Class Of Wave Equations Research Articles

Measurement and analysis of photoacoustic pressure induced by weak microsecond pulsed light

The photoacoustic effect (PAE) of weak microsecond pulsed light (WMPL) has immense potential for application to biomedical engineering. However, practical measurements and theoretical analysis of the photoacoustic pressure of WMPL are lacking. Hence, we constructed a WMPL photoacoustic pressure measurement system using an electret piezoelectric sensor and multi-wavelength parameter-adjustable pulsed light generator. Based on the system, the photoacoustic pressure interacting with the air medium induced by the WMPL with different optical parameters was measured. The measured pressure data were analyzed using the fixed variable method, and the pressure response characteristic was obtained. It was found that the measured results conform to the law of energy conversion but have a specific trend for some wavelengths. The analysis and discussion were conducted based on the classical wave equation of the PAE, and the extended wave equation was presented. It is shown that the proposed approach provides a reliable basis for the measurement and analysis of WMPL photoacoustic pressure.

Receptivity of Rayleigh-Taylor instability to acoustic pulses: Theoretical explanation of pulse propagation

Direct numerical simulation (DNS) of Rayleigh-Taylor instability (RTI) has been reported in “Three-dimensional (3D) DNS of RTI triggered by acoustic excitation - Sengupta, et al. Phys. Fluids 34(5), 054108 (2022)” using more than 4 billion points with initial time steps of 7.69 ×10−8s. The set-up consists of quiescent heavy (cold) fluid atop lighter (hot) fluid initially separated by an insulated partition. Removal of this partition creates acoustic pulses in the directions normal to the interface, which provides the receptivity route of RTI to transient acoustic pulses spread over several mega-Hz frequencies, far in excess of ultrasonic frequencies. The DNS and its analysis reveal the pulses to be severely attenuated, which cannot be explained by the classical wave equation. In the present research, after demonstrating the features of the DNS results, we report in detail, the theoretical development of a wave equation incorporating losses based on the Navier-Stokes equation without Stokes' hypothesis for a quiescent ambience. Apart from the physical properties of this altered wave equation, the numerical solution of the same is obtained. The governing partial differential equation (PDE) for the propagation of disturbances in the spectral plane, provides the dispersion relation between wavenumber and circular frequency in the dissipative medium accounting for viscous losses. The presented analysis provides physical properties using the global spectral analysis (GSA). This shows the far-field perturbation to propagate either as attenuated waves or strictly in a diffusive manner depending upon the wavenumber. The PDE is solved numerically by a high-accuracy compact scheme and the four-stage, Runge-Kutta scheme for time advancement. The computed solution is shown to match not only with the developed theoretical analysis but also explains the DNS results for RTI at early times when the computed flow field truly represents the quiescent ambience.

On peridynamic acoustics

We consider a nonlocal (peridynamic) version of the classical forced wave equation. This scalar three-dimensional equation contains a weight function (the “micromodulus”) and a length parameter (the “horizon”) that have to be selected. We investigate various properties (the locality limit as the horizon shrinks, plane waves and group velocity), paying attention to how these properties depend on the choice of the micromodulus. We solve the forced peridynamic equation in the static case (avoiding divergent integrals) and in the time-harmonic case (with a radiation condition, when needed).

Axion-photon mixing in 3D: classical equations and geometric optics

Light particle-photon mixing in magnetised plasmas plays a vital role in constraining the existence of new physics, especially axions, dark photons, and ultra-high-frequency gravitational waves. Recently, we derived an expression for the resonant conversion of axions to photons in inhomogeneous media using kinetic theory to derive photon transport equations. In this work, we show how the same expression for the conversion probability can be obtained from the classical wave equations of axion-electrodynamics by deriving an equivalent transport equation along the photon worldline. This result provides further corroboration of this expression for the resonant production of photons from light particles, which has also recently been supported by independent numerical simulations of full axion-electrodynamics. In addition, this new approach provides a more general expression that accounts for mixing away from resonance, which is integrated along the whole worldline of the photon in a way that naturally incorporates a curved photon trajectory relevant to refractive media where the photon and light-particle worldlines differ.

