Guided waves in elastic plates with Gaussian section variation: experimental and numerical results

Experimental and numerical results are presented on the behavior of guided waves in elastic plates in plane strain that include a Gaussian variation of their section, located between two areas of constant thickness. The area of varying section is wide compared to the used wavelengths, which allows wave propagation inside this area. The experimental results show that an incident Lamb wave is indeed converted into an adiabatic wave inside the varying section domain. A trapped wave in the Gaussian domain is also observed, depending on the incident mode and on the Gaussian maximum height. Outside the varying section domain, conversion into different Lamb waves is observed. This conversion phenomenon is experimentally quantified by the measurement of the Lamb wave normal displacement and of its carried energy. A numerical study, based on the Finite Elements Method is performed, and successfully compared to the experimental results.     

X-ray standing waves in a multi-trilayer system with linearly varying period

Generation of X-ray standing waves in a multi-trilayer system with a varying periodicity along its depth is described. Microstructures of a synthetic 15-period Pt/Ni/C multilayer, which has such a varying periodicity, are investigated using X-ray reflectivity (XRR) and X-ray standing wave (XSW) analysis. Microstructural parameters, e.g., thicknesses and electron densities of individual layers, interface roughnesses, varying periodicity along the depth, etc. are precisely determined by the XRR technique. The variation of period is essentially introduced in multilayer fabrication by varying the thickness of the C-layer as a function of depth. The linear increase in C-layer thickness from the bottom of the multilayer towards the surface of the multilayer structure is found by simulating the XRR data with a depth-graded multilayer model. Intensity of the XSW field is computed as a function of depth over the first order Bragg peak. Integrated intensities over the Pt- and the Ni-layers are computed and compared with the measured fluorescence yield from Pt and Ni. Sensitivity for the detection of inter-atomic diffusion across the interfaces in comparison to multi-bilayer systems, such as Pt/C multilayer, is discussed.     

Axisymmetric vibrations of an isotropic elastic non-homogeneous circular plate of linearly varying thickness

Free axisymmetric vibrations of an isotropic, elastic, non-homogeneous circular plate of linearly varying thickness have been studied on the classical theory of plates. The non-homogeneity of the material of the plate is assumed to arise due to the variation of Young's modulus and density with the radius vector whereas Poisson's ratio is assumed to remain constant. The governing differential equation of motion is solved by the method of Frobenius. The transverse displacement of the plate has been expressed as a power series in terms of the radial co-ordinate. The frequency parameters corresponding to the first two modes of vibration have been computed for different values of the non-homogeneity parameter and taper constant and for clamped and simply supported edge conditions of the plate. A comparison between the numerical results for homogeneous and non-homogeneous material of the plate is also made.     

Dynamic stability of linearly varying thickness viscoelastic rectangular plate with crack and subjected to tangential follower force

Based on the two-dimensional viscoelastic differential constitutive relation and the thin plate theory, the differential equations of linearly varying thickness viscoelastic plate with crack and subjected to uniformly distributed tangential follower force in the Laplace domain are established, and the expression of the additional rotation induced by the crack is given. The complex eigenvalue equations of linearly varying thickness viscoelastic plate constituted by elastic behavior in dilatation and the Kelvin-Voigt laws for distortion with crack and under the action of uniformly distributed tangential follower force are obtained by the differential quadrature method. The generalized eigenvalue under different boundary conditions is calculated, and the curves of real parts and imaginary parts of the first three order dimensionless complex frequencies versus uniformly distributed tangential follower force are obtained. The effects of the aspect ratio, the thickness ratio, the crack parameters and the dimensionless delay time on the dynamic stability of the viscoelastic plates are analyzed.     

Vibrations of a square plate with parabolically varying thickness

Vibration results for a square plate with a parabolically varying thickness distribution and built-in edges are presented. Frequency and mode shape predictions obtained from a finite element analysis are compared with measurements made with real-time laser holography. In general, the agreement between the two is very good except for a few of the lower modes where the predicted frequencies are about 15% high. So far, a satisfactory explanation for this has not been found. The vibration mode shapes for the plate exhibit a striking blend of radial and square symmetries that results from the axisymmetric thickness distribution and the square symmetry of the boundary frame.     

Free vibration analysis of Mindlin plates with linearly varying thickness

A method based on the variational principles in conjunction with the finite difference technique is applied to examine the free vibration characteristics of isotropic rectangular plates of linearly varying thickness by including the effects of transverse shear deformation and rotary inertia. The validity of the present approach is demonstrated by comparing the results with other solutions proposed for plates with uniform and linearly varying thickness. Natural frequencies and mode shapes of Mindlin plates with simply supported and clamped edges are determined for various values of relative thickness ratio and the taper thickness constant.     

