Relative Phase Velocity Research Articles

Tunable microwave components based on dielectric nonlinearity by using HTS/ferroelectric thin films

This study provides the possibility of developing tunable microwave components based on the dielectric substrate nonlinearity, with the conducting surfaces made of a superconductor. The displacement vector, the dipole moment, polarization, polarizability, susceptibility, and relative permittivity concepts are used for ferroelectrics; and for superconductors, Bose statistics and the Gorter and the Casimir model for a two-fluid model, London's equations, and the classical skin effect for the normal component of the current are used. A sinusoidal wave solution is found for a planar superconducting transmission line. This solution gives expressions for the phase velocity and attenuation coefficient which are used to characterize the tunability of microwave components. The measured data in the literature have been used to compute the relative phase velocities and phase shift per cm versus temperature and the dc bias electric field E (kV/cm). It is shown that with a ferroelectric film of thickness of 140 nm, with /spl epsiv//sub /spl tau//=2/spl times/10/sup 3/ and tan /spl delta/=10/sup -3/ phase shifts and attenuation of the order of tens of degrees per centimeter and 5.76/spl times/10/sup -3/ dB/cm, respectively, at 10 GHz, can be obtained with tens of millivolts at 4 K.

Temperature-dependent kinetic inductance of YBa2Cu3O7−δ thin films predicted by the coupled grain model in the strong-coupling limit

The introduction of a temperature dependence to the effective penetration depth as calculated in the strong-coupling limit of the coupled grain model is discussed. Assuming a phenomenological Gorter–Casimir behavior for the intrinsic penetration depth of YBa2Cu3O7−δ (YBCO) and a simple model for the temperature-dependent intergrain critical current density, the coupled grain model is found to predict a kinetic inductance which accurately fits the relative phase velocity, vφ(T)/vφ(T0), measured on coplanar YBCO transmission lines. This indicates that the coupled grain model provides an adequate physical interpretation of the empirical penetration depth proposed by, e.g., Rauch et al. [J. Appl. Phys. 73, 1866 (1993)] and suggests that the kinetic inductance is strongly influenced by boundaries between grains with minor relative misalignment. The average grain size and critical current density of the grain boundaries were estimated at a=140–240 nm and Jgc(0)=3.5×1010–1×1011 A m−2, respectively. We argue that the coupled grain model may prove suitable for implementation in microwave circuit design software, as thin film preparation techniques mature in the sense of providing samples with highly reproducible properties.

Convectively forced internal gravity waves: Results from two‐dimensional numerical experiments

AbstractTwo‐dimensional numerical simulations were performed to investigate the nature of tropospheric internal gravity waves of the type which are observed to occur above active thermal convection over an unstable boundary layer. These gravity waves are believed to be excited by a combination of pure thermal forcing and by the boundary layer eddies and cumulus clouds acting as obstacles to the flow in the presence of mean environmental wind shear.Large amplitude internal gravity waves were obtained in the simulations with amplitudes and horizontal scales similar to the 12 June 1984 aircraft observations over western Nebraska. This was a day with strong wind shear in the lowest 3 km above the ground and with scattered cumulus clouds topping the boundary layer. The simulations show that there is significant difference between the early time solutions (as might be predicted by linear theory) and late time solutions for the boundary layer eddy structure. A layer interaction occurs in which gravity waves of the stable layer are excited by the boundary layer convection. There is evidence to suggest that this layer interaction occurs both with and without shear but that it is stronger in the presence of low‐level shear. Results indicate that shear (or the obstacle) effect is a more efficient generator of gravity waves than is the pure thermal forcing. The simulations show that the gravity waves initially forced by the boundary layer eddies lead to a feedback mechanism that acts to organize the boundary layer eddies and the cumulus convection. The solutions suggest that the character of fair weather convection (moist or dry) is a non‐local problem involving at times the full depth of the troposphere.The clouds produced in the simulations have very little influence on the wave field or boundary layer eddy structure as they are relatively small cumuli. On the other hand the clouds are strongly influenced by the interactions between the wave and eddy fields. Upshear growth of cumulus clouds similar to that which is frequently observed in nature is reproduced in the simulations. The development of feeder (‘feeder’ is used here in a dynamical sense only) clouds on (typically) the upshear side of the cloud is found to be a result of the interaction between the gravity wave field and the dry and moist convection. The relative phase velocity between the gravity waves and the cloud plays a crucial role in determining the character of the cumulus cloud growth in the present simulations. These simulations suggest that the dynamics both internal and external to the boundary of a cumulus cloud is a complicated mix between wave dynamics and the usually considered convection dynamics. A brief discussion of the implications of the present results to cloud boundary baroclinic instability dynamics is also presented.

