Mechanical wave momentum from the first principles

Axial momentum carried by waves in a uniform waveguide is considered based on the conservation laws and a kind of the causality principle. Specifically, we examine (without resorting to constitutive data) steady-state waves of an arbitrary shape, periodic waves which speed differs from the speed of its form and binary waves carrying self-equilibrated momentum. The approach allows us to represent momentum as a product of the wave mass and the wave speed. The propagating wave mass, positive or negative, is the excess of that in the wave over its initial value. This general representation is valid for mechanical waves of arbitrary nature and intensity. The finite-amplitude longitudinal and periodic transverse waves are examined in more detail. It is shown in particular, that the transverse excitation of a string or an elastic beam results in the binary wave. The closed-form expressions for the drift in these waves functionally reduce to the Stokes’ drift in surface water waves (a half the latter by the value). Besides, based on the general representation an energy–momentum relation is discussed and the physical meaning of the so-called “wave momentum” is clarified.     

Transition wave in a supported heavy beam

We consider a heavy, uniform, elastic beam rested on periodically distributed supports as a simplified model of a bridge. The supports are subjected to a partial destruction propagating as a failure wave along the beam. Three related models are examined and compared: (a) a uniform elastic beam on a distributed elastic foundation, (b) an elastic beam in which the mass is concentrated at a discrete set of points corresponding to the discrete set of the elastic supports and (c) a uniform elastic beam on a set of discrete elastic supports. Stiffness of the support is assumed to drop when the stress reaches a critical value. In the formulation, it is also assumed that, at the moment of the support damage, the value of the ‘added mass’, which reflects the dynamic response of the support, is dropped too. Strong similarities in the behavior of the continuous and discrete-continuous models are detected. Three speed regimes, subsonic, intersonic and supersonic, where the failure wave is or is not accompanied by elastic waves excited by the moving jump in the support stiffness, are considered and related characteristic speeds are determined. With respect to these continuous and discrete-continuous models, the conditions are found for the failure wave to exist, to propagate uniformly or to accelerate. It is also found that such beam-related transition wave can propagate steadily only at the intersonic speeds. It is remarkable that the steady-state speed appears to decrease as the jump of the stiffness increases.     

Localized solutions to the nonlinear wave equation for flexural motions of an elastic beam traveling in an air-filled tube

Steady-progressive-wave solutions are sought to the nonlinear wave equation derived previously [J. Fluids Struct. 16 (2002) 597] for flexural motions of an elastic beam traveling in an air-filled tube along its center axis at a subsonic speed. Fluid-structure interactions are taken into account through aerodynamic loading on the lateral surface of the beam subjected to small but finite deflection but end effects and viscous effects are neglected. Linear dispersion characteristics are first examined by exploiting the small ratio of the induced mass to the mass of the beam per unit length. Centered around the traveling speed of the beam, there exists such a narrow range of propagation velocity that the linear steady propagation is prohibited. In this range, it is revealed that some interesting nonlinear solutions exist. The periodic wavetrain is found to exist as the exact solution. Asymptotic analysis is then made by applying the method of multiple scales and the stationary nonlinear Schrödinger equation is derived for a complex amplitude. A monochromatic solution to this equation corresponds to the exact periodic solution. Imposing undisturbed boundary conditions at infinity, it is revealed that the localized solution exists as a result of balance between the linear instability and the nonlinearity. This solution is checked by solving the nonlinear equation numerically. It is further revealed that the amplitude-modulated wavetrain exists not only in the range of the velocity mentioned above but also outside of it.     

Reflection of a beam of elastic waves by a periodic surface profile

Reflection of elastic waves from a traction-free solid-air boundary of periodic saw-tooth profile is investigated analytically and experimentally. For an incident plane wave the surface displacements on the profile are computed as the solution of a singular integral equation. The reflected field is subsequently obtained by using an integral representation. Incident beams of finite width are represented by Fourier superpositions of plane waves. The dependence of the reflected signal spectra on the incident beam width is examined closely near the fundamental surface resonance frequency. Experimental spectra which were obtained using two different diameter transducers, are compared to the corresponding theoretical spectra. It is found that the depth of the spectral minima depends on the incident beam width. Both analytical and experimental results exhibit the splitting of an incident beam of elastic waves into two reflected beams. The beam splitting is more pronounced for a narrower incident beam and for frequencies close to a resonance frequency of the profile.     

