Quasi-longitudinal Research Articles

General properties of the Anisimkin jr. plate modes

Acoustic plate modes of different orders n, having equal velocities v(n) close to that of the longitudinal BAW v(L), are numerically studied in crystals of different symmetries. Three families of the modes with v(n) ≈ v(L), each at relevant plate thickness h/λ = (h/λ)n, are found (h is the thickness, λ is the wavelength): the generalized Lamb mode with comparable longitudinal u1, shear-horizontal u2, and shear-vertical u3 displacements, the Anisimkin Jr. (AN) mode with u1 >> u2 and u3, and u1 ≈ constant ≠ 0 at any depth, and the quasilongitudinal (QL) mode with u1 > u2, and u3, but u1 ≠ constant over the plate thickness. Existence of the families does not depend on anisotropy or piezoelectric properties of the plate, but on the closeness of the mode velocity v(n) to the BAW velocity v(L), the value of the dispersion slope dv(n)/d(h/λ) at v(n) = v(L) and h/λ = (h/λ)n, and the proximity of the plate thickness (h/λ)n supporting the mode, to the thickness (h/λ)R providing transverse BAW resonance between plate faces. The Lamb modes approach v(n) = v(L) at irregular (h/λ)n far from resonance (h/λ)R and at large dv(n)/d(h/λ) ~10(3) m/s. The two other modes are characterized by lower dispersion dv(n)/d(h/λ) ≤ 10(3) m/s and regular (h/λ)n close to the resonance (h/λ)R. Because both modes have small vertical displacement on plate faces and propagate almost entirely within the crystals, they are attractive for liquid sensing.

Wave propagation in an initially stressed transversely isotropic thermoelastic solid half-space

The governing equations of thermoelasticity of transversely isotropic solid with initial stresses are formulated at uniform temperature. These equations are solved analytically in two-dimensions to show the existence of three plane quasi waves, namely, Quasi-Longitudinal (QL), Thermal (T-mode) and Quasi-Transverse (QT) waves. Reflection from a thermally insulated stress free surface of an initial stressed transversely isotropic thermoelastic solid half-space is studied. A particular model is chosen for the numerical computations of the propagation speeds, attenuation coefficients and reflection coefficients. Effects of initial stress parameter and thermal disturbances are observed on speeds of propagation, attenuation coefficients and reflection coefficients.

Peculiarities of the Anisimkin Jr.' plate modes in LiNbO3 and Te single crystals

Quasi-longitudinal (QL) Anisimkin Jr.' modes discovered recently in quartz plates are found now in 2 other piezoelectric crystals, belonging to trigonal symmetry. In 128 degrees Y, X+90 degrees-LiNbO3, 2 different QL modes may propagate simultaneously for small plate thickness h/lambda = 0 to 0.06 (h is thickness, lambda is wavelength). The velocity of the 1st mode, v1, is close to the longitudinal bulk wave velocity vL. It is varied with h/lambda and piezoelectrically stiffened (maximum K(n)(2) = 39% at h/lambda = 0.08). The velocity of the 2nd QL mode, v2, is close to the shear-horizontal bulk wave velocity vQSH, not varied with h/lambda and not stiffened (K(n)(2) = 0). On the contrary, 210 degrees Y, X-Te crystal supports only one QL-mode, but it is unusually wide-ranging and low-dispersive: the mode exists for all h/lambda from 0 to 2.5 with velocity v(n) almost permanent and equal to vL in the whole range. This mode is piezoelectrically stiffened (maximum K(n)(2) = 2% at h/lambda = 0.13). The variety of the Anisimkin Jr.' modes in different crystals makes them attractive for liquid sensors, where the amount of suitable waves is very restricted.

