Flexural Wave Motion Research Articles

Asymptotic analysis of acoustic black hole effect in cylindrical shells.

The acoustic black hole (ABH) effect is investigated within the framework of thin shell theory. Asymptotic solutions to the dispersion equation for the thin cylindrical shell are obtained, and the ABH effect is examined using analytical formulas for group velocities and anti-derivatives of the asymptotic expansions of wave numbers. It is shown that the ABH effect is achievable in thin cylindrical shells with variable thickness, in a similar manner as for beams and plates. However, it should not be expected to exist in the low-frequency range where the flexural wave motion in the wall of a shell is strongly coupled with uniform longitudinal wave motion.

Effects of topographical disturbances on flexural wave motion in a viscous fluid

Main causes of ground disturbances are geological events such as earthquakes, underwater gravity mass flows, volcanic eruptions, and bottom explosions. This paper investigates the effects of transitory ground disturbance on the generation of flexural waves in the presence of a thin floating elastic plate. The problem is formulated using the Stokes stream function and wave potential boundary value problems in a viscous fluid. The expression of the plate deflection is obtained as multiple infinite integrals using the Fourier and Laplace transforms, which is further solved by the steepest descent method. Three types of ground disturbances such as H0(x)=e(−x2/2), H0(x)=e−|x|, and H0(x)=δ(x) are considered. The deflection of the floating elastic plate is investigated in connection to ground disturbances, fluid viscosity, and structural parameters such as mass per unit length of the plate and flexural rigidity. The study reveals that with higher viscosity and flexural rigidity, the amplitude of the plate deflection is reduced. Moreover, these two parameters play a great role in the reduction of plate deflection.

The role of blocking dynamics and current on wave scattering due to finite floating elastic plate

The role of wave blocking on flexural gravity wave motion with oblique incidence and current is discussed. The oblique wave incidence due to a finite floating elastic plate is investigated using the eigenfunction expansion method. The role of different wave propagating modes on reflection coefficient is discussed. The reflection and transmission coefficients, the surface elevation, and plate deflection in the presence of current are derived. The effects of flexural rigidity and the uniform current speed on reflection and transmission coefficients are analyzed. The resonating nature of reflection coefficient for some fixed values of plate length is observed. Significant effects of different combinations of wave propagating modes of plate covered regions and open water regions on the reflection and transmission coefficients are observed. The wave energy propagation from reflection coefficient decreases whenever the Froude number increases. It has been found that the amplitude of the plate deflection increases as the value of the Froude number increases. The amplitude and wavelength of the reflection coefficient are higher for the combination of the first propagating mode of the plate covered region and the second propagating mode of the open water region. Moreover, it is found that zero reflection and full transmission occurs periodically. Co-propagating waves are found to transmit more energy while counter-propagating waves reflect more energy. Higher values of flexural rigidity result in reduced wave energy transmission.

Wave resonances and hydroelastic wave motion

Three different types of resonances, i.e. blocking, trapping and Bragg resonance, associated with flexural gravity wave motion in the presence of current and permeable bottom are discussed. Also, using the different modes of wave propagation, the interconnection among all these resonances is analysed. An uneven pattern in the plate deflection is seen inside the blocking resonance zone. The uniform convergence of the infinite series associated with the trapped mode is proved. We have found that the trapped mode does not exist if the magnitude of the opposing current is higher or whenever we move the cylinder away from the platecovered surface. It is also observed that two propagating modes within the blocking frequencies create the environment for the occurrence of trapped waves. At the same time, all three modes have a remarkable contribution to Bragg resonance. The Bragg reflection increases in the blocking zone and decreases outside of it. We have seen that the amplitude of the Bragg reflection increases with the increase in current speed and porosity parameter. Also, the discontinuities in Bragg reflection coefficients are seen at blocking frequencies. The model is also carefully validated numerically for trapped and Bragg resonances against the results available in the existing literature.