Investigation of the complex dynamical structure of bifurcation and dark soliton solutions to fractional generalized double [formula omitted]-Gordon equation

In the current research work, the classical wave equation is combined with a nonlinear sinh source term in the sinh-Gordon equation. It has been used in several scientific domains such as differential geometry theory, integrable quantum field theory, kink dynamics, and statistical mechanics. It makes more comprehensible the dynamics of strings and multi-strings in the constant curvature space. The current study has three main objectives. Examine the governing model to get novel solutions by employing the modified Kudryashov technique. Then compare it with the numerical technique of modified variational iteration method (MVIM) to calculate the error terms. Furthermore, employing bifurcation theory to produce a dynamical system. Additionally, use the dynamical system’s sensitivity analysis to investigate the model’s sensitivity. At last, for the validation of acquired results, the cryptography technique of novel image encryption and decryption is used. The research is greatly enhanced by the presentation of thorough 2D and 3D phase portraits. The field of mathematics and other sciences will benefit from these discoveries.

A Study of The Exact Solutions and Conservation Laws of The Classical Lonngren Wave Equation for Communication Signals

A Study of The Exact Solutions and Conservation Laws of The Classical Lonngren Wave Equation for Communication Signals

Transmission of waves and particles through the interface: Reversibility and coherence

We examine the transmission of quantum particles (phonons, electrons, and photons) across interfaces, identifying universal patterns in diverse physical scenarios. Starting with classical wave equations, we quantize them and derive kinetic equations. Those are matching conditions for the distribution functions of particles at the interface. We note the time irreversibility of the derived kinetic equations — an essential feature for accurately describing irreversible processes like heat transport. We identify the juncture in our derivation where the time symmetry of wave equations is disrupted, it is the assumption of the non-coherence of incident waves. Consequently, we infer that non-coherent transmission through the interface exhibits time irreversibility. We propose an experiment to validate this hypothesis.

On the critical regularity of nonlinearities for semilinear classical wave equations

In this paper, we consider the Cauchy problem for semilinear classical wave equations utt-Δu=|u|pS(n)μ(|u|)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\begin{aligned} u_{tt}-\\Delta u=|u|^{p_S(n)}\\mu (|u|) \\end{aligned}$$\\end{document}with the Strauss exponent pS(n)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$p_S(n)$$\\end{document} and a modulus of continuity μ=μ(τ),\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\mu =\\mu (\ au ),$$\\end{document} which provides an additional regularity of nonlinearities in u=0\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$u=0$$\\end{document} comparing with the power nonlinearity |u|pS(n).\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$|u|^{p_S(n)}.$$\\end{document} We obtain a sharp condition on μ\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\mu $$\\end{document} as a threshold between global (in time) existence of small data radial solutions by deriving polynomial-logarithmic type weighted Lt∞Lr∞\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$L^{\\infty }_tL^{\\infty }_r$$\\end{document} estimates, and blow-up of solutions in finite time even for small data by applying iteration methods with slicing procedure. These results imply a conjecture for the critical regularity of source nonlinearities for semilinear classical wave equations. We verify this conjecture in the 3d case.

Physics informed neural networks for an inverse problem in peridynamic models

AbstractDeep learning is a powerful tool for solving data driven differential problems and has come out to have successful applications in solving direct and inverse problems described by PDEs, even in presence of integral terms. In this paper, we propose to apply radial basis functions (RBFs) as activation functions in suitably designed Physics Informed Neural Networks (PINNs) to solve the inverse problem of computing the perydinamic kernel in the nonlocal formulation of classical wave equation, resulting in what we call RBF-iPINN. We show that the selection of an RBF is necessary to achieve meaningful solutions, that agree with the physical expectations carried by the data. We support our results with numerical examples and experiments, comparing the solution obtained with the proposed RBF-iPINN to the exact solutions.

Solvability of Vekua-type periodic operators and applications to classical equations

In this note, we investigate Vekua-type periodic operators of the form Pu=Lu−Au−Bū, where L is a constant coefficient partial differential operator. We provide a complete characterization of the necessary and sufficient conditions for the solvability and global hypoellipticity of P. As an application, we provide a comprehensive characterization of Vekua-type operators associated with classical wave, heat, and Laplace equations.

Theory and simulation of shock waves freely propagating through monoatomic non-Boltzmann gas

The effect of non-Boltzmann energy distributions on the free propagation of shock waves through a monoatomic gas is investigated via theory and simulation. First, the non-Boltzmann heat capacity ratio γ\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\gamma $$\\end{document}, as a key property for describing shock waves, is derived from first principles via microcanonical integration. Second, atomistic molecular dynamics simulations resembling a shock tube setup are used to test the theory. The presented theory provides heat capacity ratios ranging from the well-known γ=5/3\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\gamma = 5/3$$\\end{document} for Boltzmann energy-distributed gas to γ→1\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\gamma \\rightarrow 1$$\\end{document} for delta energy-distributed gas. The molecular dynamics simulations of Boltzmann and non-Boltzmann driven gases suggest that the shock wave propagates about 9% slower through the non-Boltzmann driven gas, while the contact wave appears to be about 4% faster if it trails non-Boltzmann driven gas. The observed slowdown of the shock wave through applying a non-Boltzmann energy distribution was found to be consistent with the classical shock wave equations when applying the non-Boltzmann heat capacity ratio. These fundamental findings provide insights into the behavior of non-Boltzmann gases and might help to improve the understanding of gas dynamical phenomena.Graphical abstract

