Inhomogeneous Wave Equation Research Articles

Reconstruction of a time-dependent potential from wave measurements

We add a time-dependent potential to the inhomogeneous wave equation and consider the task of reconstructing this potential from measurements of the wave field. This dynamic inverse problem becomes more involved compared to static parameters, as, e.g. the dimensions of the parameter space do considerably increase. We give a specifically tailored existence and uniqueness result for the wave equation and compute the Fréchet derivative of the solution operator, for which also show the tangential cone condition. These results motivate the numerical reconstruction of the potential via successive linearization and regularized Newton-like methods. We present several numerical examples showing feasibility, reconstruction quality, and time efficiency of the resulting algorithm.

Seismic Exploration Random Noise on Land: Modeling and Application to Noise Suppression

In seismic exploration, random noise suppression is one of the key problems in seismic data processing. For random noise attenuation, the most important thing is the understanding of seismic random noise generation and propagation. Seismic random noise is considered as temporal and spatial random processes, and it can be analyzed only qualitatively for now, due to its high variability. In this paper, we classify seismic random noise sources by their generation factors and simulate the random noise of the desert in West China. According to Green’s function, it can be assumed that seismic random noise sources are point-like sources that are distributed around geophones. A seismic random noise record is taken as the superimposed wave field exited by all the independent sources in a homogeneous isotropy half-infinite surface. Based on the wind vibration theory and preliminary study about ambient vibrations, the noise source functions are determined. We obtain the waveforms of different kinds of noise by solving the inhomogeneous wave equations and analyze the characteristics qualitatively and quantitatively. The seismic synthetic record with a 1.6-s time and 250-m distances is obtained, and the characteristics are compared between the simulated and the real noise record in time domain and space domain, respectively. The comparative results show the same characteristics of the simulated noise and the real noise, which demonstrates the feasibility of the proposed method. According to the noise modeling, it is known that the near-field cultural noise is the main component of the random noise in the desert, on the basis of which complex diffusion filtering is selected. The filtered results by complex diffusion filtering is compared with the results of time–frequency peak filtering, which is a popular filtering method of seismic random noise suppression in recent years. The comparative results show that complex diffusion filtering is more suitable for the noise of the desert in the Tarim Basin. This result proves that seismic random noise modeling can provide the guidance for noise attenuation. It lays a foundation for researching the propagation characteristics and better attenuation of seismic random noise in the future.

A Scalable Algorithm for Physically Motivated and Sparse Approximation of Room Impulse Responses With Orthonormal Basis Functions

Parametric modeling of room acoustics aims at representing room transfer functions by means of digital filters and finds application in many acoustic signal enhancement algorithms. In previous work by other authors, the use of orthonormal basis functions (OBFs) for modeling room acoustics has been proposed. Some advantages of OBF models over all-zero and pole-zero models have been illustrated, mainly focusing on the fact that OBF models typically require less model parameters to provide the same model accuracy. In this paper, it is shown that the orthogonality of the OBF model brings several additional advantages, which can be exploited if a suitable algorithm for identifying the OBF model parameters is applied. Specifically, the orthogonality of OBF models does not only lead to improved model efficiency (as pointed out in previous work), but also leads to improved model scalability and model stability. Its appealing scalability property derives from a previously unexplored interpretation of the OBF model as an approximation to a solution of the inhomogeneous acoustic wave equation. Following this interpretation, a novel identification algorithm is proposed that takes advantage of the OBF model orthogonality to deliver efficient, scalable, and stable OBF model estimates, which is not necessarily the case for nonlinear estimation techniques that are normally applied.

A note on isotropic random flights moving in mixed Poisson environments

The aim of this note is to deepen the knowledge of isotropic random flights moving in mixed Poisson media; that is we analyze the random evolution of a particle with random scattering rate Λ. By assuming Λ distributed according to some standard probability laws, we are able to explicit the probability measure of the random flights. Furthermore, we show that there exists a link between our processes and a inhomogeneous wave equation.

