Square Root Of Frequency Research Articles

Factor V Leiden (Activated Protein C Resistance) Versus Factor V Deficiency in Padua, Italy

Ten homozygous patients with factor V deficiency and 19 homozygous patients with activated protein C (APC) resistance were seen in Padua during the last 30 or 3 years, respectively. The actual numbers of heterozygous patients seen during the same periods of time were 41 and 306, respectively. All patients came from the northeastern part of Italy with a population of 10,000,000 people. The probable prevalence of homozygous patients was obtained by multiplying the actual number by four. The theoretical prevalence of heterozygous patients was obtained using the formula 2 qp, where q is the frequency of the abnormal gene as calculated by the square root of frequency of the homozygotes and p is the frequency of the normal allele as calculated by the formula p = (1 - q). Using such calculations, the number of heterozygotes probably exist ing in northeastern Italy is 39,840 and 56,497, respectively for the factor V deficiency and for APC resistance. After correction for the different observation periods (3 years vs. 30 years), there are 56,497 per 10 million people (5.6% of the population) for factor V Leiden. The prevalence of factor V deficiency is 0.39% of the population. Therefore the factor V Leiden muta tion responsible for APC resistance appears to be more preva lent than factor V deficiency on the basis of actual figures and accepted theoretical calculations.

A class of low-frequency modes in range-independent fluid–solid media

A unified treatment is given of the class of low-frequency P–SV modes whose complex phase velocities do not approach zero as fast as the frequency when the frequency is decreased toward zero. Utilizing the analyticity of the dispersion function, a complete characterization is given of these modes and the linearly viscoelastic fluid–solid media in which they occur. The two Lamb plate modes and the classical interface waves (Rayleigh, Scholte, Stoneley) all appear in a general setting, and asymptotic low-frequency expressions are given for modal slownesses and mode forms. The corresponding phase velocities will either tend to a nonzero constant or approach zero like the square root of frequency. In addition, slow modes appear whose phase velocities approach zero like three other powers of frequency: 1/3, 3/5, and 2/3. Concerning 1/3, a correction is made to a paper by Ferrazzini and Aki; 3/5 and 2/3 appear for certain bending-type waves that are slower than the classical bending wave. A less known type of interface wave also emerges from the analysis, and precise conditions for existence and uniqueness are obtained. The results are extended to interface conditions with slip. In an extreme case, yet another kind of interface wave is obtained.

Anomalous viscous loss in emulsions.

We propose a model for concentrated emulsions based on the speculation that a macroscopic shear strain does not produce an affine deformation in the randomly close-packed droplet structure. The model yields an anomalous contribution to the complex dynamic shear modulus that varies as the square root of frequency. We test this prediction using a novel light scattering technique to measure the dynamic shear modulus, and directly observe the predicted behavior over six decades of frequency and a wide range of volume fractions.

Electromagnetic modelling of buried line conductors using an integral equation

An integral equation is derived to represent the electromagnetic response of line conductors buried below the surface of a horizontally stratified earth. By permitting several finite-length line conductors of arbitrary topology, this representation is free of the limitations imposed by analytic solutions. The integral equation is formulated in terms of excess electric current (scattering current) flowing along the line conductor and is solved numerically by dividing the line conductor into many small segments. In general, the excess electric current is controlled by both the internal and external impedance of the line conductor. The internal impedance is the longitudinal resistance of the line conductor. The external impedance is caused by galvanic and inductive coupling between the line conductor and its local environment. The galvanic resistance to current channelling is spatially variable with a minimum in the centre of a uniform line conductor and is determined by conductor geometry and the host conductivity structure. The inductive external impedance is proportional to frequency and quadrature dominant. It is a function of line-conductor geometry, cross-section and burial environment. The inductive impedance effectively reduces the spatial dependence of the external impedance at high frequency by presenting a large reactance (which is uniform at all points along the conductor) to the exciting electric field. Within the quasi-static limit (i.e. where displacement current can be neglected), electromagnetic excitation by either horizontal electric or vertical magnetic dipoles produces a constant primary electric field at high frequencies (far field). The excess electric current in the line conductor will always be inversely proportional to frequency for these types of sources at sufficiently high frequencies where the inductive external impedance is dominant. Horizontal magnetic dipole and vertical electric dipole sources generate primary electric fields that are proportional to the square root of frequency in the high-frequency limit of the quasi-static domain and the excess electric current excited by such sources will decrease as the inverse of the square root of frequency.