Axisymmetric vibrations of a circular plate of linearly varying thickness on an elastic foundation according to Mindlin theory

Free axisymmetric vibrations of an elastic circular plate of linearly varying thickness on an elastic foundation have been studied on the basis of shear theory [1,2]. The transverse displacement and local rotation are expressed as an infinite series. The frequencies corresponding to the first two modes of vibrations are obtained for a circular plate with clamped and simply supported edge conditions for various values of the taper constant and the foundation modulus. The results have been compared with those of reference [3].     

Vibration and stability of a circular plate of unidirectionally varying thickness

An analysis is presented for the vibration and stability of an elastically restrained circular plate of unidirectionally varying thickness subjected to an in-plane force. For this purpose, the transverse deflection of a circular plate of variable thickness is written in a series of the deflection functions of a uniform circular plate without the action of a force. The dynamical energies of the plate are evaluated analytically, and the frequency equation of the plate is derived by the Ritz method. The analysis is applied to circular plates of unidirectionally tapered or stepped thickness; the natural frequencies and the divergence loads are calculated numerically, and the effects of the varying thickness and edge conditions on the vibration and stability are studied.     

Propagation of thickness-twist waves in elastic plates with periodically varying thickness and phononic crystals

We study the propagation of thickness-twist (TT) waves in a crystal plate of AT-cut quartz with periodically varying, piecewise constant thickness. The scalar differential equation by Tiersten and Smythe is employed. The problem is found to be mathematically equivalent to the motion of an electron in a periodic potential field governed by Schrodinger’s equation. An analytical solution is obtained. Numerical results show that the eigenvalue (frequency) spectrum of the waves has a band structure with allowed and forbidden bands. Therefore, for TT waves, plates with periodically varying thickness can be considered as phononic crystals. The effects of various parameters on the frequency spectrum are examined.     

Most probable ballistic waves in random media: a weak-fluctuation approximation and numerical results

A statistical approach to describe the time-space pulse evolution in single realizations of two- and three-dimensional random media is presented. For this purpose, we construct the ensemble-averaged wavenumber of a plane wave. The wavenumber is obtained by a combination of the Rytov and Bourret approximations for the log-amplitude and phase increment, both related by the Kramers-Kronig relations. The real part of the wavenumber is related to the phase velocity, whereas its imaginary part denotes the attenuation coefficient of the ballistic wavefield. The validity range is limited by the weak-fluctuation regime, but has practically no restrictions in the frequency domain. We show that this wavenumber is a partially self-averaged quantity. This means that the wavenumber is a Gaussian-distributed quantity exhibiting decreasing relative fluctuations while the wave travels a finite distance such that it is still in a weak-fluctuation regime. Then, the ensemble-averaged wavenumber carries the information of maximum probability realizations. The Green's function constructed with this wavenumber corresponds to the most probable ballistic wave. Using this Green's function, it is possible to characterize single-wavefield realizations in single realizations of random media. We verify the concept and analytic results with the help of finite-difference simulations of waves propagating in two-dimensional random media. We demonstrate the partial self-averaging property of the wavefield attributes by computing their relative standard deviations as a function of propagation distance L. Within the weak-fluctuation regime, these relative standard deviations decrease as 1/√L, indicating that the process of self-averaging takes place. Reconstructing the joint probability density of amplitudes and phases of ballistic waves allows us to identify most probable ballistic waves. They show a good agreement with the analytic results.     

Guided waves in elastic rectangular bars:A solution using partial plate mode decomposition

Similar to Auld's solution for Lamb waves,the wave modes in elastic rectangular bar are solved by partial wave decomposition method.The partial waves are composed of plate modes with the same wavenumber component in waveguide longitudinal direction,thus free boundary conditions on one pair of opposite surfaces are automatically satisfied.Based on completeness assumption and orthogonality of the plate modes,four independent eigenequations are eventually derived for dispersion curve and mode shape investigation.Numerical evaluation shows the calculated results are in consistent with the FEM results.It is then verified that the plate modes which obliquely bounced back and forth between the two opposite surfaces compose the guided modes traveling in the rectangular waveguides with certain wave numbers in transversely resonant cases.     

Guided waves in a stratified half-space

The dispersion and excitation mechanisms and the energy distribution of guided waves in a stratified half-space are studied. All possible guided waves excited by a symmetric point source in two or three-layer medium models and their relation to the medium parameters are analyzed in detail. The excitation and propagation characteristics, as well as the energy distribution along the depth direction, of all modes of the surface waves and trapped waves are numerically investigated and analyzed thoroughly not only in the case when the shear wave velocity increases from up to down layers but also when a low-velocity layer is contained in halfspace, especially when the shear wave velocity decreases from up to down layers. It is found that there exist many guided wave modes in the case where the shear wave velocity of each layer increases from up to down layers. However, there is less than one guided wave mode in the case where the shear wave velocity of each layer decreases from up to down layers. The trapped waves exist and propagate along the low-velocity structure in the stratified half-space. It is also found that the characteristic of a mode is related to the source frequency. It is possible that a surface wave at one value of frequency is like a trapped wave at another value of frequency. Finally, the relation of the characteristics of all guided waves (surface waves and trapped waves) to the parameters of media is studied.     

