One-dimensional Linear Wave Equation Research Articles

Аналитическая модель одномерной нестационарной динамики одиночной частицы в акустическом и электрическом полях

This paper presents the results of calculations of the dynamics of an electrically neutral and electrically charged particle in a closed channel. A mathematical model of the dynamics of a single particle in an acoustic field is presented, in which the potential of the acoustic field is given by a one-dimensional linear wave equation. The dynamics of the particle was described by a differential equation that puts the dependence of the particle velocity on the forces applied to it - the Stokes force acting from the side of the acoustic field, and the Coulomb force acting from the side of the electric field. It is revealed that the amplitude of oscillations of an electrically neutral particle is less than the amplitude of oscillations of an electrically charged particle in an electric field. It is determined that at a small amplitude of the piston oscillations, the amplitude of the change in the speed of an electrically charged particle under the action of electric field forces exceeds the amplitude of the change in the speed of an electrically neutral particle. In this paper, we analyzed the influence on the dynamics of an electrically charged particle in an acoustic field for stationary and nonstationary distributions of the electric field potential. The paper presents calculations of the dynamics of an electrically charged particle in a periodic electric field with different frequencies of electric field variation. It was found that when a particle is exposed to an alternating electric field that is in phase with the gas velocity field, the particle velocity increases, if the frequency of the alternating electric field is in antiphase with the gas velocity, then the particle velocity decreases. It is determined that if a stationary electric field acts on a particle, then the particle drifts to that part of the channel to which a potential is applied, the sign of which is opposite to the sign of the particle charge.

Geometric Optimization for Structure-Preserving Model Reduction of Hamiltonian Systems

Classical model reduction methods disregard the special symplectic structure associated with Hamiltonian systems. A key challenge in projection-based approaches is to construct a sym-plectic basis that captures the essential system information. This necessitates the computation of a so-called proper symplectic decomposition (PSD) of a given sample data set. The PSD problem allows for a canonical formulation as an optimization problem on the symplectic Stiefel manifold. However, as with their Euclidean counterparts, symplectic projectors only depend on the underlying symplectic subspaces and not on the selected symplectic bases. This motivates to tackle the PSD problem as a Riemannian optimization problem on the symplectic Grassmann manifold, i.e., the matrix manifold of symplectic projectors. Initial investigations on this manifold feature in a recent preprint of the authors. In this work, we investigate the feasibility and performance of this approach on two academic numerical examples. More precisely, we calculate an optimized PSD for snapshot matrices that stem from solving the one-dimensional linear wave equation and the one-dimensional nonlinear Schrödinger equation.

Propagation of singularities for the stochastic wave equation

We study the existence and propagation of singularities of the solution to a one-dimensional linear stochastic wave equation driven by an additive Gaussian noise that is white in time and colored in space. Our approach is based on a simultaneous law of the iterated logarithm and general methods for Gaussian processes.

Comparative Review on Computational Performance of Multistep Schemes in Solving One-Dimensional Linear Wave Equation

Among several numerical methods used to solve the hyperbolic model of the linear wave equation, single-step algorithms can be the more popular ones. However, these algorithms are time-consuming while incurring numerical inaccuracy. Thus, multistep methods can be a suitable option as it has a high order of accuracy. This study aims to investigate and compare the computational performance of these multistep schemes in solving hyperbolic model based on one-dimensional linear wave equation. The techniques studied in this paper comprise the two-step Lax-Wendroff method, MacCormack method, second-order upwind method, Rusanov-Burstein-Mirin method, Warming-Kutler-Lomax method, and fourth-order Runge-Kutta method. Finite difference method is applied in discretisation. Our simulation found that although higher-order multistep methods are more stable than single-step algorithm, they suffer numerical diffusion. The two-step Lax-Wendroff method outperforms other schemes, although it is relatively simple compared with the other three and four steps schemes. The second-order upwind method is attractive as well because it is executable even with a high Courant number.

Exact controllability for a one-dimensional degenerate wave equation in domains with moving boundary

In this paper, we study one-dimensional linear degenerate wave equations in domains with moving boundary. We deduce the exact controllability for degenerate wave equation by Hilbert Uniqueness Method when the control acts on the moving boundary. Moreover, an explicit expression for the controllability time is given.