Slow dynamics of linear relaxation systems

Linear relaxation, occurring in dielectrics, viscoelastic fluids, and many other systems, is often characterized by a broad continuous spectrum. We show that the relaxation behavior may be analyzed effectively by means of N-point Padé approximants, applied in the complex plane of the square root of frequency. The method leads to a compact analytic expression for the Laplace transform of the relaxation function, characterized by a small number of poles and their residues in the square root of frequency plane. We study the method in detail for a model of diffusion in three dimensions with a single or double radial potential barrier, and demonstrate its use in the analysis of the viscoelastic relaxation spectrum of polyisobutylene.

Electromagnetic response of a discretely grounded circuit—An integral equation solution

We derive an integral equation to describe the electromagnetic response of a discretely grounded circuit. This investigation is relevant to the study of man‐made structures such as metallic fences, grounded powerlines, and pipelines, all of which may fall into the class of discretely grounded conductors. The solution developed here is an extension to existing circuit theory and takes into account the self and mutual interaction of the circuit elements. It is possible to ignore these interactions at low frequencies where the grounding impedances dominate the effective impedance of the circuit. However, at frequencies where the electromagnetic skin depth is comparable to the length between adjacent grounding points, the effective impedance of the circuit is proportional to frequency, and the inductance of the circuit dominates its electromagnetic response. Within the quasi‐static limit (i.e., where displacement currents can be neglected) electromagnetic excitation by either horizontal electric or vertical magnetic dipoles produces a constant primary electric field at high frequencies (far‐field). Thus, the electric current in the discretely grounded circuit will always be inversely proportional to frequency for these types of sources. Horizontal magnetic dipole or vertical electric dipole sources generate primary electric fields that are proportional to the inverse square root of frequency in the high frequency limit of the quasi‐static domain, and thus the current in a circuit excited by such sources will decrease as the inverse of square root of frequency. The integral equation solution derived here can be used to investigate the influence from cultural conductors on actual electromagnetic surveys and also provides further insights into the current channeling response of surficial conductors.

Long-time collective motion of rigid bodies immersed in a viscous fluid

In linear approximation a number of rigid bodies, immersed in a viscous incompressible fluid, to which at time zero arbitrary small translational and rotational impulses are imparted, move collectively at long times with a common translational velocity which slowly decays to zero. The proof of this statement is based on a study of the low frequency behavior of the friction and mobility matrices of the set of bodies. It is shown that the linear term in a Taylor expansion of the friction matrix in powers of the square root of frequency may be expressed in terms of the zero frequency friction matrix. The corresponding term for the mobility matrix is even simpler. It is independent of the shape or size of the rigid bodies. Its particular form gives rise to the theorem stated above. Related consequences for the theory of Brownian motion are discussed. For example, the long-time tail of the velocity autocorrelation function of a selected particle in a polydisperse suspension of spheres has universal character. In particular it is independent of the concentration of the suspension.

Slow wave propagation in air-filled permeable solids

The propagation of slow compressional waves in air-saturated permeable solids was studied by experimental means between 10 and 500 kHz. The velocity and attenuation coefficient were measured as functions of frequency from the insertion delay and loss of airborne ultrasonic waves transmitted through thin slabs of 1–5 mm in thickness. Porous ceramics of 2–70 Darcy and natural rocks of 200–700 mDarcy permeability were tested. In the low-frequency (diffuse) regime, the experimental results are consistent with theoretical predictions; the phase velocity and attenuation coefficient are essentially determined by the permeability of the specimen and both increase proportionally to the square root of frequency. In the high-frequency (propagating) regime, the experimental results are consistent with the theoretical predictions for the phase velocity but not for the attenuation coefficient. The phase velocity asymptotically approaches a maximum value determined by the tortuosity of the specimen while the attenuation coefficient becomes linearly proportional to frequency instead of the expected square-root relationship. It is suggested that the observed discrepancy is due to the irregular pore geometry that significantly reduces the high-frequency dynamic permeability of the specimens.

