Acoustic properties of the film/plate layered structure

A new propagation medium-a layered structure composed of a film and a plate-is suggested and studied, using c-oriented ZnO and AlN films on (001), {100}-Si plate, as two opposite examples of slow-on-fast and fast-on-slow material combinations. For both structures, the modes belonging to Lamb, quasi-longitudinal (QL), and Anisimkin Jr.' (AN) families are found. For each family, the velocities v(n), displacement profiles, and electromechanical coupling coefficients K(n)2 for 4 electrode configurations are numerically calculated by the matrix method as a function of the mode order n = 0 to 8, plate thickness H/λ = 0 to 2.0, and film thickness h/λ = 0.02 to 0.04 (H and h are thicknesses; λ is the wavelength). Some high-order modes in the structure have K(n)2 = 0 for any H/λ, h/λ, and electrode configuration. Other modes possess variable K(n) 2 with a maximum value larger than the coupling coefficient for the Rayleigh SAWs in ZnO and AlN single crystals or layered structures using the same films and semi-infinite silicon substrate. There are also QL-modes having high velocity v(n), large K(n)2, and low propagation loss caused by liquid loading. These modes are well suited for liquid sensors.     

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°Y,X+90°-LiNbO3, 2 different QL modes may propagate simultaneously for small plate thickness h/λ = 0 to 0.06 (h is thickness, λ is wavelength). The velocity of the 1st mode, v1, is close to the longitudinal bulk wave velocity vL. It is varied with h/λ and piezoelectrically stiffened (maximum Kn² = 39% at h/λ = 0.08). The velocity of the 2nd QL mode, v2, is close to the shear-horizontal bulk wave velocity vQSH, not varied with h/λ and not stiffened (Kn² = 0). On the contrary, 210°Y,X-Te crystal supports only one QL-mode, but it is unusually wideranging and low-dispersive: the mode exists for all h/λ from 0 to 2.5 with velocity vn almost permanent and equal to vL in the whole range. This mode is piezoelectrically stiffened (maximum Kn² = 2% at h/λ = 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.     

Analytical solutions for sagittal plane waves in three-layer composites

The analytic dispersion equations for the symmetric and antisymmetric saggital plane plate modes of a three-layer composite system are presented. The composite consists of a solid isotropic plate sandwiched between two acoustically thin (<0.02 lambda) isotopic solid layers, where lambda is the acoustic wavelength. The thin layers are considered either as the mass loading or the chemical selective coating layers for the plate wave sensors. Explicit formulas which identify the contributions of the elasticity and inertia effects for the phase velocity and mass loading sensitivity of the lowest symmetric (S(0)) and antisymmetric (A(0)) mode for the case where the thickness of the composite plate is much less than lambda are obtained. The amounts by which the elasticity of the thin layer and the inertia decrease the mass loading sensitivity is found for both sensors. It is also found that the sensitivity of the A(0) mode significantly depends on the operating frequency but that of the S(0) mode does not. Specific examples are given for the case of a fused silica plate sandwiched by two thin lucite layers.     

Attenuation of acoustic normal modes in piezoelectric plates loaded by viscous liquids

Attenuation cac versus viscosity eta of adjacent liquid is measured for each normal mode n generated in 30 different plates of commercially available, piezoelectric crystals with thickness-to-wavelength ratio in the range h/lamda = 0.6 - 2.5. Two modes with an optimal combination of sensitivity (0.1 dB/mm x cP), insertion loss (<35 dB), and stop-band rejection (>15 dB) are found in liquid-loaded 128 degrees Y,X + 90 degrees - LiNbO3 with h/lamda = 1.67. Both modes are suited for viscosity measurements and other sensing tasks in viscous liquids. They have predominantly longitudinal displacement and large propagation velocity v(n), about 15,000 m/s.     

