Finite-amplitude Waves Research Articles

New soliton solutions and modulation instability analysis of fractional Huxley equation

In this research, the new auxiliary equation method (NAEM) for higher order nonlinear fractional Huxley equation is being employed to extricate the novel soliton solutions using Beta and M-Truncated fractional derivatives. For waves of finite amplitude, the Huxley equation demonstrates a substantial transfer of spectrum energy. A comparison of the solutions of the model with both fractional derivatives is also included in this research. Various kinds of solitary traveling wave solutions, such as trigonometric, hyperbolic, exponential, rational functions, etc., are found. These types of solutions demonstrate the superiority of the novelty of the method. This method’s key advantage over others is that it provides more broad solutions with certain flexible parameters. 3D and 2D graphs are used graphically to demonstrate the dynamical structures of the solutions. The results are presented in a way that demonstrate the usefulness and competence of the approach used to handle various nonlinear fractional partial differential equations. Lastly, we investigate the comparison of the gain spectra for modulation instability and the depiction of certain noteworthy outcomes by illustratively depicting the 2D figures produced by carefully considering the parameters

Finite-amplitude elastic waves in viscoelastic channel flow from large to zero Reynolds number

Using branch continuation in the FENE-P model, we show that finite-amplitude travelling waves borne out of the recently discovered linear instability of viscoelastic channel flow (Khalid et al., J. Fluid Mech., vol. 915, 2021, A43) are substantially subcritical reaching much lower Weissenberg ( $Wi$ ) numbers than on the neutral curve at a given Reynolds ( $Re$ ) number over $Re \in [0,3000]$ . The travelling waves on the lower branch are surprisingly weak indicating that viscoelastic channel flow is susceptible to (nonlinear) instability triggered by small finite-amplitude disturbances for $Wi$ and $Re$ well below the neutral curve. The critical $Wi$ for these waves to appear in a saddle node bifurcation decreases monotonically from, for example, $\approx 37$ at $Re=3000$ down to $\approx 7.5$ at $Re=0$ at the solvent-to-total-viscosity ratio $\beta =0.9$ . In this latter creeping flow limit, we also show that these waves exist at $Wi \lesssim 50$ for higher polymer concentrations, $\beta \in [0.5,0.97)$ , where there is no known linear instability. Our results therefore indicate that these travelling waves, found in simulations and named ‘arrowheads’ by Dubief et al. (Phys. Rev. Fluids, vol. 7, 2022, 073301), exist much more generally in $(Wi,Re, \beta )$ parameter space than their spawning neutral curve and, hence, can either directly, or indirectly through their instabilities, influence the dynamics seen far away from where the flow is linearly unstable. Possible connections to elastic and elasto-inertial turbulence are discussed.

Three-dimensional simulations of sound propagation in a trumpet with accurate mouthpiece shank geometry

The length and bore geometry of musical instruments directly influences the quality of sound that can be produced. In brass instruments, nonlinear effects from finite-amplitude wave propagation can lead to wave distortion giving sounds a brassy timbre [3,5,16,22,29]. In this paper, we propose a three-dimensional model to describe nonlinear wave propagation in a trumpet and investigate the importance of the mouthpiece shank geometry. The time pressure waveforms corresponding to B3b and B4b notes at the mouthpiece were recorded at the mouthpiece shank are recorded and used as inputs for our model. To incorporate compressibility and nonlinear effects, the compressible Euler equations are used to describe the evolution of the nonlinear waves. The system was solved numerically using the discontinuous Galerkin method. The numerical results were compared with the waveforms of the musical notes, which were also measured outside the bell of the trumpet, to validate our approach. Simulations were run for both notes in computational trumpets where different bore geometries were modelled. Our results demonstrate that the shape of the narrow region near the mouthpiece greatly influences the wave propagation and must be considered in a trumpet model.