A sound extrapolation method for aeroacoustics far-field prediction in presence of vortical waves

Off-surface integral solutions to an inhomogeneous wave equation based on acoustic analogy could suffer from spurious wave contamination when volume integrals are ignored for computation efficiency and vortical/turbulent gusts are convected across the integration surfaces, leading to erroneous far-field directivity predictions. Vortical gusts often exist in aerodynamic flows and it is inevitable their effects are present on the integration surface. In this work, we propose a new sound extrapolation method for acoustic far-field directivity prediction in the presence of vortical gusts, which overcomes the deficiencies in the existing methods. The Euler equations are rearranged to an alternative form in terms of fluctuation variables that contains the possible acoustical and vortical waves. Then the equations are manipulated to an inhomogeneous wave equation with source terms corresponding to surface and volume integrals. With the new formulation, spurious monopole and dipole noise produced by vortical gusts can be suppressed on account of the solenoidal property of the vortical waves and a simple convection process. It is therefore valid to ignore the volume integrals and preserve the sound properties. The resulting new acoustic inhomogeneous convected wave equations could be solved by means of the Green’s function method. Validation and verification cases are investigated, and the proposed method shows a capacity of accurate sound prediction for these cases. The new method is also applied to the challenging airfoil leading edge noise problems by injecting vortical waves into the computational domain and performing aeroacoustic studies at both subsonic and transonic speeds. In the case of a transonic airfoil leading edge noise problem, shocks are present on the airfoil surface. Good agreements of the directivity patterns are obtained compared with direct computation results.

Uniqueness for a seismic inverse source problem modeling a subsonic rupture

ABSTRACTWe consider an inverse source problem for an inhomogeneous wave equation with discrete-in-time sources, modeling a seismic rupture. The inverse source problem, with an arbitrary source term on the right-hand side of the wave equation, is not uniquely solvable. Here we formulate conditions on the source term that allow us to show uniqueness and that provide a reasonable model for the application of interest. We assume that the source term is supported on a finite set of times and that the support in space moves with subsonic velocity. Moreover, we assume that the spatial part of the source is singular on a hypersurface, an application being a seismic rupture along a fault plane. Given data collected over time on a detection surface that encloses the spatial projection of the support of the source, we show how to recover the times and locations of sources microlocally and then reconstruct the smooth part of the source assuming that it is the same at each source location.

Conditions of dynamic change in radiation mode

The inhomogeneous wave equation for determining the previously experimentally detected change in the radiation phase in an accelerated cavity with stationary content is studied. It is shown that the radiation phase is independent of a homogeneous medium filling the accelerated cavity with unchanged phase structure. The necessary and sufficient conditions of the existence of nontrivial solutions of the equation are discussed.

Influence of ice accretion on the noise generated by an airfoil section

We investigate the noise generated by an airfoil section. Three cases are considered, one with a clean airfoil and two cases with airfoils with ice accretion. The amount of ice is the same in the two cases with ice accretion, but the surface of the accreted ice layer is smoother in one of them. The noise is computed using a hybrid approach. First the flow and the acoustic sources are computed. Second, the noise propagation is predicted by solving an inhomogeneous wave equation. The results indicate that in this case the accreted ice layer leads to a decrease of the radiated noise levels, especially in the lower frequency range.

On Solutions for Linear and Nonlinear Schrödinger Equations with Variable Coefficients: A Computational Approach

In this work, after reviewing two different ways to solve Riccati systems, we are able to present an extensive list of families of integrable nonlinear Schrödinger (NLS) equations with variable coefficients. Using Riccati equations and similarity transformations, we are able to reduce them to the standard NLS models. Consequently, we can construct bright-, dark- and Peregrine-type soliton solutions for NLS with variable coefficients. As an important application of solutions for the Riccati equation with parameters, by means of computer algebra systems, it is shown that the parameters change the dynamics of the solutions. Finally, we test numerical approximations for the inhomogeneous paraxial wave equation by the Crank-Nicolson scheme with analytical solutions found using Riccati systems. These solutions include oscillating laser beams and Laguerre and Gaussian beams.

