Pure Transverse Waves Research Articles

Dispersive transverse waves for a strain-limiting continuum model

It is well known that propagation of waves in homogeneous linearized elastic materials of infinite extent is not dispersive. Motivated by the work of Rubin, Rosenau, and Gottlieb, we develop a generalized continuum model for the response of strain-limiting materials that are dispersive. Our approach is based on both a direct inclusion of Rivlin–Ericksen tensors in the constitutive relations and writing the linearized strain in terms of the stress. As a result, we derive two coupled generalized improved Boussinesq-type equations in the stress components for the propagation of pure transverse waves. We investigate the traveling wave solutions of the generalized Boussinesq-type equations and show that the resulting ordinary differential equations form a Hamiltonian system. Linearly and circularly polarized cases are also investigated. In the case of unidirectional propagation, we show that the propagation of small-but-finite amplitude long waves is governed by the complex Korteweg–de Vries (KdV) equation.

ZnO piezoelectric coatings used for the ultrasonic excitation of longitudinal-transverse waves and their application for smart bolts

The accurate measurement of the prestress for bolt has been intensively investigated in recent years. Many research studies have been conducted to improve the accuracy of prestress measurement using ultrasound techniques; in particular, the real-time prestress of bolt can be obtained without the requirements to calibrate the original load when ultrasonic longitudinal and transverse waves are applied simultaneously. However, there are few studies that have focused on ultrasonic transducers that can generate longitudinal and transverse waves simultaneously. Furthermore, research on coatings deposited directly on the top of the bolts for the excitation of longitudinal and transverse waves has not been reported. ZnO coating is widely used for the ultrasonic transducer due to its excellent piezoelectric property and stable structure in high temperatures. Therefore, it is an outstanding candidate for the excitation of both longitudinal and transverse waves and has a good application prospect in the field of bolt prestress measurement by ultrasonic sound wave with high accuracy. In this paper, ZnO piezoelectric coatings were deposited on (100)-oriented Si substrates by radio frequency sputtering techniques. The morphology, structure, and echo signal characteristics of the coatings were characterized by SEM, AFM, XRD, and ultrasonic measurement instruments. The experimental results showed that Ar/O2 ratio and deposition position exhibited significant effects on the excitation waveform of the ZnO piezoelectric coating. When the Ar/O2 ratio reduced to 1:3, the coating could excite both transverse and longitudinal waves. However, when the deposition position gradually moved away from the sputtering center, the transverse wave gradually enhanced. The coating could excite a pure transverse wave when the sample was at the edge of the effective sputtering area, which showed that the type of the excitation wave and the relative intensity of each wave could be well controlled by the Ar/O2 ratio and deposition position. These research results have a good application prospect in bolt stress measurement.

Intrinsic spin of elastic waves

Unveiling spins of physical systems usually gives people a fundamental understanding of the geometrical properties of waves from classical to quantum aspects. A great variety of research has shown that transverse waves can possess nontrivial spins and spin-related properties naturally. However, until now, we still lack essential physical insights about the spin nature of longitudinal waves. Here, demonstrated by elastic waves, we uncover spins for longitudinal waves and the mixed longitudinal-transverse waves that play essential roles in spin-momentum locking. Based on this spin perspective, several abnormal phenomena beyond pure transverse waves are attributed to the hybrid spin induced by mixed longitudinal-transverse waves. The unique hybrid spin reveals the complex spin essence in elastic waves and advances our understanding about their fundamental geometrical properties. We also show that these spin-dependent phenomena can be exploited to control the wave propagation, such as nonsymmetric elastic wave excitation by spin pairs, a unidirectional Rayleigh wave, and spin-selected elastic wave routing. These findings are generally applicable for wave cases with longitudinal and transverse components.