Effect of rotation and imperfection on reflection and transmission of plane waves in anisotropic generalized thermoelastic media

The present investigation is concerned with the propagation of plane waves at an imperfectly bonded interface of two orthotropic generalized thermoelastic rotating half-spaces with different elastic and thermal properties. The thermoelastic theory with one relaxation time developed by Lord and Shulman [A generalized dynamical theory of thermoelasticity, J. Mech. Phys. Solids 15 (1967) 299–309] is used to study the problem. The reflection and transmission coefficients of Quasi Longitudinal (QL-) wave, Quasi Thermal (T-mode) wave and Quasi Transverse (QT-) wave have been derived. The effect of rotation has been studied on the velocities of different waves. Some special cases of boundaries i.e. normal stiffness, transverse stiffness, thermal contact conductance, slip boundary and welded contact boundary have been deduced from an imperfect one. Impact of different boundaries has been studied graphically. It is observed that thermal properties, rotation and imperfect boundary have significant effect on the propagation of waves.

Interfacial imperfection on reflection and transmission of plane waves in anisotropic micropolar media

This study is concerned with the reflection and transmission of plane waves at an imperfectly bonded interface between two orthotropic micropolar elastic half-spaces with different elastic and micropolar properties. There exist three types of coupled waves in xy-plane. The reflection and transmission coefficients of quasi-longitudinal (QLD) wave, quasi-coupled transverse microrotational (QCTM) wave and quasi-coupled transverse displacement (QCTD) wave have been derived for different incidence waves and deduced for normal force stiffness, transverse force stiffness, transverse couple stiffness and perfect bonding. The numerical values of modules of the reflection and transmission coefficients are presented graphically with the angle of incidence for orthotropic micropolar medium (MOS) and isotropic micrpolar medium (MIS). Some particular cases of interest have been deduced from the present investigation.

Reflection of piezothermoelastic waves from the charge and stress free boundary of a transversely isotropic half space

The present paper concentrates on the study of propagation and reflection characteristics of waves from the stress free, thermally insulated/isothermal boundary of a piezothermoelastic half space. The non-classical (generalized) theories of linear piezo-thermoelasticity have been employed to investigate the problem. In the two-dimensional model of the transversely isotropic piezothermoelastic medium, there are three types of plane waves quasi-longitudinal (QL), quasi-transverse (QT) and thermal wave ( T-mode), whose velocities depend on the angle of incidence and frequency. These waves are dispersive in character and are also affected by piezoelectric as well as pyroelectric properties of the materials. The low and high frequency approximations for the speeds of propagation and the attenuation coefficients of these waves have been obtained. The quasi-longitudinal (QL), quasi-transverse (QT) and thermal wave ( T-mode) incident cases at the stress free, thermally insulated or isothermal open circuit boundary of a transversely isotropic piezothermoelastic half space are considered to discuss the reflection characteristics of various waves. The amplitude ratios of reflected waves to that of incident one in each case have been obtained. The special cases of normal and grazing incidence are also derived and discussed. Finally, the numerical computations of reflection coefficients are carried out for cadmium Selenide (CdSe) material by using Gauss elimination procedure. In addition the phase velocities and attenuation coefficients are also computed along various directions of wave propagation. The obtained results in each case are presented graphically.

Linearized approximations for phase velocities of elastic waves in weaklyanisotropic media

This paper focuses on the development of linearized formulae for the phasevelocities of quasi-longitudinal (QL) and quasi-transverse (QT) waves in weaklyanisotropic media. The formulae for quasi-transverse waves derived fromfirst-order perturbation theory possess the property of rotational invariance in aplane perpendicular to the wave normal. Based on this property, a planarcoordinate system, describing the possible polarization of the two degeneratetransverse waves, can be conveniently selected. The relationships between thefirst-order formulae and other related existing approximate expressions areclarified. For a symmetry plane of orthorhombic media, we extended Gassmann'sformula for the QL-wave phase velocity in transversely isotropic media to QLand QT waves. Three forms of phase velocity expressions are discussed. Fornon-symmetry planes of weakly orthorhombic media, we propose a unified formfor the QL-wave phase velocity and in a very simple way derive the linearizedThomsen- and Gassmannn-like formulae. Furthermore we suggest QT-wave phasevelocity expressions which are linear functions of the elastic constantsassuming that wave propagation has a slight deviation from a symmetry plane oforthorhombic media. Finally, numerical results in a symmetry plane show thatthe set of first-order approximate formulae performs best for weaklyanisotropic materials.

