Search Results (181)

We study the dynamics of observables that are conserved under the Hamiltonian evolution of a closed quantum system, but cease to be conserved when the system is coupled to a Markovian environment and described by a Lindblad master equation. Starting from the adjoint Lindblad equation, we derive elementary expressions for the time derivatives of the expectation value and second moment of an observable O, with particular emphasis on the case $[H,O]=0$ but ${\mathcal{L}}^{†}\left(O\right)\ne 0$. These formulae provide a direct assessment of how collapse operators break Hamiltonian conservation laws and generate fluctuations of formerly conserved quantities. The discussion is illustrated by analytic examples: one-qubit amplitude damping, a two-qubit excitation-number model, a momentum-diffusion model in which the mean is conserved while the variance grows, and the Jaynes–Cummings model. The latter also shows the complementary case of a reservoir coupled through a conserved quantity, where dephasing can occur without changing the statistics of that quantity. We finally comment on the relation between Lindblad source terms and idealized wave-function reduction models in which local conservation may hold only statistically. Full article

Hereditary damping and fractional attenuation are widely used to model wave propagation in complex media, but the variational and spectral origin of the corresponding nonlocal-in-time operators is often left implicit. In this work, we derive such operators from a minimal conservative field–reservoir model. A real scalar field is coupled locally to a continuum of harmonic reservoir modes, which are then eliminated exactly. The resulting reduced dynamics is a causal wave equation with a memory-friction term acting on the field velocity. The memory kernel is generated by the reservoir coupling spectrum through a cosine-transform relation, establishing a direct spectrum-to-kernel correspondence. This relation provides both a physical interpretation of hereditary damping and a practical admissibility criterion: macroscopic attenuation and dispersion arise from the delayed back-action of unresolved internal modes, while physically admissible kernels are constrained by the non-negativity of the underlying spectral density. The framework unifies several standard damping regimes. A broadband reservoir recovers the Markovian locally damped wave equation, reservoirs with a finite characteristic time generate finite-memory relaxation and frequency-dependent dispersion, and scale-free reservoir spectra produce power-law memory kernels. In the latter case, the hereditary damping operator reduces to a Caputo-type fractional derivative, showing that fractional wave attenuation can emerge as an effective reduced dynamics rather than being postulated phenomenologically. We further analyze dispersion, attenuation, causality, stability, and admissibility conditions in terms of the reservoir spectrum. The main contribution of the work is therefore to provide a variational and spectral derivation of hereditary and fractional wave damping, linking the structure of unresolved reservoir modes to macroscopic nonlocal wave dynamics. Full article

This work is concerned with the numerical treatment of a wave equation with fractional Kelvin–Voigt damping, where the viscoelastic contribution is described by a Caputo derivative in time acting on the elliptic part of the model. Such models are of interest because memory effects produce hereditary damping and reduced regularity near the initial time, which makes both the analysis and the numerical discretization more delicate than in the classical wave equation. We study the problem on a bounded convex domain under homogeneous Dirichlet boundary conditions and derive a solution representation that is suitable for regularity analysis. Based on this representation, we establish stability and smoothing estimates for both homogeneous data and forcing terms, with particular attention to the influence of nonsmooth initial data. For the spatial discretization, we employ a continuous Galerkin finite element method with piecewise linear elements and prove error estimates that are explicit in the regularity of the initial displacement, initial velocity, and source term. We show that the fully discrete approximation inherits the regularity-dependent behavior of the continuous problem and achieves optimal convergence in space together with second-order accuracy in time under appropriate assumptions on the data. Several numerical experiments are presented to illustrate the theoretical findings and to confirm the predicted convergence rates, thereby supporting the effectiveness of the proposed space–time discretization. Full article