Wave resonances in the presence of current and the frequency and time-domain interconnection

The current work investigates the relationship between frequency and time domain problems associated with flexural gravity wave motion with oblique incidence and current. Different types of wave resonances such as buckling resonance, primary and secondary blocking resonance, Bragg resonance and interaction among them are discussed. Three propagating wave modes are observed between the primary and secondary blocking points, two of which are associated with positive group velocities and the third one is related to negative group velocities. Again, the Bragg resonance is analyzed for wave scattering problems. The complex porosity parameter is found to have a significant effect on the buckling and blocking resonance but does not affect Bragg resonance. The discontinuity of Bragg reflection is observed at primary and secondary blocking points and the peak values of Bragg reflection increase in the blocking zone. Again, we establish the interconnection between the frequency and time domain by deriving the time-domain solution from the frequency-domain solution and vice versa. The derivation of time-domain solutions using the dispersion relation and method of stationary phase is an intriguing finding. Wave energy and wave packets move faster in co-propagating waves than in counter-propagating waves and shallow water than in finite water depth. To validate the results, time-domain simulation and spectral methods are used.

Vibration self-suppression of spinning fluid-conveying pipes composed of periodic composites

In this paper, a novel fluid-conveying phononic crystal (PC) pipe model is proposed. The pipe is composed of different materials arranged alternately, and an axially spinning motion is considered. The flexural wave motions along the orthogonally transverse directions trigger a two-dimensional (2D) PC structure. A planar spectral element (SE) model of the system is established, and the transverse free vibration and wave attenuation performance of such spinning periodic structure are explored thereby. The transfer matrix method is also utilized for validation. It is found that different pseudo Bragg band gaps (BGs) exist in the two transverse directions, while the effective BGs are actually located in their coupled regions, in which the vibration is truly self-suppressed. Such peculiar BG characteristic has not been theoretically revealed previously. Additionally, the spinning motion will reduce the effective BG regions of the periodic pipe. The impacts of the number of cells, flow velocity and component geometry on the natural frequencies and coupled BGs are also studied. The results obtained will provide theoretical basis and design reference for the potential applications of PC-type fluid-conveying devices.

Elastic Bottom Effects on Ocean Water Wave Scattering by a Composite Caisson-Type Breakwater Placed Upon a Rock Foundation in a Two-Layer Fluid

The association of oblique surface gravity waves with a caisson-type multi-chamber porous breakwater fitted with a perforated front wall in a two-layer fluid is studied in finite ocean depth with an elastic bottom. This study focuses on the influence of porous parameters of the interface-piercing structure on wave attenuation in surface and interfacial modes. The flexural gravity wave motion establishes the influence of the elastic bottom. The reflection coefficients for waves in both modes are evaluated to show their effects on the free surface and interface elevations and the waveloads. Consequently, the appropriateness of various configurations of the structure on the wave scattering is studied. Due to wave dissipation by the structure, less waveload is detected on the stiff wall and less elevation is noticed in the porous zone. The structure’s multi-chamber division allows it to have more dissipative and reflective properties. Adjustment of the structure’s height, breadth, and porous parameter leads to achieving good amount of wave reflection and maximum energy dissipation. An optimal width can be determined for a suitable configuration of the structure so that a breakwater can be built with an acceptable level of reflection and dissipation characteristics. The shear force and bottom deflection show how elastic parameters of the sea-floor affect wave scattering.

Interaction of oblique water waves with a single chamber caisson type breakwater for a two-layer fluid flow over an elastic bottom

