One-dimensional Wave Equation Research Articles

Resonant-State Expansion Applied to Non-Relativistic Wave Equation in One-Dimension

Study’s Excerpt The rigorous resonant-state expansion (RSE) method is extended to the non-relativistic one-dimensional wave equation. Resonant states (RSs) wave numbers as the unperturbed basis were employed to confirm the RSE's convergence to exact solutions. RSE's has potentials for systematically calculating RSs in complex quantum systems with multi-well potentials. Full Abstract The resonant-state expansion (RSE), a rigorous perturbation theory recently developed in electrodynamics, is here applied to the non-relativistic wave equation in one-dimension. The resonant states (RSs) wave numbers for the double well system are analytically calculated and used as the unperturbed basis for calculating the RSE. We demonstrate the efficiency of the RSE by verifying its convergence to the exact solution for a triple well potential. We show that for the chosen perturbations (i.e., for and ), the method is particularly suitable for calculating all the RSs within the spectrum.

Solution of certain problem with nonlocal boundary condition for one-dimensional wave equation

In this paper, we study a problem with nonlocal boundary condition for one-dimensional wave equation. To prove the solvability of the problem we construct biortogonal basis consisting of eigen- and adjoint-functions of not self adjoint operator.

Longitudinal wave propagation in FG rods under impact force

A well-known fact is that one-dimensional wave analysis is the theoretical basis of the famous Hopkinson bar dynamic testing technique. The current one-dimensional wave theory is mostly confined to the slender rods of isotropic materials. It is not easy to obtain an analytical solution to the wave equation of an anisotropic rod. In this work, rods with both elastic modulus and density graded in the length direction are presented and analyzed. The one-dimensional variable coefficient wave equation corresponding to the functionally graded rod is constructed and converted into a second-order variable coefficient partial differential equation using the Laplace approach. Then, the details of solving the partial differential equation of the second-order variable coefficients are given. It is worth noting that here we construct a variable coefficient equation that satisfies the form of Euler's equation. It is still difficult to obtain analytical solutions for other equations that do not satisfy this form. Subsequently, studies of the wave propagation characteristics of rods with different graded configurations are carried out. The theoretical results show that the wave propagation behavior and post-impact vibration of the rod are significantly influenced by the graded configuration. It is possible to adjust not only the impact response at the end but also the impact response in the middle of the rod by optimizing the design of the rod's graded configuration.

On the local linearization of the one-dimensional stochastic wave equation with a multiplicative space-time white noise forcing

On the local linearization of the one-dimensional stochastic wave equation with a multiplicative space-time white noise forcing

The time horizon for stochastic homogenization of the one-dimensional wave equation

The wave equation with stochastic rapidly oscillating coefficients can be classically homogenized on bounded time intervals; solutions converge in the homogenization limit to solutions of a wave equation with constant coefficients. This is no longer true on large time scales: Even in the periodic case with periodicity ε, classical homogenization fails for times of the order ε − 2 . We consider the one-dimensional wave equation with random rapidly oscillation coefficients on scale ε and are interested in the critical time scale ε − β from where on classical homogenization fails. In the general setting, we derive upper and lower bounds for β in terms of the growth rate of correctors. In the specific setting of i.i.d. coefficients with matched impedance, we show that the critical time scale is ε − 1 .

Classical solution to mixed problems from the theory of longitudinal impact on an elastic semi-infinite rod in the case of separation of the impacting body after the collision

In this work, we consider two coupled initial-boundary value problems, which, based on the Saint-Venant theory, model the longitudinal impact phenomena in a semi-infinite rod. The mathematical formulation of the problem is two mixed problems for the one-dimensional wave equation with conjugation conditions. The Cauchy conditions are specified on the spatial half-line. The initial condition for the partial derivative with respect to the time variable has a discontinuity of the first kind at one point. The boundary condition, which includes the unknown function and its first- and second-order partial derivatives, is specified on the time half-line. The solution is constructed by the method of characteristics in an explicit analytical form. The uniqueness of the solution is proved, and the conditions under which a piecewise-smooth solution exists are established. The classical solution to a mixed problem with matching conditions is considered.

Periodic output regulation for 1-D wave equation subject to non-collocated control

In this paper, we focus on the periodic output regulation for a one-dimensional wave equation where the performance output is non-collocated with the control input. The exosystems considered here are multi-periodic time-varying, capable of generating periodic reference signals and disturbances within the domain and at the boundaries, each with distinct periods. Initially, a backstepping-based state feedback regulator is designed by tackling an ill-posed periodic regulator equation. Subsequently, leveraging an external matrix, we design a novel time-varying observer tailored for the transformed wave equation and the exosystems. An output feedback regulator is then obtained by replacing the states with their estimations. The tracking error is shown to decay exponentially and the closed-loop system is proved to be bounded. Some numerical simulations are presented to validate the results.