Effects of structural joints on sound scattering

This paper addresses the effects of conditions at attachment points of coupled elastic structures on sound scattering. This is done by examining the scattering of plane sound waves by an infinite elastic plate loaded with fluid on one side and a semi-infinite plate on the other side. Different attachment conditions between the two plates are considered, ranging from clamped connections that transmit both forces and moments to various pinned connections, each of which allows free motions in one or two dimensions at the attachment point. It is shown that in the low- and intermediate-frequency domain, attachments supporting normal forces to the fluid-loaded plate cause much more acoustic scattering than those that only transmit tangential forces and bending moments. The ratio of contributions to the scattered acoustic field, respectively from bending moments and normal forces, is found to be proportional to a (13/5)th power of the plate Mach number, defined by the square root of frequency normalized by the plate coincidence frequency.

Peak current—frequency relationship in adsorptive stripping square-wave voltammetry

A reversible redox reaction between the adsorbed reactant and the dissolved product or the amalgam is analysed. The relationship between the square-wave (SW) stripping peak current and the SW frequency is a curve defined by Δφ max = 0, for f sw = 0, and an asymptote Δφ max = kf SW + z. The intercept z depends on the duration t 0 of the adsorptive accumulation and apparently vanishes when t 0 > 30 s. Consequently, the ratio Δφ max / f SW may not be constant for all frequencies. This effect is caused by the additional adsorption of the reactant after the end of the accumulation period. The SW peak potentials depend linearly on the logarithm of frequency, with a slope which approaches −2.3 RT/2 nF and depends on both the accumulation time and the step increment. If only the product is adsorbed, the SW peak currents depend linearly on the square root of frequency. The peak potentials are a linear function of the logarithm of frequency, with slope 2.3 RT/2 nF.

Acoustics of handbells.

Although handbells date back at least several centuries B.C., tuned handbells of the type played today were developed in England in the 18th century. One early use of handbells was to provide tower bellringers with a convenient means to practice change ringing. In more recent years, handbell choirs have become popular in schools and churches−some 2000 choirs are reported in the USA alone. In the so-called English tuning of handbells, followed by most handbell makers in England and the United States, the (3,0) mode is tuned to three times the frequency of the (2,0) mode. The fundamental (2,0) mode radiates a rather strong second harmonic partial, however, so that the sound spectrum has prominent partials at the first three harmonics. Modes of vibration of handbells have been studied by means of holographic interferometry, scanning the near-field sound with a microphone, and by modal analysis with impact excitation. Handbells are usually scaled so that the diameter is inversely proportional to the square root of frequency, except in the smallest bells in which it varies inversely with the cube root of frequency. The thickness h is then adjusted so that h/d2 is nearly proportional to the frequency.

Vibrational modes of piano soundboards

The vibrational modes of the soundboards have been compared in two small upright pianos: one of solid spruce and one of three layers (two thin layers of spruce with a core of poplar). The model frequencies are quite similar, although the mode shapes are somewhat different. In the spruce soundboard, the bending wave velocities for the low-frequency modes appear to increase with the square root of frequency, which suggests that the stiffness of the soundboard plus the ribs is approximately the same along and across the grain (i.e., the soundboard is fully compensated). In the layered soundboard, on the other hand, the nodal lines tend to follow the ribs, indicating that the soundboard is overcompensated. Mode damping does not appear to be substantially different in the two soundboards.