A finite ocean depth is considered with the bottom possessing elasticity and subsequently, the present investigation for an interface-piercing structure focusses on the influence of porous parameters of the structure on attenuation of waves in both surface and interfacial modes for a two-layer fluid flow. Further, the influence of the elastic bottom is studied by using the dispersion relation to analyze the flexural gravity wave motion with the influence of the flexural wave motion factored into the solution. The analytical results are graphically illustrated. Further, a different linear algebraic technique is adopted, and a comparison of the two methods supports our numerical finding. The reflection coefficients for waves in surface and interfacial modes is examined, along with their effects on the free surface and interface elevations and the wave-loads on the structure. Consequently, the appropriateness of various configurations of the structure on the scattering of surface waves is studied. As per the findings, an optimal width can be determined for a suitable configuration of the porous structure in order to build a breakwater with an acceptable performance of reflection and dissipation characteristics. Such an analysis is likely going to be immensely helpful for a variety of coastal systems in terms of reflection and dissipation of wave energy at continental shelves that is influenced by a stratified fluid, which is being sculptured during this work, as a two-layer fluid for a matter of convenience. The shear force and bottom deflection are discussed to examine how elastic parameters of the sea floor affect the scattering process. It also aids in the analysis of the transformation of physical properties of water waves as well as the structure’s effectiveness. Analyzing the results with respect to the sea floor helps to reduce the waveload on the structure which consequently protects the shore. The analytical solution approximation findings are consistent with previous results from theoretical methods with an impermeable bottom. This research presents a basic and elegant method for dealing with water wave encountering submerged porous structures, and these results can be used as a reliable substitute technique for preliminary engineering analysis. Additionally, an identical geometrical structure is also considered which contains a perforated front wall. Comparison of results for those two completely different structures points towards an additional effectiveness of the model.

Elastic and viscoelastic flexural wave motion in woodpecker-beak-inspired structures

This paper comprehensively investigates elastic and viscoelastic flexural wave propagation in structures that are inspired by the unique suture configurations present in a woodpecker’s beak, in order to understand their ability to attenuate velocity amplitudes and wave speeds. Waveguides characterized by sinusoidal depth variations, both plain and variously graded along the length, mimicking the suture geometry, are considered in this work. Elastic and viscoelastic wave propagation analyses, along with prior static and free vibration studies, are carried out using a novel superconvergent finite element formulation. In elastic wave propagation analysis, firstly the attenuation characteristics are appraised in relation to the high amplitude and frequency waves of three different plain waveguides with differing depth profile orientations. This prompted us to next consider waveguides of hybrid configurations derived from them. Further, waveguides with lengthwise graded sinusoidal segments, as observed in nature, are studied for better wave attenuation properties compared to plain waveguides. This is followed by viscoelastic wave propagation analysis. Regarding the important role of the suture geometry, which is the focus of this work, the results from the elastic analyses revealed the nature of the reduction in wave speeds and amplitudes, both qualitatively and quantitatively, in such waveguides, and their dependence on the orientation and magnitude of the sinusoidal depth variation. Some waveguide configurations with remarkable wave attenuation characteristics, in terms of both wave speeds and amplitudes, are presented, along with their implications regarding impact mitigation applications.

Time dependent wave motion in a permeable bed

The present study deals with the transient flexural gravity wave motion associated with floating elastic plate in the presence of permeable bottom. Integral form of the floating plate deflection is obtained analytically using Laplace–Fourier transform method and the asymptotic solution is derived for large time using stationary phase method. Also, using fast Fourier transform method the time domain simulation of floating plate deflection is obtained. The effect of current, compressive force and porosity parameter on phase and group velocities and floating plate deflection are analyzed. It is observed that in the absence of current, due to the presence of porosity parameter there exist two points where there is a situation of no wave propagation. It is also observed that the current and compressive force have significant effect on transient flexural gravity wave motion whereas the porosity parameter has small impact on the gravity wave motion associated with floating elastic plate. The properties of the roots of the dispersion relation associated with the flexural gravity wave motion in the presence of permeable bed are discussed analytically and numerically.

Transient Axi-Symmetric Disturbances in Two-Layer Fluid

The present manuscript deals with the generation of flexural gravity waves due to transient axi-symmetric disturbances in two-layer fluid. Roles of structural rigidity and compressive force on the transient flexural gravity wave motion are analyzed under the assumption of small amplitude water wave theory and structural responses in the case of finite water depth. Using Laplace and Hankel transforms, integral forms of the velocity potentials, structural deflection, surface and interface elevations are derived for the flexural gravity wave motion. Asymptotic solutions for surface and interface elevations are derived in special cases using method of stationary phase for large time and space.