Lifespan estimates for 1d damped wave equation with zero moment initial data

In this manuscript, a sharp lifespan estimate of solutions to semilinear classical damped wave equation is investigated in one dimensional case when the Fourier 0th moment of sum of initial position and speed is 0. Especially, it is shown that the behavior of lifespan changes with p=3/2 with respect to the size of the initial data.

Wave Propagation in High-Contrast Media: Periodic and Beyond

Abstract This work is concerned with the classical wave equation with a high-contrast coefficient in the spatial derivative operator. We first treat the periodic case, where we derive a new limit in the one-dimensional case. The behavior is illustrated numerically and contrasted to the higher-dimensional case. For general unstructured high-contrast coefficients, we present the Localized Orthogonal Decomposition and show a priori error estimates in suitably weighted norms. Numerical experiments illustrate the convergence rates in various settings.

Multiplicity of the large periodic solutions to a super-linear wave equation with general variable coefficient

<abstract><p>In this paper, we were concerned with the multiplicity of the large periodic solutions to a super-linear wave equation with a general variable coefficient. In general, the variable coefficient $ \rho(\cdot) $ needs to be satisfied $ \text{ess inf}\, \eta_\rho(\cdot) > 0 $ with $ \eta_\rho(\cdot) = \frac{1}{2}\frac{\rho''}{\rho}-\frac{1}{4}\big(\frac{\rho'}{\rho}\big)^2 $. Especially, the case $ \eta_\rho(\cdot) = 0 $ is presented as an open problem in [Trans. Amer. Math. 349: 2015-2048, 1997]. Here, without any restrictions on $ \eta_{\rho}(\cdot) $, we established the multiplicity of large periodic solutions for the Dirichlet-Neumann boundary condition and Dirichlet-Robin boundary condition when the period $ T = 2\pi\frac{2a-1}{b} $ with $ a, b \in \mathbb{N}^+ $. The key ingredient of the proof is the combination of the variational method and an approximation argument. Since the sign of $ \eta_\rho(\cdot) $ can change, our results can be applied to the classical wave equation.</p></abstract>

Classical Wave Equation: A Gross Error in Mathematics and Physics

A detailed proof of the incorrectness of the classical wave equation is proposed. The correct methodological basis for the proof is the unity of formal logic and rational dialectics. The proof leads to the following irrefutable statement: the classical wave equation and the derivation of the classical wave equation are a gross error in mathematics and physics. The proof of this statement is based on the following irrefutable main results: (1) The first gross error is the following approximate relationship: “sine of angle is approximately equal to tangent of angle; cosine of angle is approximately equal to 1”. This relationship means (implies) that the quantity of the angle and the right-angled triangle do not exist. Consequently, the relationship between the tangent of the angle and the derivative of the displacement with respect to coordinate does not exist; (2) The second gross error is that the second-order derivative of the displacement with respect to coordinate does not exist, because the dimensions of displacement and coordinate are “meter”, the first-order derivative of the displacement with respect to coordinate is dimensionless quantity (i.e. the first-order derivative of the displacement respect to coordinate has no the dimension “meter”); (3) The third gross error is that the first-order derivative of the displacement with respect to coordinate cannot be expanded into the Taylor series, because the second-order derivative of the displacement with respect to coordinate does not exist; (4) The fourth gross error is as follows: (a) the left side of the wave equation contains the condition that time is a variable quantity, and the coordinate is a constant quantity; (b) the right side of the wave equation contains the condition that time is a constant, and the coordinate is a variable. This means (implies) that the equation contains contradictory conditions (propositions). Therefore, the equation represents a violation of the formal-logical law of the lack (absence) of contradiction; (5) The fifth gross error is that the standard derivation of the equation relies on the following false theories: negative number theory, complex numbers theory, trigonometry, vector calculus, differential calculus, and Newton's second law. Thus, the classical wave equation does not satisfy the criterion of truth and is not a scientific achievement.

