Small-amplitude Waves Research Articles

Dynamics of a pile-moored fish cages in current and waves: A numerical study

We propose a numerical study of the dynamics of a double-constrained cylindrical pile-moored fish-farm cage, to be possibly installed at the foundations of dismissed offshore structures. Our model, derived from the net-truss model of Kristiansen and Faltinsen (2012), takes into account the elasticity of the net material, the real size of the net and is applied with and without mesh-grouping approaches, this allowing us to evaluate the range of validity of mesh-grouping techniques. First, cage dynamics have been studied with small-scale tests, in which the numerical model accurately reproduces the behaviour of the cage in both sea currents (velocity in the range 0.5−1m/s) and waves of small amplitude (with height of 1 m and period in the range 4−32 s). Use of the net real size for the computation grid of small-scale tests allows for identification of the main phenomena influencing the load distribution and the volume loss, which are: the dominant sail-shape deformation in the streamwise direction, the flattened deformations in the cross flow direction, and high-frequency oscillations in the vertical direction in long waves. Subsequently, a prototype-scale test is run to simulate the dynamics of a realistic installation of a double-constrained cylindrical pile-moored fish-farm cage in the Adriatic sea (considering both 1m/s sea current and storm sea state characterized by a peak period Tp=4 s and significant wave height Hs=1.5 m), showing that the major loads are a compression load on the pile up to 1 kN and a bending moment at the pile base up to 9.1 kNm. The proposed configuration always showed a small net relative volume loss VL<5%.

Incremental equations and propagation of small amplitude waves for isotropic incompressible elastic bodies. The case the Hencky strain tensor is a function of the Cauchy stress tensor

For a relatively new class of constitutive equation for incompressible elastic bodies, wherein the Hencky strain tensor is a function of the Cauchy stress tensor, incremental equations are obtained, where it is assumed the presence of an initial time-independent stress that causes large deformations, to which a small time-dependent stress tensor is added. That stress tensor is assumed to cause a small (incremental) time-dependent deformation. For the case of infinite media the incremental equations are solved assuming travelling waves, and the speed of such waves is obtained for different problems considering initial homogeneous distributions of stresses and strains. Some numerical calculations of the speed of such waves are presented, considering a particular constitutive equation for rubber published recently in the literature. The speed of such small amplitude waves are compared with the predictions of the classical theory of nonlinear elasticity for similar problems.

NOVEL TRAVELING-WAVE SOLUTIONS OF SPATIAL-TEMPORAL FRACTIONAL MODEL OF DYNAMICAL BENJAMIN–BONA–MAHONY SYSTEM

This paper investigates the dynamics of exact traveling-wave solutions for nonlinear spatial and temporal fractional partial differential equations with conformable order derivatives arising in nonlinear propagation waves of small amplitude including nonlinear fractional modified Benjamin–Bona–Mahony equation, fractional Zakharov–Kuznetsov–Benjamin–Bona–Mahony equation and fractional (2+1)-dimensional Kadomtsev–Petviashvili–Benjamin–Bona–Mahony equation as well. By utilizing the Sine-Gordon expansion method (SGEM), new real- and complex-valued exact traveling-wave solutions are reported by preferring suitable values of physical free parameters. The nonlinear governing equations are reduced into auxiliary nonlinear ordinary differential equations with aid of fractional traveling-wave transformation, in which the fractional derivative is evaluated in a conformable sense. The productivity process of the proposed method for predicting the desirable solutions is also provided. Some of the obtained solutions are simulated graphically in 3D and contour plots. Meanwhile, the effects of the fractional parameter [Formula: see text] in the space and the time direction are illustrated in 2D plots to ensure the novelty, applicability and credibility of the SGEM. These results reveal that the suggested method is general and adequate for dealing with nonlinear models featuring fractional derivatives and can be employed to analyze wide classes of complex phenomena of partial differential equations occurring in engineering and nonlinear dynamics.

