A decoupled elastic wave equation for CO2 monitoring and its numerical simulation

Numerical investigation of an explicit weak Galerkin scheme for a class of weakly damped semi-linear wave equations

Cauchy problem for stochastic regularized nonlinear dispersive wave equations

Wave equations with nonlocal conditions appear in many scientific and engineering applications, such as, the population dynamics, the mathematical biology, and the materials science. The numerical challenge mainly stems from nonlocal terms, whose global coupling degrades the efficiency and stability of classical methods. In recent years, physics-informed neural networks (PINN) have achieved notable success in solving partial differential equations. In this paper, we propose an enhanced physics-informed neural network for wave equations subject to nonlocal conditions, termed NPINN+. By exploiting an equivalent transformation of the nonlocal condition, the original problem is reformulated into a wave equation satisfying Neumann boundary conditions with an additional integral-form source term. NPINN+ employs a single neural network to provide a unified representation of the spatiotemporal solution, while incorporating the governing equation, derivative information, initial and boundary conditions, and nonlocal constraints into a unified physics-informed loss function, enabling effective capture of the underlying physical features. Furthermore, a residual-based dynamic sampling strategy and a SoftAdapt-driven adaptive loss weighting mechanism are introduced to enhance accuracy and training robustness. Numerical experiments on regular domains demonstrate the effectiveness of the proposed method, and its extension to star-shaped domains is achieved via a polar coordinate transformation. Comparative results with PINN, APINN, and RAR-PINN show that NPINN+ consistently achieves superior accuracy and stability.

Abstract Magnetoelastic coupling provides a direct link between magnetic and elastic properties and forms the basis of a wide class of sensors and resonant devices. In this work, we present a simple and didactic approach to magnetoelastic resonance in amorphous ferromagnetic ribbons, suitable for undergraduate teaching and low-cost laboratory activities. The longitudinal dynamics of the resonator are described using classical wave equations with damping, leading to analytical expressions for the resonance frequency, dispersion relation, and electrical impedance. Experimental results demonstrate the viscous damping, mass loading, and ΔE effect, which describes the change in the Young’s modulus of magnetostrictive materials when a magnetic field is applied. The clear correspondence between mechanical perturbations and measurable electrical signals makes magnetoelastic resonators an effective platform for teaching concepts from waves, elasticity, magnetism, and experimental physics within a single integrated experiment.

This paper is devoted to the study of the global behavior of solutions to a class of strongly damped wave equations involving the p–Laplacian operator and logarithmic source terms. More precisely, we consider the problem u tt − Δ u t − Δ p u = λ | u | α − 2 uln ⁡ ( 1 + | u | ) + γ | ∇u | β ln ⁡ ( 1 + | ∇u | ) , ( x , t ) ∈ R N × ( 0 , + ∞ ) , where Δ p u = div ( | ∇u | p − 2 ∇u ) with p>2, and 0 $ ]]> λ , γ , α , β > 0 are given constants. The presence of logarithmic nonlinearities combined with nonlinear diffusion and strong damping introduces significant analytical difficulties, since the logarithmic terms exhibit a growth behavior that lies between polynomial and exponential nonlinearities. To overcome these challenges, we employ a rescaled test function method together with suitable logarithmic inequalities. Under appropriate conditions on the spatial dimension and the nonlinear exponents, we prove the nonexistence of nontrivial global weak solutions. Furthermore, we derive an explicit upper bound for the lifespan of local weak solutions in terms of the initial data and the parameters of the problem. The obtained results illustrate how strong damping, p–Laplacian diffusion, and logarithmic source terms interact with each other. In particular, this interaction leads to the appearance of a critical threshold phenomenon that differs from the one observed in the classical power-type nonlinear case.

Abstract The one-dimensional dynamic contact problem describing a rigid obstacle collided by an elastic bar is considered in gravitational field. The collision problem is non-smooth with respect to velocity and axial strain, and formulated as a variational inequality. For the corresponding wave equation subject to complementary conditions which are imposed at the contact boundary, its nonlinear solution is expressed with the help of Fourier Series representing longitudinal vibrations. Moreover, before the bar rebound, an analytical solution comprising piece-wise quadratic function is constructed on a partition of the rectangular space-time domain along characteristics. The analytical benchmark is implemented for low initial speeds in numerical experiments solving the variational inequality over uniform space-time triangulation with the Space-Time Primal-Dual Active Set method of semi-smooth Newton type.

Fast frequency-domain symbolic coherence total focusing method for array electromagnetic acoustic transducer.