Transverse Wave Propagation in [ab0] Direction of Silicon Single Crystal

The speed and oscillation directions of elastic waves propagating in the [ab0] direction of a silicon single crystal were obtained by solving Christoffel's equation. It was found that the quasi waves propagate in the off-principal axis, and hence, the directions of the phase and group velocities are not the same. The maximum deviation of the two directions was <TEX>$7.2^{\circ}$</TEX>. Two modes of the pure transverse waves propagate in the [110] direction with different speeds, and hence, two peaks were observed in the pulse echo signal. The amplitude ratio of the two peaks was dependent on the initial oscillating direction of the incident wave. The pure and quasi-transverse waves propagate in the [210] direction, and the oscillation directions of these waves are perpendicular to each other. The skewing angle of the quasi wave was calculated as <TEX>$7.14^{\circ}$</TEX>, and it was measured as <TEX>$9.76^{\circ}$</TEX>. The amplitude decomposition in the [210] direction was similar to that in the [110] direction, since the oscillation directions of these waves are perpendicular to each other. These results offer useful information in measuring the crystal orientation of the silicon single crystal.

Reflection ofP andSV waves at the free surface of a monoclinic elastic half-space

The propagation of plane waves in an anisotropic elastic medium possessing monoclinic symmetry is discussed. The expressions for the phase velocity ofqP andqSV waves propagating in the plane of elastic symmetry are obtained in terms of the direction cosines of the propagation vector. It is shown that, in general,qP waves are not longitudinal andqSV waves are not transverse. Pure longitudinal and pure transverse waves can propagate only in certain specific directions. Closed-form expressions for the reflection coefficients ofqP andqSV waves incident at the free surface of a homogeneous monoclinic elastic half-space are obtained. These expressions are used for studying numerically the variation of the reflection coefficients with the angle of incidence. The present analysis corrects some fundamental errors appearing in recent papers on the subject.

Direct determination of group velocity surfaces in a cuspidal region in zinc

A direct determination of group velocity surfaces of zinc from the group velocities measured in various directions within and without a cuspidal region is described. A zinc crystal disk oriented in the symmetry direction [0001] is used as a specimen. Quasilongitudinal (QL) and quasitransverse (QT) elastic pulses propagating in a wide range of directions are generated by a broadband pointlike source activated by the fracture of a tiny glass capillary on the surface of the specimen and they are detected at epicentral and off-epicentral positions on the opposite face by a miniature capacitive transducer which senses a displacement normal to the surface. The group velocities associated with QL and QT modes are directly measured from the detected signals. The QL mode arrival is easily identified as the first point from which the signal jumps from the noise level. In the cuspidal region, there are three branches of QT group velocity: fast QT (FQT) ray branch, intermediate speed QT (IQT) ray branch, and slow QT (SQT) ray branch. It is observed that the arrivals of FQT, IQT, and SQT rays correspond to the first zero crossing, to the first negative minimum, and to the second local minimum, respectively, in the displacement signal. In the epicentral signal, both IQT and SQT rays coincide, whose arrivals correspond to single sharp negative minimum, and the FQT ray arrival is found at the first zero crossing after the arrival of a longitudinal wave. Outside the cuspidal region, both FQT and IQT branches disappear and so do their corresponding characteristics, i.e., the zero crossing and sharp negative minimum in the displacement curve. The SQT ray arrival outside the cuspidal region is observed at the first sharp minimum in the displacement signal. Finally, the group velocities of pure transverse waves with shear horizontal polarization propagating in various directions of a [0001] zonal section within and without the cuspidal region are measured with the miniature piezoelectric shear transducer (PZT) source and shear PZT detector.

Satellite observations of Pi 2 activity at synchronous orbit

Pi 2 magnetic pulsations are frequently observed at synchronous orbit by the UCLA fluxgate magnetometer on ATS 6. Events that occurred in September 1974 have been studied by using digital power spectra and coherency analysis to determine wave characteristics. From examination of wave form and application of spectral analysis, these Pi 2 events can be divided into three types. The first is a superposition of a 100‐s oscillation and a large‐amplitude, higher‐frequency Pi 1 activity. The second is a 100‐s wave unaccompanied by Pi 1. Both types have a significant compressional component. The third is a pure transverse wave in the azimuthal component. The pure transverse waves are quite rare, while the compressional waves occur during almost every substorm. In general, if the satellite is in the local time sector 1900–0300 LT, a Pi 2 burst accompanies every onset. In addition, onsets at the satellite are associated with ground Pi 2 bursts. The peak occurrence time of the satellite Pi 2 is 2100 LT. An examination of Pi 2 polarization at the satellite suggests that a polarization reversal occurs around midnight during quiet magnetic conditions (Kp ≤ 3+), left‐handed premidnight and right‐handed postmidnight. This result is similar to that obtained from ground‐based studies at stations equatorward of the auroral electrojet. The initial perturbation in the azimuthal component of a Pi 2 event is in the same sense as the perturbations caused by substorm‐associated, field‐aligned currents, positive (eastward) premidnight, negative (westward) postmidnight. This observation indicates that there may be a very close association between their causative mechanisms.