Approximation of Small Amplitude Waves Short in Vertical in the Atmosphere Taking Into Account the Average Wind

Using the method of different scales, formulas for the hydrodynamic fields of acoustic-gravity waves (AGWs) with vertical wavelengths small compared to the scales of changes in the background temperature and wind fields are derived. These formulas are equivalent to the conventional WKB approximation, but explicitly include the vertical gradients of the background fields. The conditions for the applicability of the obtained formulas for describing the propagation of AGWs from the troposphere to the thermosphere are formulated and analyzed. The absence of singular points (critical levels) in the equations for wave modes in the analyzed height range is one of the conditions for the applicability of approximate formulas. For the wind from the empirical HWM model, singular points are often located below 200 km and are typical for internal gravity waves (IGWs), with lengths of the order of 10 km. As the wavelength increases, the number of singular points decreases. For IGWs with scales on the order of 300 km or more, there are usually no singular points. It is shown that IGWs with periods of less than 20 min propagating upward from tropospheric heights usually have one turning point in the altitude range from 100 to 130 km. The obtained formulas are useful, in particular, for parametrization of AGW effects in numerical models of atmospheric dynamics and energy.

Generation of solar chromosphere heating and coronal outflows by two-fluid waves

Context. It is known that Alfvén and magnetoacoustic waves both contribute to the heating of the solar chromosphere and drive plasma outflows. In both cases, the thermalization of the wave energy occurs due to ion-neutral collisions, but the obtained rates of plasma heating cannot explain the observational data. The same is true for the magnitudes of the outflows. Aims. The aim of the present paper is to reexamine two-fluid modeling of Alfvén and magnetoacoustic waves in the partially ionized solar chromosphere. We attempt to detect variations in the ion temperature and vertical plasma flows for different wave combinations. Methods. We performed numerical simulations of the generation and evolution of coupled Alfvén and magnetoacoustic waves using the JOANNA code, which solves the two-fluid equations for ions (protons)+electrons and neutrals (hydrogen atoms), coupled by collision terms. Results. We confirm that the damping of impulsively generated small-amplitude waves negligibly affects the chromosphere temperature and generates only slow plasma flows. In contrast, waves generated by large-amplitude pulses significantly increase the chromospheric temperature and result in faster plasma outflows. The maximum heating occurs when the pulse is launched from the center of the photosphere, and the magnitude of the related plasma flows increases with the amplitude of the pulse. Conclusions. Large-amplitude coupled two-fluid Alfvén and magnetoacoustic waves can significantly contribute to the heating of the solar chromosphere and to the generation of plasma outflows.

Canonical conformal variables based method for stability of Stokes waves

AbstractWe study the stability of Stokes waves in an ideal fluid of infinite depth. The perturbations that are either coperiodic with a Stokes wave (superharmonics) or integer multiples of its period (subharmonics) are considered. The eigenvalue problem is formulated using the conformal canonical Hamiltonian variables and admits numerical solution in a matrix‐free manner. We find that the operator matrix of the eigenvalue problem can be factored into a product of two operators: a self‐adjoint operator and an operator inverted analytically. Moreover, the self‐adjoint operator matrix is efficiently inverted by a Krylov‐space‐based method and enjoys spectral accuracy. Application of the operator matrix associated with the eigenvalue problem requires only flops, where N is the number of Fourier modes needed to resolve a Stokes wave. Additionally, due to the matrix‐free approach, storage for the matrix of coefficients is no longer required. The new method is based on the shift‐invert technique, and its application is illustrated in the classic examples of the Benjamin–Feir and the superharmonic instabilities. Simulations confirm numerical results of preceding works and recent theoretical work for the Benjamin–Feir instability (for small amplitude waves), and new results for large amplitude waves are shown.

Global bifurcation structure and some properties of steady periodic water waves with vorticity

This paper studies the classical water wave problem with vorticity described by the Euler equations with a free surface under the influence of gravity over a flat bottom. Based on fundamental work [5], we first obtain two continuous bifurcation curves which meet the laminar flow only one time by using the modified analytic bifurcation theorem. They are symmetric waves whose profiles are monotone between each crest and trough. Moreover, we show a connection between the concavity and convexity of wave profile and the monotonicity of the vertical velocity component v along the free surface. As an important application, we make up the missing major aspect on the behavior of v as mentioned in [4, Sect 4.4] for small amplitude waves. In addition, for favorable vorticity, we prove that the vertical displacement of water waves decreases with depth.

