Frequency-dependent Phase Velocity Research Articles

Relationships of the phase velocity with the microarchitectural parameters in bovine trabecular bone in vitro: Application of a stratified model

The present study aims to provide insight into the relationships of the phase velocity with the microarchitectural parameters in bovine trabecular bone in vitro. The frequency-dependent phase velocity was measured in 22 bovine femoral trabecular bone samples by using a pair of transducers with a diameter of 25.4 mm and a center frequency of 0.5 MHz. The phase velocity exhibited positive correlation coefficients of 0.48 and 0.32 with the ratio of bone volume to total volume and the trabecular thickness, respectively, but a negative correlation coefficient of −0.62 with the trabecular separation. The best univariate predictor of the phase velocity was the trabecular separation, yielding an adjusted squared correlation coefficient of 0.36. The multivariate regression models yielded adjusted squared correlation coefficients of 0.21–0.36. The theoretical phase velocity predicted by using a stratified model for wave propagation in periodically stratified media consisting of alternating parallel solid-fluid layers showed reasonable agreements with the experimental measurements.

Analysis of Vibration and Sound Propagation in a Fluid-Filled Pipe

Pipelines are used widely to convey fluids and gases in the petrochemical, water, and energy sectors. Vibration, noise and fatigue failure of pipelines are serious problems in many engineering situations, especially in ship and ocean engineering areas. In this paper, the acoustic propagation and attenuation characteristics of axisymmetrical eigenmodes in a cylindrical, elastic, fluid-filled pipe are investigated analytically. Pipe equations for n=0 axisymmetrical wave motion are derived, and analytical expressions of wavenumber are obtained for s=1 and s=2 wave, which correspond to a fluid dominated wave and an axial shell wave, respectively. The numerical results for wavenumber of pipe and fluid are obtained and discussed. It shows that the frequency dependent phase velocity of this mode depend strongly upon the shell thickness/radius ratio and the density of the contained fluid. It seems that this analysis method is simple and effective.

Imaging a shallow salt diapir using ambient seismic vibrations beneath the densely built-up city area of Hamburg, Northern Germany

Salt diapirs are common features of sedimentary basins. If close to the surface, they can bear a significant hazard due to possible dissolution sinkholes, karst formation and collapse dolines or their influence on ground water chemistry. We investigate the potential of ambient vibration techniques to map the 3-D roof morphology of shallow salt diapirs. Horizontal-to-vertical (H/V) spectral peaks are derived at more than 900 positions above a shallow diapir beneath the city area of Hamburg, Germany, and are used to infer the depth of the first strong impedance contrast. In addition, 15 small-scale array measurements are conducted at different positions in order to compute frequency-dependent phase velocities of Rayleigh waves between 0.5 and 25 Hz. The dispersion curves are inverted together with the H/V peak frequency to obtain shear-wave velocity profiles. Additionally, we compare the morphology derived from H/V and array measurements to borehole lithology and a gravity-based 3-D model of the salt diapir. Both methods give consistent results in agreement with major features indicated by the independent data. An important result is that H/V and array measurements are better suited to identify weathered gypsum caprocks or gypsum floaters, while gravity-derived models better sample the interface between sediments and homogeneous salt. We further investigate qualitatively the influence of the 3-D subsurface topography of the salt diapir on the validity of local 1-D inversion results from ambient vibration dispersion curve inversion.

Experimental characterization of dielectric properties of carbon nanotube networks

This paper explores the characterization of dielectric and conductive properties of carbon nanotube (CNT) networks. This is carried out by building planar transmission lines where conventional metallic traces are replaced by CNT networks. The proposed transmission lines with CNT networks are presented. Experimental realization and repeated two-port microwave measurements of proposed transmission lines enable the accurate extractions of their fundamental parameters showing percolation effects due to presence of CNT networks. The frequency-dependent phase velocity characteristics show a dramatic reduction compared to the speed of light in vacuum. The large magnitude of extracted complex permittivity for CNT networks also exhibits its percolation performance. The effects of CNTs' bulk density on measured and calculated parameters are explained. The results presented in this paper demonstrate the feasibility and the potential of building transmission lines and radio-frequency (RF) circuits elements using CNT networks.