Energy and momentum transport in string waves

Formulas are derived for the energy, momentum, and angular momentum transmitted by waves of arbitrary shape in an inextensible string and by pure transverse waves in a suitably extensible string. The analysis is based on Tait’s procedure of viewing the wave from a moving frame of reference, and its application to the inextensible string, at least, is simple enough for use in elementary physics courses. The same method is used to find the elastic condition on an extensible string necessary for the propagation of mutually coherent shape and density waves. It is shown that the perfect transverse string waves referred to in textbooks can only be propagated in strings of the ideal Slinky spring type, and that they are incapable of carrying net momentum.

Self-Consistent Electromagnetic Waves in Relativistic Vlasov Plasmas

It is shown that the treatment of relativistic superluminous waves can be simplified by making a Lorentz transformation to a new frame in which the spatial variable no longer appears. Self-consistent solutions of the relativistic Vlasov-Maxwell equations can be constructed in the new frame. These solutions correspond to nonlinear traveling waves in a finite temperature plasma, and reduce to the usual linear results in the small-amplitude limit. A particular example is considered in which the nonlinear waves propagate through a relativistic Maxwellian plasma, and it is shown that pure transverse waves cannot exist in such a case, but that a coupled longitudinal field necessarily appears. For this case the nonlinear effects are evaluated correct to second order in the transverse field amplitude, thus obtaining a nonlinear dispersion relation and a measure of the coupling. In addition, solutions involving pure longitudinal waves can be found, and are also investigated. In limiting cases the results agree closely with previous calculations based on hydrodynamic models.

Propagation of a magnetospheric compressional wave to the ground

A detailed analysis indicates that a pure compressional wave in the magnetosphere at the equator was observed on the ground near the northern conjugate area as an attenuated nearly pure transverse wave. The plane of polarization of the wave was tilted down ∼12° from the ground plane. Ground-based observations suggest that no proposed theory readily explains the occurrence of the magnetospheric wave.

Propagation of waves perpendicular to the magnetic field in a two-component warm plasma

Propagation of small amplitude waves perpendicular to the magnetic field in a two component magnetoactive, unbounded warm plasma having arbitrary mass ratio has been examined using Maxwell's equations for the electromagnetic field and the first three moment equations for each component. The full pressure tensor equation (neglecting the heat flow tensor) has been used and the effect of momentum and pressure relaxation mechanisms has been included in the analysis. Dispersion relations for pure transverse waves are obtained and discussed in detail for both low and high frequency propagation. The effect of thermal motions and collisions is examined for transverse and coupled waves. It has been found that the pressure relaxation mechanism contributes significantly in the damping of oscillations at the harmonic cyclotron frequency.

Stability of Waves in Relativistic Plasma

AbstractThe dispersion relation of electromagnetic waves propagating perpendicular to an applied uniform magnetic field B0 in relativistic plasma is derived. Waves propagating perpendicular to the uniform applied magnetic field can be separated into two modes ‐ one is the linearly polarized transverse wave and the other is a hybrid mode. In the present analysis, dispersion relation of the first mode i. e., for a pure transverse wave is analysed under the assumption that the wavelength is much longer than the cyclotron radii of the electrons. A stability criterion which limits the thermal energy of the electrons along B0 is obtained.