Numerical analysis of the limits of effective medium theory for nonlinearly propagating dispersive waves

Acoustic metameterials are an increasingly popular approach to control both linear and nonlinear sound propagation. In this approach, heterogeneous structures are engineered to behave as predetermined continuous media. A key requirement for a system to be accurately approximated as a homogeneous material for wave motion is that the smallest length scale of the waveform be much larger than any length scale associated with the microscale structure of the system. However, the fact that the characteristic length scales of a finite-amplitude waveform can change due to nonlinear effects means that care must be taken when analyzing such signals in an acoustic metamaterial. This talk presents recent work toward understanding the breakdown of effective medium theory as finite-amplitude waves propagate in a heterogeneous system and weak shocks or solitons begin to form. A finite-difference time-domain code was used to model the propagation of a pulse of sound in a one-dimensional propagation domain using both a full description of the system and an effective-medium description accounting for effective mass density, compressibility, coefficient of nonlinearity, and dispersion. The difference between the solutions of the two models provides a method by which one can estimate the error associated with using effective properties.

Quantifying Eddy Generation and Dissipation in the Jet Response to Upper- versus Lower-Level Thermal Forcing

Abstract The relative roles of upper- and lower-level thermal forcing in shifting the eddy-driven jet are investigated using a multilevel nonlinear quasigeostrophic channel model. The numerical experiments show that the upper-level thermal forcing is more efficient in shifting the eddy-driven jet. The finite-amplitude wave activity diagnostics of numerical results show that the dominance of the upper-level thermal forcing over the lower-level thermal forcing can be understood from their different influence on eddy generation and dissipation that affects the jet shift. The upper-level thermal forcing shifts the jet primarily by affecting the baroclinic generation of eddies. The lower-level thermal forcing influences the jet mainly by affecting the wave breaking and dissipation. The former eddy response turns out to be more efficient for the thermal forcing to shift the eddy-driven jet. Furthermore, two quantitative relationships based on the imposed thermal forcing are proposed to quantify the response of both eddy generation and eddy dissipation, and thus to help predict the shift of eddy-driven jet in response to the vertically nonuniform thermal forcing. By conducting the overriding experiments in which the response of barotropic zonal wind is locked in the model and a multiwavenumber theory in which the eddy diffusivity is decomposed to contributions from eddies and mean flow, we find that the eddy generation response is sensitive to the vertical structure of the thermal forcing and can be quantified by the imposed temperature gradient in the upper troposphere. In contrast, the response of eddy diffusivity is almost vertically independent of the imposed forcing, and can be quantified by the imposed vertically averaged thermal wind. Significance Statement Climate models predict enhanced warming over tropical upper troposphere and Arctic surface in response to greenhouse gas increases, which has competing effects on the latitudinal shift of eddy-driven jet and thus requires a better understanding of the relative roles of upper- and lower-tropospheric thermal forcing for future climate projection. We make a new quantitative comparison on responses of eddy generation and dissipation that sustain the jet shift and relate them to the imposed thermal forcing. This approach extends the fundamental eddy closure topic from vertically uniform situation to vertically nonuniform forcing. These quantitative relationships are also helpful for better understanding and predicting the jet and storm-track variabilities under other forms of thermal forcing (e.g., SST front, aerosols, latent heat release).

Numerical simulation of solitary gravity waves on deep water with constant vorticity

We present a numerical study of essentially nonlinear dynamics of surface gravity waves on deep water with constant vorticity using governing equations in conformal coordinates. The dispersion relation of surface gravity waves on shear flow is known to have two branches; one of which is weakly dispersive for long waves. Weakly nonlinear evolution of the waves of this branch can be described by the Benjamin–Ono equation, which is integrable and has soliton and multi-soliton solutions. Currently, the extent to which the properties of such solitary waves obtained within the weakly nonlinear model are preserved in the exact Euler equations is unknown. We investigate the behavior of this class of solitary waves without the restrictive assumption of weak nonlinearity by using the exact Euler equations. The evolution of localized initial perturbations leading to the formation of single or multiple solitary waves is modeled, and the properties of finite-amplitude solitary waves are discussed. We show that within the framework of the exact equations, two-soliton collisions are almost elastic, but in contrast to solutions of the Benjamin–Ono equation, the waves receive a phase shift as a result of the interaction.