We develop a fast Chebyshev spectral collocation method for a coupled system of nonlinear Klein–Gordon equations augmented by Caputo-type fractional memory integrals. The governing equations retain the classical second-order time derivative as the leading operator and incorporate weakly singular convolution integrals modelling viscoelastic memory damping. The spatial discretisation employs Chebyshev–Gauss–Lobatto collocation, while the temporal integration uses a Newmark scheme (${\beta }_{\mathrm{NM}}=1/4$) combined with an implicit–explicit linearisation in which the linear spatial operator is treated implicitly and the nonlinear terms are treated explicitly through a second-order extrapolation. This linearisation eliminates the need for Newton–Raphson iterations at each time step. To overcome the dense memory bottleneck arising from two distinct fractional orders $\alpha \ne \beta$, the convolution memory kernels are compressed by independent sum-of-exponentials approximations obtained from a double-exponential quadrature of the kernel’s integral representation, which significantly reduces the computational complexity of the history term. A rigorous stability estimate and a global convergence bound are established using a discrete Grönwall inequality. Numerical experiments confirm the theoretical temporal and spatial convergence rates and demonstrate the practical speed-up afforded by the sum-of-exponentials acceleration. A solitary wave collision scenario illustrates the method’s capability to capture asymmetric dispersive wakes generated by the fractional memory. Full article

A time-dependent nonlinear model is presented to describe internal gravity waves propagating upwards in the F-region of the Earth’s ionosphere. The model is based on a configuration where the background neutral velocity is constant, the geomagnetic field is approximately constant, and the angular gyrofrequency of the ions is much larger than the ion-neutral collision frequency, which is in turn larger than the angular frequency of the gravity waves. For small-amplitude waves the equations are linearized, and a time-dependent analytical solution is obtained for the special case corresponding to the limit of zero vertical-to-horizontal aspect ratio. This analytical solution and the linear numerical results for general aspect ratio show that in the limit of infinite time the linear solution approaches a steady state in which the ion damps the wave amplitude in the vertical direction. For the more general configuration that includes larger amplitude waves, time-dependent nonlinear numerical simulations show that, in the presence of the ion drag, there are wave mean-flow interactions even in the absence of vertical shear in the background neutral flow. With time, the perturbation develops a zero-wavenumber component corresponding to a wave-induced mean flow acceleration, which depends on the dip angle of the geomagnetic field and on the aspect ratio. Full article

This paper develops an analytical treatment of vibro-acoustic wave propagation in a cylindrical waveguide containing two clamped elastic membranes and a central flexible-shell segment. The acoustic field obeys the time-harmonic Helmholtz equation, the shell motion is described by Donnell–Mushtari thin-shell theory under axisymmetric loading, and the membrane response is governed by classical membrane theory and incorporated through a tailored Galerkin scheme. The resulting coupled fluid–structure boundary-value problem is solved by the Mode-Matching Method: the acoustic potentials are expanded in orthogonal radial eigenfunctions within each subregion, and continuity of pressure, normal velocity, and structural displacement are enforced at every interface. The mirror symmetry of the configuration is exploited by an exact decomposition into symmetric and anti-symmetric sub-problems, each of which reduces to a truncated linear algebraic system of dimension $4N+4$ for the unknown modal amplitudes. Acoustic power-balance identities provide a quantitative consistency check on the numerical implementation and diagnose convergence with respect to the truncation order; structural damping is accommodated through complex-modulus substitutions for the shell and the membrane tension without altering the algebraic structure of the system. The numerical results demonstrate that the dual-membrane configuration delivers transmission-loss values exceeding $25\mathrm{dB}$ across the low-frequency band relevant to HVAC and automotive applications, with a representative plateau near $13\mathrm{dB}$ at the reference geometry, through resonance-driven modal coupling between the acoustic field and the compliant interfaces. Parametric studies identify the excitation frequency, the inner-membrane radius, the shell radius, and the chamber length as effective design parameters for tuning the attenuation. The formulation furnishes a unified and computationally efficient analytical tool for predicting and optimising noise attenuation in flexibly coupled cylindrical duct systems. Full article