ABSTRACT Non-contact real-time monitoring of cardiopulmonary signals can achieve early warning and diagnosis and treatment of cardiovascular diseases. However, the signal transmitted from the human body lying on the bed to the sensor through the monitoring pad will produce loss, resulting in a weak cardiopulmonary signal monitored by the sensor. Therefore, it is of great significance to study the propagation mechanism of cardiopulmonary vibration signals. This paper proposes the Kelvin-Voigt model and derives the wave equation to determine the attenuation coefficient of cardiopulmonary physiological vibration signals propagating through viscoelastic materials to FBG sensors. Based on viscoelastic structures and the principle of energy conservation, the propagation mechanism of cardiopulmonary vibration signals is investigated. A physical model is established for the transmission of these signals through monitoring pads to FBG sensors, alongside a strain transfer model for the sensors. This reveals the strain transfer patterns of cardiopulmonary dynamic signals through the sensors, with finite element simulation analysis employed to evaluate stress-strain values.

Conservation laws, modulation instability and analysis of the Rosenau regularized long wave equation via Lie symmetry

The inverse problem of designing materials to achieve desired acoustic functionality while strictly adhering to acoustic principles remains an unresolved scientific challenge. This work introduces a generative adversarial network based on the wave equation (Wave-GAN), in which the governing equation is directly embedded into the training process to iteratively optimize the material density distribution. The physics-guided framework enables the model to learn acoustic patterns directly from sound and generate material distributions with specified functionalities. As a verification example, a speaker recognition task was conducted. The results demonstrate that Wave-GAN produces physically consistent materials, achieving a recognition accuracy of 95.6%. This opens a promising direction for fully physics-driven material design at the interface of acoustics and machine learning.

Steady hydroelastic cnoidal wave on a thin flexural plate floating on a shallow water surface

Efficient linear implicit invariant-preserving fourth-order numerical scheme for damped nonlinear fourth-order wave equation

A family of finite spherical pulses for the wave equation

Abstract We present the first rigorous experimental numerical and experimental investigation of the propagation of Gaussian vortex beams, which are Gaussian beams imprinted with a spiral phase, under the realistic condition of an aperture-limited optical system. By combining approximate analytic results with a numerical solution of the paraxial wave equation, we produce a simple empirical model that describes the radius of the vortex singularity and its relationship with the imprinted orbital angular momentum ℓ and the spatial bandwidth of the optical system. We compare our empirical model with experimental data and show that we achieve excellent agreement for beams with ℓ = 1, 2, 5, and 10 over a range of spatial bandwidths.

Global existence and blow-up for a p-Laplacian wave equation with logarithmic nonlinearity and singular dissipation

Numerical study of solitary wave equations of fractional order via modified cubic B-spline collocation technique

• Comparative analysis of Finite Difference, Finite Volume, and Spectral Elements methods for full 3D time-domain acoustic wave propagation modelling. • Benchmarks ranging from simplified geometries that enable analytical comparison, to complex heterogeneous domains. • Implementation of a dedicated Finite volume-based acoustic solver in OpenFOAM with absorbing boundaries. • Comparison of omnidirectional and directional sources to analyze directivity effects on the resulting acoustic wave filed. • Best applicability range of each numerical method for near- and far- field acoustic prediction. A comparative study of three numerical methods - Finite Difference (FD), Finite Volume (FV), and Spectral Element Method (SEM) - for modeling underwater acoustic propagation is presented. The time-domain acoustic wave equation is solved using an in-house FD code, the open-source SPECFEM3D software for SEM, and a newly developed FV-based acoustic solver implemented and released within the OpenFOAM framework, extending a software environment traditionally used for computational fluid dynamics to underwater acoustics applications. The methods are systematically assessed through benchmark problems, ranging from homogeneous unbounded and semi-infinite domains to the Pekeris waveguide and a Gaussian canyon. Comparisons with analytical solutions demonstrate that all solvers accurately reproduce monopole and dipole radiation in simplified configurations. However, the analysis reveals that directional sources introduce non-trivial numerical sensitivities, even in simple environments. These effects manifest as spurious reflections and dispersion-related distortions, whose severity depends on the source implementation and the numerical scheme. The results show that SPECFEM3D generally provides the highest accuracy and robustness in heterogeneous and geometrically complex environments, while the in-house FD code and FV-based solver are more sensitive to dispersion but can recover accuracy through increased spatial resolution. Strategies to mitigate source-related artifacts, such as non-reflective hard sources and reduced source regions, are discussed. A preliminary investigation of moving sources highlights their straightforward implementation in FD and FV solvers, while requiring additional care within the SPECFEM3D framework. Overall, this work provides practical guidance on the accuracy, robustness, and applicability of different solvers for simulating underwater noise in near- and far-field conditions, while laying the ground for future source–propagation coupling within acoustic analogy frameworks in OpenFOAM.

Rotating waves for nonlinear wave equations with angular velocities on a positive-measure set

Stability of the wave equation with a saturated dynamic boundary control