Excitation of linearly stable waves in a multispecies plasma

Excitation of electrostatic modes as a result of the injection of an argon beam into a multispecies plasma is investigated. It is shown that the injection of a warm artificial beam into a thermal plasma may excite waves of significant amplitude with a range of phase velocities which are linearly stable. These waves appear as a result of the correlation between motions of an individual particle in a Vlasov fluid. When the free energy is derived from a warm beam and the linear instability diminishes due to the large temperature of the beam, the competing mechanism of thermal fluctuations is responsible for the main excitation of waves. This mechanism is valid for the various modes in a multispecies plasma with a higher enhancement at the lower hybrid than at the ion-ion hybrid modes     

Guided circumferential waves in orthotropic cylindrical curved plate and the mode conversion by the end-reflection

In ultrasonic nondestructive inspection of large-diameter pipes and curved plate, longitudinal cracks are detected more efficiently by using guided circumferential waves. In the present, the study of guided circumferential waves and their application in detecting longitudinal defect were relative adequate when pipe material is isotropic. Based on linear three-dimensional elasticity, an orthogonal polynomial series expansions approach is used for determining the guided circumferential waves dispersion curves in homogeneous infinitely long orthotropic hollow cylinders. Results are compared with those published earlier and with the finite element simulation to check up the accuracy and range of applicability of this polynomial approach. Through the analysis of the displacements distributions and finite element simulation, the mode conversion of guided circumferential waves by end-reflection in cylindrical curved plate is discovered.     

Guided ionization waves: Theory and experiments

This review focuses on one of the fundamental phenomena that occur upon application of sufficiently strong electric fields to gases, namely the formation and propagation of ionization waves–streamers. The dynamics of streamers is controlled by strongly nonlinear coupling, in localized streamer tip regions, between enhanced (due to charge separation) electric field and ionization and transport of charged species in the enhanced field. Streamers appear in nature (as initial stages of sparks and lightning, as huge structures—sprites above thunderclouds), and are also found in numerous technological applications of electrical discharges. Here we discuss the fundamental physics of the guided streamer-like structures—plasma bullets which are produced in cold atmospheric-pressure plasma jets. Plasma bullets are guided ionization waves moving in a thin column of a jet of plasma forming gases (e.g., He or Ar) expanding into ambient air. In contrast to streamers in a free (unbounded) space that propagate in a stochastic manner and often branch, guided ionization waves are repetitive and highly-reproducible and propagate along the same path—the jet axis. This property of guided streamers, in comparison with streamers in a free space, enables many advanced time-resolved experimental studies of ionization waves with nanosecond precision. In particular, experimental studies on manipulation of streamers by external electric fields and streamer interactions are critically examined. This review also introduces the basic theories and recent advances on the experimental and computational studies of guided streamers, in particular related to the propagation dynamics of ionization waves and the various parameters of relevance to plasma streamers. This knowledge is very useful to optimize the efficacy of applications of plasma streamer discharges in various fields ranging from health care and medicine to materials science and nanotechnology.     

Double-quantum filtration under rotational-resonance conditions: numerical simulations and experimental results

The dependence of the performance of a recently introduced pulse sequence to achieve double-quantum excitation under the n = 1 rotational-resonance condition (T. Karlsson, M. Edén, H. Luthman, and M. H. Levitt, 2000, J. Magn. Reson. 145, 95–107) on different spin-system properties is investigated by means of numerical simulations and ¹³C MAS NMR experiments. For spin systems where chemical shielding anisotropies amount to only an insignificant fraction of the isotropic chemical shielding difference, high efficiencies are found for large and small dipolar coupling interactions. In the presence of significant chemical shielding anisotropies the overall efficiencies decrease and become strongly dependent on the duration of the excitation period. It is demonstrated that those spin-system parameters which are sensitively encoded in the lineshapes of a conventional n = 1 rotational-resonance spectrum are similarly sensitively encoded in the corresponding rotational-resonance double-quantum-filtered lineshapes and may be quantitatively recovered by iterative lineshape-fitting approaches. In certain favorable circumstances, the in-built selectivity of the rotational-resonance double-quantum-filtration approach permits successful application of the experiment on spin systems with more than two spins.     

Nonlinear lamb waves in a metal plate with defects

Experimental results on the propagation of finite-amplitude Lamb waves in a solid plate made of polycrystalline aluminum alloy with defects are presented. The Lamb waves are recorded and visualized using a scanning laser vibrometer. The dependences of the higher harmonic amplitudes, both averaged over the plate surface and measured at a point far from the defect and at the site of the defect, on the fundamental frequency amplitude are studied. A threshold character of the higher harmonic generation and a power-law behavior of their amplitudes are revealed, the latter feature being unconventional for the nonlinearity associated with the anharmonicity of the crystal lattice of the material. The possibility of locating the distribution of individual defects from the measured spatial distribution of structural nonlinearity in the material under study is experimentally demonstrated. The results of the experiments are explained in terms of the bilinear medium model.