A Hybrid Method for Solving the One-Dimensional Wave Equation of Tapered Sucker-Rod Strings

A Hybrid Method for Solving the One-Dimensional Wave Equation of Tapered Sucker-Rod Strings

Dynamics for wave equations connected in parallel with nonlinear localized damping

Abstract This study investigates the properties of solutions about one-dimensional wave equations connected in parallel under the effect of two nonlinear localized frictional damping mechanisms. First, under various growth conditions about the nonlinear dissipative effect, we try to establish the decay rate estimates by imposing minimal amount of support on the damping and provide some examples of exponential decay and polynomial decay. To achieve this, a proper observability inequality has been proposed and constructed based on some refined microlocal analysis. Then, the existence of a global attractor is proved when the damping terms are linearly bounded at infinity, a special weighting function has been used in this part, which eliminates undesirable terms of the higher order while contributing lower-order terms. Finally, we establish that the long-time behavior of solutions of the nonlinear system is completely determined by the dynamics of large finite number of functionals.

Development of the Rodsim numerical simulator for the study of a sucker rod pump, comparing the Gibbs (1963) and Lea (1990) models

It is of great importance to be fully cognizant of the sucker rod pump system in order to make reliable production forecasts and to evaluate potential strategies for optimizing production processes. A frequently employed methodology in such circumstances is numerical simulation. A numerical sucker road pump simulator can provide valuable insights into production performance by generating dynamometric charts that represent the efficiency of this activity. Since the rod column behaves like a thin rod, it is possible to represent the mathematical model by a one-dimensional wave equation that describes the forces acting on this rod column. Lea (1990), proposing an improvement of the work of Gibbs (1963), considers the effects of the viscosity of the fluid produced in a vertical well without heat exchange, the piston diameter, the production column and the rods to obtain the damping factor, thus being more complete and producing a more accurate result in the dynamometric charts. Both models are implemented and discussed herein. The equations are discretized and solved in the Matlab® environment. The code includes a graphical user interface that generates the internal Rodsim application. The application is used to solve a typical case of a sucker rod pump and to obtain the surface and downhole dynamometric plots for certain scenarios.

Polynomial stabilization of the wave equation with a time varying delay term in the dynamical control

We consider the one-dimensional wave equation with a time-varying delay term in the dynamical control. Under suitable assumptions, we show the well posedness of the problem. These results are obtained by using semi-group theory. Combining the multiplier method with a non linear integral inequality, a rational energy decay result of the system is established. A fundamental aspect of this paper is that our involved operator is time dependent, therefore the standard frequential method cannot be invoked.

Active disturbance rejection control approach to the output tracking of PDEs with general disturbances: A new perspective

In recent years, significant progress has been made in output regulation for systems described by partial differential equations (PDEs) through the active disturbance rejection control approach (ADRC). However, in all these studies, the disturbance typically manifests in a single channel and requires estimation, often necessitating additional measured signals. This is because the disturbance must be identified through the output. In this paper, we approach this problem from a novel perspective: disturbances occur in all channels, yet few measured signals are required. In most cases, only the tracking error is utilized as the measured signal. This marks a significant advancement in the application of ADRC to PDEs. We primarily demonstrate this approach by addressing the output tracking problem for a multi-dimensional heat equation, considering general disturbances across all possible channels. Additionally, we briefly discuss some other one-dimensional problems studied in the literature, such as the one-dimensional wave equation.

Study of Support Motion of a Finite Bar with a Boundary Damper and a Spring Using Analytical Approaches and FEM

In this paper, we solve the vibration problem of a finite bar with a viscously damped boundary in conjunction with a spring on the same side subject to the support motion on the other side. To avoid the computation of complex modes and the difficulty of orthogonal conditions for the partial differential equation (PDE) of a continuous system, two alternatives are employed. One is the analytical derivation by using the diamond rule of the method of characteristics. The advantage is that this method can yield an analytical solution. However, the disadvantage is that the method can be only applied to some simple problems such as a linear system and the one-dimensional (1D) wave equation. The other numerical method, finite element method (FEM), is incorporated into a general-purpose program to solve this problem. The advantage of the FEM is that this methodology can be applied to solve various problems such as different PDEs. Therefore, the solution obtained by using the FEM is compared with that of the analytical solution. Two special cases, only a spring and a damper alone, are also considered by using the FEM. Interestingly, the same silent area is captured in the displacement profile by using the FEM for the three cases. We also find that the displacement profiles have slope discontinuity which only occurs on the characteristic line. After the wave front arrives at the boundary, various responses appear due to different boundaries, spring, damper and both. It is found that the displacement amplitude of the general case is smaller than the other two special cases. This result matches the application of engineering practice.