Long waves with a small amplitude on the surface of the water behave dynamically in nonlinear lattices on a non-dimensional grid

Approximation and analysis are used for investigating accurate soliton solutions of the ill-posed Boussinesq (IPB) equation. The investigated model explains shallow-water gravitational waves. It examines one-dimensional nonlinear strings and lattices. IPB explains small-amplitude surface waves on nonlinear strings and lattices. We provide unique analytical solutions to analyze numerical beginning and boundary conditions. A solution’s quality is judged by its divergence from analytical predictions. Physical wave properties are illustrated.

Oblique-mode breakdown of the vertical buoyancy layer

The O-type transition caused by a pair of small-amplitude oblique waves in a vertical buoyancy layer of a fluid with Prandtl number $0.71$ at a Reynolds number of $200$ is investigated using linear stability analysis and three-dimensional direct numerical simulation. The small-amplitude oblique waves experience linear growth and undergo nonlinear interactions to generate streamwise vortices/streaks, two-dimensional streamwise waves and harmonic oblique waves. The streamwise vortices/streaks and two-dimensional streamwise waves have twice the spanwise or streamwise wavenumber of the original perturbation, respectively. Unlike the O-type transition in isothermal flat-plate incompressible and compressible boundary layers where streaks dominate the transition, in the vertical buoyancy layer, either streaks or two-dimensional streamwise waves can dominate the flow field during the early stages of oblique transition. The growth rates of streaks and two-dimensional waves are dependent on the wavenumber of the initial oblique waves. Streaks dominate the flow for high streamwise wavenumbers, while two-dimensional streamwise waves dominate the flow for low streamwise wavenumbers. Analysis of the turbulent kinetic energy production and the Reynolds stresses reveals that the early stages of the transition differ depending on the wavenumber of the oblique waves. An increase in the initial amplitude of the oblique waves causes a faster transition from laminar flow; however, the growth rates of the streaks and two-dimensional streamwise waves are independent of the initial amplitude. Even though different modes are dominant during the early stages of the O-type transition, the onset of chaotic flow is caused by the breakdown of streak modes.

Asymptotic behaviour of a linearized water waves system in a rectangle1

We consider the asymptotic behaviour of small-amplitude gravity water waves in a rectangular domain where the water depth is much smaller than the horizontal scale. The control acts on one lateral boundary, by imposing the horizontal acceleration of the water along that boundary, as a scalar input function u. The state z of the system consists of two functions: the water level ζ along the top boundary, and its time derivative ∂ ζ ∂ t . We prove that the solution of the water waves system converges to the solution of the one dimensional wave equation with Neumann boundary control, when taking the shallowness limit. Our approach is based on a special change of variables and a scattering semigroup, which provide the possiblity to apply the Trotter–Kato approximation theorem. Moreover, we use a detailed analysis of Fourier series for the dimensionless version of the partial Dirichlet to Neumann and Neumann to Neumann operators.

Dissipation of Gravity Waves Due to Submerged Porous Plate Coupled With Porous Structures

Abstract The present study focuses on wave trapping due to the submerged horizontal porous plate combined with the bottom-standing porous structure and surface-piercing porous structure. The submerged plate thickness is considered to be negligible as compared to the incident wavelength and water depth, and the porous structure is considered to be of finite width. The study is performed based on the eigenfunction expansion method, and the wave interaction with the combined structure is investigated using the small amplitude wave theory. The orthogonal mode-coupling relation is used to analyze the wave interaction with the combined structure. The reflection, transmission, and dissipation coefficients along with wave force on the porous structure are investigated to analyze the hydrodynamic performance of the composite porous breakwater system. Further, the effect of porosity of submerged plate and structure, submergence depth of plate and structure, angle of incidence, and the submerged plate length are investigated to analyze the effective wave dissipation by the composite breakwater. In addition, the comparative study of the numerical method is performed with the results available in the literature. The study noted that the wave damping due to the submerged porous plate backed by surface-piercing porous structure is more as compared to the submerged porous plate backed by the bottom-standing porous structure. The study performed will be helpful to scientists and engineers in the design of suitable composite breakwater systems and also assists in selecting the best structural configuration for attenuation of wave height and to protect the offshore facility from high waves in the coastal region.