Symplectic time-domain finite element method (STD-FEM) for room acoustic modeling

A new, extended symplectic time-domain finite element method (STD-FEM) is proposed for room acoustic modeling. As a mathematical model for sound propagation and reflection, the classical time-domain wave equations are used, which can be extended with air absorption. Frequency-dependent locally reactive boundary conditions are also introduced to the model. Spatially, a third-order tensor product type spectral element discretization is applied, which allows us to use explicit time steps. For this purpose, a partitioned Runge-Kutta method is employed, which is an extension of a third-order symplectic time-discretization. The method is validated on a number of examples and the performance and stability benefits are presented. We present the implementation of the method in the soundy.ai application.

Love-type wave propagation in a coated fluid-saturated fractured poro-viscoelastic layer with sliding contacts and point source effect

The present paper studies Love-type wave propagation in a fluid-saturated fractured poro-viscoelastic layer sandwich between a coated elastic layer and a non-homogeneous anisotropic Voigt-type viscoelastic half-space. The interfaces between the layers and the half-space are considered as sliding contacts with two different sliding parameters. An impulsive point source is located at a varying depth in the intermediate poro-viscoelastic layer in the form of the Dirac-delta function. The vertical non-homogeneity in the half-space is assumed to be a combination of linear and exponential functions that vary with depth in the medium. Using the Fourier transforms and Green's functions, the equations of motion, in the form of partial differential equations, are solved analytically. An appropriate boundary condition is used to accomplish this. The closed form of the complex dispersion equation has been obtained by assuming that the wavenumber is in the form of a complex quantity. The validity of our model has been examined using several specific cases and compared to the classical Love wave equation. Numerical and graphical analyses of dispersion and displacement were performed using MATLAB software. It has been found that a single porous medium has a higher Love-type phase velocity than a double porous medium. Additionally, the sliding parameters increase the range of wavenumber when there is no phase velocity. An inverse relationship was found between the depth of the point source and the particle displacement in the coated layer. In addition to contributing to our understanding of geophysical phenomena and subsurface explosions, these findings may provide insight into the inner structure of the planet.

A numerical method for solving uncertain wave equation

Partial differential equations are usually used to deal with natural or physical problems, and wave equations are an important branch of them. Considering the fluctuation phenomena with uncertain noise in real life, this paper will focus on the study of the uncertain wave equation. The uncertain wave equation is a special kind of partial differential equation excited by the Liu process, which is widely used to describe wave propagation phenomena. However, not all uncertain wave equations can be solved analytically, so numerical solutions are of particular importance. The main objective of this paper is to study the numerical solution of uncertain wave equations. We first introduce the concept of α-path and, based on it, prove an important formula that reveals the connection between the uncertain wave equation and the classical wave equation. Based on this finding, we design a new numerical solution method to solve the uncertain wave equation. To verify the feasibility and accuracy of the method, we conduct numerical experiments. In addition, as an application, we obtain the uncertainty distribution, the inverse uncertainty distribution and the expected value of the solution of the equation based on the important formula. The originality and novelty of the conclusions of this work are demonstrated by comparing the results with those of other researchers.

Impact of imperfectness on the love-type wave propagation in a viscoelastic composite structure with couple stress

The present paper is intended to frame-out the analytical approach to study the transference of Love-type waves in a viscoelastic sandy layer sandwiched between a size-dependent consistent couple stress to take care of earth’s interior microstructural nature and an orthotropic viscoelastic half-space. An additional parameter termed as characteristic length which is used to investigate the internal microstructural structure of the earth’s interior, which provides us the deeper insights of the analysis of wave propagation aspects. With the help of analytical variable separable technique, the closed version of dispersion equation for the Love-type waves has been derived. The impacts of different material parameters like characteristic length (l), sandy parameter ( η 2 ′ ) , viscoelastic parameters (F1, F2, F3), thickness ratio (H3), and imperfectness parameter (σ1, σ2) which are involved in this study are encapsulated by means of graphs on the phase velocity of Love-type waves. Finally, the obtained result is well match up with the classical Love wave equation. This approach of employing micro continuum theories to analyze seismic waves can bring some useful adjustments to previously established facts about seismic waves, as well as improve the applications of micro continuum theories in the field of seismology.

High-frequency magnetohydrodynamics

We consider the case that the displacement current is not neglected in the classical MHD equations, as it is usually done. This amounts to cast them in the relativistic framework of a finite speed of light. We show some consequences in describing magnetic reconnection phenomena and for hydromagnetic waves. In the first case, the equation for the magnetic induction is changed from (formally) parabolic to (formally) hyperbolic, in the second case both, the perturbed magnetic field and the particle velocity, obey to a certain third-order in time partial differential equation, rather than to the classical wave equation. We stress the role of two typically small but nonzero parameters, the magnetic diffusivity, η (corresponding to large values of the Lundquist number), and ε:=c−2.