Convectively forced internal gravity waves: Results from two-dimensional numerical experiments

Two-dimensional numerical simulations were performed to investigate the nature of tropospheric internal gravity waves of the type which are observed to occur above active thermal convection over an unstable boundary layer. These gravity waves are believed to be excited by a combination of pure thermal forcing and by the boundary layer eddies and cumulus clouds acting as obstacles to the flow in the presence of mean environmental wind shear. Large amplitude internal gravity waves were obtained in the simulations with amplitudes and horizontal scales similar to the 12 June 1984 aircraft observations over western Nebraska. This was a day with strong wind shear in the lowest 3 km above the ground and with scattered cumulus clouds topping the boundary layer. The simulations show that there is significant difference between the early time solutions (as might be predicted by linear theory) and late time solutions for the boundary layer eddy structure. A layer interaction occurs in which gravity waves of the stable layer are excited by the boundary layer convection. There is evidence to suggest that this layer interaction occurs both with and without shear but that it is stronger in the presence of low-level shear. Results indicate that shear (or the obstacle) effect is a more efficient generator of gravity waves than is the pure thermal forcing. The simulations show that the gravity waves initially forced by the boundary layer eddies lead to a feedback mechanism that acts to organize the boundary layer eddies and the cumulus convection. The solutions suggest that the character of fair weather convection (moist or dry) is a non-local problem involving at times the full depth of the troposphere. The clouds produced in the simulations have very little influence on the wave field or boundary layer eddy structure as they are relatively small cumuli. On the other hand the clouds are strongly influenced by the interactions between the wave and eddy fields. Upshear growth of cumulus clouds similar to that which is frequently observed in nature is reproduced in the simulations. The development of feeder (‘feeder’ is used here in a dynamical sense only) clouds on (typically) the upshear side of the cloud is found to be a result of the interaction between the gravity wave field and the dry and moist convection. The relative phase velocity between the gravity waves and the cloud plays a crucial role in determining the character of the cumulus cloud growth in the present simulations. These simulations suggest that the dynamics both internal and external to the boundary of a cumulus cloud is a complicated mix between wave dynamics and the usually considered convection dynamics. A brief discussion of the implications of the present results to cloud boundary baroclinic instability dynamics is also presented.

Low‐frequency transmission losses and effective flexural rigidity of cracked ice cover

Upward refraction in the artic causes substantial interactions of underwater acoustic waves with the ice cover. Although several research papers focused on the effects of ice cover roughness on transmission losses, apparently no studies have been reported on the effects of cracks in the ice cover on reducing the flexural rigidity of the ice plate in relation to leaky Lamb waves and energy exchange between the water and the ice. Coupling between the acoustic waves in the water and the elastic waves in the ice plate depends on the relative densities and phase velocities of the waves. Depending on the number, size, and spatial distribution of the cracks, the flexural wave velocity of a low‐frequency component may be higher or lower than the water compressional velocity. The steep sloped region of the flexural wave dispersion curve makes the low‐frequency transmission characteristics very sensitive to the presence of cracks which reduce the flexural rigidity. Laboratory ultrasonic models are used to provide physical insights into a potentially important low‐frequency transmission loss mechanism caused by the effective reduced flexural rigidity of the cracked ice cover. Thin plates with machined multiple shallow cracks are utilized. The crack spacing is about half a wavelength and the crack depth is less than 1/6 the plate thickness. [Work supported by DREP.]

