Dispersive Wave Propagation Research Articles

On some new travelling wave solutions and dynamical properties of the generalized Zakharov system.

This study examines the extended version of the Zakharov system characterizing the dispersive and ion acoustic wave propagation in plasma. The genuine, non-dispersive field depicts a shift in plasma ion density from its equilibrium state, whereas the complex, dispersive field depicts the fluctuating envelope of a highly oscillatory field of electricity. The main focus of the analysis is on employing the expanded Fan sub-equation approach to achieve some novel travelling wave structures including the explicit, periodic, linked wave, and other new exact solutions are developed for different values of this parameter. Three dimensional graphs are utilised to examine the properties of the obtained solutions. Furthermore, ideas from planar dynamical theory are applied in this work to analyse the intricate behaviour of the analysed model. Sensitivity analysis, multistability, quasi-periodic and chaotic patterns, Poincaré map, and the Lyapunov characteristic exponent are used to analyse the dynamical features.

Low-cost instrumentation for high frequency ultrasonic guided wave laboratory research in free rock bolts

Guided wave ultrasound offers the ability to inspect slender structures from a single transducer location. However, the propagation of guided waves is more complex than the propagation of bulk waves used in conventional ultrasonic testing, and the experimental equipment required is more sophisticated. Typical equipment is expensive and may limit research and industrial application. This paper investigates what can be achieved by using low-cost instrumentation for laboratory measurements. The application is the measurement of guided waves in rock bolts (cylindrical rods) at relatively high frequencies (5 MHz). The dispersive wave propagation requires the use of narrowband excitation signals. The use of a bench-top waveform generator and oscilloscope is compared to the use of a compact and low-cost USB oscilloscope, which includes an arbitrary waveform generator. The introduction of a pre-amplifier mitigated the poorer noise performance of the USB oscilloscope, and comparable results were obtained from the two setups. It is demonstrated that waves propagated 9 m in a free rock bolt, with 10 dB/m attenuation, could be detected and analyzed with a measurement setup costing approximately USD 1110.

Ultra-wideband longitudinally coupled-resonator filters on lithium niobate using periodically slotted SiO2 as an acoustic coupler

This paper discusses the applicability of periodically slotted SiO2 as a coupler for ultra-wideband longitudinally coupled-resonator filters. It is shown that a −3 dB fractional bandwidth of more than 31% can be achieved by using lower-order three resonance modes. It is also shown that a periodically slotted SiO2 structure can also control the frequency dispersion of wave propagation along the surface, and transverse modes are suppressed well for one particular mode under a proper setting. Then the piston design is applied for the remaining two modes, and the spurious-free and flat-passband triple-mode filter is synthesized.

Solitons, dispersive shock waves and Noel Frederick Smyth

Noel Frederick Smyth (NFS), a Fellow of the Australian Mathematical Society and a Professor of Nonlinear Waves in the School of Mathematics at the University of Edinburgh, passed away on February 5, 2023. NFS was a prominent figure among applied mathematicians who worked on nonlinear wave theory in a broad range of areas. Throughout his academic career, which spanned nearly forty years, NFS developed mathematical models, ideas, and techniques that have had a large impact on the understanding of wave motion in diverse media. His major research emphasis primarily involved the propagation of solitary waves, or solitons, and dispersive shock waves, or undular bores, in various media, including optical fibers, liquid crystals, shallow waters and atmosphere. Several approaches he developed have proven effective in analyzing the dynamics and modulations of related wave phenomena. This tribute in the journal of Wave Motion aims to provide a brief biographical sketch of NFS, discuss his major research achievements, showcase his scientific competence, untiring mentorship and unwavering dedication, as well as share final thoughts from his former students, colleagues, friends, and family. The authors had a special connection with NFS on both on personal and professional levels and hold deep gratitude for him and his invaluable work. In recognition of his achievements in applied mathematics, Wave Motion hosts a Special Issue entitled “Modelling Nonlinear Wave Phenomena: From Theory to Applications,” which presents the recent advancements in this field.