Propagation of the Anisimkin Jr.' and quasi-longitudinal acoustic plate modes in low-symmetry crystals of arbitrary orientation

The Anisimkin Jr. (AN) acoustic plate mode having dominant and depth-independent longitudinal displacement (u(1) > u(2), u(3); u(1) ≈ constant) is numerically found in tetragonal 4mm Li(2)B(4)O(7) crystal with one of the low-symmetry orientations (Euler angles 89°, 37°, 104°), as an example. The quasi-longitudinal (QL) modes with dominant and depth-dependent longitudinal displacement (u(1) > u(2), u(3); u(1) ≠ constant) are experimentally detected along several propagation directions Θ in 128°y-LiNbO(3) plate, where Θ = 0°, 30°, 60°, and 90° with respect to the x-axis. Compared with more symmetrical plate materials and orientations, the displacement profiles of the AN and QL modes in lower-symmetry counterparts are qualitatively the same, but their phase profiles are more complicated. Moreover, like any acoustic wave, all plate modes in anisotropic crystals suffer from beam steering, in general. The power flow angles of the modes propagating in a fixed direction are different and depend on the mode order n.     

Propagation of QSH (quasi shear horizontal) acoustic waves in piezoelectric plates

The characteristics of QSH (quasi shear horizontal) acoustic waves propagating in thin plates of Y-cut, X-propagation lithium niobate are investigated theoretically and experimentally. The fractional velocity change (Deltanu/nu) produced by electrical shorting of the surface is calculated as a function of the normalized plate thickness h/lambda (h=plate thickness, lambda=acoustic wavelength). It was found that values of Deltanu/nu as high as 0.18 could be obtained. Experimental measurements show good agreement with theory. The properties of QSH waves propagating in the presence of a perfectly conducting electrode separated from the piezoelectric plate by a small air gap have been studied theoretically and experimentally. It was found that by varying the height of the gap, the phase shift through a 3.2-MHz QSH wave delay line can be varied by more than 230 degrees . We have also theoretically investigated the influence of a thin layer of arbitrary conductivity on the velocity and attenuation of the QSH wave. Calculations show that the variations in these parameters can be as high as 18% and 5 dB per wavelength for a change in layer surface conductance from 10(-7) to 10(-5) S. Results obtained in this paper confirm the attractive properties of QSH waves for a variety of sensing and signal processing applications.     

Question of solving nonlinear problems under the combined action of heat conduction and radiation

The radiant heat transfer of an annular plate to the environment is examined. A simple analytic solution is obtained that describes the stationary temperature field.Notation T running temperature - T_i temperature of the inner edge of the ring - r running radius - emissivity - Stefan-Boltzmann constant - ₀ heat conductivity of the material - b thickness of the annular plate - relative temperature - R=r/r₀ relative running radius Translated from Inzhenerno-Fizicheskii Zhurnal, Vol. 42, No. 1, pp. 119–122, January, 1982.     

On the solution of nonstationary heat-conduction problems with variable heat-transfer coefficient

The article describes an exact method for calculating the temperature field in solids when they are heated in a medium with a variable heat-transfer coefficient and a nonuniform initial temperature distribution.Notation temperature - L thickness of plate - x space coordinate - a thermal diffusivity - thermal conductivity - heat-transfer coefficient - t time - X=x/L dimensionless coordinate - Fo=at/L² Fourier number - Bi(Fo)=(Fo)L/ Biot number Translated from Inzhenerno-Fizicheskii Zhurnal, Vol. 20, No. 5, pp. 921–924, May, 1971.     

Five-mode frequency spectra of x3-dependent modes in AT-cut quartz resonators

We study straight-crested waves and vibration modes with variations along the x(3) direction only in an AT-cut quartz plate resonator near the operating frequency of the fundamental thickness-shear mode. Mindlin's two-dimensional equations for anisotropic crystal plates are used. Dispersion relations and frequency spectra of the five relevant waves are obtained. It is found that, to avoid unwanted couplings between the resonator operating mode and other undesirable modes, in addition to certain known values of the plate length/thickness ratio that need to be avoided, an additional series of discrete values of the plate length/thickness ratio also must be excluded.     