Transient flexural gravity waves in two-layer fluid

In the present manuscript, flexural gravity wave generation due to an initial disturbance in the case of two-layer fluid of finite depth in the presence of current is analyzed in three dimensions. From the plane progressive wave solution, effect of wave and current direction and the combined effect of current and compressive force on flexural gravity wave motion are discussed. Using Laplace and Fourier transform, the solution of the transient wave structure interaction problem is obtained and integral forms of velocity potentials, structural deflection and interface elevations are derived. Assuming the initial disturbances is on the plate covered surface asymptotic solutions for structural deflection and interface elevations are obtained using method of stationary phase for large time and space. The study reveals that the effect of opposing current and critical compressive force in the presence of current has significant impact on the plate deflection.

Hydroelastic analysis of very large floating structure over viscoelastic bed

Effect of viscoelastic bed on the hydroelastic response analysis of very large floating structures is studied using the linear water wave theory and small amplitude structural response in finite water depth. The floating structure is modeled using Euler–Bernoulli beam equation and the bottom bed is assumed to be viscoelastic in nature and is based on the Voigt’s model. The dispersion relation, phase speed and response amplitude of the floating structure as well as viscoelastic bed surface, pressure distribution along water depth are analyzed to study the effect of viscoelastic bed parameters, flexural rigidity of the floating structure, time period on flexural gravity wave motion. The study reveals that structural response of the floating structure can be mitigated for moderate thickness of the viscoelastic layer. Moreover, both shear modulus and viscosity of the viscoelastic layer play dominant role in reducing the structural response compared to the flexural rigidity of the structure. Further, pressure distribution within the viscoelastic bed decreases at a higher rate compared to the inviscid fluid layer irrespective of shear modulus and viscosity. The present study will be of immense help in the site selection of very large floating structures in the coastal water and installation of various marine facilities over muddy bed.

Experimental and simulation study of wave motion upon railway overhead wire systems

To study the railway overhead wire's flexural wave motion that significantly affects the quality of current collection between pantograph and catenary, dynamic responses of a real catenary excited by different excitations are measured and analyzed by using a set of noncontact photogrammetric devices together with the finite-element simulation. Based on the measured and simulated results, the wave motion along the contact wire is investigated and some findings have been obtained. First, it is the first two dominant modal components in the catenary displacement responses that induce a beat phenomenon, and this beat phenomenon is increasingly obvious near the registration arm. Moreover, with the simulated displacement contour, catenary local vibration characteristics affected by the travelling wave can be clearly interpreted. Furthermore, the wave group velocity of contact wire in catenary is determined and verified to be approximately 139.18 m/s; in addition, treating the contact wire in the overhead wire system as an ideal string or a tensioned Euler beam will overestimate the corresponding group velocity with a relative error by about 13.21%.

Flexural gravity wave motion over poroelastic bed

Using Biot’s consolidation theory, effect of poroelastic bed on flexural gravity wave motion is analyzed in both the cases of single-layer and two-layer fluids. The model for the flexural gravity waves is developed using linear water wave theory and small amplitude structural response in finite water depth. The effects of permeability and shear modulus of poroelastic bed and time period on flexural gravity wave motion are studied by analyzing the dispersion relation, phase speed, plate deflection, interface elevation and pressure distribution along water depth. Various results for surface gravity waves are analyzed as special cases. The study reveals that bed permeability retards the hydrodynamic pressure distribution along the water depth significantly compared to shear modulus whilst, floating plate deflection decreases significantly with change in shear modulus compared to permeability of the poroelastic bed. The present study can be generalized to analyze various wave–structure interaction problems over poroelastic bed.

Measurement-based determination of the irrotational part of the structural intensity by means of test functional series expansion

The irrotational part of the structural intensity for flexural wave motion in plate-like structures is calculated from measurement data of kinematic quantities, such as out-of-plane velocities, by a new method that makes use of a test functional series expansion. The computation of the structural intensity and its irrotational part, which allows to effectively assess vibrational sources and sinks, is less sensitive to measurement noise as compared to standard approaches.