Singularity formation for the inhomogeneous nonlinear wave equation with damping

It is well known that the damping term will give more smooth effect to obtain global solutions. In this paper, we consider the effect of damping term on the solutions to system of inhomogeneous wave equation with damping term. We can obtain the singularity that will be formed in finite time for some large initial data. Copyright © 2016 John Wiley & Sons, Ltd.

Mixed problem for the inhomogeneous wave equation with a summable potential

The resolvent approach to the Fourier method is used to examine the behavior of the formal solution to the mixed problem for an inhomogeneous wave equation with a summable potential. A classical solution is obtained under minimum conditions imposed on the initial function. It is shown that, in the case of square summable initial and driving functions, the series sum of the formal solution is a weak solution of the mixed problem.

Hyperbolic self-gravity solver for large scale hydrodynamical simulations

A new computationally efficient method has been introduced to treat self-gravity in mesh based hydrodynamical simulations. It is applied simply by slightly modifying the Poisson equation into an inhomogeneous wave equation. This roughly corresponds to the weak field limit of the Einstein equations in general relativity, and as long as the gravitation propagation speed is taken to be larger than the hydrodynamical characteristic speed, the results agree with solutions for the Poisson equation. The solutions almost perfectly agree if the domain is taken large enough, or appropriate boundary conditions are given. Our new method can not only significantly reduce the computational time compared with existent methods, but is also fully compatible with massive parallel computation, nested grids and adaptive mesh refinement techniques, all of which can accelerate the progress in computational astrophysics and cosmology.

A Remark on Regularity of Solutions to Wave Equations

We prove a generalization of the Strichartz estimate for the inhomogeneous wave equation $\square u(t,x) =f(t,x)$ in the space-time $\boldsymbol{R}^{1+n}$. We estimate the solution in vector-valued homogeneous Besov spaces $\dot{B}^\theta_{q,2}(\boldsymbol{R}; \dot{B}^\sigma_{r,2}(\boldsymbol{R}^n))$. Such an estimate shows the time differentiability of the solution of fractional order.

Separable wave equations for gravitoelectromagnetic perturbations of rotating charged black strings

Rotating charged black strings are exact solutions of four-dimensional Einstein–Maxwell equations with a negative cosmological constant and a non-trivial spacetime topology. According to the AdS/CFT correspondence, these black strings are dual to rotating thermal states of a strongly interacting quantum field theory with nonzero chemical potential that lives in a cylinder. The dynamics of linear fluctuations in the dual field theory can be studied from the perturbation equations for classical fields in a black-string spacetime. With this motivation in mind, we develop here a completely gauge and tetrad invariant perturbation approach to deal with the gravitoelectromagnetic fluctuations of rotating charged black strings in the presence of sources. As usual, for any charged black hole, a perturbation in the background electromagnetic field induces a metric perturbation and vice versa. In spite of this coupling and the non-vanishing angular momentum, we show that linearization of equations of the Newman–Penrose formalism leads to four separated second-order complex equations for suitable combinations of the spin coefficients, the Weyl and the Maxwell scalars. Then, we generalize the Chandrasekhar transformation theory by the inclusion of sources and apply it to reduce the perturbation problem to four decoupled inhomogeneous wave equations—a pair for each sector of perturbations. The radial part of such wave equations can be put into Schrödinger-like forms after Fourier transforming them with respect to time. We find that the resulting effective potentials form two pairs of supersymmetric partner potentials and, as a consequence, the fundamental variables of one perturbation sector are related to the variables of the other sector. The relevance of such a symmetry in connection to the AdS/CFT correspondence is discussed, and future applications of the pertubation theory developed here are outlined.