Doubly rotated contoured quartz resonators

Doubly rotated contoured quartz resonators are used in the design of temperature-compensated stable clocks and dual-mode sensors for simultaneous measurements of pressure and temperature. The design of these devices is facilitated by models that can predict frequency spectra associated with the three thickness modes and temperature and stress-induced frequency changes as a function of crystalline orientation. The Stevens-Tiersten technique for the analysis of the C-mode of a doubly rotated contoured quartz resonator is extended to include the other two thickness modes. Computational results for harmonic and anharmonic overtones of all three thickness modes of such resonators help in optimizing the radius of curvature of the contour and electrode shape for suppression of unwanted modes and prevention of activity dips. The temperature and stress-induced changes in thickness-mode resonator frequencies are calculated from a perturbation technique for small dynamic fields superposed on a static bias. The static bias refers to either a temperature or stress-induced static deformation of the resonator plate. Phenomenological models are also used for calculating the temperature and stress-induced changes in resonant frequencies as a function of crystalline orientation. Results for the SBTC-cut quartz plate with a spherical convex contour of 260 mm indicate that normal trapping occurs for the third (n=3) and fifth (n=5) harmonic of the A-mode, the fundamental (n=1) and third (n=3) harmonic of the B-mode, and the fundamental (n=1) and fifth (n=5) harmonic of the C-mode     

Collisional Damping and Resonance Behavior of Coupled Scissors Modes of a Bose–Einstein Condensate

We investigate the nonlinear coupling between the lowest three scissors modes of a Bose–Einstein condensate at zero temperature. Using a variational approach with a general variational wave function we determine, solely from the parity of the scissors modes, the nonvanishing coupling terms. In agreement with a similar previous calculation with a Gaussian variational wave function, which is a special case of our general function, we find two resonance conditions at trap anisotropy ratios = 1 and = 7. We use the latter condition to explain the observed resonance in the collisional damping of scissors modes. In addition, we investigate the higher order scissors modes and the eigenmodes and eigenfrequencies for isotropic traps.     

Characteristics of ultrasonic Lamb waves in 128° rotated Y-cutlithium niobate

The characteristics of ultrasonic Lamb waves propagating along the X-direction of a 128° rotated Y-cut lithium niobate plate are investigated. The first higher-order antisymmetric mode, the A₁ mode, is found to exhibit an anomalous behavior. The velocity of this mode remains nearly constant for all values of h/λ, where h is the plate thickness and λ is the acoustic wavelength. The particle displacement of the mode tends towards that of a pure shear horizontal (SH) wave as the ratio h/λ tends to zero. The electromechanical coupling coefficient of the wave has a value of k²=0.78×10^-2 at h/λ≅0. The coupling decreases as h/λ increases, becoming negligible for h/λ>1. The velocity and coupling coefficient of the mode have been measured for various values of h/λ, and are found to be in fair agreement with theoretical calculations     

Experimental study on the characteristic of the NS-GT cut quartz crystal resonator oscillating in the sub-resonant frequency

We previously reported that the dynamic photo-elastic method was a very effective measuring technique for the stress distribution of vibrating quartz crystal resonators. The existence of a twisted asymmetrical vibration mode has been verified experimentally when the NS-GT cut quartz crystal resonator was vibrating in the main resonant frequency (MRF). A MRF and a sub-resonant frequency (SRF) of the NS-GT cut quartz resonator were defined as follows. If a mechanical standing wave was in the x' or y' direction of the resonator, the former was MRF vibration and the latter was SRF vibration, respectively. In this paper, stress distributions of two samples of the NS-GT cut quartz crystal resonator, one of which had a thickness of 80 mum and the other 150 mum, were measured by the dynamic photo-elastic method when the resonators were vibrating in each SRF. Thereafter, vibration modes of those resonators were estimated by the experimental data of stress distributions. We find that the vibration mode of the 80-mum resonator had a simple mechanical standing wave on the y' direction and the vibration mode of the 150-mum resonator was combined with a shearing mode in the SRF vibration. From the experiment, we decided that vibration modes of the NS-GT cut quartz crystal resonator were composed of the longitudinal stress T(3)' belonging to the z' direction of the plate and of the shearing stress T(5)' when the plate thickness was thickened and the resonator was oscillating in the SRF.     