New Travelling Wave Solutions of Conformable Cahn-Hilliard Equation

In this article, two methods are proposed to solve the fractional Cahn-Hilliard equation. This model describes the process of phase separation with nonlocal memory effects. Cahn-Hilliard equations have numerous applications in real-world scenarios, e.g., material sciences, cell biology, and image processing. Different types of solutions have been obtained. For this, the fractional complex transformation has been used to convert fractional differential equation to ordinary differential equation of integer order. As a result, these solutions are new solutions that do not exist in the literature.

An integral equation for solitary surface gravity waves of finite amplitude

An integral equation for solitary surface gravity waves of finite amplitude

Traveling wave solutions of conformable Duffing model in shallow water waves

The Duffing equation is a nonlinear second-order differential equation. The equation describes the motion of a damped oscillator with a more complicated potential than in simple harmonic motion; in physical terms, it models, for example, a spring pendulum whose spring stiffness does not exactly obey Hooke’s law. It is also an example of a dynamical system that exhibits chaotic behavior. Nonlinear equations, such as Duffing model, exhibit significant spectral energy transfer for finite amplitude waves in shallow areas above the flat seafloor. In this paper, a method is proposed to solve nonlinear conformable Duffing model. The solutions found are hyperbolic function solutions. These solutions are new solutions.

High-order strongly nonlinear long wave approximation and solitary wave solution

We consider high-order strongly nonlinear long wave models expanded in a single small parameter measuring the ratio of the water depth to the characteristic wavelength. By examining its dispersion relation, the high-order system for the bottom velocity is found stable to all disturbances at any order of approximation. On the other hand, systems for other velocities can be unstable and even ill-posed, as signified by the unbounded maximum growth. Under the steady assumption, a new third-order solitary wave solution of the Euler equations is obtained using the high-order strongly nonlinear system and is expanded in an amplitude parameter, which is different from that used in weakly nonlinear theory. The third-order solution is shown to well describe various physical quantities induced by a finite-amplitude solitary wave, including the wave profile, horizontal velocity profile, particle velocity at the crest and bottom pressure. For numerical computations, the first- and second-order strongly nonlinear systems for the bottom velocity are considered. It is shown that finite difference schemes are unstable due to truncation errors introduced in approximating high-order spatial derivatives and, therefore, a more accurate spatial discretization scheme is necessary. Using a pseudo-spectral method based on finite Fourier series combined with an iterative scheme for the inversion of a non-local operator, the strongly nonlinear systems are solved numerically for the propagation of a single solitary wave and the head-on collision of two counter-propagating solitary waves of finite amplitudes, and the results are compared with previous laboratory measurements.

Large internal solitary waves on a weak shear.

Large amplitude solitary internal waves of permanent form propagating in a stratified shallow fluid between the free surface and a horizontal bottom are described by the amplitude equation obtained by a regular asymptotic procedure, which incorporates a complicated nonlinearity and Korteweg-de Vries (KdV) dispersion. It is discussed how the structure of stratification and shear affects wave properties. The particular case of a constant buoyancy frequency and a quadratic polynomial for the ambient shear for the flow under free surface is considered in detail analytically. It is shown that for such profiles, the equation for the wave amplitude reduces to the mixed-modified KdV equation and finite amplitude waves obey it up to the breaking level. Rogue waves could appear in this case, and the condition for their generation is identified. More complicated shear profiles lead to higher-order nonlinearities, which produce the multiscaled pyramidal wave patterns, asymmetric bores, and various instabilities. Such wave structures are studied numerically. An analytical bore-like solution having both exponential and algebraic asymptotes is presented.

New solutions for the high-dimensional fractional BLMP equations

Nonlinear Boiti-Leon-Manna-Pempinelli equation (BLMPE) exhibit significant spectral energy transfer for finite amplitude waves in shallow areas above the flat seafloor. In this study, we given solutions to (2+1), (3+1) and (4+1)-dimensional time-fractional BLMP equations representing the wave propagation through incompressible fluids. We found the exact solutions of the high-dimensional time-fractional BLMP equations. Fractional derivative solutions of these equations have not been found in the literature until now. With this study, we have obtained the new exact solutions of both (2+1) dimensional time fractional derivative, (3+1) dimensional time fractional derivative and (4+1) dimensional time fractional derivative BLMP equations.