Transverse-inertia-induced wave dispersion in plate–shell chains with a variable Poisson ratio

One-dimensional shell chain structures can cause stress wave dispersion, and the dispersion ability is mainly affected by the apparent elastic Poisson ratio of the unit cell composing the chain. In this study, a novel plate–shell chain structure with a variable Poisson ratio is proposed by combining bent plates and arc shells (BPAS). The dispersive wave propagation features in the BPAS chains under striker impact are studied theoretically, numerically, and experimentally. An equivalent continuum model of the unit cell under compression in a linear-elastic deformation regime is established, which contains two material parameters, i.e., the apparent elastic modulus and the apparent elastic Poisson ratio. It is found that the unit cell can reach a relatively large Poisson ratio by adjusting its structural parameters. A continuum-based wave propagation model considering transverse inertia is developed, and an explicit analytical solution is obtained to represent the dispersion wave propagation behavior within a linear-elastic deformation regime. The model can reasonably predict the force and velocity evolution features of the chains and is verified by the numerical/experimental tests. The results show that the BPAS chain with a large Poisson ratio exhibits superior wave dispersion performances. The underlying mechanism is that more impact energy is dispersed by the transverse movement of the cells in the BPAS chain. Increasing the aspect ratio of the unit cell can improve the wave dispersion ability of the BPAS chains, achieving a high attenuation of the stress wave and is helpful for impact mitigation.

Nonlocal elasticity of Klein‐Gordon type with internal length and time scales: Constitutive modelling and dispersion relations

AbstractIn this work, a nonlocal elasticity theory with nonlocality in space and time is presented by considering nonlocal constitutive equations with a dynamical scalar nonlocal kernel. Based on the proposed theory, we consider the isotropic nonlocal elasticity of Klein‐Gordon type including a characteristic internal time scale parameter in addition to the characteristic internal length scale parameter. The dispersion relations for homogeneous isotropic media in the framework of nonlocal elasticity of Klein‐Gordon type are analytically determined. The obtained results reveal the ability of the presented nonlocal elasticity model to predict, for the first time in the framework of nonlocal elasticity, in addition to the acoustic modes (low‐frequency modes), optic modes (high‐frequency modes) as well as frequency band‐gaps. The phase and group velocities for all four modes (acoustic and optic branches of longitudinal and transverse waves) are determined showing that all four modes exhibit normal dispersion with positive group velocity. The presented model allows for physically realistic dispersive wave propagation.

Boundless Metamaterial Experimentation: Physical Realization of a Unidirectional Virtual Periodic Boundary Condition

We experimentally implement a one-directional virtual geometric periodicity in an elastic metamaterial. First, unwanted boundary reflections at the domain ends are cancelled through the iterative injection of the polarity-reversed reflected wave field. The resulting boundless experimental state allows for a much better analysis of the influence of metamaterials on the propagating wave field. Subsequently, the propagating wave field exiting on one end of the structure is reintroduced at the opposite end, creating a virtual geometric periodicity in one propagation direction. We find that the experimentally observed band gap converges to the analytical solution through the introduction of the virtual periodicity. The established work flow introduces an approach to the experimental investigation and validation of metamaterial prototypes in the presence of strongly dispersive wave propagation and internal scattering. The fully data-driven ad hoc treatment of boundary conditions in metamaterial experimentation with arbitrary mechanical properties enables reflection suppression, virtual periodicity, and the introduction of more general fictitious boundaries.

The study of nonlinear dispersive wave propagation pattern to Sharma–Tasso–Olver–Burgers equation

This paper discusses the wave propagation to the nonlinear Sharma–Tasso–Olver–Burgers (STOB) equation which is used as the governing model in different fields. Natural phenomena are typically complex and nonlinear, defying simple linear superposition. Researchers have been studying a wide range of natural phenomena in depth, and nonlinear science has gradually become a part of people’s consciousness. One of the most significant research questions in nonlinear science centers around the nonlinear evolution equation and its precise solution. We have secured different shapes of the solitary wave solutions including kink-type, shock-type and combined solitary wave solutions with the assistance of recently developed integration tool, namely the new extended direct algebraic method (NEDAM). Additionally, the solutions for the hyperbolic, exponential and trigonometric functions are retrieved. Moreover, based on a comparison of our results to those that are well known, the study indicates that our solutions are innovative. Using proper parameters in numerical simulations and physical explanations, it is possible to demonstrate the significance of the results. The results of this research can improve the nonlinear dynamic behavior of a system and indicate that the methodology employed is adequate. It is proposed that the offered method can be utilized to support nonlinear dynamical models applicable to a wide variety of physical situations. We hope that a wide spectrum of engineering model professionals will find this study to be beneficial.

Solitary Waves in Two-Layer Fluid with Piecewise Exponential Stratification

The problem on internal stationary waves in a two-layer fluid with density, depending exponentially on the depth inside the layers and having a jump at the interface, is considered. A non-linear equation of the second-order long-wave approximation is derived, and their asymptotic submodels, describing solitary waves of finite amplitude, are discussed. Dispersive properties and wave propagation regimes depending on dimensionless parameters of the background piecewise constant flow are studied.