Quasi-static finite element modeling of seismic attenuation and dispersion due to wave-induced fluid flow in poroelastic media

[1] The finite element method is used to solve Biot's equations of consolidation in the displacement-pressure (u − p) formulation. We compute one-dimensional (1-D) and two-dimensional (2-D) numerical quasi-static creep tests with poroelastic media exhibiting mesoscopic-scale heterogeneities to calculate the complex and frequency-dependent P wave moduli from the modeled stress-strain relations. The P wave modulus is used to calculate the frequency-dependent attenuation (i.e., inverse of quality factor) and phase velocity of the medium. Attenuation and velocity dispersion are due to fluid flow induced by pressure differences between regions of different compressibilities, e.g., regions (or patches) saturated with different fluids (i.e., so-called patchy saturation). Comparison of our numerical results with analytical solutions demonstrates the accuracy and stability of the algorithm for a wide range of frequencies (six orders of magnitude). The algorithm employs variable time stepping and an unstructured mesh which make it efficient and accurate for 2-D simulations in media with heterogeneities of arbitrary geometries (e.g., curved shapes). We further numerically calculate the quality factor and phase velocity for 1-D layered patchy saturated porous media exhibiting random distributions of patch sizes. We show that the numerical results for the random distributions can be approximated using a volume average of White's analytical solution and the proposed averaging method is, therefore, suitable for a fast and transparent prediction of both quality factor and phase velocity. Application of our results to frequency-dependent reflection coefficients of hydrocarbon reservoirs indicates that attenuation due to wave-induced flow can increase the reflection coefficient at low frequencies, as is observed at some reservoirs.

Rational approximation for estimation of quality Q factor and phase velocity in linear, viscoelastic, isotropic media

The article presents a numerical inversion method for estimation of quality Q factor and phase velocity in linear, viscoelastic, isotropic media using reconstruction of relaxation spectrum from measured or computed complex velocity or modulus of the medium. Mathematically, the problem is formulated as an inverse problem for reconstruction of relaxation spectrum in the analytic Stieltjes representation of the complex modulus using rational approximation. A rational (Pade) approximation to the relaxation spec trum is derived from a constrained least squares minimization problem with regularization. The recovered stress-strain relaxation spectrum is applied to numerical calculation of frequency-dependent Q factor and frequency-dependent phase velocity for known analytical models of a standard linear viscoelastic solid (Zener) model as well as a nearly constant-Q model which has a continuous spectrum. Numerical results for these analytic models show good agreement between theoretical and predicted values and demonstrate the validity of the algorithm. The proposed method can be used for evaluating relaxation mechanisms in seismic wavefield simulation of viscoelastic media. The constructed lower order Pade approximation can be used for determination of the internal memory variables in time-domain finite difference numerical simulation of viscoelastic wave propagation.

Generalization of Biot’s equations with allowance for shear relaxation of a fluid

On the basis of the generalized variational principle for dissipative continuum mechanics, a system of generalized Biot’s equations is derived to describe the wave propagation in a two-phase porous permeable medium in the presence of shear relaxation in the pore-filling fluid. It was shown that the inclusion of shear viscoelasticity of the fluid leads to the appearance of two transverse modes in addition to two longitudinal modes described by the Biot theory. One of the transverse modes is an acoustic mode, whereas the other is a diffusion mode characterized by the linear frequency dependence of phase velocity and attenuation coefficient in the low-frequency region.

The self-consistent dynamic problem of estimating the damage of a material by an acoustic method

An approach is proposed for formulating the self-consistent problem on the basis of equations that describe the dynamics of a material and the state of its defects. It is shown that the damage of the material gives rise to frequency-dependent attenuation and phase velocity dispersion of an ultrasonic wave propagating in it. This makes it possible to estimate the damage by means of the acoustic method. The applied deformation field leads to accumulation of defects. A kinetic equation is derived for describing the damage growth, which proves to be an exponential process. The parameters of the system for which the accumulation of defects can approximately be considered as linear are estimated.