In this study, the energy harvesting mechanism of a two-degree-of-freedom (2-DOF) oscillating foil under near-surface conditions and the underlying influence of structural parameters are systematically investigated. Numerical simulations are conducted using the open-source CFD platform OpenFOAM and the waves2Foam toolbox. The free surface is captured using the volume of fluid (VOF) method, while the heave and pitch motions of the foil are simulated via the overWaveDyMFoam solver, coupling 6-DOF dynamic equations with the overset grid technique. The results demonstrate that the periodic evolution and shedding of the leading-edge vortex (LEV) fundamentally drive the self-sustained oscillation of the foil. Moreover, the phase synchronization between the fluid-induced force and the kinematic response serves as the core mechanism for efficient energy extraction. Structural parameters critically regulate these characteristics: stiffness coefficients dictate the natural frequency and phase coordination, thereby modulating the overall motion response. Notably, a local resonance occurs when the system’s natural frequency approaches the fluid’s vortex shedding frequency, inducing the maximum kinematic response. Within the investigated parameter space, the system achieves a peak energy harvesting efficiency of 45.6% and a maximum average power coefficient of 1.15. Finally, the damping coefficients are found to primarily govern the response amplitude and the viscous dissipation of the system. Full article

Weight reduction and dynamic performance optimization are critical for airborne direct-drive VICTS satellite communication antennas, which require lightweight, high-speed, and high-precision rotation. Traditional vibration suppression methods, such as uniform support layout and added damping, rely heavily on empirical trial and error, lack targeted modal control, and cannot balance lightweight design with dynamic stiffness. To address these issues, this paper proposes a wave-theory-based dynamic modeling and rapid optimization method for multi-layer rotating components in direct-drive VICTS antennas. The kinematic model of the rotating ring and ball revolution excitation are derived using the annular wave equation and bearing kinematics. A Modal Blocking Mechanism is established: placing support balls at positions satisfying the half-wavelength constraint suppresses target mode shapes via wave interference, achieving vibration attenuation at the source. A homogenization equivalent method based on RVE is developed for irregular cross-section rings, yielding analytical expressions for in-plane equivalent elastic modulus and out-of-plane equivalent shear modulus. These parameters are integrated into the wave equation to analytically solve vibration modes, avoiding iterative finite element computations. A rapid multi-objective optimization framework is then constructed, minimizing the structural weight and maximizing the modal separation interval under dynamic stiffness and excitation frequency constraints. Numerical simulations, FE analysis, and prototype tests validate the method: the maximum analytical error is only 3.1%. Compared with uniform support designs, the optimized structure achieves a 40% weight reduction, a 40% increase in minimum modal separation, and a 65% reduction in the RMS tracking error. This work provides an efficient, deterministic dynamic design method for large-diameter ring structures, transforming vibration control from empirical adjustment into a precise, physics-informed optimization. Full article

This paper presents a computational study of wave–bathymetry and wave–structure interaction problems using advanced numerical techniques based on high-fidelity, two-phase Navier–Stokes (TpNS) flow and reduced-order, fully nonlinear potential flow models. For high-fidelity simulations, the TpNS equations are discretized using the finite-element method, with free-surface evolution captured through a hybrid level-set (LS) and volume-of-fluid (VOF) formulation. A monolithic, phase-conservative LS equation is introduced to mitigate mass loss and interface smearing, combined with a semi-implicit projection scheme. Hydrodynamic forces are resolved using a high-order, phase-resolving cut finite-element method (CutFEM), which enables the representation of complex solid geometries within a fixed background mesh. An equivalent polynomial of Heaviside and Dirac distributions ensures accurate evaluation of surface and volume integrals. Hence, no explicit generation of cut cell meshes, adaptive quadrature, or local refinement is required. For reduced-order modeling, a fast regularized boundary integral method (RBIM) is employed to solve the fully nonlinear potential flow. Singular and near-singular integrals are treated using a subtract-and-addition technique based on auxiliary functions derived from Stokes’ theorem, allowing direct application of high-order quadrature without conventional boundary element discretization. An arbitrary Lagrangian–Eulerian (ALE) formulation is adopted to enforce free-surface boundary conditions while avoiding excessive mesh distortion. The proposed approaches are applied to investigate highly nonlinear wave transformation over complex bathymetry and wave-induced dynamics of floating structures, including eddy-making damping effects. Numerical results are validated against experimental measurements. These two modeling approaches represent complementary levels of physical fidelity and computational efficiency, and their systematic comparison clarifies the trade-offs between computational accuracy, efficiency, and cost for practical marine problems. Full article