A new reductive perturbation formalism for ion acoustic cnoidal waves

A new formalism for the derivation of the cnoidal wave solution is presented by introducing a new set of initial conditions and a subacoustic moving frame. The resulting set of solutions is also constructed in a way that ensures the conservation of number density. The solutions are illustrated in a variety of graphical representations, and the effect of the amplitude's magnitude on the cnoidal wave form is presented. Interestingly, it is shown that the wavelength of the solutions decreases with amplitude. In addition, it is shown that the small-amplitude cnoidal wave solutions converge to linear waves in the small-amplitude limit.

The higher-order nonlinearity and parametric effects on dust-ion-acoustic shock waves

In the case of obliquely propagated small-amplitude shock waves, in this study, we investigate the impacts of higher-order nonlinearity as well as various parameters (such as dust concentrations, viscosity, trapping parameters, etc) on the shock wave structures. The considered magnetized plasma system consists of three components, such as inertial positive ions (mobile), trapped electrons, and immobile negatively charged dust particles. The modified Burgers equation with a dominating dissipative term (in which the viscous effect is significant) is derived initially to examine the lower-order nonlinear and dissipative effects, and then, to the best of our knowledge, the modified Burgers-type linear inhomogeneous equation is derived for the first time to observe the higher-order nonlinear effects on shock waves while the plasma contains trapped electrons. The reductive perturbation method is used for the derivation of the equations, whereas the Abel’s theorem and the method of variation of parameters are used for adding the higher-order effect. From the theoretical investigation, we observe that the higher-order nonlinearity has an increasing effect on the shock amplitude. Furthermore, the viscosity and dust concentration increase the shock width and the phase speed, respectively.

Nonlinear air dynamics of a surface effect ship in small-amplitude waves

In many existing works, the seakeeping motions and air dynamics of a surface effect ship (SES) were assumed to be linear under small-amplitude waves (wave amplitude to wave length ratio ≤ 5%) to enhance the computational efficiency. However, according to SES model test results, it was found that even in small-amplitude waves, the fluctuating air cushion pressure shows significantly nonlinear effects. To precisely reveal this distinctive feature, the origin of nonlinearity was carefully investigated and the air leakage was considered as the main source of nonlinearity based on mathematical analysis in this paper. The reason is that the variance of clearance height under seals is comparable to the clearance height at equilibrium state in small-amplitude waves, which makes the air leakage area intermittently equal to zero without any harmonic variance. Therefore, an efficient partial nonlinear numerical model for the SES dynamics was proposed by combining a linear frequency-domain hydrodynamic model based on the efficient 2.5D methods with a nonlinear time-domain air dynamic model. The nonlinear parts of numerical results from the partial nonlinear model, including the fluctuating air pressure and midship accelerations, agree well with experimental results. The results demonstrate the effectiveness of the partial nonlinear model on the SES seakeeping performance prediction, and confirm that its nonlinearity mainly originates from the air leakage.

Instability of small-amplitude periodic waves from fold-Hopf bifurcation

We study the existence and stability of small-amplitude periodic waves emerging from fold-Hopf equilibria in a system of one reaction–diffusion equation coupled with one ordinary differential equation. This coupled system includes the FitzHugh–Nagumo system, caricature calcium models, and other models in real-world applications. Based on the recent results on the averaging theory, we solve periodic solutions in related three-dimensional systems and then prove the existence of periodic waves arising from fold-Hopf bifurcations. Numerical computation by Tsai et al. [SIAM J. Appl. Dyn. Syst. 11, 1149–1199 (2012)] once suggested that the periodic waves from fold-Hopf bifurcations in a caricature calcium model are spectrally unstable, yet without a proof. After analyzing the linearization about periodic waves by the relatively bounded perturbation, we prove the instability of small-amplitude periodic waves through a perturbation of the unstable spectra for the linearization about the fold-Hopf equilibria. As an application, we prove the existence and stability of small-amplitude periodic waves from fold-Hopf bifurcations in the FitzHugh–Nagumo system with an applied current.

