature methods based on asymptotic properties of the integral, accurate approximations in the high frequency case can be obtained with little computational effort.

Q‐bursts: A comparison of experimental and computed ELF waveforms

Experimental waveforms of natural ELF transient signals (Q‐bursts) are compared with computations based on the analytical time domain solution. The computations were performed for the uniform Earth‐ionosphere cavity model with a linear frequency dependence of the ELF propagation constant. The vertical electric field was recorded in the frequency band from 1 Hz to 11 kHz by a ball antenna in fair weather conditions at Kochi (geographic coordinates: 33.3°N and 133.4°E) during the period of 2003–2004. Waveforms of discrete pulses were compared, each showing a high similarity. This confirms the validity of the model and the consistency of the source‐observer distance found experimentally, directly in the time domain.

Analytical Design Equations for Class-E Power Amplifiers

This paper presents the analytical solution in time domain for the ideal single-ended class-E power amplifier (PA). Based on the analytical solution a coherent noniterative procedure for choosing the circuit parameters is presented for class-E PA's with arbitrary duty-cycle and finite dc-feed inductance (e.g., continuously ranging from class-E with small finite drain inductance to class-E with RF choke). The obtained analysis results link all known class-E PA design equations as well as presenting new design equations. The result of the analysis gives more degrees of freedom to designers in their design and optimization by further expanding the design space of class-E PA.

Recent studies of Schumann resonance and ELF transients

Schumann resonance (SR) is the global electromagnetic phenomenon occurring in the Earth – ionosphere cavity at frequencies of 8, 14, 20 Hz, etc (the ELF band). Phenomenon was named after the author of the pioneer publication on the planetary resonance. Since natural electromagnetic signal originates from lightning strokes distributed worldwide, parameters of oscillations reflect the average condition of ionospheric plasma, spatial distribution and current intensity of thunderstorms. ‘Regular’ lightning strokes occur at a rate of 50 events per second, and their electromagnetic radiation forms the SR background. An irregular or transient component exists caused by relatively rare powerful strokes (the rate is about one event per minute or even less) that provide pulses ten times higher than an ordinary discharge. Three orthogonal field components (vertical electric and two crossed horizontal magnetic fields) are recorded at a Schumann resonance field site, and the power spectra are obtained in the frequency band up to 40 Hz averaged over a few minutes with the typical frequency resolution of 0.1 Hz. The main objective of report is to review the recent results in the Schumann resonance studies. The report consists of two parts; one of them summarizes the experimental data obtained during last 3 – 5 years, while the second describes model studies of ELF radio propagation. We try to mention the most important results in the Schumann resonance. The report surveys the data obtained by using ‘classical’ power spectra acquired at different observatories. Power spectra give information on the properties of the lower ionosphere boundary and on the dynamics of the global thunderstorms. For instance, the sudden changes can be detected in the space weather such as Solar Proton Events (SPE); the regular seasonal motion of thunderstorms is monitored, which is associated with the ground surface temperature; unusual changes might be found in the SR spectra prior to disastrous earthquakes, etc. The cross-spectra of both magnetic fields and magnetic and electric fields (Poynting vector) might be obtained when three field components are recorded. Cross-spectra enhance the coherent fields and reduce the noise thus presenting a more exact information on both the source location and the SR parameters. Elliptical polarization of horizontal magnetic field is measured via the cross-spectra technique, and recent comparison with the model computations proved the detection of Schumann resonance line splitting. Accumulated experimental data present a wide area for resolving the inverse electromagnetic problems directed to establishing an unknown instant distribution of the global lightning activity. Resonance parameters measured worldwide allowed for constructing the effective globally averaged models of the lower ionosphere and the dispersion relation for ELF radio waves. Recent model developments are connected with applications of finite element methods to the resonance problem. Simultaneously, the ‘classical’ spectral field representation was transformed into the time domain providing an effective computational algorithm. Analytical time domain solution allowed generating the quasi–random pulse successions modeling the ELF radio noise. The model time domain ELF signals might be synthesized with predetermined properties. In the conclusion of review, we note a renewal of interest towards the extra-terrestrial Schumann resonance studies. The Huygens probe mission on Titan stimulated interest to distant detection of feasible lightning strokes at the planets and their global location by using the planetary resonance concurrently with studies of ionospheres.