Integrability and new periodic, kink-antikink and complex optical soliton solutions of [formula omitted]-dimensional variable coefficient DJKM equation for the propagation of nonlinear dispersive waves in inhomogeneous media

This article deals with the (3+1)-dimensional variable coefficient Date–Jimbo–Kashiwara–Miwa equation which describes the behavior of waves as they travel through nonlinear dispersive media. The integrability of this equation has been evaluated by mean of Painlevé property. An auto-Bäcklund transformation of the considered equation is being produced with the aid of Painlevé analysis. To get the analytic solutions of the considered equation, the auto-Bäcklund transformation method is employed. Three new analytical solution families including complex solution have been derived successfully via the auto-Bäcklund transformation method. All the results have been plotted in three-dimension by taking different parameter values and functions. These solutions depict the kink-antikink wave, periodic wave, complex periodic wave and complex kink-antikink wave solutions for the considered equation.

UNRAVELING THE COMPLEX DYNAMICS OF FLUID FLOW IN POROUS MEDIA: EFFECTS OF VISCOSITY, POROSITY, AND INERTIA ON THE MOTION OF FLUIDS

This study investigates novel solitary wave solutions of the Gilson–Pickering ([Formula: see text]) equation, which is a model that describes the motion of a fluid in a porous medium. An analytical scheme is applied to construct these solutions, utilizing the extended Khater method in conjunction with the homogenous balance technique. The derived expressions for the solitary wave solutions are exact and are presented in terms of hyperbolic functions. The [Formula: see text] equation is valuable for a wide range of applications, including oil and gas reservoir engineering, groundwater flow, and flow in biological tissues. Additionally, this model is employed to describe the behavior of waves in various physical systems such as fluids and plasmas. Specifically, it models the propagation of dispersive waves in a media that exhibits both dispersion and dissipation. To ensure the accuracy of the constructed solutions, a numerical scheme is employed. The properties of the solitary wave solutions are analyzed, and their physical implications are explored. The results of this investigation reveal a rich variety of solitary wave solutions that exhibit interesting behaviors, including oscillatory and non-oscillatory behavior, which are elucidated through various types of distinct graphs. Consequently, this study provides significant insights into the behavior of fluid flow in porous media and its applications in various fields, including oil and gas reservoir engineering and groundwater flow modeling. The analytical and numerical methods employed in this investigation demonstrate their potential for studying nonlinear evolution equations and their applications in the physical sciences.

The scaled boundary finite element method for dispersive wave propagation in higher‐order continua

AbstractThe classical theory of elasticity relies on the perfect structure of matter. No matter how small a medium is, it always stays homogeneous—but the reality is different. Even though the classical continuum mechanics suffices well for describing phenomena that evolve at super‐microscopic scales, it ceases to reproduce reasonable responses when the size effects are pronounced. It is due to the local constitutive equations of classical continuum mechanics, which are devoid of any length scales associated with the underlying microstructure. The gradient‐dependent theory of elasticity redresses this shortcoming by incorporating the kinematic quantities of the corresponding representative volume element by enriching the differential equations with higher‐order spatial and temporal derivatives. In this study, the scaled boundary finite element method is formulated for the equations of motion with higher‐order inertia terms. To this end, the semi‐discretized scaled boundary finite element equations of the medium are derived by introducing the scaled boundary transformation of geometry to the gradient‐enriched equations of motion and applying the Galerkin method of weighted residuals. It is shown that the well‐established available solution methods are incapable of handling the frequency‐domain representation of the derived formulation. Accordingly, a numerical solution method based on the shooting technique is proposed. The solution procedure is formulated for general numerical integration schemes via the infinite Taylor series. The evolution of the impedance‐diffusion matrix and contribution of inter‐subdomain forces and tractions are extracted. Four different numerical integration methods are employed to describe the solution procedure. In addition, a comparison regarding their computational efficiency is presented. For the sake of verification, the scaled boundary finite element solutions of three numerical examples are compared with reference solutions that are obtained using finite element models with extremely fine meshes. The convergence trends are also presented for both ‐ and ‐refinement methods. The results demonstrate the capability of the proposed formulation in reproducing accurate responses.