Asymptotic normal modes of a laterally heterogeneous Earth

In the geometrical optics limit on an aspherical earth, the 2l + 1 singlet eigenfunctions of an isolated multiplet nSl, or nTl with n ≪ l are associated with the stable closed orbits of propagating surface waves. The long‐term evolution of surface‐wave trajectories is in turn governed by the great circular average of the local relative phase velocity perturbation ϵ = δωlocal/ω0 where ω0 is the degenerate multiplet eigenfrequency and δωlocal is the local eigenfrequency perturbation. Correct to first order in ϵ, the normal to a surface wave orbital plane, averaged over an orbit, satisfies the fast gyroscope equation , where Δ is the angular arc length along the orbit. This equation stipulates that precesses around the contours of at a rate proportional to the contour spacing. The only paths that remain closed in the presence of the perturbation ϵ are those corresponding to the critical or stationary points where = 0. Stable closed paths correspond to local minima and maxima of , while unstable paths correspond to saddle points. A singlet eigenfunction is composed of waves propagating around all the great circle paths belonging to a single precessing family, and the associated asymptotic eigenfrequency perturbation δω/ω0 is just the value of the particular contour of around which the normal is precessing. Since the contours are closed about the local minima and maxima of , the singlet eigenfunctions are confined to the vicinity of the locally slow and fast great circle paths on the surface of the earth. Caustics that are the envelopes of the precessing trajectories separate regions of oscillatory behavior from regions of exponential decay. The quantization condition that selects the eigenfrequencies or eigencontours from the continuum of contours of is derived from the requirement that the WKBJ singlet eigenfunctions be single valued and properly connected along the caustics. Correct to first order in ϵ the condition is that the length of either caustic be equal to . Alternatively, if is regarded as the orbital angular momentum of a precessing surface wave, the condition is that 〈Lz〉 = m where the axis coincides with a local minimum or maximum of and 〈Lz〉 is averaged over a full precession cycle. This latter form of the quantization condition emphasizes the gyroscopic analogy and obviously resembles the definition of angular momentum eigenstates in quantum mechanics. All but one of the 2l + 1 asymptotic eigenfrequencies on a nonrotating earth are doubly degenerate because of the even parity of , but this degeneracy is removed by the earth's rotation, which splits each doublet by an additional amount Δω ≈ 2mβΩz, where β is the rotational splitting parameter and Ω2 is the component of the angular velocity of rotation.

Axially symmetric elastic waves in a laminated compressible composite material with initial stresses

From an analysis of the above results for ζ=1 we can conclude the following: 1. The initial stresses have a substantial effect on the phase velocities of the generated waves. 2. There are frequencies at which the relative phase velocity is independent of the initial stresses. 3. Each mode has a frequency range in which the variation of the phase velocity caused by the initial stresses is strongly dependent on the frequency. 4. As the ratio of layer thicknesses varies, there is also a variation both in the critical frequencies and in the nature of the dependence of the phase velocity on the frequency and on the initial stresses.

Hold-up and flooding in a vibrating plate extractor

The mean volumetric hold-up of the dispersed phase and the limiting flow rates of phases in five sizes of experimental vibrating plate extractors VPE were measured and the results compared with the relations of Richardson and Zaki, Míšek and Ishii and Zuber for the relative phase velocity. The experiments were performed with the system toluene-water, toluene dispersed. All the relations were found to describe the dependence of hold-up on flow rates very well. They also allowed a satisfactory estimation of the limiting phase velocities. Within the range of the experimental conditions the characteristic velocity as well as the limiting phase velocities could be approximated by linear functions of the frequency of vibrations. For the columns studied no influence of column size on limiting velocities was detected. The deviations of individual columns from geometrical similarity, however, brought about an appreciable effect both on the hold-up and on the flooding rates. This phenomenon has been explained in terms of longitudinal hold-up profiles, the shape of which depends on the boundary conditions in the particular column. This explanation was corroborated by case studies using a polydisperse mathematical model which show that even in the presence of hold-up profiles the effect of phase velocities on the mean hold-up can be correlated by means of the investigated relations with constant parameters and that the parameters depend on the hold-up profiles form.