Unique cuspidal features in the group-velocity surface of barium sodium niobate and Rochelle salt

Crystals that have sufficient elastic anisotropy can exhibit cuspidal features in the group-velocity surface so that the group velocity is double- or triple-valued. In nonpiezoelectric crystals such cuspidal features occur only in the quasi-transverse (QT) modes when the wave vectors are restricted to the symmetry planes of the crystal. Sufficient piezoelectric stiffening of the elastic constants, however, can generate cuspidal features in other modes when the wave vectors are restricted to these same symmetry planes. In barium sodium niobate (BSN) there is insufficient elastic anisotropy to generate cuspidal features when piezoelectric stiffening is neglected. The addition of piezoelectric stiffening changes only the quasi-longitudinal (QL) and QT modes in the (100) and (010) planes, and changes only the pure transverse (T) mode in the (001) plane. There is sufficient piezoelectric stiffening, however, to generate small cuspidal features in the QL mode about the [001] and \([00\bar 1]\) axes in the (100) and (010) planes. In Rochelle salt, however, there is sufficient elastic anisotropy to generate cuspidal features in the QT modes about collinear axes along nonsymmetry directions in the (100) and (010) planes. The addition of piezoelectric stiffening changes only the T mode in the (100), (010) and (001) symmetry planes. This piezoelectric stiffening is large enough, however, to generate large cupsidal features in the T mode about new collinear axes along nonsymmetry directions in the (010) and (001) planes.

Determination of all elastic constants of orthotropic plate specimens from group velocity data

This paper presents a novel method of nondestructively determining all nine elastic constants of fiber-reinforced orthotropic plate specimens. Group velocities of bulk quasi-longitudinal (QL), pure transverse (PT), and surface skimming pseudo-longitudinal modes are measured in glass-fiber (GF) and carbon-fiber (CF) reinforced Poly Ether Ether Ketone (PEEK) specimens by a point-source/point-receiver (PS/PR) technique. First, the pure index longitudinal moduliC 11,C 22, andC 33, are obtained from the longitudinal (L) group velocities measured in three principal directions. Next, the pure index shear moduli,C 44,C 55, andC 66, are determined by measuring the shear horizontally (SH) polarized PT group velocities in the symmetry planes. Finally, the mixed index elastic constantsC ij (i ≠ j) are calculated from QL group velocities measured in the symmetry planes and using analytic formulas.

Determination of All Elastic Constants of Orthotropic Plate Specimens from Group Velocity Data

Abstract This paper presents a novel method of nondestructively determining all nine elastic constants of fiber-reinforced orthotropic plate specimens. Group velocities of bulk quasi-longitudinal (QL), pure transverse (PT), and surface skimming pseudo-longitudinal modes are measured in glass-fiber (GF) and carbon-fiber (CF) reinforced Poly Ether Ether Ketone (PEEK) specimens by a point-source/point-receiver (PS/PR) technique. First, the pure index longitudinal moduli C 11, C 22, and C 33, are obtained from the longitudinal (L) group velocities measured in three principal directions. Next, the pure index shear moduli, C 44, C 55, and C 66, are determined by measuring the shear horizontally (SH) polarized PT group velocities in the symmetry planes. Finally, the mixed index elastic constants C ij (i ≠ j) are calculated from QL group velocities measured in the symmetry planes and using analytic formulas.

