Inhomogeneous Wave Equation Research Articles

Assessment of methodologies for the solution of the Ffowcs Williams and Hawkings equation using LES of incompressible single-phase flow around a finite-size square cylinder

The acoustic analogy represents a powerful tool for the prediction of noise generated by the interaction between the flow and a moving body. It is based on decoupling the acoustic problem from the fluid dynamic one: the velocity and pressure fields, obtained through a separate numerical simulation, are used as source terms in an inhomogeneous wave equation whose solution reconstructs the noise in the far field. When the method is based on the fundamental Ffowcs Williams and Hawkings (FW-H) equation, different solving methodologies may be adopted.The present work considers the original FW-H equation and gives the advective formulation of the volume integral terms. The results are compared with those obtained with the Curle and porous formulations.To account for volume integrals, the assumption of compact noise source is needed. This assumption is common in literature, however, in the present work, a dimensional analysis is proposed, in order to indicate in a rigorous way the cases in which the compressibility delays can be avoided. The dimensional analysis is tested in the case of an acoustic monopole field. Successively, the FW-H porous formulation is compared with the original FW-H equation in the case of an irrotational advected vortex. This example puts in evidence the different response of the two methods in the case of a vortex crossing the acoustic domain.Then, different solution strategies of the FW-H are evaluated using a fluid dynamic dataset obtained through large eddy simulation of a turbulent flow around a finite-length cylinder with square section. The analysis allows to point out the strengths and drawback of the different techniques and to achieve, through the comparison of the different solutions, an accurate understanding of the noise source mechanisms taking place in the flow. Finally, a mixed procedure, merging the advantages of the porous formulation with the direct evaluation of the volume integral terms is proposed. It may be used in presence of significant time delays. Overall, the present study is oriented to the analysis of very low Mach number flows, although the complete porous method might be applicable in a more general framework. This aspect will require additional research in the future.

An Operator Method for the Stability of Inhomogeneous Wave Equations

In this paper, we will apply the operator method to prove the generalized Hyers-Ulam stability of the wave equation, u t t ( x , t ) − c 2 ▵ u ( x , t ) = f ( x , t ) , for a class of real-valued functions with continuous second partial derivatives. Finally, we will discuss the stability more explicitly by giving examples.

Analytical method to study a mathematical model of wave processes under two­point time conditions

Research and analysis of dynamic processes in oscillatory systems are closely connected to the establishment of exact or approximate analytical solutions to the problems of mathematical physics, which model such systems. The mathematical models of wave propagation in oscillatory systems under certain initial conditions at a fixed time are well known in the literature. However, wave processes in lengthy structures subject to an external force only and at the assigned states of the process at two points in time have been insufficiently studied. Such processes are modeled by a two-point time problem for the inhomogeneous wave equation in an unbounded domain t>0, x∈ℝ s . The model takes into consideration the assignment of a linear combination with unknown amplitude of oscillations and the rate of its change at two points in time. A two-point problem, generally speaking, is the ill-posed boundary value problem, since the respective homogeneous problem has non-trivial solutions. A class of quasi-polynomials has been established as the class of the existence of a single solution to the problem. This class does not contain the non-trivial elements from the problem's kernel, which ensures the uniqueness of solution to the problem. We have proposed a precise method to build the solution in the specified class. The essence of the method is that the problem's solution is represented as the action of a differential expression, whose symbol is the right-hand side of the equation, on some function of parameters. The function is constructed in a special way using the equation and two-point conditions, and has special features associated with zeroes of the denominator – the characteristic determinant of the problem. The method is illustrated by the description of oscillatory processes within an infinite string and a membrane. The main practical application of the constructed method is the possibility to adequately mathematically model the oscillatory systems, which takes into consideration a possibility to control the system's parameters. Such a control over parameters makes it possible to perform optimal synthesis and design of parameters for the relevant technical systems in order to analyze and account for special features in the dynamic modes of oscillations

On the Relationship between the Inhomogeneous Wave and Helmholtz Equations in a Fractional Setting

We study convergence of solutions of a space and time inhomogeneous fractional wave equation on the quarter-plane to the stationary regime described by solutions of the Helmholtz equation.