Exponential traveling-wave on a viscous fluid flowing down an inclined plane

An exponential traveling-wave solution of the free surface equation for a viscous film flowing down an inclined plane is presented. The free surface equation, valid for long waves of finite amplitude, is obtained from the general equation for waves of arbitrary amplitude, which is developed by the method of multiple-scales and considering a series solution expanded at two orders higher than the series used by Lin in 1974. Thus, confining ourselves to the case of weakly nonlinear wave motion, we use the Ince transformation method to find an exponential travelling-wave solution that depend of the physical parameters of the viscous fluid , the angle θ of the inclined plane and the wave parameters, which are subjected to three constraints between them. This solution show a free surface with a dynamic gap of thickness between the top and the bottom of the inclined plane. Also, we find a particular numerical example, for laminar flow, that may represent the evolution of the free surface h(x,t) of a viscous film flowing down an inclined plane, when a small quantity, of the same fluid, is added constantly to the primary fluid in the top of the inclined plane. An interesting result of this case show that the final state, when t → ∞, is also stationary and has the same behavior as the unperturbed primary flow, however the final layer of fluid has a greater thickness.

Plane-polarised finite-amplitude shear waves in deformed incompressible materials

We investigate how two finite-amplitude, transverse, plane body waves may be superposed to propagate in a deformed hyperelastic incompressible solid. We find that the equations of motion reduce to a well-determined system of partial differential equations, making the motion controllable for all solids. We find that in deformed Mooney–Rivlin materials, they may travel along any direction and be polarised along any transverse direction, an extension of a result by Boulanger and Hayes (Quart. J. Mech. Appl. Math. 45 (1992) 575). Furthermore, their motion is governed by a linear system of partial differential equations, making the Mooney–Rivlin special in that respect. We select another model to show that for other materials, the equations are nonlinear. We use asymptotic equations to reveal the onset of nonlinearity for the waves, paying particular attention to how close the propagation direction is to the principal axes of pre-deformation.

Two-dimensional stability analysis of finite-amplitude interfacial gravity waves in a two-layer fluid

We explore basic mechanisms for the instability of finite-amplitude interfacial gravity waves through a two-dimensional linear stability analysis of the periodic and irrotational plane motion of the interface between two unbounded homogeneous fluids of different density in the absence of background currents. The flow domains are conformally mapped into two half-planes, where the time-varying interface is always mapped onto the real axis. This unsteady conformal mapping technique with a suitable representation of the interface reduces the linear stability problem to a generalized eigenvalue problem, and allows us to accurately compute the growth rates of unstable disturbances superimposed on steady waves for a wide range of parameters. Numerical results show that the wave-induced Kelvin–Helmholtz (KH) instability due to the tangential velocity jump across the interface produced by the steady waves is the major instability mechanism. Any disturbances whose dominant wavenumbers are greater than a critical value grow exponentially. This critical wavenumber that depends on the steady wave steepness and the density ratio can be approximated by a local KH stability analysis, where the spatial variation of the wave-induced currents is neglected. It is shown, however, that the growth rates need to be found numerically with care and the successive collisions of eigenvalues result in the wave-induced KH instability. In addition, the present study extends the previous results for the small-wavenumber instability, such as modulational instability, of relatively small-amplitude steady waves to finite-amplitude ones.

On validity of quasi-linear theory for non-resonant pitch-angle diffusion by finite amplitude parallel propagating Alfvén waves

Pitch-angle diffusion of charged particles via non-resonant interaction with low-frequency magnetohydrodynamic waves is discussed. Even in the presence of finite amplitude waves, a quasi-linear theory in which the normalization using wave energy is carried out agrees well with numerical results of test particle simulations. The quasi-linear theory for non-resonant diffusion is also in good agreement with the numerical results, even if the spectra of the waves are not broad.

How well is Rossby wave activity represented in the PRIMAVERA coupled simulations?