Imaging of group velocity surfaces in a cuspidal region of zinc by laser-generated ultrasonic waves

An image of group velocity surfaces within and outside a cuspidal region of a (0001) oriented zinc crystal is obtained with a pointlike scanning laser source and a fixed longitudinal (L) mode piezoelectric detector. Because of axisymmetric excitation and sensing, the group velocity surface of the pure transverse (PT) mode, being shear horizontally (SH) polarized, is absent in the obtained image. In the cuspidal region the quasi transverse (QT) mode has three group velocity branches in a given direction: the fast QT (FQT) branch, the intermediate speed QT (IQT) branch, and the slow QT (SQT) branch. The laser imaging clearly shows the wave fronts of the quasi longitudinal (QL), IQT, and SQT rays in the cuspidal region. Near the epicentral [0001] direction, the arrivals of the head wave (HW) and SQT rays are almost simultaneous and their wave fronts are not distinct. However, farther from the epicentral direction, they separate and outside the cuspidal region the imaging shows the group velocity surfaces of the QL, HW, and SQT rays.

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.

Group velocity of whistlers in a two‐ion plasma

Computations of the group velocity of whistlers in a two‐ion plasma show that a quasi‐longitudinal (QL) right‐handed polarized wave (electron wave) has a minimum group velocity at the so‐called crossover frequency; a QL left‐handed polarized wave has a maximum group velocity at the same frequency. A quasi‐transverse (QT) electron wave has a minimum group velocity at the ion hybrid frequency. In an oxygen/hydrogen plasma this minimum occurs only for almost perpendicular waves, whereas in a helium/hydrogen plasma it occurs also for waves propagating off perpendicular to the magnetic field. We also find that a small concentration of helium in a hydrogen plasma introduces a ‘stop band’ for quasi‐transverse waves with frequencies just above the helium gyrofrequency. The theoretical results are used to interpret satellite observations of quasi‐transverse low‐frequency whistlers.

On the propagation of ionospheric whistlers at low latitude

The K-9M-26 rocket was launched from Kagoshima Space Center in Japan (geomag. lat. 20°N) to study the propagation characteristics of ionospheric or short fractional-hop whistlers coming from the local Earth-ionosphere waveguide. Their dispersion and amplitude characteristics are investigated in detail, and they are checked against the propagation theory. An analysis of their dispersions shows that the quasi-longitudinal ( QL) approximation is pertinent above the E-region even at such low latitudes as 20°. Then it is found from the study of wave field intensities that the observed ionospheric whistlers are identified as having propagated up to the rocket after propagation in the Earth-ionosphere waveguide over considerable distances.

Conditions for Orthogonality Between the Geomagnetic Field and an HF Sky‐Wave Ray

At HF or higher frequencies, in computing the polarization conditions along a radio ray traveling in the ionosphere, it is often sufficient to express the results in terms of the Faraday rotation of the plane of polarization, obtained as an integral (along the path) of the phase difference between the ordinary and extraordinary components. The computation simplifies when the quasilongitudinal (QL) approximation of magneto‐ionic theory can be applied. The QL approximation is invalid at any point where the ray becomes transverse to the geomagnetic field, and in some regions around that point. In 1963 George A. Dulk showed that neglect of this effect may lead to major errors. In considering an oblique path with sky‐wave reflection, it is easy to be misled into thinking that a transverse condition is unlikely except on east‐west paths, particularly in equatorial regions. On the contrary, transverse conditions are likely to occur under a wide range of conditions at all latitudes and for all propagation directions. This presentation describes conditions of ray geometry and magnetic latitude, which result in transversality at some point along the path. The method for defining the conditions is shown to be applicable even when the ray is permitted to follow a twisted path in three dimensions, and the geomagnetic field is defined under very general assumptions. The basic results are also directly applicable to problems concerned with the reflection of HF signals by field‐aligned ionization.

Wave propagation in a transversely isotropic heat‐conducting elastic material

A transversely isotropic elastic material can transmit three body waves in each direction, a quasi-longitudinal (QL) wave, a quasi-transverse (QT) wave, and a purely transverse wave. When the material is able to conduct heat the properties of small amplitude QL and QT waves are modified and we consider here the analysis of such thermo-elastic interactions in plane harmonic disturbances. The modified QL and QT waves are both found to exhibit frequency-dependent dispersion and damping of the kind known to affect dilatational waves in isotropic heat-conducting elastic materials, and in addition we show that the particle paths in the associated motions are ellipses with their axes inclined to the wave normal. This latter effect is peculiar to body waves travelling in anisotropic heat-conducting elastic materials and seems not to have been studied in detail hitherto. Numerical results referring to the propagation of plane harmonic body waves in a single crystal of zinc are presented and discussed.