High-order adapter schemes for cell-centered finite difference method

The present paper proposes that reconstruction scheme and interpolation scheme can be converted into each other through two series of adapter schemes, which include reconstruction-to-interpolation (RI) adapter schemes and interpolation-to-reconstruction (IR) adapter schemes. For the high-order spatial discretization of the compressible Navier-Stokes equations, the RI adapter schemes can be used to derive interpolation schemes for the interpolation-based cell-centered finite difference method from the available optimized reconstruction schemes. The main advantage of the interpolation-based cell-centered finite difference method is the capability to realize high-order discretization on curvilinear grids with both shock-capturing capability and satisfaction of the geometric conservation law. In the present paper, we first derive the IR adapter schemes by comparing the difference schemes with their strong conservative forms. We then develop the corresponding RI adapter schemes by inversing the IR adapter schemes. Thereafter, the applications to the one-dimensional linear wave equation and the one-dimensional inviscid Burgers' equation have been briefly discussed. Finally, to demonstrate the application to three-dimensional Navier-Stokes equations, three highly optimized nonlinear reconstruction schemes are adapted into the corresponding interpolation ones through RI adapter schemes, which include WENO-CU6, WGVC-WENO7 and OMP6 schemes. The new interpolation schemes from adapters are compared with their original reconstruction ones through several benchmark cases. No noticeable robustness loss or accuracy loss has been found in these cases, indicating the effectiveness of the adapter schemes. No obvious increase in time cost has been observed, indicating the efficiency of the adapter schemes.

Chaotic behaviors of one dimensional wave equations with van der Pol nonlinear boundary conditions

For one-dimensional linear wave equations with van der Pol boundary conditions, there have existed several different ways in the literature to characterize the complexity of their solutions. In this paper, we give a new way to characterize the dynamical behaviors of the systems, including almost equicontinuity and Devaney chaos. In order to do this, we consider a one-dimensional system (I, f) and its functional envelope (L1(I,I),Hf) and show that Hf is weakly mixing if and only if f is; Hf is Devaney chaotic if f is weakly mixing; Hf is almost equicontinuous if f has an attracting 2-period orbit. Those abstract results have their own significance and can be applied to the linear wave equations with van der Pol boundary conditions.

Comparison of solutions of linear and non-linear shallow water wave equations using homotopy perturbation method

PurposeThis paper aims to solve linear and non-linear shallow water wave equations using homotopy perturbation method (HPM). HPM is a straightforward method to handle linear and non-linear differential equations. As such here, one-dimensional shallow water wave equations have been considered to solve those by HPM. Interesting results are reported when the solutions of linear and non-linear equations are compared.Design/methodology/approachHPM was used in this study.FindingsSolution of one-dimensional linear and non-linear shallow water wave equations and comparison of linear and non-linear coupled shallow water waves from the results obtained using present method.Originality/valueCoupled non-linear shallow water wave equations are solved.

Null controllability of some degenerate wave equations

This paper is devoted to a study of the null controllability problems for one-dimensional linear degenerate wave equations through a boundary controller. First, the well-posedness of linear degenerate wave equations is discussed. Then the null controllability of some degenerate wave equations is established, when a control acts on the non-degenerate boundary. Different from the known controllability results in the case that a control acts on the degenerate boundary, any initial value in state space is controllable in this case. Also, an explicit expression for the controllability time is given. Furthermore, a counterexample on the controllability is given for some other degenerate wave equations.

Long wave propagation and run-up in converging bays

Analytical solutions are derived to describe two-dimensional wave evolution in converging bays. Three bay types of different cross-sections are studied: U-shaped, V-shaped and cusped bays. For these bays, the two-dimensional linear shallow water equations can be reduced to one-dimensional linear dispersive wave equations if the transverse flow acceleration inside them is assumed to be small. The derived solutions are characterized as the leading-order plane-wave solutions with higher-order corrections for two-dimensionality due to wave refraction. Wave amplitude longitudinally increases with different rates for the three bay types, whereas it exhibits weak parabolic variations in the transverse direction. Wave refraction significantly affects relatively short waves, contributing to wave energy transfer to the inner bay in a different manner depending on the bay type. The perturbation analysis of very high-order wave celerity suggests that the solutions are valid only when the ratio of the bay width to the wavelength is smaller than a certain limit that differs with bay type. Beyond the limit, the higher-order effect is no longer a minor correction, implying that wave behaviours become highly two-dimensional and possibly cause total reflection. The higher-order effect on the run-up height at the bay head is found to be small within the applicable range of the solution, and thus, the run-up formula neglecting the transverse flows has a wide validity. We also discuss the limitation of run-up height by wave breaking on the basis of a breaking criterion from previous studies.