The Diffusional (Warburg) Impedance at a Sinusoidal Shape Electrode

The surface average, diffusional impedance at a sinusoidal shape electrode is calculated using a perturbation solution to the transient‐diffusion equation for the concentration field, with the perturbation parameter being the amplitude‐to‐length ratio of the sinusoid, A. Terms in the perturbation expansion up to and including A4 are included in the solution, and the error in is determined to be less than 2% for A less than 0.15. It is shown that at low and high frequency (ω) is proportional to , the standard planar electrode dependency, but with a different proportionality coefficient in each limit. At intermediate frequencies, deviation from this ideal is predicted with the occurrence of an amplitude‐dependent maximum in the phase lag. The local diffusional impedance at high frequency is spatially uniform, whereas at low frequency it is the most nonuniform, even though , which is the area weighted average, is identical to a planar electrode in this limit. It is noted that, more generally, the diffusional impedance at an electrode at which the surface profile may be expressed as a Fourier series will also show a proportionality to the square root of frequency in the low and high asymptotic limit.

Errata - Reflection function of a plane sound wave in a cylindrical tube

As dissipation and dispersion of plane waves in a cylindrical tube are proportional to the square root of frequency, the form of the impulse response can be obtained analytically. We analyse the reflection function at the input of the tube, i.e. the inverse Fourier transform of the reflection coefficient, especially the important distortions due to a numerical treatment and the property of stability through convolution. Using this property, the reflection function is determined for a sequence of tubes as well as for a connection of three tubes. An extension of the formulas to the case of rigid-walled closed cylindrical tube is given in Appendix A Etude de la fonction de reflexion a l'entree d'un tuyau et des importantes distorsions dues au traitement numerique ainsi que de la propriete de stabilite par convolution. Fonction de reflexion pour plusieurs tubes en serie et pour un branchement de 3 tubes

Reflection function of a plane sound wave in a cylindrical tube

As dissipation and dispersion of plane waves in a cylindrical tube are proportional to the square root of frequency, the form of the impulse response can be obtained analytically. We analyse the reflection function at the input of the tube, i.e. the inverse Fourier transform of the reflection coefficient, especially the important distortions due to a numerical treatment and the property of stability through convolution. Using this property, the reflection function is determined for a sequence of tubes as well as for a connection of three tubes. An extension of the formulas to the case of rigid-walled closed cylindrical tube is given in Appendix A Etude de la fonction de reflexion a l'entree d'un tuyau et des importantes distorsions dues au traitement numerique ainsi que de la propriete de stabilite par convolution. Fonction de reflexion pour plusieurs tubes en serie et pour un branchement de 3 tubes

Measurement of Phase Transitions by Photothermal Radiometry: The Semiconductor-To-Metal Transition of Vanadium(IV) Oxide, VO2

A new experimental method of investigating phase transitions of solids, photothermal radiometry, is described. The effects of modulation frequency, laser intensity, and heating-cooling rates on the first-order semiconductor-to-metal phase transition of a microcystalline vanadium dioxide sample have been studied. The PTR signal increases with temperature in both phases, and a negative peak occurs at the phase transition. The intensity of signal is inversely proportional to the square root of frequency at laser-beam chopping frequencies lower than 100 Hz, indicating that the sample is thermally thick, and the signal is controlled by the thermal conductivity and the heat capacity and is independent of the optical properties of the material. The relative peak intensity decreases slightly in the low-frequency range with increasing frequency and more sharply at higher frequencies. The measured temperature of the phase transition decreases linearly with the intensity of illuminating light, from which the static component of the temperature rise at the surface can be determined. From the increase of linewidth with the laser intensity one may estimate the amplitude of the temperature modulation at the surface. The measured temperature of the phase transition increases with increasing heating rate and decreases with increasing cooling rates. The effects of variable light intensity and heating-cooling rates can be eliminated by extrapolation, and the exact temperature of the phase transition is obtained.