Perturbation method for analyzing mass sensitivity of planarmultilayer acoustic sensors

A perturbation method is developed to analyze the mass loading sensitivity of planar composite acoustic gravimetric sensors. The sensitivity formulas are obtained in explicit forms for the two lowest sagittal (D₁ and D₂) modes, the lowest shear horizontal (SH₀) mode and high-order SH_m modes in a two-layer isotropic composite plate sensor. The composite plate consists of a plate of thickness b coated by a film of thickness h on which the mass loading layer of infinitesimal thickness is deposited. This coating can be a chemically selective film which is assumed to be acoustically thin (h≪λ), where λ is the acoustic wavelength. For Love modes supported by a film coated on a semi-infinite substrate and for Rayleigh modes on a semi-infinite substrate, the sensitivity formulas are expressed in analytical form. These formulas specify the contribution of each material parameter in the substrate and film, and of the elasticity of the mass loading layer for each planar sensor, and provide a general guide for enhancing the sensor sensitivity     

On the accuracy of Mindlin plate predictions for the frequency-temperature behavior of resonant modes in AT- and SC-cut quartz plates

The frequency spectra of resonant modes in AT- and SC-cut quartz plates and their frequency-temperature behavior were studied using Mindlin first- and third-order plate equations. Both straight-crested wave solutions and two-dimensional plate solutions were studied. The first-order Mindlin plate theory with shear correction factors was previously found to yield inaccurate frequency spectra of the modes in the vicinity of the fundamental thickness-shear frequency. The third-order Mindlin plate equations without correction factors, on the other hand, predict well the frequency spectrum in the same vicinity. In general, the frequency-temperature curves of the fundamental thickness-shear obtained from the first-order Mindlin plate theory are sufficiently different from those of the third-order Mindlin plate theory that they raise concerns. The least accurately predicted mode of vibration is the flexure mode, which results in discrepancies in its frequency-temperature behavior. The accuracy of other modes of vibrations depends on the degree of couplings with the flexure mode. Mindlin first-order plate theory with only the shear correction factors is not sufficiently accurate for high frequency crystal vibrations at the fundamental thickness-shear frequency. Comparison with measured resonant frequencies and frequency-temperature results on an AT-cut quartz plate shows that the third-order plate theory is more accurate than the first-order plate theory; this is especially true for the technically important fundamental thickness shear mode in the AT-cut quartz plate.     

Mutual friction and vortex motion near the superfluid transition

The mutual friction parametersB and B for a moving vortex are calculated near the superfluid transition. They are proportional to the kinetic coefficient associated with the order parameter and, asT , diverge as (T – T)^–1/3, in agreement with experiment. The nonlinear couplings between the order parameter and the entropym, both the reversible one and the one in the free energy, are found to be crucial in the mutual friction near the point. These couplings were neglected in a previous paper by the author. First, the reversible coupling in the dynamic equations makes the chemical potential deviation long-ranged and causes the dissipation to take place only near the vortex core. Second,B can diverge asT T only in the presence of the coupling of the formm||² in the free energy. Thus theE model of Halperin et al., where the latter coupling is absent, cannot explain the critical anomaly ofB. The helical mode of a single vertex line is also investigated and its dispersion relation is found to be quite different from that at low temperatures. This mode has the same time scale as that of the second-sound mode when the wave vectors are of the order of the inverse correlation length, thus obeying the usual dynamic scaling law. The time correlation functions of the displacement fluctuations of a vortex line are also obtained. The force acting on a moving vortex is calculated and is found to be equal to the classical Magnus force.     