A Dilation Invariance Method and the Stability of Inhomogeneous Wave Equations

We apply the method of a kind of dilation invariance to prove the generalized Hyers-Ulam stability of the (inhomogeneous) wave equation with a source, u t t ( x , t ) − c 2 ▵ u ( x , t ) = f ( x , t ) , for a class of real-valued functions with continuous second partial derivatives in each of spatial and the time variables.

Mathematical Modeling of Impact of Possible Shock Dynamic Loads on the Modern Composite Materials

In this paper, we study the dynamic processes in materials reinforced with fibers, that can be represented as composite rods. There has been developed a mathematical model of wave propagation under the impact of a shock pulse in semi-infinite composite rods. It is believed that the considered composite rod consists of two layers formed by simpler rods of different isotropic materials with different mechanical properties. The cross sections of such rods are considered to be constant and identical. When such composite materials are impacted by dynamic loads, a significant part of the energy is dissipated due to the presence of friction forces between the contact surfaces of the rods. In this regard, we study the propagation of waves in an elastic fiber-rod, the layers of which interact according to Coulomb law of dry friction. The case of instantaneous excitation of rods by step pulses is investigated. The blow is applied to a rod made of a harder material. In the absence of slippage, the friction force gets a value not exceeding the absolute value of the limit. In the absence of slippage, the friction force takes a value not exceeding the absolute value of the limit. Let us consider the value of the friction force constant. Normal stresses and velocities satisfy the equations of motion and Hooke’s law. The problem statement results in the solution of inhomogeneous wave equations by the method of characteristics in different domains, which are the lines of discontinuities of the solution. Solutions are found in all constructed domains. On the basis of the analysis of the obtained solution, qualitative conclusions are made and curves are constructed according to the obtained ratios. From the found analytical solution of the problem it is possible to obtain ratios for stresses and strain rates in composite rods and composite materials.

Some globally conserved currents from generalized Killing vectors and scalar test fields

In this article, I discuss the construction of some globally conserved currents that one can construct in the absence of a Killing vector. One is based on the Komar current, which is constructed from an arbitrary vector field and has an identically vanishing divergence. I obtain some expressions for Komar currents constructed from some generalizations of Killing vectors which may in principle be constructed in a generic spacetime. I then present an explicit example for an outgoing Vaidya spacetime which demonstrates that the resulting Komar currents can yield conserved quantities that behave in a manner expected for the energy contained in the outgoing radiation. Finally, I describe a method for constructing another class of (non-Komar) globally conserved currents using a scalar test field that satisfies an inhomogeneous wave equation, and discuss two examples; the first example may provide a useful framework for examining the arrow of time and its relationship to energy conditions, and the second yields (with appropriate initial conditions) a globally conserved energy- and momentumlike quantity that measures the degree to which a given spacetime deviates from symmetry.

Safety of active catheters in MRI: Termination impedance versus RF-induced heating.

To investigate the effect of the termination impedance on the RF-induced heating of active catheters using analytical modeling. Interaction of an arbitrary electric (E) field and an isolated transmission line (TL) embedded in cascaded lossy media was analytically modeled. Termination impedances at the tip and the input sides were expressed as distinct parameters in the current and voltage distribution formulae that are obtained by solving the inhomogeneous wave equations using the Green's function approach. The tip specific absorption rate (SAR) was calculated for different E field configurations. The tip SAR was displayed on a color-coded Smith chart in terms of the normalized input reflection coefficient. Results of the analytical calculations were compared to transfer function (TF) measurements. An input impedance control unit that is integrated to the interface circuit was introduced. TFs from analytical model and measurements exhibited similar behaviors. Color-coded Smith charts shows that the analytical model and measured TF-based tip SAR depends strongly on the input reflection coefficient. Both for measured and analytical TFs, SAR can deviate up to 70% from the mean value for different input impedance values. This study shows that it is possible to change the RF-induced heating characteristics of an active catheter by adjusting the input impedance, and the presented analytical model is in good agreement with TF measurements.