Independent scattering model and velocity dispersion in trabecular bone: comparison with a multiple scattering model

Speed of sound measurements are used clinically to assess bone strength. Trabecular bone is an attenuating composite material in which negative values of velocity dispersion have been measured; this behavior remaining poorly explained physically. The aim of this work is to describe the ultrasonic propagation in trabecular bone modeled by infinite cylinders immersed in a saturating matrix and to derive the physical determinants of velocity dispersion. An original homogenization model accounting for the coupling of independent scattering and absorption phenomena allows the computation of phase velocity and of dispersion while varying bone properties. The first step of the model consists in the computation of the attenuation coefficient at all frequencies. The second step of the model corresponds to the application of the general Kramers-Krönig relationship to derive the frequency dependence of phase velocity. The model predicts negative values of velocity dispersion in agreement with experimental results obtained in phantoms mimicking trabecular bone. In trabecular bone, only negative values of velocity dispersion are predicted by the model, which span within the range of values measured experimentally. However, the comparison of the present results with results obtained in Haiat etal. (J Acoust Soc Am 124:4047-4058, 2008) assuming multiple scattering indicates that accounting for multiple scattering phenomena leads to a better prediction of velocity dispersion in trabecular bone.

THz surface wave collapse on coated metal surfaces

The Zenneck THz surface wave (Z-TSW) on metals is discussed with respect to its difficulty in generation and measurement. The spatial collapse of the extent of the Z-TSW evanescent field, upon the addition of a sub-wavelength dielectric layer on the metal surface, is explained by a simple model, in good agreement with exact analytical theory. Experimental measurements of the THz surface wave on an aluminum surface covered with a 12.5 microm thick dielectric layer have completely characterized the resultant single-mode dielectric layer THz surface wave (DL-TSW). The measured frequency-dependent exponential fall-off of the evanescent wave from the surface agrees well with theory. The DL-TSW frequency-dependent absorption coefficient, phase velocity, group velocity and group velocity dispersion have been obtained. These guided-wave parameters compare favorably with other guided wave structures.

A new experimental method to estimate viscoelastic properties from ultrasonic wave transmission measurements

We develop a new experimental method to estimate viscoelastic properties of a sample by ultrasonic wave transmission. Viscoelastic properties are generally described by the frequency dependent phase velocity v ( f ) and the quality factor Q ( f ) . In order to estimate v ( f ) and Q ( f ) from waveform data obtained by the ultrasonic wave transmission method, it is necessary to correct the data for the effects of dynamic response of the source and receiver transducers and of the wave diffraction caused by finite size source. A practical method to perform these corrections based on the theoretical models is developed. The validity of the present method is demonstrated by applying it to the experimental data for stainless steel sample ( Q ⪢ 200 ), acrylic plastic sample ( Q ∼ 100 ), and partially molten organic polycrystal sample ( Q < 10 ). In comparison with the other existing methods, the present method has the advantage of not requiring any reference waveform. Using the results of this study, we discuss the validity of the reference method.

The dependencies of phase velocity and dispersion on volume fraction in cancellous-bone-mimicking phantoms

Frequency-dependent phase velocity was measured in eight cancellous-bone-mimicking phantoms consisting of suspensions of randomly oriented nylon filaments (simulating trabeculae) in a soft-tissue-mimicking medium (simulating marrow). Trabecular thicknesses ranged from 152 to 356 mum. Volume fractions of nylon filament material ranged from 0% to 10%. Phase velocity varied approximately linearly with frequency over the range from 300 to 700 kHz. The increase in phase velocity (compared with phase velocity in a phantom containing no filaments) at 500 kHz was approximately proportional to volume fraction occupied by nylon filaments. The derivative of phase velocity with respect to frequency was negative and exhibited nonlinear, monotonically decreasing dependence on volume fraction. The dependencies of phase velocity and its derivative on volume fraction in these phantoms were similar to those reported in previous studies on (1) human cancellous bone and (2) phantoms consisting of parallel nylon wires immersed in water.

Comparison between independent and multiple scattering models to predict velocity dispersion in trabecular bone.