Extended water wave systems of Boussinesq equations on a finite interval: Theory and numerical analysis

Considered here is a class of Boussinesq systems of Nwogu type. Such systems describe propagation of nonlinear and dispersive water waves of significant interest such as solitary and tsunami waves. The initial-boundary value problem on a finite interval for this family of systems is studied both theoretically and numerically. First, the linearization of a certain generalized Nwogu system is solved analytically via the unified transform of Fokas. The corresponding analysis reveals two types of admissible boundary conditions, thereby suggesting appropriate boundary conditions for the nonlinear Nwogu system on a finite interval. Then, well-posedness is established, both in the weak and in the classical sense, for a regularized Nwogu system in the context of an initial-boundary value problem that describes the dynamics of water waves in a basin with wall-boundary conditions. In addition, a new modified Galerkin method is suggested for the numerical discretization of this regularized system in time, and its convergence is proved along with optimal error estimates. Finally, numerical experiments illustrating the effect of the boundary conditions on the reflection of solitary waves by a vertical wall are also provided.

Surface similarity parameter: A new machine learning loss metric for oscillatory spatio-temporal data

Supervised machine learning approaches require the formulation of a loss functional to be minimized in the training phase. Sequential data are ubiquitous across many fields of research, and are often treated with Euclidean distance-based loss functions that were designed for tabular data. For smooth oscillatory data, those conventional approaches lack the ability to penalize amplitude, frequency and phase prediction errors at the same time, and tend to be biased towards amplitude errors. We introduce the surface similarity parameter (SSP) as a novel loss function that is especially useful for training machine learning models on smooth oscillatory sequences. Our extensive experiments on chaotic spatio-temporal dynamical systems indicate that the SSP is beneficial for shaping gradients, thereby accelerating the training process, reducing the final prediction error, increasing weight initialization robustness, and implementing a stronger regularization effect compared to using classical loss functions. The results indicate the potential of the novel loss metric particularly for highly complex and chaotic data, such as data stemming from the nonlinear two-dimensional Kuramoto–Sivashinsky equation and the linear propagation of dispersive surface gravity waves in fluids.

Nonlinear dispersive wave propagation pattern in optical fiber system

Nonlinear fractional-order partial differential equations are an important tool in science and engineering for explaining a wide range of nonlinear processes. We consider the nonlinear space-time fractional modified Korteweg de Vries and sine-Gordon equations in this article and extract diverse sorts of traveling wave as well as soliton solutions using a typical approach, namely the generalized G′/G-expansion approach. These equations are used to model a wide range of nonlinear phenomena, including plasma physics, high-intensity laser-generated plasma, ultra-small electronic devices, optical fibers, control theory, turbulence, acoustics, and others. The fractional derivative is defined in the sense of the beta-derivative established by Atangana and Baleanu. Some standard shapes of waveforms, including kink type, singular-kink type, bell-shaped, periodic-type, single soliton, multiple soliton type, and several other types of solitons, have been established. To validate the physical compatibility of the results, the 3D, contour, and vector plots have been delineated using consistent values of the parameters. The approach used in this study to extract inclusive and standard solutions is approachable, efficient, and speedier in computing.

Nonlocal elasticity of Klein–Gordon type: Fundamentals and wave propagation

A nonlocal elasticity theory with nonlocality in space and time is developed by considering nonlocal constitutive equations with a dynamical scalar nonlocal kernel function. The proposed theory is specified to isotropic nonlocal elasticity of Klein–Gordon type, which is an extension of nonlocal elasticity of Helmholtz type, and it possesses one characteristic internal length scale parameter as well as one characteristic internal time scale parameter in addition to the elastic material parameters. Dynamical nonlocal kernels are analytically derived as retarded Green functions of the Klein–Gordon operator. The dispersion relations of the considered isotropic nonlocal elasticity model of Klein–Gordon type are analytically determined. The obtained results reveal the advantage of the proposed nonlocal elasticity model, that is, its ability to predict for the first time in the framework of nonlocal elasticity in addition to the acoustic modes (low-frequency modes), optic modes (high-frequency modes) as well as frequency band-gaps between the acoustic and optic modes. A qualitative examination of the phase and group velocities for all four modes (acoustic and optic branches of longitudinal and transverse waves) shows that the proposed model allows for physically realistic dispersive wave propagation.

Lie symmetries, invariant solutions and phenomena dynamics of Boiti–Leon–Pempinelli system