Fundamental solutions for transient heat transfer by conduction and convection in an unbounded, half-space, slab and layered media in the frequency domain

Analytical Green's functions in the frequency domain are presented for the three-dimensional diffusion equation in an unbounded, half-space, slab and layered media. These proposed expressions take into account the conduction and convection phenomena, assuming that the system is subjected to spatially sinusoidal harmonic heat line sources and do not require any type of discretization of the space domain. The application of time and spatial Fourier transforms along the two horizontal directions allows the solution of the three-dimensional time convection-diffusion equation for a heat point source to be obtained as a summation of one-dimensional responses. The problem is recast in the time domain by means of inverse Fourier transforms using complex frequencies in order to avoid aliasing phenomenon. Further, no restriction is placed on the source time dependence, since the static response is obtained by limiting the frequency to zero and the high frequency contribution to the response is small. The proposed functions have been verified against analytical time domain solutions, known for the case of an unbounded medium, and the Boundary Element Method solutions for the case of the half-space, slab and layered media.

ELF sub‐ionospheric pulse in time domain

We present an analytical time domain solution for ELF electromagnetic waves propagating in the Earth‐ionosphere cavity. The workability of the formula obtained is demonstrated by computations of the pulse waveforms originating from a point vertical electric source.

Transient response of three-phase slurry reactors

Analytical time domain solutions are provided for the transient response of three-phase slurry reactors, with diffusion and first-order irreversible ch input of the reactant in the inlet gas stream; and (b) when the catalyst particles are suddenly introduced into the reactor which has otherwise reached obtained for the general case of continuous slurry reactors but can be easily reduced for application to semi-batch slurry reactors in which there is n and are expected to be useful for evaluating the rate and equilibrium constants from dynamic response experiments. Asymptotic solutions have also been obtained for the case of large Thiele parameter (φ ≥ 3) and some other simpler expressions have been derived as only for gaseous reactants of moderate or high solubility in the liquid phase and are not applicable to those with low solubility.

On some properties of stochastic information processes in neurons and neuron populations. Mathematical model approach.

The information in the nervous spike trains and its processing by neural units are discussed. In these problems, our attention is focused on the stochastic properties of neurons and neuron populations. There are three subjects in this paper, which are the spontaneous type neuron, the forced type neuron and the reciprocal inhibitory pairs. 1. The spontaneous type neuron produces spikes without excitatory inputs. The mathematical model has the following assumptions. The neuron potential (NP) has the fluctuation and obeys the Ornstein-Uhlenbeck process, because the N P is not so perfectly random as that of the Wiener process but has an attraction to the rest value. The threshold varies exponentially and the NP has the constant lower limit. When the NP reaches the threshold, the neuron fires and the NP is reset to a certain position. After a firing, an absolute refractory period exists. In discussing the stochastic properties of neurons, the transition probability density function and the first passage time density function are the important quantities, which are governed by the Kolmogorov's equations. Although they can be set up easily, we can rarely obtain the analytical solutions in time domain. Moreover, they cover only simple properties. Hence the numerical analysis is performed and a good deal of fair results are obtained and discussed. 2. The forced type neuron has input pulse trains which are assumed to be based on the Poisson process. Other assumptions and methods are almost the same as above except the diffusion approximation of the stochastic process. In this case, we encounter the inhomogeneous process due to the pulse-frequency-modulation, whose first passage time density reveals the multimodal distribution. The numerical analysis is also tried, and the output spike interval density is further discussed in the case of the periodic modulation. 3. Two types of reciprocal inhibitory pairs are discussed. The first type has two excitatory driving inputs which are mutually independent. The second type has one common excitatory input but it advances in two ways, one of which has a time lag. The neuron dynamics is the same as that of the forced type neuron and each neuron has an identical structure. The inputs are assumed to be based on the Poisson process and the inhibition occurs when the companion neuron fires. In this case, the equations of the probability density functions are not obtained. Hence the computer simulation is tried and it is observed that the stochastic rhythm emerges in spite of the temporally homogeneous inputs. Furthermore, the case of inhomogeneous inputs is discussed.