Optimal shaping of acoustic resonators for the generation of high-amplitude standing waves

Within this paper, optimal shaping of acoustic resonators for the generation of high-amplitude standing waves through the use of evolutionary algorithms is discussed. The resonator shapes are described using sets of control points interconnected with cubic-splines. Positions of the control points are calculated by means of an evolutionary algorithm in order to maximize acoustic pressure amplitude at a given point of the resonator cavity. As an objective function for the optimization procedure, numerical solution of one-dimensional linear wave equation taking into account boundary-layer dissipation is used. Resonator shapes maximizing acoustic pressure amplitude are found in case of a piston, shaker, or loudspeaker driving. It is shown that the optimum resonator shapes depend on the method of driving. In all the cases, acoustic field attains higher amplitude in the optimized resonators than in simple-shaped non-optimized resonators of similar dimensions. Theoretical results are compared with experimental data in the case of a loudspeaker driving, good agreement of which is achieved.

Nonlinear wave run-up in bays of arbitrary cross-section: generalization of the Carrier–Greenspan approach

AbstractWe present an exact analytical solution of the nonlinear shallow water theory for wave run-up in inclined channels of arbitrary cross-section, which generalizes previous studies on wave run-up for a plane beach and channels of parabolic cross-section. The solution is found using a hodograph-type transform, which extends the well-known Carrier–Greenspan transform for wave run-up on a plane beach. As a result, the nonlinear shallow water equations are reduced to a single one-dimensional linear wave equation for an auxiliary function and all physical variables can be expressed in terms of this function by purely algebraic formulas. In the special case of a U-shaped channel this equation coincides with a spherically symmetric wave equation in space, whose dimension is defined by the channel cross-section and can be fractional. As an example, the run-up of a sinusoidal wave on a beach is considered for channels of several different cross-sections and the influence of the cross-section on wave run-up characteristics is studied.

Analytical Closed-Form Solution for 1D Linear Kinematic Overland Flow under Moving Rainstorms

AbstractAn analytical solution for overland flow under (upstream and downstream) moving storms that uses Laplace transformation to solve the one-dimensional (1D) linear kinematic wave equation is presented. This solution, which corresponds to a single continuous function for the total space-time domain of the overland hydrograph, enables evaluation of the discharge over time for the total drainage plane surface. The result was compared with another analytical solution, a numerical simulation, and experimental runs using a laboratory flume. The comparison showed very good fit, and the proposed analytical solution was thus regarded as validated. By applying the model to hypothetical catchments and storm patterns, distinct hydrologic responses for upstream and downstream moving storms were identified.

Distributed feedback design for systems governed by the wave equation

This article deals with the geometric control of a one-dimensional non-autonomous linear wave equation. The idea consists in reducing the wave equation to a set of first-order linear hyperbolic equations. Then, based on geometric control concepts, a distributed control law that enforces the exponential stability and output tracking in the closed-loop system is designed. The presented control approach is applied to obtain a distributed control law that brings a stretched uniform string, modelled by a wave equation with Dirichlet boundary conditions, to rest in infinite time by considering the displacement of the middle point of the string as the controlled output. The controller performances have been evaluated in simulation by considering both tracking and disturbance rejection problems. The robustness of the controller has also been studied when the string tension is subjected to sudden variations.

Homogenization of the variable-speed wave equation

The existence of traveling waves in strongly inhomogeneous media is reviewed in the framework of the one-dimensional linear wave equation with a variable speed. Such solutions are found by using a homogenization, in which the variable-coefficient wave equation is transformed to a constant-coefficient Klein–Gordon equation. This transformation exists if and only if the spatial variations of the variable speed satisfy a constraint expressed by a second-order ordinary differential equation with two arbitrary parameters. All solutions of the constraint are found in explicit form, and our results obtained by this systematic procedure include many previous results found in the literature. Further, we show that the wave equation under the same constraint on the variable speed admits a two-parameter Lie group of nontrivial commuting point symmetries.