We analyze the spectral stability of travelling waves in a $\delta$-regularized dissipative sine-Gordon equation modelling refined long Josephson junction dynamics. Linearization about a wave yields a singularly perturbed fourth-order spectral problem with intrinsic slow–fast spatial structure. Using an Evans-function formulation on a domain of consistent spatial splitting, we establish a local factorization separating slow and fast modes and prove that the $\delta$-induced fast subsystem remains uniformly hyperbolic and does not generate an additional point spectrum near $\lambda =0$. Hence, the local point spectrum coincides with that of the classical dissipative sine-Gordon equation. Numerical computations of the essential spectrum and Evans winding numbers confirm the analysis and show that the higher-order terms enhance high-frequency damping without altering low-frequency spectral stability. Full article

In this contribution, we propose and study long-time behaviors of a new class of N-dimensional delayed Kirchhoff–Carrier-type problems with variable transfer coefficients involving a logarithmic nonlinearity. We take into account the dependence of diffusion and damping coefficients on the position and direction, as well as the presence of different types of delays. This class of nonlocal anisotropic and nonlinear wave-type equations with multiple time-varying mixed delays and dampings, of a fairly general form, containing several arbitrary functions and free parameters, is of the following form: ${\displaystyle \frac{{\partial }^{2}u}{\partial t^2}-{div(\mathcal{K}(\Vert \sigma \nabla u\Vert }_{L^2(\Omega )}^2)A_{\sigma }{(\mathbf{x})\nabla u)+\mathcal{M}(\Vert u\Vert }_{L^2(\Omega )}^2)u-div(\zeta (t)A_{\sigma }(\mathbf{x})\nabla \frac{\partial u}{\partial t})+d_0(t)\frac{\partial u}{\partial t}+{\mathbb{D}}_r(\mathbf{x},t;\frac{\partial u}{\partial t})=\mathbb{G}(u),}$ where $u(\mathbf{x},t)$ is the state function, $\mathcal{M}$ and $\mathcal{K}$ are the nonlocal Kirchhoff operators and the nonlinear operator $\mathbb{G}(u)$ corresponds to a logarithmic source term. The symmetric tensor $A_{\sigma }$ describes the anisotropic behavior and processes of the system, and the operator ${\mathbb{D}}_r$ represents the multiple time-varying mixed delays related to velocity ${\displaystyle \frac{\partial u}{\partial t}}$. Our problem, which encompasses numerous equations already studied in the literature, is relevant to a wide range of practical and concrete applications. It not only considers anisotropy in diffusion, but it also assumes that the strong damping can be totally anisotropic (a phenomenon that has received very little mathematical attention in the literature). We begin with the reformulation of the problem into a nonlinear system coupling a nonlocal wave-type equation with ordinary differential equations, with the help of auxiliary functions. Afterward, we study the local existence and some necessary regularity results of the solutions by using the Faedo–Galerkin approximation, combining some energy estimates and the logarithmic Sobolev inequality. Next, by virtue of the potential well method combined with the Nehari manifold, conditions for global in-time existence are given. Finally, subject to certain conditions, the exponential decay of global solutions is established by applying a perturbed energy method. Many of the obtained results can be extended to the case of other nonlinear source terms. Full article

Owing to its substantial implications for black hole spectroscopy, spectral instability has attracted considerable attention in the literature. While the emergence of such instability is attributed to the non-Hermitian nature of the gravitational system, it remains sensitive to various factors. In this work, we conduct a focused analysis of black hole spectral instability using the Pöschl–Teller potential as a toy model. We investigate the dependence of the resulting spectral instability on the magnitude, spatial scale, and localization of deterministic and random perturbations in the effective potential of the wave equation, and discuss the underlying physical interpretations. It is observed that small perturbations in the potential initially have a limited impact on the less damped black hole quasinormal modes, with deviations typically around their unperturbed values, a phenomenon first derived by Skakala and Visser in a more restrictive context. In the higher-overtone region, the deviation propagates, amplifies, and eventually gives rise to spectral instability and, inclusively, bifurcation in the quasinormal mode spectrum. While deterministic perturbations give rise to a deformed but well-defined quasinormal spectrum, random perturbations lead to uncertainties in the resulting spectrum. Nonetheless, the primary trend of the spectral instability remains consistent, being sensitive to both the strength and location of the perturbation. However, we demonstrate that the observed spectral instability might be suppressed for perturbations that are physically appropriate. Full article