Gravity travelling waves for two superposed fluid layers,one being of infinite depth: a new type of bifurcation

In this paper, we study the travelling gravity waves in a system of two layers of perfect fluids, the bottom one being infinitely deep, the upper one having a finite thickness h. We assume that the flow is potential and the dimensionless parameters are the ratio between densities rho = rho(2)/rho(1) and lambda = gh/c(2). We study special values of the parameters such that lambda(1 - rho) is near 1(-), where a bifurcation of a new type occurs. We formulate the problem as a spatial reversible dynamical system, where U = 0 corresponds to a uniform state (velocity c in a moving reference frame), and we consider the linearized operator around 0. We show that its spectrum contains the entire real axis (essential spectrum), with, in addition, a double eigenvalue in 0, a pair of simple imaginary eigenvalues +/-ilambda at a distance O(1) from 0, and for lambda(1 - rho) above 1, another pair of simple imaginary eigenvalues tending towards 0 as lambda(1 - rho) --> 1(+). When lambda(1 - rho) 相似文献   

Mass sensitivities of shear horizontal waves in three-layer platesensors

Explicit velocity and mass sensitivity formulas for shear-horizontal (SH) plate wave sensors loaded symmetrically on both sides of a plate are presented. The sensor geometry is a composite plate which consists of a central isotropic plate sandwiched symmetrically between two identical layers of isotropic solids. It is demonstrated that if the side layers are considered as the mass loading, for the lowest SH mode (SSo) the sensitivity decreases by a factor of (1-(μ ₂/ρ₂)/(μ₁/ρ₁)) due to the elasticity and by a factor of (1+ρ₂h/ρ₁d)^-1 due to the inertia of the mass loading layers, where μ₁,μ₂ ; ρ₁, ρ₂ and 2d, h are the shear moduli, densities and thickness of the central plate and of the side layers, respectively. For higher order modes, the behavior of sensors which are operated near cutoff frequency is analyzed. The mass loading decreases the cutoff frequencies of the higher order modes and near the cutoff frequencies the mass sensitivities are very high but decrease dramatically as the mass loading increases. Specific examples are given for the case of a fused silica plate sandwiched between two thin lucite layers     

Accuracy of crystal plate theories for high-frequency vibrations inthe range of the fundamental thickness shear mode

The first-order plate theories with correction factors are generally assumed to predict accurately the plate modes which have half wavelengths greater than the plate thickness, and at frequencies up to 20% higher than the fundamental thickness shear frequency. This assumption is assessed by comparing the straight crested wave solutions of the plate theories with those of the three-dimensional elastic equations of motion. The frequency spectra for bandwidths of resonant frequencies versus the aspect ratio of length to thickness of plate are compared for three sets of plate equations: the first-order Mindlin plate equations, the third-order Mindlin plate equations, and the third-order Lee and Nikodem plate equations. The finite element results for a quartz SC-cut strip with free edges show that Mindlin's first-order plate equations, and Lee and Nikodem's third-order plate equations yield less accurate frequency spectra of the modes in the vicinity of the fundamental thickness shear mode than the third-order Mindlin plate equations without correction factors. The degree of inaccuracy increases with the ratio of plate length to thickness, and the slope of the modal branches in the frequency spectra. For a plate length to thickness ratio of 31 to 33, the first-order plate theory is found to yield accurate frequency spectra for normalized frequencies less than 0.3, which is lower than previously assumed. At normalized frequencies greater than 0.3, deviations are seen in the frequency spectra, starting with the modal branches which are more steeply inclined     

Two-dimensional laminar boundary layer in flow of thermodynamically equilibrated water vapor

This paper examines the boundary problem of a laminar boundary layer in flow of thermodynamically equilibrated water vapor. An approximate method of solution is proposed, based on an approximation for the density and the coefficient of dynamic viscosity across the layer.Notation u and v longitudinal and transverse velocity components - x and y longitudinal and transverse coordinates - density - p pressure - T temperature - h enthalpy - V specific volume - Pr Prandtl number - z dryness level - adiabatic exponent of the heated vapor - , ,gu coefficients of thermal conductivity, dynamic and kinematic viscosity Translated from Inzhenerno-Fizicheskii Zhurnal, Vol. 35, No. 5, pp. 789–795, November, 1978.