Rays and modes for plane wave coupling into a large open-ended circular waveguide

Generalizing an earlier study of high-frequency plane wave coupling into a wide semi-infinite parallel-plane waveguide, this investigation is concerned with coupling into a large semi-infinite circular waveguide. Emphasis is placed on the interplay between ray fields and mode fields, on the excitation mechanism of each, and on their properties when the field incidence angle is strongly oblique, thereby introducing marked azimuthal asymmetries. The tools of analysis are the geometrical theory of diffraction (GTD), physical optics (PO) and hybrid ray-mode theory. The results reveal that coupling from edge-diffracted rays into a waveguide mode occurs when such rays emanating from the flash points on the edge line up with rays in the modal congruence. However, the diffracted ray caustics between initial reflections do not coincide with the cylindrical caustic of the mode, and only after many reflections is there synchronism between the edge-diffracted and modal rays. Complete synchronism throughout obtains for each angular harmonic constituent of the field, for which the incident wave is conical. Under these conditions, one may also distinguish a multiple reflected truncated conical geometric optical beam, the analog of the geometric optical sheet beam in plane parallel geometry. A variety of other ray and modal characteristics are elucidated as well from different perspectives, thereby granting insight into the ray-mode phenomenology in the circular wave-guide environment. Such insight is expected to be useful when one attempts to deal with nonideal but strongly guiding configurations.     

Investigations on flexural wave propagation of a periodic beam using multi-reflection method

The flexural wave propagation in a periodic beam with a propagating disturbance is studied by the use of the multi-reflection method. A propagating wave is incident upon a discontinuity and gives rise to transmitted and reflected waves. Here all of the transmitted and reflected waves of given flexural wave incident upon the beam at some specified location are found and superposed, and the method is extended to the case of incident evanescent wave. The results of incident waves at some location between discontinuities in a periodic beam are concerned. The relation between the wave-field of incident waves and the wave-field of resulting waves on any segments is expressed. As an example, the application of the results to the analysis of a finite periodic beam with a propagating disturbance is then demonstrated. The influences of the number of cells on the energy associated with propagating waves are considered.     

Scattering of a Rayleigh wave by an elastic wedge whose angle is less than 180°

The steady-state problem of scattering of an incident Rayleigh wave by an elastic wedge whose angle is less than 180° is considered. The problem is reduced to the numerical solution of a pair of Fredholm integral equations of the second kind whose kernels are continuous functions. Numerical results are given for the amplitude and phase of the Rayleigh waves transmitted and reflected by the corner.     

Periodic responses and chaotic behaviors of an axially accelerating viscoelastic Timoshenko beam

This paper investigates the steady-state periodic response and the chaos and bifurcation of an axially accelerating viscoelastic Timoshenko beam. For the first time, the nonlinear dynamic behaviors in the transverse parametric vibration of an axially moving Timoshenko beam are studied. The axial speed of the system is assumed as a harmonic variation over a constant mean speed. The transverse motion of the beam is governed by nonlinear integro-partial-differential equations, including the finite axial support rigidity and the longitudinally varying tension due to the axial acceleration. The Galerkin truncation is applied to discretize the governing equations into a set of nonlinear ordinary differential equations. Based on the solutions obtained by the fourth-order Runge–Kutta algorithm, the stable steady-state periodic response is examined. Besides, the bifurcation diagrams of different bifurcation parameters are presented in the subcritical and supercritical regime. Furthermore, the nonlinear dynamical behaviors are identified in the forms of time histories, phase portraits, Poincaré maps, amplitude spectra, and sensitivity to initial conditions. Moreover, numerical examples reveal the effects of various terms Galerkin truncation on the amplitude–frequency responses, as well as bifurcation diagrams.     

The Asymptotic Perturbation Method for Nonlinear Continuous Systems

We apply the asymptotic perturbation (AP) method to the study of the vibrations of Euler--Bernoulli beam resting on a nonlinear elastic foundation. An external periodic excitation is in primary resonance or in subharmonic resonance in the order of one-half with an nth mode frequency. The AP method uses two different procedures for the solutions: introducing an asymptotic temporal rescaling and balancing the harmonic terms with a simple iteration. We obtain amplitude and phase modulation equations and determine external force-response and frequency-response curves. The validity of the method is highlighted by comparing the approximate solutions with the results of the numerical integration and multiple-scale methods.     

Load motion along a beam on a viscoelastic half-space

An expression is derived for equivalent foundation of a viscoelastic half-space interacting with an Euler–Bernoulli beam. It is shown that this equivalent viscoelastic foundation depends on frequencies and wave numbers of the waves in the beam. The real and imaginary part of it substantially varies for phase velocities in between the Rayleigh and shear waves velocities. Radiation of elastic waves occurs for velocities larger than some velocity in that interval. The steady-state beam displacements due to a uniformly moving constant load are calculated for different velocities. The maximum displacement under the load takes place for a velocity of order of the Rayleigh waves velocity.     