Aiming at the limited applicability of traditional empirical formulas in roll prediction for offshore wind turbine installation vessels (WTIVs), this study proposes a collaborative verification method that integrates model tests with Computational Fluid Dynamics (CFD) simulations. This approach reveals the influence mechanism of the fluid trapped in the leg-spudcan well region on the roll period and damping, facilitating high-precision prediction. A numerical model of the WTIV in a jack-up operating condition was established, and a CFD method based on the RANS equations was employed alongside experimental data for synergistic analysis. The results demonstrate that the fluid in the leg-spudcan well generates a significant additional moment of inertia, which reduces the natural roll period by approximately 7% and increases the damping coefficient by approximately 58%. Furthermore, an increase in leg length leads to a linear increase in damping and a linear decrease in the roll period. The motion response transfer functions derived from tests and the motion response errors of key structures in irregular waves are generally less than 10%. On this basis, a motion response conversion method applicable to any location on the entire ship is derived, providing a reliable numerical analysis tool for WTIV seakeeping evaluation and operational window assessment. Full article

A temporally periodic model is presented to describe the vertical profile of internal gravity waves in the F region of the Earth’s ionosphere where the waves are subject to a magnetic force due to the high concentration of ions. The configuration studied is representative of the situation where the geomagnetic field is approximately constant and is so strong that the angular gyrofrequency of the ions is very large compared with the ion-neutral collision frequency, which is in turn larger than the angular frequency of the gravity waves. We examine the situation where the gravity wave amplitude is small enough that the equations for the neutral fluid flow can be linearized. This allows for the description of wave propagation in terms of a system of coupled equations that include the effects of ion drag on waves for any orientation of the magnetic field. It is assumed that the background neutral fluid flow is nonzero and horizontal, but there is no vertical shear, and that the wave amplitude depends on altitude only, and an exact analytical solution is readily found. This dynamical model captures some essential features of ionospheric gravity waves that are consistent with observational measurements. In particular, the ion drag acts to damp the waves in the direction of vertical propagation and increase their vertical wavelength relative to the corresponding wavelength in the neutral atmosphere. The vertical damping rate and the vertical wavelength both depend on the dip angle of the magnetic field. When the magnetic field acts in the direction of the gravity lines of constant phase, there is no damping, and the vertical wavelength is the same as that of the corresponding waves in the neutral atmosphere. The dip angles that produce stronger damping also result in waves with greater wavelengths. Full article

This study examines outflow boundary conditions (BCs) in computational fluid dynamics (CFD) simulations of a transition duct with and without guide vanes that converts supersonic flow exiting a rotating detonation combustor (RDC) to subsonic flow to drive a turbine. Since the flow exiting the transition duct has swirling shock waves with significant spatial and temporal variations in pressure, temperature, and Mach number, imposing proper BCs poses a challenge. To ensure all swirling shock waves exit the transition duct without creating non-physical reflected waves at its outlet, this study examined three outflow BCs: (1) the average pressure imposed at the duct’s outlet, (2) a nonreflecting BC (NRBC) with a specified average pressure imposed at the duct’s outlet, (3) the average pressure imposed at the outlet of an extension duct made up of a buffer layer and a sponge layer. This study is based on the three-dimensional, unsteady density-weighted-ensemble-averaged continuity, Navier–Stokes, and energy equations for a thermally perfect gas closed by the realizable k–ε model and “enhanced” wall functions. The results obtained show that imposing an average pressure at the transition duct’s outlet produces spurious waves that degrade the physical meaningfulness of the solution. When the NRBC was applied, swirling shock waves exited the duct’s outlet without creating spurious waves. However, its usage requires the gas to be thermally, as well as calorically, perfect, which this study shows could be a concern. By imposing the average pressure at the outlet of an extension duct, the gas does not need to be calorically perfect. The results obtained show the effects of the sponge layer’s length and coarsening ratio on damping nonuniformities in non-physical reflected waves to ensure the flow exiting the transition duct’s outlet can do so as if there are no boundaries present and has the desired average pressure—even though the BC is applied at the extension duct’s outlet. Full article