Research on combustion instability in liquid propellant rockets

Progress in rocket development has resulted in a shift of interest from spontaneous to triggered instability. Hence, the primary objective of research has moved into the field of nonlinear phenomena. Analytical methods of dealing with such phenomena are under development. Results are discussed for particular chamber and nozzle geometries using the heuristic n−τ combustion model, already extensively used in linear instability studies and in practical instability correlations. They consider either longitudinal or pure transverse waves, with or without shock waves. Final wave shapes are calculated in the regions of linear instability. In welldefined portions of the region of linear stability, nonlinear instability is predicted with the corresponding wave shapes and triggering criterion. A physically more-realistic combustion model, based on droplet evaporation, was also treated, and the corresponding results are discussed. Preliminary experimental results are presented obtained from combustion chambers reproducing the conditions of the analytical study, the ultimate objective of these experiments being to check the validity of the theoretical assumptions. A more-direct check on the dynamics of the combustion phenomena is also being carried out through a kind of shock-tube technique applied to the actual combustion zone. Preliminary results from this apparatus are also presented. Another important field of research concerns the ways of improving the dissipation of the oscillation energy so as to favorably affect the energy balance responsible for instability. This problem has recently attracted wide attention in rocket development. The best known ways of dissipating oscillatory energy are through the exhaust nozzle itself, through baffles, and through acoustic resonators. Theoretical and/or experimental results on the three damping devices are discussed. The exhaust nozzle can only provide damping when a longitudinal component is present in the oscillation. Baffles can be quite effective if properly designed, but the control of their damping quality is not easy. Acoustic resonators seem to provide the most effective and easily controllable damper, if properly used. Probably the best design is not in the form of an “acoustic liner” but in more concentrated forms. Combinations of baffles and acoustic resonators may provide the best solutions.

WAVE PROPAGATION IN MAGNETOPLASMAS

Wave propagation in a fully ionized unbounded magnetoplasma is considered taking into account the momentum transfer and energy exchange due to collisions. Dispersion relations are examined for wave propagation along and perpendicular to the magnetic field. It is found that there is no additional damping due to temperature relaxation for pure transverse wave propagation. For longitudinal waves dispersion relations are presented which include damping due to temperature relaxation and momentum transfer due to collisions.

Wave Motion in a Plasma with Anisotropic Pressure

The moment equations, neglecting the heat-flow tensor, have been used to study the wave propagation in a plasma consisting of electrons and singly charged ions, in a magnetic field, with anisotropic pressure tensors for both components. Two cases of wave propagation along the magnetic field and perpendicular to it have been worked out separately. In the first case of propagation along the magnetic field, the waves separate into a pure longitudinal and a pure transverse wave, the second of these only being affected by the anisotropy of the pressure tensor. Conditions are deduced under which the anisotropy of the pressure tensor can alter the propagation conditions of the waves. Attention is also given to the possibility that it can give rise to an instability. One important result which follows for magnetohydrodynamic waves is that an anisotropy in the electron pressure can give rise to an instability of the Alfvén waves. In the other case of the propagation across the field, the waves separate into a pure transverse wave and a wave which represents a combined longitudinal and transverse wave. The second of these is unaffected by the anisotropy of the pressure tensor. For the pure transverse wave, conditions are obtained under which instability can arise.

ELASTISCHE WELLEN IM ANISOTROPEN MEDIUM NACH EINER MAKROSEISMISCHEN UNTERTAGE‐MESSUNG*

AbstractIn some mines of the Siegerland, besides dilatational and shear waves, an additional direct wave has sometimes been found the velocity of which was somewhat lower than that of the shear wave. Investigations were carried out with the object to establish the directions of oscillations and the velocities of all three kinds of waves. It was the purpose to draw conclusions as to the character and the causes of the appearence of the third wave.By these investigations the wave having the lowest velocity can be identified as a second shear wave. This points to the fact that double refraction of the shear wave is taking place, the wave of higher velocity being a pure transverse wave and the wave of lower velocity being a quasi‐transverse wave.As a result of these measurements the examined medium is shown to have a transverse isotropy due to stratification. It is, however, not possible to decide whether there exists also an anisotropy due to schistosity.Some further calculations show that the wave of lowest velocity can not be explained as a surface wave propagated on the walls of the gallery.