Nonconforming Finite Elements Based on Nitsche-Type Mortaring for Inhomogeneous Wave Equation

We propose the nonconforming Finite Element (FE) method based on Nitsche-type mortaring for efficiently solving the inhomogeneous wave equation, where due to the change of material properties the wavelength in the subdomains strongly differs. Therewith, we gain the flexibility to choose for each subdomain an optimal grid. The proposed method fulfills the physical conditions along the nonconforming interfaces, namely the continuity of the acoustic pressure and the normal component of the acoustic particle velocity. We apply the nonconforming grid method to the computation of transmission loss (TL) of an expansion chamber utilizing micro-perforated panels (MPPs), which are modeled by a homogenization approach via a complex fluid. The results clearly demonstrate the superiority of the nonconforming FE method over the standard FE method concerning pre-processing, mesh generation flexibility and computational time.

The Cauchy problem for linear inhomogeneous wave equations with variable coefficients

In this paper, we present a new analytical formula for the Cauchy problem of the linear inhomogeneous wave equation with variable coefficients. The formula gives a much simpler solution than that given by the classical Poisson formula. The derivation is based on Duhamel’s Principle and the theory of pseudodifferential operator. An example is solved by using the formula to illustrate the feasibility.

Massive vector field perturbations on extremal and near-extremal static black holes

We discuss a new perturbation method to study the dynamics of massive vector fields on (near-)extremal static black hole spacetimes. We start with, as our background, a rather generic class of warped product metrics, and classify the field variables into the vector(axial)- and scalar(polar)-type components. On this generic background, we show that for the vector-type components, the Proca equation reduces to a single master equation, whereas the scalar-type components remain to be coupled. Then, focusing on the case of (near-)extremal static black holes in four-dimensions, we consider the near-horizon expansion of both the background geometry and massive vector field by a scaling parameter $\lambda$ with the leading-order geometry being the so called near-horizon geometry. We show that on the near-horizon geometry, thanks to its enhanced symmetry, the Proca equation for the scalar-type components also reduces to a set of two mutually decoupled homogeneous wave equations for two variables, plus a coupled equation through which the remaining variable is determined. Therefore, together with the vector-type master equation, we obtain the set of three decoupled master wave equations, which govern the three independent dynamical degrees of freedom of the massive vector field in four-dimensions. We further expand the geometry and massive vector field with respect to $\lambda$ and show that at each order, the Proca equation for the scalar-type components can reduce to a set of decoupled inhomogeneous wave equations whose source terms consist only of the lower-order variables, plus one coupled equation that determins the remaining variable. Therefore, if one solves the master equations on the leading-order near-horizon geometry, then in principle one can successively solve the Proca equation at any order.

Approximate fields of an ultra-short, tightly-focused, radially-polarized laser pulse in an under-dense plasma: a Bessel-Bessel light bullet

Provided the intensity is not too high (for example, with I << 1018 W/cm2, for a wavelength of 1 μm), response of an under-dense plasma to the fields of a laser pulse can still be considered linear, and inhomogeneous wave equations for the vector and scalar potentials A and Φ, respectively, may be derived from Maxwell’s equations. A rigorous, but approximate, solution to the wave equation satisfied by a one-component, azimuthally symmetric, vector potential is developed using a Fourier transform method. It is found that an ultra-short and tightly-focused, radially-polarized laser pulse, described by this vector potential, propagates in the plasma like a laser bullet. The pulse is termed a Bessel-Bessel bullet because, to leading order in a power-series expansion, the vector potential, from which the pulse fields E and B are derived, is expressed in terms of a Bessel function of the first kind J0 and a spherical Bessel function of the first kind j0.