The present work deals with highly nonlinear dispersive water equation named Boiti–Leon–Pempinelli (BLP) system, which is a well known two dimensional generalization of sinh-Gordon equation. The BLP system describes interaction of horizontal wave component propagating in plane in infinite narrow channel with constant depth. The Lie symmetry method is employed to derive infinitesimal generators under invariance conditions of one-parameter transformations. The invariance conditions ensure the reducibility of independent variables and preserve the characteristics of system. The commutative relations of infinitesimal vectors show infinite dimensional algebra of symmetries. Thereafter, similarity variables are derived using infinitesimal generators, which lead to first symmetry reductions. Thus, a repeated process of symmetry reductions provide equivalent system of ODEs. These ODEs are integrated and thus some invariant solutions of BLP system are constructed under parametric constraints. The derived solutions are general than previous researches [J Q Yu et al 2010 Exact solutions and conservation laws of (2 + 1)-dimensional Boiti–Leon–Pempinelli equation, Appl. Math. Comput. 216 2293–2300; M Kumar, R Kumar, 2014 On new similarity solutions of the boiti–leon–pempinelli system, Commun. Theor. Phys. 61 121–126; M Kumar et al 2015 Some more similarity solutions of the (2 + 1)-dimensional BLP system, Comput. Math. Appl. 70 212–221; J Fei et al 2015 Symmetry reduction and explicit solutions of the (2 + 1)-dimensional boiti–leon–pempinelli system, Appl. Math. Comput. 268 432–438; M Kumar et al 2021 Some more invariant solutions of (2 + 1)-water waves, Int. J. Appl. Comput. Math. 7 18] as they contain all four arbitrary functions involved in infinitesimals and several free parameters as well. The deduction of previous results obtained by taking particular choices of arbitrary functions and constants, ensure novelty and importance of these solutions as well as the authenticity and effectiveness of Lie symmetry method. To demonstrate the distinct physical structures of phenomena, these solutions are expanded via numerical simulation. Thus, doubly soliton, line soliton, multi soliton, soliton fusion and fission nature have been analyzed. The graphical representations are seemed useful to understand the dynamics of BLP system and propagation of dispersive water waves in infinite narrow channel.

Lax–Wendroff Schemes with Polynomial Extrapolation and Simplified Lax–Wendroff Schemes for Dispersive Waves: A Comparative Study

One of the features of Boussinesq-type models for dispersive wave propagation is the presence of mixed spatial/temporal derivatives in the partial differential system. This is a critical point in the design of the time marching strategy, as the cost of inverting the algebraic equations arising from the discretization of these mixed terms may result in a nonnegligible overhead. In this paper, we propose novel approaches based on the classical Lax–Wendroff (LW) strategy to achieve single-step high-order schemes in time. To reduce the cost of evaluating the complex correction terms arising in the Lax–Wendroff procedure for Boussinesq equations, we propose several simplified strategies which allow to reduce the computational time at fixed accuracy. To evaluate these qualities, we perform a spectral analysis to assess the dispersion and damping error. We then evaluate the schemes on several benchmarks involving dispersive propagation over flat and nonflat bathymetries, and perform numerical grid convergence studies on two of them. Our results show a potential for a CPU reduction between 35 and 40% to obtain accuracy levels comparable to those of the classical RK3 method.

On Lie symmetries and invariant solutions of Broer–Kaup–Kupershmidt equation in shallow water of uniform depth

The dynamics of atmosphere and ocean can be examined under different circumstances of shallow water waves like shallow water gravity waves, Kelvin waves, Rossby waves and inertio-gravity waves. The influences of these waves describe the climate change adaptation on marine environment and planet. Therefore, the present work aims to derive symmetry reductions of Broer–Kaup–Kupershmidt equation in shallow water of uniform depth and then a variety of exact solutions are constructed. It represents the propagation of nonlinear and dispersive long gravity waves in two horizontal directions in shallow water. The invariance of test equations under one parameter transformation leads to reduction of independent variable. Therefore, twice implementations of symmetry method result into equivalent system of ordinary differential equations. Eventually, the exact solutions of these ODEs are computed under parametric constraints. The derive results entail several arbitrary constants and functions, which make the findings more admirable. Based on the appropriate choice of existing parameters, these solutions are supplemented numerically and show parabolic nature, intensive and non-intensive behavior of solitons.

Elastic wave propagation in nonlinear two-dimensional acoustic metamaterials

This study describes the wave propagation in a periodic lattice which is formed by a spring-mass two-dimensional structure with local Duffing nonlinear resonators. The wave propagation characteristics of the system are evaluated by using the perturbation method to determine the dispersion relationships and wave propagation characteristics in the nonlinear two-dimensional acoustic metamaterials. A quantitative study of wave amplitude is carried out to determine the maximum allowable wave amplitude for the whole structures under the assumption of small parameters. In particular, the harmonic balance method is introduced to investigate the frequency response and effective mass of the nonlinear systems. We find that the dispersion relations and group velocity of unit cell are related to wave amplitude. Furthermore, the dual-wave vector is observed in the nonlinear systems. Numerical simulations validate the dispersion analytical results. The results can be used to tune wave propagation in the nonlinear acoustic metamaterials and provide some ideas for the study of nonlinear metamaterials.