Ultrasonic waves in layered anisotropic media: characterization of multidirectional composites

An efficient and stable recursive compliance/stiffness matrix algorithm is presented to model wave propagation in multidirectional composites. The models are applied to clarify angle beam transmission through a multidirectional composite and to process ultrasonic data for determination of the elastic properties of a composite lamina (single ply) from measurements on a multidirectional composite. Ultrasonic characterization of composites using double-through-transmission and time-resolved line focus acoustic microscopy has been addressed. The double-through-transmission measurements and simulations show that the transmission amplitude is highly dependent on ply orientation and angle of incidence. The transmission amplitude decreases rapidly with incident angle deviation from the normal; however, a transmission window is found in the incident angle range 45–60° at frequencies below 2.25 MHz. The time-delay measurements by the double-through-transmission technique have been used to reconstruct lamina properties using the Floquet wave concept. A unidirectional lamina elastic properties measurement using line focus acoustic microscopy of a multidirectional composite sample is also briefly discussed. The effective elastic properties for the composites are determined from the lamina properties by a Floquet wave dynamic homogenization method.     

Ultrasonic waves in layered anisotropic media: characterization of multidirectional composites

An efficient and stable recursive compliance/stiffness matrix algorithm is presented to model wave propagation in multidirectional composites. The models are applied to clarify angle beam transmission through a multidirectional composite and to process ultrasonic data for determination of the elastic properties of a composite lamina (single ply) from measurements on a multidirectional composite. Ultrasonic characterization of composites using double-through-transmission and time-resolved line focus acoustic microscopy has been addressed. The double-through-transmission measurements and simulations show that the transmission amplitude is highly dependent on ply orientation and angle of incidence. The transmission amplitude decreases rapidly with incident angle deviation from the normal; however, a transmission window is found in the incident angle range 45–60° at frequencies below 2.25 MHz. The time-delay measurements by the double-through-transmission technique have been used to reconstruct lamina properties using the Floquet wave concept. A unidirectional lamina elastic properties measurement using line focus acoustic microscopy of a multidirectional composite sample is also briefly discussed. The effective elastic properties for the composites are determined from the lamina properties by a Floquet wave dynamic homogenization method.     

双介质耦合刚性基弹性层平面应变型导波模式及界面散射能量分配

过渡段动力稳定性问题已成为制约400 km/h及以上高铁路基设计的关键难题,亟需从波动和能量的角度探究由基础非均匀引发的线路系统动力响应放大机理.文章将轨下基础简化为上表面自由、底端固定的刚性基弹性层,将高铁过渡段车致弹性波传播问题提炼为非均匀介质刚性基弹性层中波的散射问题,建立双介质耦合刚性基弹性层平面应变模型,优化该类波导结构频散方程在复平面求根方法,并结合岩土类介质特征展开刚性基弹性层频散分析,以明确其多模式导波特性及散射能量分配,最后,围绕弹性层厚度、刚度比等影响因素开展对比分析.结果表明:刚性基弹性层各模式导波均具有截止频率,弹性层厚度越小,杨氏模量越大,各阶导波模式的截止频率越高;入射波在双介质刚性基弹性层发生散射后,透射场基阶模式导波会占据主体能量,随着高阶导波模式被逐一激发,反射场及透射场高阶模式能量占比会在全频率范围呈现“此消彼长”状态;交换两侧弹性层材料,改变弹性层厚度及两弹性层刚度比不会显著改变能量分布规律,但总体来看,能量更易集中在较软侧弹性层中,各模式导波在激发初始频段会更为活跃,可分配到更多能量.     

Dynamic crack growth under Rayleigh wave

A nonuniform crack growth problem is considered for a homogeneous isotropic elastic medium subjected to the action of remote oscillatory and static loads. In the case of a plane problem, the former results in Rayleigh waves propagating toward the crack tip. For the antiplane problem the shear waves play a similar role. Under the considered conditions the crack cannot move uniformly, and if the static prestress is not sufficiently high, the crack moves interruptedly. For fracture modes I and II the established, crack speed periodic regimes are examined. For mode III a complete transient solution is derived with the periodic regime as an asymptote. Examples of the crack motion are presented. The crack speed time-period and the time-averaged crack speeds are found. The ratio of the fracture energy to the energy carried by the Rayleigh wave is derived. An issue concerning two equivalent forms of the general solution is discussed.     