Coupling processes in the D- and E-regions at low and very low frequencies—II Frequencies greater than critical: a study of the night-time ionosphere

The generalised thin-film optical method is used to analyze propagation phenomena in a typical night-time ionosphere at frequencies in the range 10–300 kHz, the ‘critical’ frequency being 24 kHz. Propagation phenomena in the VLF and lower LF range take place in conditions of ‘limiting polarisation’, so that the ‘composite’ or physical waves pass quite smoothly through the X = 1 level and other regions of rapidly varying polarization, even at the critical frequency. In such cases the inter-mode coupling coefficients attain very large values which are just sufficient to produce the requisite ‘mode reshuffling’ to maintain composite waves of almost constant polarisations and amplitudes. The low values of electron density in the D-layer lead to large departures from quasilongitudinal (Q.L.) propagation conditions, which are generally reestablished however in the region of large and rapid increase of electron density at the top of the D-layer. This leads to ‘mode reshuffling’, and if the X = 1 level is at a somewhat greater altitude the coupled ordinary wave may then be reflected at the X = 1 level to produce effects analogous to the Z-coupling echo and the Z-trace. In the upper LF range the inter-mode coupling processes associated with the rapid variation of wave polarisation at the X = 1 level are suppressed by an interference process so that ‘mode reshuffling’ is eliminated, and an initial extraordinary wave follows the characteristic polarisation reversal at the X = 1 level with little loss.

Electric Field Observations of Ionospheric Whistlers

A Nike‐Tomahawk sounding rocket was launched near local noon on November 8, 1966, from Wallops Island with a triaxial electric field experiment using long cylindrical dipoles for sensors. Twenty ELF whistlers with dispersion constants ranging from 0.15 to 0.85 sec1/2 were detected and identified as having propagated up to the rocket through the E region from the earth‐ionosphere waveguide. The data extend to the smallest dispersion ever encountered and the lowest frequencies observed on ionospheric whistlers (60 Hz). The waves were eliptically polarized, becoming more nearly circular at higher altitudes. Analysis showed good agreement between the experimental results and quasi‐longitudinal (QL) propagation theory for amplitude, dispersion, and polarization characteristics of the wave once it is away from the D‐ to E‐region boundary. No appreciable effects of negative ions or protons were observed in the dispersion of the whistlers.

Instability near Electron Cyclotron Harmonic Frequencies in Beam-Plasma System

When a weak electron beam is injected into a plasma in a thermal equilibrium, there appear the instabilities, which result in the excitation of the following waves; (i) the electrostatic cyclotron harmonic waves (CHW), (ii) the upper hybrid resonance wave, (iii) the fine structure near the cyclotron harmonic frequencies, (iv) the harmonic waves of half the cyclotron frequency and (v) the quasi-longitudinal (QL) waves in a low frequency range. The intensity of the microwave emission resulting from these instabilities is very strong and is larger by several tens d B than that emitted from the plasma in the absence of the electron beam. The excitation of CHW is ascribed to the interaction of CHW in plasma with those in the electron beam. The fine structure is due to the standing waves of CHW confined in the plasma column. The instability in the l f range is ascribed to a coupling of the QL waves in the plasma with the space charge waves in the beam.

An improved whistler-mode refractive index equation

Introduction. The Appleton-Hartree equation is often approximated for computational and conceptual purposes. Most often used, and the most valuable of these approximations, is the quasi-longitudinal (QL) [Booker, 1935]. We propose here a modification of this approximation for the whistler mode which results in the correct location of the refractive index infinity and a closer approximation to the position of the points of inflection.