Forced Flexural Gravity Wave Motion in Two-Layer Fluid

Generation of flexural gravity waves in a two-layer fluid due to the forced motion of a vertical rigid wavemaker is studied in both finite and infinite water depths. The two-dimensional (2D) fluid domain having an interface is covered by a semi-infinite ice sheet, which is modeled as an elastic beam. As an application of the wavemaker problem, flexural gravity wave reflection by a vertical cliff is analyzed. Under the assumptions of small amplitude water wave theory and structural response, the mathematical models are solved using a recently developed expansion formulae and the associated orthogonal mode-coupling relations as appropriate for finite and infinite water depths. Effect of three types of edges such as free edge, simply supported edge, and built-in edge on the wave reflection by the vertical cliff is analyzed whilst, for the wavemaker, the floating ice sheet is assumed to have free edge. Effect of various physical parameters on the wave motion is studied by analyzing the reflection coefficients, deflection of the ice sheet, interface elevation, strain and shear force on the floating ice sheet.

Time dependent flexural gravity waves in the presence of current

In the present study, a combined effect of current and compressive force on time dependent flexural gravity wave motion in both the cases of single and two-layer fluids is analyzed in finite and infinite water depths in two dimensions. The roots of the dispersion relation associated with the plane flexural gravity waves are analyzed via contour plots and by plotting various terms of the dispersion relation separately. The characteristic of plane flexural gravity waves is studied by analyzing the phase and group velocities along with the law of conservation of energy flux to understand the combined effect of current and compressive force on the wave motion. The integral form of the time dependent Green's function in the presence of current is obtained using the Laplace transform method and used in Green's identity to derive the time dependent velocity potential for the flexural gravity wavemaker problem. The time harmonic Green's function and velocity potentials are obtained as a special case from the time dependent problems. Numerical results are computed and analyzed in particular cases using the method of stationary phase to obtain the asymptotic results for Green's function and the deflection of ice sheet. The integral form of Green's function derived here will be suitable to deal with physical problem when the roots of the dispersion relation for the flexural gravity wave problem coalesces which were otherwise not possible in the eigenfunction expansion method used for time harmonic problems.

Wave dispersion under finite deformation

We derive exact dispersion relations for axial and flexural elastic wave motion in a rod and a beam under finite deformation. For axial motion we consider a simple rod model, and for flexural motion we employ the Euler–Bernoulli kinematic hypothesis and consider both a conventional transverse motion model and an inextensional planar motion model. The underlying formulation uses the Cauchy stress and the Green–Lagrange strain without omission of higher order terms. For all models, we consider linear constitutive relations in order to isolate the effect of finite motion. The proposed theory, however, is applicable to problems that also exhibit material nonlinearity. For the rod model, we obtain the exact analytical explicit solution of the derived finite-deformation dispersion relation, and compare it with data obtained via numerical simulation of nonlinear wave motion in a finite rod. For the beam model, we obtain an approximate solution by standard root finding. The results allow us to quantify the deviation in the dispersion curves when exact large deformation is considered compared to following the assumption of infinitesimal deformation. We show that incorporation of finite deformation following the chosen definitions of stress and strain raises the frequency branches for both axial and flexural waves and consequently also raises the phase and group velocities above the nominal values associated with linear motion. For the beam problem, only the inextensional planar motion model provides an accurate description of the finite-deformation response for both static deflection and wave dispersion; the conventional transverse motion model fails to do so. Our findings, which represent the first derivation of finite-strain, amplitude-dependent dispersion relations for any type of elastic media, draw attention to (1) the tangible effect of finite deformation on wave dispersion and consequently on the speed of sound in an elastic medium and (2) the importance of incorporating longitudinal–transverse motion coupling in both static and dynamic analysis of thin structures subjected to large nonlinear deformation.

Effects of Axial Load and Elastic Matrix on Flexural Wave Propagation in Nanotube With Nonlocal Timoshenko Beam Model

In this paper, the effects of the axial load and the elastic matrix on the flexural wave in the carbon nanotube are studied. Based on the nonlocal continuum theory and the Timoshenko beam model, the equation of the flexural wave motion is derived. The dispersion relation between the frequency and the wave number is illustrated. The characteristics of the flexural wave propagation in the carbon nanotube embedded in the elastic matrix with the axial load are analyzed. The wave frequency and the phase velocity are presented with different wave numbers. Furthermore, the small scale effects on the wave properties are discussed.