On the ionospheric parameters which govern high‐latitude ELF propagation in the earth‐ionosphere waveguide

An approximate wave solution is obtained for the propagating ELF mode at high latitudes in the earth‐ionosphere waveguide. A simple approximate expression for the complex propagation constant emerges from the solution. The propagation constant depends on four parameters, two altitudes and a scale height associated with each altitude. The lower altitude is the height at which the conduction current parallel to the magnetic field becomes equal to the displacement current. The associated scale height is the local scale height of the parallel conductivity. Under daytime ionospheric conditions, the upper altitude is the height at which the local wave number becomes equal to the reciprocal of the local scale height of the refractive index. The associated scale height is the local scale height of the refractive index. Under the simplest nighttime conditions, the second set of parameters is replaced by the altitude of the E‐region bottom and the local wave number just inside the E region. The relative phase velocity depends, in first approximation, only on the ratio of the two altitudes. The attenuation rate depends on the other two parameters as well. The two principal attenuation mechanisms are Joule heating by longitudinal currents in the vicinity of the lower altitude and energy leakage of the whistler component of the ELF wave at the upper altitude.

Some differences in diurnal phase and amplitude variations for VLF signals

The daily change in illumination over a VLF propagation path produces a regular variation in the phase and amplitude of a received signal. Significant differences have been reported for such variations when measured at the same frequency over propagation paths of similar length but different orientation. Past attempts have been made to explain these observations in terms of stellar X-ray sources and a major variation in VLF propagation parameters at the magnetic equator. This paper shows that a minor azimuthal variation in the relative phase velocities of interfering waveguide modes propagating at night can account for the observations in some detail.

Dispersion in Shielded Planar Transmission Lines on Two-Layer Composite Substrate

The singular integral equation technique has been used to analyze a shielded planar transmission line, which allows one to calculate the dispersion characteristics of shielded microstrips on two-layer substrates as well as the effect of shielding on coplanar waveguides. Dispersion curves for suspended substrate microstrips and the variation of the relative phase velocity, with frequency, of coplanar waveguide (CPW) on alumina substrates of finite thicknesses and variable ground plane positions are presented. The results of computations with the lowest order 4 x 4 determinant show good agreement with the available data.

Transverse surface waves on a piezoelectric material carrying a metal layer of finite thickness

Theory of the propagation of a surface wave related both to Bleustein-Gulyaev waves and Love waves is developed. The wave has unidirectional particle motion perpendicular to the direction of propagation and parallel to the surface of a piezoelectric material which is covered with a finite-thickness layer of an isotropic conducting material. An equation relating phase velocity to material costants is solved in closed form for a piezoelectric material of class 6mm, and conditions for the existence of various modes are presented. Piezoelectricity allows a nonleaky but dispersive wave to exist under conditions is which no love wave is possible, namely, when the shear wave velocity in the layer is greater than that in the substrate. Numerical results are presented for aluminium, gold, and zinc layers on PZT 4 ceramic.

Electrostatic Microstrip Analysis Programs - MICRO and INFSTR (Computer Program Descriptions)

Each program determines the electrostatic capacitance of microstrip transmission lines. MICRO also returns characteristic impedance and relative phase velocity, INFSTR the charge density distribution.

Power transfer between Goubau lines with unequal phase velocities

Results from experimental measurements and theoretical calculations are compared for coupling between surface-wave lines with unequal relative phase velocities. The agreement is good for small differences in relative phase velocities but is less satisfactory as the differences become larger. It is seen that coupling between Goubau lines with unequal relative phase velocities is practical where complete power transfer is not required. The allowable differences in the relative phase velocities depend on the length of the coupled section and, of course, on the application intended.