Analysis of axis symmetric circular crested elastic wave generated during crack propagation in a plate: A Helmholtz potential technique

This paper presents cylindrical coordinate solutions of axis symmetric circular crested elastic waves that appear due to sudden energy release during incremental crack propagation in a plate. Axis symmetric assumption decouples the elastic wave problem to Lamb (P+SV) and shear (SH) horizontal waves. Helmholtz decomposition principle was used to decompose displacement field in to unknown scalar and vector potentials; and body force vectors to known excitation scalar and vector potentials respectively. Therefore, Navier–Lame equations yield a set of four inhomogeneous wave equations of unknown potentials Φ, Hr, Hθ, Hz and known excitation potentials A*,Br*,Bθ*,Bz*. There are two types of potentials exist in a plate for axis symmetric circular crested Lamb wave: pressure potentials Φ, A* and shear potentials Hθ, Bθ*. Inhomogeneous wave equations for Φ and Hθ were solved due to generalized excitation potentials A* and Bθ* in a form, suitable for numerical calculation. The theoretical formulation shows that elastic waves generated in a plate using excitation potentials follow the Rayleigh-Lamb equations. The resulting solution is a series expansion containing the superposition of all the Lamb wave modes existing for the particular frequency-thickness combination under consideration. In addition, bulk wave solution is also recovered due to the effect of the excitation potentials. The numerical studies modeled the two-dimensional (2D) (circular crested) AE elastic wave propagation in order to simulate the out-of-plane displacement that would be recorded by an AE sensor placed on the plate surface at some distance away from the source. Parameter studies were performed to evaluate: (a) the effect of the pressure and shear potentials; (b) the effect of the thickness-wise location of the excitation potential sources varying from mid-plane to the top surface (source depth effect); (c) the effect of peak time (d) the effect of propagating distance away from the source. A Gaussian pulse is used to model the growth of the excitation potentials during the AE event; as a result, the actual excitation potential follows the error function variation in the time domain. The numerical studies show that the peak amplitude of A0 signal is higher than the peak amplitude of S0 signal and the peak amplitude of bulk wave is not significant compared to S0, A0 peak signals.

A Helmholtz Potential Approach to the Analysis of Guided Wave Generation During Acoustic Emission Events

This paper addresses the predictive simulation of acoustic emission (AE) guided waves that appear due to sudden energy release during incremental crack propagation. The Helmholtz decomposition approach is applied to the inhomogeneous elastodynamic Navier–Lame equations for both the displacement field and body forces. For the displacement field, we use the usual decomposition in terms of unknown scalar and vector potentials, Φ and H. For the body forces, we hypothesize that they can also be expressed in terms of excitation scalar and vector potentials, A* and B*. It is shown that these excitation potentials can be traced to the energy released during an incremental crack propagation. Thus, the inhomogeneous Navier–Lame equation has been transformed into a system of inhomogeneous wave equations in terms of known excitation potentials A* and B* and unknown potentials Φ and H. The solution is readily obtained through direct and inverse Fourier transforms and application of the residue theorem. A numerical study of the one-dimensional (1D) AE guided wave propagation in a 6 mm thick 304-stainless steel plate is conducted. A Gaussian pulse is used to model the growth of the excitation potentials during the AE event; as a result, the actual excitation potential follows the error function variation in the time domain. The numerical studies show that the peak amplitude of A0 signal is higher than the peak amplitude of S0 signal, and the peak amplitude of bulk wave is not significant compared to S0 and A0 peak amplitudes. In addition, the effects of the source depth, higher propagating modes, and propagating distance on guided waves are also investigated.

Polarized optical scattering by inhomogeneities and surface roughness in an anisotropic thin film.

We extend the theory of Kassam etal. [J. Opt. Soc. Am. A12, 2009 (1995)JOAOD60740-323210.1364/JOSAA.12.002009] for scattering by oblique columnar structure thin films to include the induced form birefringence and the propagation of radiation in those films. We generalize the 4×4 matrix theory of Berreman [J. Opt. Soc. Am.62, 502 (1972)JOSAAH0030-394110.1364/JOSA.62.000502] to include arbitrary sources in the layer, which are necessary to determine the Green function for the inhomogeneous wave equation. We further extend first-order vector perturbation theory for scattering by roughness in the smooth surface limit, when the layer is anisotropic. Scattering by an inhomogeneous medium is approximated by a distorted Born approximation, where effective medium theory is used to determine the effective properties of the medium, and strong fluctuation theory is used to determine the inhomogeneous sources. In this manner, we develop a model for scattering by inhomogeneous films, with anisotropic correlation functions. The results are compared with Mueller matrix bidirectional scattering distribution function measurements for a glancing-angle deposition (GLAD) film. While the results are applied to the GLAD film example, the development of the theory is general enough that it can guide simulations for scattering in other anisotropic thin films.