Boundary controllability for variable coefficients one-dimensional wave equation with interior degeneracy

Abstract In this paper, we study boundary controllability for the linear extension problem of a wave equation with space-dependent coefficients and having an internal degeneracy. For this purpose, we mainly focus on the well-posedness and the boundary null controllability of a relaxed version of the original problem, namely, to some degenerate transmission problem. The key ingredient is to derive direct and inverse inequalities for the associated homogeneous degenerate adjoint problem. By these inequalities, we deduce that the transmission problem has a unique solution by transposition and this solution is null controllable. Moreover, we give an explicit formula of the controllability time.

Multiple solutions and conserved vectors of a shallow water wave equation arising in fluid mechanics; Lie group analysis

Investigation has shown that shallow water wave model equations constitute a class of hyperbolic partial differential equations describing the movement below a pressure surface in a fluid, and this makes them highly useful to be studied. Therefore, in this paper we analytically investigate a one-dimensional shallow water wave equation that is highly applicable in fluid dynamics. We leverage the robust nature of the Lie group technique to provide symmetries to the equation. Commutator and adjoint representations of symmetries were generated and presented in tabular form. Sequel to that, we perform reductions of the shallow water wave model under study via its point symmetries so that its exact solutions can be found. Thus, to achieve this, various combinations of the symmetries are made. Besides, we examine the travelling wave solutions of the model. The direct integration method is used to obtain elliptic integral functions as well as elliptic solutions in terms of the Weierstrass function. In addition, we utilize standard techniques such as (G′/G)-expansion, power series, and Kudryashov’s approaches to generate more closed-form solutions to the model. Other interesting solutions achieved include hyperbolic, trigonometric, rational, and exponential functions. In search of the physical meaning associated with the results, graphical demonstrations are included, and these are given in 2D, 3D, and density plots. Thereafter, the construction of conserved vectors for the aforementioned model is done with the use of the standard multiplier approach and Ibragimov’s theorem. These techniques give the opportunity to furnish various forms of conserved currents.

Synthetically non-Hermitian nonlinear wave-like behavior in a topological mechanical metamaterial

Topological mechanical metamaterials have enabled new ways to control stress and deformation propagation. Exemplified by Maxwell lattices, they have been studied extensively using a linearized formalism. Herein, we study a two-dimensional topological Maxwell lattice by exploring its large deformation quasi-static response using geometric numerical simulations and experiments. We observe spatial nonlinear wave-like phenomena such as harmonic generation, localized domain switching, amplification-enhanced frequency conversion, and solitary waves. We further map our linearized, homogenized system to a non-Hermitian, nonreciprocal, one-dimensional wave equation, revealing an equivalence between the deformation fields of two-dimensional topological Maxwell lattices and nonlinear dynamical phenomena in one-dimensional active systems. Our study opens a regime for topological mechanical metamaterials and expands their application potential in areas including adaptive and smart materials and mechanical logic, wherein concepts from nonlinear dynamics may be used to create intricate, tailored spatial deformation and stress fields greatly transcending conventional elasticity.

Exact Null Controllability of String Equations with Neumann Boundaries

This article focuses on the exact null controllability of a one-dimensional wave equation in noncylindrical domains. Both the fixed endpoint and the moving endpoint are Neumann-type boundary conditions. The control is put on the moving endpoint. When the speed of the moving endpoint is less than the characteristic speed, we can obtain the exact null controllability of this equation by using the Hilbert uniqueness method. In addition, we get a sharper estimate on controllability time that depends on the speed of the moving endpoint.

Global existence and asymptotic profile for a damped wave equation with variable-coefficient diffusion

We considered a Cauchy problem of a one-dimensional semilinear wave equation with variable-coefficient diffusion, time-dependent damping, and perturbations. The global well-posedness and the asymptotic profile are given by employing scaling variables and the energy method. The lower bound estimate of the lifespan to the solution is obtained as a byproduct.
 For more information see https://ejde.math.txstate.edu/Volumes/2024/04/abstr.html

Substructure elimination and binding method for vibration systems governed by a one-dimensional wave equation

Substructure elimination and binding method for vibration systems governed by a one-dimensional wave equation

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

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