Abstract. This work aims to assess the performance of state-of-the-art global climate models in representing the upper-tropospheric Rossby wave pattern in the Northern Hemisphere and over the European–Atlantic sector. A diagnostic based on finite-amplitude local wave activity is used as an objective metric to quantify the amplitude of Rossby waves in terms of Rossby wave activity. This diagnostic framework is applied to a set of coupled historical climate simulations at different horizontal resolutions, performed in the framework of the PRIMAVERA project and compared with observations (ERA5 reanalysis). At first, the spatio-temporal characteristics of Rossby wave activity in the Northern Hemisphere are examined in the multimodel mean of the whole PRIMAVERA set. When examining the spatial distribution of transient wave activity, only a minimal improvement is found in the high-resolution ensemble. On the other hand, when examining the temporal variability of wave activity, a higher resolution is beneficial in all models apart from one. In addition, when examining the Rossby wave activity time series, no evident trends are found in the historical simulations (at both standard and high resolutions) and in the observations. Finally, the spatial distribution of Rossby wave activity is investigated in more detail focusing on the European–Atlantic sector, examining the wave activity pattern associated with weather regimes for each model. Results show a marked inter-model variability in representing the correct spatial distribution of Rossby wave activity associated with each regime pattern, and an increased horizontal resolution improves the models' performance only for some of the models and for some of the regimes. A positive impact of an increased horizontal resolution is found only for the models in which both the atmospheric and oceanic resolution is changed, whereas in the models in which only the atmospheric resolution is increased, a worsening model performance is detected.

A unifying model of weakly nonlinear elastic waves; large on large theory.

This article reconsiders traditional topics in nonlinear elastic waves and nonlinear ultrasonics. Herein, higher-order coupling between finite initial deformation and finite amplitude waves are considered. To allow for coupling, a large-on-large deformation model is developed and used to generate the equations of motion relative to the deformed and undeformed material configurations. Thus, the equations of motion provide a single setting to describe topics in nonlinear elastic waves such as acoustoelasticity, second harmonic generation, and coupling relations between these topics. The model is evaluated to recover the traditional linearized acoustoelastic relations and predicted second harmonic amplitudes. Then, the so-called large acoustoelasticity theory is developed for anisotropic materials with specific results given for isotropic materials. Last, the stress influence on second harmonic generation is presented.

Burgers equation and convexification method: Propagation of wave packets and narrowband noise.

The Burgers equation is a standard model for the propagation of progressive finite amplitude waves in a lossy medium. This paper is mainly devoted to the one-dimensional Burgers equation and its solution deduced from a convexification method. Its starting point is the Hopf-Cole solution in the inviscid limit, then the Legendre-Fenchel transform and the convex envelope construction are employed to obtain the single-value waveform. A fractional step method is used to numerically solve the Burgers equation in a segregated manner (nonlinearity and dissipation). Numerical results are obtained for nonlinear propagation of a wave packet with a Gaussian envelope and a narrowband noise (deterministic and non-deterministic signals). The convexification method provides a tool to solve the problems of nonlinear propagation, especially in the case of noise propagation with the formation and interaction (coalescence) of a large number of discontinuities (shocks).

Finite-amplitude acoustic responses of large-amplitude vibration objects with complex geometries in an infinite fluid

High-intensity acoustic waves existing commonly in aeronautical and aerospace vehicles manifest nonlinear propagation behaviors. Large-amplitude vibration and irregular shape of the aerospace vehicles further complicate the acoustic responses. This paper is concerned with numerical analysis of finite-amplitude acoustic responses of complex-shaped vibration objects. The time-dependent effect of the solid boundary position due to the large-amplitude vibration of the objects is considered. A set of first-order differential equations is derived to govern the finite-amplitude acoustic wave. A fourth-order dispersion-relation-preserving finite difference formulation is employed to solve the nonlinear acoustic equations on a fixed Cartesian grid. Acoustic responses of the fluid and the vibration of the complex-shaped object are coupled by considering the compatibility conditions on the fluid-solid interface. A ghost-cell sharp-interface immersed boundary method is utilized to relax the conformity requirement between the computational grid and solid boundary. Numerical filters are employed in the computational procedure to suppress numerical oscillations. The present method is validated through several numerical tests. Numerical analysis of finite-amplitude acoustic responses of a complex-shaped object is performed. The nonlinear effect of a finite-amplitude acoustic wave, the time-dependent effect of solid boundary position, and the coupling effect between them on the propagation behaviors of nonlinear acoustic waves are discussed.