Solution of the inhomogeneous Maxwell's equations using a Born series.

An algorithm for the numerical solution of the inhomogeneous Maxwell's equations is presented. The algorithm solves the inhomogeneous vector wave equation of the electric field by writing the solution as a convergent Born series. Compared to two dimensional finite difference time domain calculations, solutions showing the same accuracy can be calculated more than three orders of magnitude faster.

A mixed problem for an inhomogeneous wave equation with a summable potential

A mixed problem for an inhomogeneous wave equation with fixed ends in the case of a summable potential is studied. Using the Krylov method for acceleration of the convergence of Fourier series, a classical solution under minimal conditions on the initial data and a generalized solution in the case of quadratic summable initial data and perturbing function are obtained.

Fields of an ultrashort tightly focused radially polarized laser pulse in a linear response plasma

Analytical expressions for the fields of a radially polarized, ultrashort, and tightly focused laser pulse propagating in a linear-response plasma are derived and discussed. The fields are obtained from solving the inhomogeneous wave equations for the vector and scalar potentials, linked by the Lorenz gauge, in a plasma background. First, the scalar potential is eliminated using the gauge condition, then the vector potential is synthesized from Fourier components of an initial uniform distribution of wavenumbers, and the inverse Fourier transformation is carried out term-by-term in a truncated series (finite sum). The zeroth-order term in, for example, the axial electric field component is shown to model a pulse much better than its widely used paraxial approximation counterpart. Some of the propagation characteristics of the fields are discussed and all fields are shown to have manifested the expected limits for propagation in a vacuum.

Effective one-dimensional approach to the source reconstruction problem of three-dimensional inverse optoacoustics.

The direct problem of optoacoustic signal generation in biological media consists of solving an inhomogeneous three-dimensional (3D) wave equation for an initial acoustic stress profile. In contrast, the more defiant inverse problem requires the reconstruction of the initial stress profile from a proper set of observed signals. In this article, we consider an effectively 1D approach, based on the assumption of a Gaussian transverse irradiation source profile and plane acoustic waves, in which the effects of acoustic diffraction are described in terms of a linear integral equation. The respective inverse problem along the beam axis can be cast into a Volterra integral equation of the second kind for which we explore here efficient numerical schemes in order to reconstruct initial stress profiles from observed signals, constituting a methodical progress of computational aspects of optoacoustics. In this regard, we explore the validity as well as the limits of the inversion scheme via numerical experiments, with parameters geared toward actual optoacoustic problem instances. The considered inversion input consists of synthetic data, obtained in terms of the effectively 1D approach, and, more generally, a solution of the 3D optoacoustic wave equation. Finally, we also analyze the effect of noise and different detector-to-sample distances on the optoacoustic signal and the reconstructed pressure profiles.

Acoustic damping optimization of the injector based on dynamic mode decomposition in a subscale combustor

The acoustic tuning of a swirl coaxial jet injector mounted upstream a combustion chamber is investigated in the aspect of stabilization of unstable combustion. The injector, which has a primary function to mix the oxidizer injected through the central passage and the fuel injected through the peripheral holes, is designed to be used as a half-wave resonator. The least damped modes are extracted by applying Dynamic mode decomposition (DMD) and the injector length is tuned to damp the second longitudinal mode. Then, the sensitivity of the heat release perturbation to the velocity perturbation at the frequency of the second longitudinal mode is evaluated by flame transfer function calculated by DMD. The relation between these two quantities is achieved by combining two equations, that is, the Crocco’s model and the inhomogeneous wave equation, and enables us to compute Flame transfer function (FTF) at a specific mode. The gain field of FTF shows that the sensitivity of the heat release fluctuation to the inlet velocity fluctuation is minimal with the optimal injector length. The proposed approach for application of dynamic mode decomposition would be a valuable tool in tuning a resonator acoustically, computing efficiently flame transfer function, and quantifying the relative stability of a combustion chamber.