Waves in isotropic dispersive elastic solids

The equations of nonlinear elastodynamics have attracted the attention of pure and applied mathematicians since the second half of the last century. Nevertheless, many aspects of this theory are still obscure or need a more systematic analysis. In this paper we study the wave propagation in an isotropic elastic solid by taking into account the effects due to dispersion. Specifically, we determine a class of global (in both time and space) solutions, find exact solutions for longitudinal travelling waves within the fourth-order theory of elasticity, and derive two asymptotic reductions of the governing equations for waves of small amplitudes.

An analytical study of scattering of water waves by a surface-piercing bottom-mounted compound porous cylinder placed on a porous sea-bed

The scattering problem of a bottom-mounted surface-piercing compound porous cylinder located on a porous sea-bed is theoretically investigated under the assumption of small amplitude wave theory. The compound cylinder is comprised of an impermeable upper cylinder and a porous lower cylinder beneath it. The boundary conditions on the porous boundaries follow Darcy’s law by assuming fine pores in the porous structure. The whole fluid region is split into three bounded and an unbounded sub-regions, within which the individual velocity potentials are found by using eigenfunction expansion technique. Further, utilization of the matching conditions along the boundaries of individual successive regions leads to a semi-analytical solution of the proposed problem. The impact of the non-dimensional porous-effect parameter of the cylindrical wall, the draft ratio, radius ratio and the sea-bed porosity on waveloads and wave run-ups are studied. The results show that suitable consideration of porosity and structure parameters enhance the efficiency of the proposed compound cylinder in mitigating wave impact. Furthermore, the hydrodynamic waveload acting on the lower as well as upper cylinder can be reduced by the suitable positioning of the annular spacing of the system, which will provide explicit information for the purpose of engineering design in the coastal area.

In solid physics equations, accurate and novel soliton wave structures for heating a single crystal of sodium fluoride

This paper analyzes the analytical and numerical solutions’ structure of the combined mKdV equation and KdV equation (mKdV+KdV equation) using the Khater II (Khat. II) method and three accurate B-spline numerical schemes. ExCBS, SBS and TQBS numerical schemes are the numerical systems used. The handled model describes many distinct phenomena such as wave propagation of bounded particles with a harmonic force in a one-dimensional nonlinear lattice, propagation of ion-acoustic waves of small amplitude without Landau damping in plasma physics, and propagation of thermal pulse through a single sodium fluoride crystal in solid physics. Numerous examples show the relationship between quick and slow soliton, which generates phase shift. This phase shift is shown in a contour map to show the modest and colossal energy density along the path of fast and slow colliding solitons. Calculating the difference between analytical and numerical solutions shows whether they match spline-connected and distribution graphs.

On the Initiation of Bubble Detonation by Small-Amplitude Waves

On the Initiation of Bubble Detonation by Small-Amplitude Waves

Representing the stress and strain energy of elastic solids with initial stress and transverse texture anisotropy

Real-world solids, such as rocks, soft tissues and engineering materials, are often under some form of stress. Most real materials are also, to some degree, anisotropic due to their microstructure, a characteristic often called the ‘texture anisotropy’. This anisotropy can stem from preferential grain alignment in polycrystalline materials, aligned micro-cracks or structural reinforcement, such as collagen bundles in biological tissues, steel rods in pre-stressed concrete and reinforcing fibres in composites. Here, we establish a framework for initially stressed solids with transverse texture anisotropy. We consider that the strain energy per unit mass of the reference is an explicit function of the elastic deformation gradient, the initial stress tensor and the texture anisotropy. We determine the corresponding constitutive relations and develop examples of nonlinear strain energies that depend explicitly on the initial stress and direction of texture anisotropy. As an application, we then employ these models to analyse the stress distribution of an inflated initially stressed cylinder with texture anisotropy and the tension of a welded metal plate. We also deduce the elastic moduli needed to describe linear elasticity from stress reference with transverse texture anisotropy. As an example, we show how to measure the stress with small-amplitude shear waves.