Observables in terms of connection and curvature variables for Einstein’s equations with two commuting Killing vectors

Einstein’s equations with two commuting Killing vectors and the associated Lax pair are considered. The equations for the connection A ( ς , η , γ )= Ψ , γ Ψ −1 , where γ the variable spectral parameter are considered. A transition matrix T = A ( ς , η , γ ) A −1 ( ξ , η , γ ) for A is defined relating A at ingoing and outgoing light cones. It is shown that it satisfies equations familiar from integrable PDE theory. A transition matrix on ς = constant is defined in an analogous manner. These transition matrices allow us to obtain a hierarchy of integrals of motion with respect to time, purely in terms of the trace of a function of the connections g , ς g −1 and g , η g −1 . Furthermore, a hierarchy of integrals of motion in terms of the curvature variable B = A , γ A −1 , involving the commutator [ A (1), A (−1)], is obtained. We interpret the inhomogeneous wave equation that governs σ = lnN , N the lapse, as a Klein–Gordon equation, a dispersion relation relating energy and momentum density, based on the first connection observable and hence this first observable corresponds to mass. The corresponding quantum operators are ∂/∂ t , ∂/∂ z and this means that the full Poincare group is at our disposal.

Fundamental laser modes in paraxial optics: from computer algebra and simulations to experimental observation

We study multi-parameter solutions of the inhomogeneous paraxial wave equation in a linear and quadratic approximation which include oscillating laser beams in a parabolic waveguide, spiral light beams, and other important families of propagation-invariant laser modes in weakly varying media. A "smart" lens design and a similar effect of superfocusing of particle beams in a thin monocrystal film are also discussed. In the supplementary electronic material, we provide a computer algebra verification of the results presented here, and of some related mathematical tools that were stated without proofs in the literature. We also demonstrate how computer algebra can be used to derive some of the presented formulas automatically, which is highly desirable as the corresponding hand calculations are very tedious. In numerical simulations, some of the new solutions reveal quite exotic properties which deserve further investigation including an experimental observation.

An inverse source problem for a variable speed wave equation with discrete-in-time sources

We consider a time-dependent inverse source problem for the inhomogeneous wave equation, with sources that are delta-like in time, and (roughly speaking) non-negative and of limited oscillation in space. We take the displacement of the wave on a fixed detection surface, over some time interval, as our data. We then use time-reversal from the boundary to generate a related waveform, and discuss how to identify sources (as well as artifacts) from this.

Multi-parameter laser modes in paraxial optics

By computer algebra methods we study multi-parameter solutions of the inhomogeneous paraxial wave equation in a linear and quadratic approximation which include oscillating laser beams in a parabolic waveguide and spiral light beams in weakly varying media. A similar effect of superfocusing of particle beams in a thin monocrystal film is also discussed.

A fractional Fourier transform analysis of the scattering of ultrasonic waves.

Many safety critical structures, such as those found in nuclear plants, oil pipelines and in the aerospace industry, rely on key components that are constructed from heterogeneous materials. Ultrasonic non-destructive testing (NDT) uses high-frequency mechanical waves to inspect these parts, ensuring they operate reliably without compromising their integrity. It is possible to employ mathematical models to develop a deeper understanding of the acquired ultrasonic data and enhance defect imaging algorithms. In this paper, a model for the scattering of ultrasonic waves by a crack is derived in the time–frequency domain. The fractional Fourier transform (FrFT) is applied to an inhomogeneous wave equation where the forcing function is prescribed as a linear chirp, modulated by a Gaussian envelope. The homogeneous solution is found via the Born approximation which encapsulates information regarding the flaw geometry. The inhomogeneous solution is obtained via the inverse Fourier transform of a Gaussian-windowed linear chirp excitation. It is observed that, although the scattering profile of the flaw does not change, it is amplified. Thus, the theory demonstrates the enhanced signal-to-noise ratio permitted by the use of coded excitation, as well as establishing a time–frequency domain framework to assist in flaw identification and classification.

The variants of energy integral induced by the vibrating string

The equation which expresses the principle of energy conservation for the vibrating string can be derived from the homogeneous or inhomogeneous wave equation. In this article, we have checked the variants of energy equation induced by the vibrating string.