CHAOTIC VIBRATIONS OF THE ONE-DIMENSIONAL MIXED WAVE SYSTEM

In this paper, we consider the initial-boundary value problem of the one-dimensional linear mixed wave equation ωtt - dωtx - c2ωxx = 0 (d ∈ ℝ, c > 0) on an interval, where the boundary condition at the left endpoint is linear, pumping energy into the system, while the boundary condition at the right endpoint has odd-degree nonlinearity. This problem is said to be the one-dimensional mixed wave system. The solution of the one-dimensional mixed wave system corresponds to the iteration of an interval map h. Thus, the mixed wave system is said to be chaotic if the interval map h is chaotic in the sense of Li–Yorke. In this paper, we show that the mixed wave system is chaotic under some conditions.

Distributed Geometric Control of Wave Equation

An approach for the geometric control of a one-dimensional non-autonomous linear wave equation is presented. The idea consists in reducing the wave equation to a set of first-order linear hyperbolic equations. Then based on geometric control concepts, a distributed control law that enforces stability and output tracking in the closed-loop system is designed. The presented control approach is applied to obtain a distributed control law that brings a stretched uniform string, modeled by a wave equation with Dirichlet boundary conditions, to rest in infinite time by considering the displacement of the middle point of the string as the controlled output. The controller performances have been evaluated in simulation.

Very high-order compact finite difference schemes on non-uniform grids for incompressible Navier–Stokes equations

This article presents a family of very high-order non-uniform grid compact finite difference schemes with spatial orders of accuracy ranging from 4th to 20th for the incompressible Navier–Stokes equations. The high-order compact schemes on non-uniform grids developed in Shukla and Zhong [R.K. Shukla, X. Zhong, Derivation of high-order compact finite difference schemes for non-uniform grid using polynomial interpolation, J. Comput. Phys. 204 (2005) 404] for linear model equations are extended to the full Navier–Stokes equations in the vorticity and streamfunction formulation. Two methods for the solution of Helmholtz and Poisson equations using high-order compact schemes on non-uniform grids are developed. The schemes are constructed so that they maintain a high-order of accuracy not only in the interior but also at the boundary. Second-order semi-implicit temporal discretization is achieved through an implicit Backward Differentiation scheme for the linear viscous terms and an explicit Adam–Bashforth scheme for the non-linear convective terms. The boundary values of vorticity are determined using an influence matrix technique. The resulting discretized system with boundary closures of the same high-order as the interior is shown to be stable, when applied to the two-dimensional incompressible Navier–Stokes equations, provided enough grid points are clustered at the boundary. The resolution characteristics of the high-order compact finite difference schemes are illustrated through their application to the one-dimensional linear wave equation and the two-dimensional driven cavity flow. Comparisons with the benchmark solutions for the two-dimensional driven cavity flow, thermal convection in a square box and flow past an impulsively started cylinder show that the high-order compact schemes are stable and produce extremely accurate results on a stretched grid with more points clustered at the boundary.

Chaotic vibration of the one-dimensional linear wave equation with a van der Pol nonlinear boundary condition

The one-dimensional linear wave equation with a van der Pol nonlinear boundary condition is one of the simplest models that may cause isotropic or nonisotropic chaotic vibrations. It characterizes the nonisotropic chaotic vibration by means of the total variation theory. Some results are derived on the exponential growth of total variation of the snapshots on the spatial interval in the long-time horizon when the map and the initial condition satisfy some conditions.

A new characterization of nonisotropic chaotic vibrations of the one-dimensional linear wave equation with a van der Pol boundary condition

The one-dimensional linear wave equation with a van der Pol nonlinear boundary condition is one of the simplest models that may cause isotropic or nonisotropic chaotic vibrations (Trans. Amer. Math. Soc. 350 (1998) 4265–4311, Internat. J. Bifur. Chaos 8 (1998) 423–445, Internat. J. Bifur. Chaos 8 (1998) 447–470, J. Math. Phys. 39 (1998) 6459–6489, Internat. J. Bifur. Chaos 12 (2002) 535–559). In this paper, we characterize nonisotropic chaotic vibration by means of the total variation theory. We obtain the classification results on the growth of the total variation of the snapshots on the spatial interval in the long-time horizon with respect to two parameters entering different regimes in R 2.