Reciprocity and power-flow theorems for the scattering of plane elastic waves in a half space

In this paper we present six field-reciprocity relations, and an additional six power-reciprocity relations for the far-zone scattered field if time-harmonic plane elastic waves are incident upon an obstacle in a half space. The incident waves are successively selected as to be either a P-wave, a SV-wave or a Rayleigh surface wave, each propagating in a prescribed direction. In the derivations we employ an explicit integral representation for the far-zone scattered field amplitude. The latter is obtained by expanding the half-space Green's displacement tensor in the far-zone region. Then, starting with the general Betti-Rayleigh theorem, the reciprocity identities are systematically inferred by inspection. We also present energy-conservation relations due to a single incident P-, SV- or Rayleigh wave.     

Interface separation caused by a plane elastic wave of arbitrary form

The paper considers the separation between two contacting solids caused by an incident elastic wave. The wave is assumed to be plane, but may have an arbitrary form. The unilateral interface between the solids is taken as frictionless and incapable of transmitting tension. If the disturbance propagates along the interface with a speed that is supersonic with respect to both solids, the problem can be solved in closed form, and simple results for the extent of the separation zones and the respective gaps are obtained. Several specific examples are included.     

A time-domain energy theorem for the scattering of plane elastic waves

A time-domain energy theorem for the scattering of plane elastic waves by an obstacle of bounded extent is derived. The obstacle is embedded in a homogeneous, isotropic, perfectly elastic medium. As to the elastodynamic behavior of the obstacle no assumptions have to be made; so, lossy, non-linear and time-variant behavior is included. As to the wave motion, three different kinds of time behavior are distinguished: (a) transient, (b) periodic, and (c) pertuating, but with finite mean power flow density. For these cases, the total energy (case (a)) or the time-averaged power (cases (b) and (c)) that is both absorbed and scattered by the obstacle is related to a certain time interaction integral of the incident plane wave (P or S) and the spherical-wave amplitude of the scattered wave of the same type (P or S) in the far-field region, when observed in the direction of propagation of the incident wave.     

弹性地基上转动FGM梁自由振动的DTM分析

基于Euler-Bernoulli梁理论,利用广义Hamilton原理推导得到弹性地基上转动功能梯度材料(FGM)梁横向自由振动的运动控制微分方程并进行无量纲化,采用微分变换法(DTM)对无量纲控制微分方程及其边界条件进行变换,计算了弹性地基上转动FGM梁在夹紧-夹紧、夹紧-简支和夹紧-自由三种边界条件下横向自由振动的无量纲固有频率,再将控制微分方程退化到无转动和地基时的FGM梁,计算其不同梯度指数时第一阶无量纲固有频率值,并和已有文献的FEM和Lagrange乘子法计算结果进行比较,数值完全吻合。计算结果表明,三种边界条件下FGM梁的无量纲固有频率随无量纲转速和无量纲弹性地基模量的增大而增大;在一定无量纲转速和无量纲弹性地基模量下,FGM梁的无量纲固有频率随着FGM梯度指数的增大而减小;但在夹紧-简支和夹紧-自由边界条件下,一阶无量纲固有频率几乎不变。     

Localized waves in a string of infinite length lying on a damaged elastic base under finitely many impacts

Asymptotic solutions of the problem of dynamics of an infinitely long string lying on an elastic base with prescribed damage under the action of finitely many periodic impacts are constructed in the two cases of small and large damage of the elastic base. The condition of resonance origination in the string is obtained in the case of small damage when the standing wave is localized in the region of damage. At the final stage of the damage growth in the elastic base, when its value is close to the critical one, the localized mode and the resonance are absent, and only a traveling wave exists in the string.     

Destabilization of long-wavelength Love and Stoneley waves in slow sliding

Love waves are dispersive interfacial waves that are a mode of response for anti-plane motions of an elastic layer bonded to an elastic half-space. Similarly, Stoneley waves are interfacial waves in bonded contact of dissimilar elastic half-spaces, when the displacements are in the plane of the solids. It is shown that in slow sliding, long-wavelength Love and Stoneley waves are destabilized by friction. Friction is assumed to have a positive instantaneous logarithmic dependence on slip rate and a logarithmic rate weakening behavior at steady-state.Long-wavelength instabilities occur generically in sliding with rate- and state-dependent friction, even when an interfacial wave does not exist. For slip at low rates, such instabilities are quasi-static in nature, i.e., the phase velocity is negligibly small in comparison to a shear wave speed. The existence of an interfacial wave in bonded contact permits an instability to propagate with a speed of the order of a shear wave speed even in slow sliding, indicating that the quasi-static approximation is not valid in such problems.