Investigation of supersonic phenomena in a two- phase liquid-gas tunnel

Homogeneous two-phase flows of dispersed liquid and gas having gas-to-liquid volume ratios around 1:1 exhibit the characteristics of a continuum flow with a greatly reduced sound propagational velocity that approaches 66 ft/sec at atmospheric pressure, and that is reduced further in value as the square root of the pressure. Flows of such mixtures at velocities in excess of the local velocity of sound can produce shock phenomena similar to that experienced in supersonic gaseous media. A supersonic two-phase tunnel was designed and built with such versatility and precision that normal and oblique shock structures can be photographed and analyzed in the absence of boundary-layer interference. The applicability of the isothermal continuum theory to such flows is confirmed empirically for volume ratios near 1:1, and the theory is mathematically extended for both normal and oblique shocks over a wide range of volume ratios centered about the 1:1 value. Auxiliary flow devices were constructed for the measurement of such difficult flow parameters as the relative phase velocity, local void ratio, coefficient of friction, and stagnation pressure. A general change in the flow model matrix was found at volume ratios approaching 1:1. Pressure gradients and relative phase velocities were correlated with the proposed flow models with generally good agreement. The coefficient of friction measured for supersonic flow was found to be a simple function of the local void ratio. Stagnation pressures measured for a wide range of flow conditions approximate an isentropic relation for a substantial part of the lower velocity spectrum. At higher velocities, the stagnation pressure closely approaches the normal shock plus isentropic slowdown theory. Considerable photographic information pertaining to shock structure and phase movement is obtained over the spectrum of flow conditions with Mach numbers ranging from 2 to 20.

The Variation of Pi 2 Micropulsation Waveform with Time

Pi 2 micropulsations observed ‘simultaneously’ at two widely separated low‐latitude stations: M'Bour, Senegal, in West Africa, and Makerere, Uganda, in East Africa have been compared. The variation in the waveforms has been measured in terms of an average damping coefficient and the number of excursions taken for events to reach their peak amplitude. The differences in the waveforms are discussed in terms of dispersion varying with distance from the Pi 2 source and the damping coefficient varying with time. Shorter period activity has been noted in association with some Pi 2 events and an estimate of the relative apparent phase velocity of the higher‐frequency component has been made.

Phase velocity on a conical two-armed logarithmic foil-type spiral antenna

From an experimental investigation of the foil-type kind of a conical, logarithmic spiral antenna, the relative phase velocity along the antenna arm as a function of the cone angle has been found. It is probable that a reduction of the antenna length by one half can be achieved by using a special dielectric loading.

Axially symmetric wave propagation in a two-layered cylinder

The linear theory of elasticity is used to investigate axially symmetric wave propagation in an infinitely long two-layered cylinder. Each material is taken to be homogeneous and isotropic. A perfect bond is assumed at the interface, while the inner and outer boundaries of the composite cylinder are treated as traction-free. The dispersion determinant relating phase velocity and wave number for a harmonic train of waves satisfying these boundary conditions is presented. The character of the dispersion equation is investigated analytically and numerically. Stress and displacement distributions are also presented for the numerical example. Comparisons are made with an approximate solution of the same problem obtained by means of a thin shell theory incorporating thickness-shear deformation of each layer.

An analysis of waves in a two-layer composite plate and its application to surface wave propagation experiments on roads

In this paper a derivation is given of the theoretical dispersion curves relating phase velocity and wavelength of waves in a two-layer composite plate in which stresses and displacements are continuous across the interface. Computations are carried out for different ratios of shear moduli and thickness of the two media in the composite plate using a computer programme which is capable of deriving the real roots of the dispersion equation of waves in layered plates or a layered half-space with up to thirty layers. It is shown that part of the dispersion curve of flexural waves in the composite plate can be derived to an accuracy of 5 % in the wavelength by adding the wavelengths of the antisymmetric or flexural modes of the individual plates at each velocity. This approximation applies over a fairly wide range of shear modulus ratios (4:1 to 1: 2) and for a ratio of thickness of 2:7. The computations of the dispersion curves in free composite plates are shown to give an acceptable approximation to many features of the experimental flexural wave branch of the dispersion curves obtained on roads which have two upper layers of considerably higher shear moduli than the underlying media. Thus, the computational analysis provides a basis for obtaining the thicknesses and elastic properties of the upper layers of many types of road construction from the experimental dispersion curves.

Periodic Interferometer for Measuring Relative Phase Velocity of Sound in Fluids

Sound of short wavelength (18 in. in water) is caused to travel a pathlength that is varied periodically. Velocity of phase propagation in the fluid is inversely proportional to the total phase difference over minimum and maximum pathlengths. With the aid of quadrature processing and phase comparison between sent and received signals, the total phase angle can be read from an oscilloscopic display, or converted by digital processing to a direct velocity readout.