Acoustical impedance models for outdoor ground surfaces

A theory which predicts the acoustical characteristics of rigid porous materials in terms of four physical parameters is used to provide impedance versus frequency models for various types of ground surface. It is found possible to obtain tolerable agreement with measurements of surface normal impedance for grass-covered ground, bare ground and layered forest floor when measured values of flow resistivity and porosity and estimated values of the other two parameters are used. The agreement with measured acoustical data is shown to be superior to that obtainable with the single-parameter, semi-empirical model that is widely used to predict ground effect even when the single parameter is adjusted for best-fit. A low frequency/high flow resistivity approximation is derived in a form that requires the adjustment of only a single parameter when fitting impedance versus frequency data. This model predicts that the real part of the surface normal impedance, or resistance, and imaginary part, or reactance, are equal and decrease as the inverse square root of frequency. The low frequency/high flow resistivity approximation makes possible the derivation of a two-parameter impedance versus frequency model for a medium with a porosity that decreases exponentially with depth. If the two parameters are considered to be adjustable for best-fit, then some measurements of surface impedance are found to be explicable by means of such a model. The low frequency/high flow resistivity approximation is used, together with the assumption of layer thickness small compared with internal wavelength, to derive another simple two-parameter model for grounds that behave acoustically as thin hard-backed porous layers. Examples of the validity of this model are provided. Both of the two-parameter models predict values of reactance which are greater than those of resistance over the audio-frequency range. Methods of adjusting the two parameters of each model in fitting short-range excess-attenuation data are suggested as alternatives to the single-parameter method, and as the basis for predicting ground effect at long range and low frequencies.

Geotomography in oil shale

Geophysical tomographs were made of oil shale at the Occidental Oil Shale, Inc., mine near DeBeque, Colorado. These tomographs mapped the in situ radio frequency electromagnetic attenuation of oil shale between 1 and 40 MHz in a plane 27 by 12 m. Measured attenuation rates (inverse skin depth) ranged from 0.1 to 0.5 m−1 with the attenuation rate approximately proportional to the square root of frequency. At 20 MHz, the measured relative permittivity of oil shale is about 9. Geotomographs were made in another plane 16.8 by 12 m, before and again after the shale in this region was explosively rubbled. Measured electromagnetic attenuation rates at 20 MHz were generally larger in the rubble than in the undisturbed shale, suggesting that geotomography may be useful in mapping a characteristic of the rubbled zone such as void content or void size distribution. Model calculations of the rubble using a theory of dielectric mixtures predict a smaller attenuation rate in the rubble, implying that the measurements result from phenomena such as scattering or oil shale anisotropy, which are not accounted for in the model.

Airborne ultrasound characteristics of open-cell polyurethane foam layers

Transmission and reflection characteristics at normal incidence have been measured for 1-cm-thick foam layers of 98% porosity in the frequency range 10–200 kHz. The ’’boundary frequency’’ is determined from these measurements. Below this frequency, (1) the transmitted field has no irregularity, (2) the transmission loss is proportional to the square root of frequency, (3) the reflection is specular, and (4) the reflected wave consists of two components which are reflected from the front and back surfaces of the layer and interfere with each other. The boundary frequency is 60 to above 200 kHz, for the layers with cell densities of 14 to 50 cells/in. It is found that, for each cell density, the boundary frequency is almost equal to the frequency at which one-half a wavelength of sound is equal to the average diameter of cells.

Traveling wave optical modulator using a directional coupler LiNbO<inf>3</inf>waveguide

Fabrication methods for low drive voltage and broad-band LiNbO <inf xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">3</inf> waveguide directional coupler optical modulator are described. Optical waveguides were prepared by conventionaly Ti in-diffusion into LiNbO <inf xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">3</inf> <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">c</tex> -cut plate. To obtain wide-band frequency response, traveling wave electrodes were used. Electrode characteristic impedance measured by time domain reflectometry method coincided well with the calculated value by conformal mapping. Measured electrode conductor losses followed square root of frequency. To reduce electrode conductor losses, asymmetric and 3-μm thick Al electrodes were used. Directional coupler optical modulator frequency response was analyzed, using the phase difference average value along the propagation directions. Calculated value by this method coincided well with measured value obtained by a swept frequency technique. The characteristics of this modulator at the 1.317-μm wavelength are as follows: 100 percent modulation voltage is 4 V, extinction ratio is 17 dB, optical insertion loss is 5.4 dB, 3-dB bandwidth is 3.6 GHz, and rise time is about 400ps.