Transverse transmission devices are used clinically to assess fracture risk. However, the physical interaction between trabecular bone and ultrasound remains unclear. Unlike most soft tissues, negative values of velocity dispersion have been measured in trabecular bone. The origin of negative dispersion is still a matter of debate. The aim of this paper is to propose a model predicting the frequency dependence of phase velocity as well as its physical determinants. A two‐dimensional homogenization model accounting for the coupling of independent scattering effects with viscoelastic absorption is employed to calculate the dependence of phase velocity on frequency, bone volume fraction (BV/TV), and trabecular thickness (TbTh). A simple biphasic description of TB in which the trabeculae are assumed to be identical and parallel cylinders are considered. The first step of the method is to compute the attenuation coefficient in the entire frequency range using a suitable decomposition via Bessel functions. Then, the Kramers–Kronig relationships are used to derive the frequency dependence of phase velocity. The results obtained assuming, independent scattering are compared with a model accounting for multiple scattering [Haiat et al., J. Acoust. Soc. Am. (2008)], which is shown to be in better agreement with experimental results obtained in the literature.

Ultrasonic velocity dispersion in bovine cortical bone: An experimental study

Cortical bone quality is determinant in bone fragility and its ultrasonic evaluation has become possible in clinical practice. However, the interaction between a broadband ultrasonic pulse and this complex multiscale medium remains poorly understood. The frequency dependence of phase velocity, which may impact clinical measurements, has been sparsely investigated. Our objective is to evaluate the determinants of the frequency dependence of phase velocity in bovine femoral cortical bone samples using an in vitro ultrasonic transmission device. The apparent phase velocity varies quasilinearly on a 1 MHz restricted bandwidth around 4 MHz. After compensating for diffraction effects, significant differences in velocity dispersion are obtained according to the anatomical location. The microstructure of each sample is determined using an optical microscope, which allows assessing the dependence of dispersion on the type of bone microstructure. Mostly positive but also negative values of dispersion are measured. Negative dispersion is mainly obtained in samples constituted of mixed microstructure, which may be explained by phase cancellation effects due to the presence of different microstructures within the same sample. Dispersion is shown to be related to broadband ultrasonic attenuation values, especially in the radial direction. Results are compared with the local Kramers-Kronig relationships.

Mechanisms of Self-Sustained Oscillations Induced by a Flow Over a Cavity

Abstract This paper discusses the phenomena of flow-acoustic coupling between an unstable shear layer across a slot covering a cavity, thereby forming a Helmholtz resonator. The flow was caused by a turbulent boundary layer flowing over the slot. The physical phenomena implicated were responsible for the generation of high-amplitude sound level that occurred at the resonant frequencies of the system. It concerns a wide variety of applications, especially in the transport industry. Cavity oscillations have been studied steadily for some 50years and there is by now a vast and conflicting literature on the measured tonal frequencies, the physical explanations, and the predictive models for them. A theory was presented to predict the onset of the instability. It was based on the identification of the main parameters that played a role in the shear layer. Particularly it was suggested to use the frequency dependent phase velocity instead of the flow velocity. The theory was then confronted with results obtained from an experimental study carried out in a wind tunnel on several models. This validation underlined the sturdiness and the efficiency of the model, which made it useful for engineering applications.

Ultrasonic velocity dispersion in bovine cortical bone

The evaluation of cortical bone quality has become possible in clinical practice, but the interaction between a broadband ultrasonic pulse and this complex multiscale medium remains poorly understood. Specifically, the frequency dependence of phase velocity has been sparsely investigated. This study aims at evaluating the determinants of the frequency dependence of phase velocity in bovine femoral cortical bone samples using an in vitro ultrasonic transmission device. Phase velocity is shown to vary quasi linearly in a 1 MHz restricted bandwidth around 4 MHz, which enables dispersion evaluation. Axial dispersion is significantly higher than radial and tangential dispersions. Significant differences in dispersion are obtained according to the anatomical location. The microstructure of each sample is determined using an optical microscope, which allows assessing the dependence of dispersion on the type of bone microstructure. Mostly positive, but also negative values of dispersion are measured. Negative dispersion is obtained mostly in samples constituted of mixed microstructure, which may be explained by phase cancellation effects due to the presence of different microstructures within the same sample. Dispersion is shown to be related to broadband ultrasonic attenuation values, especially in the radial direction. This dependence is compared with results derived from the local Kramers‐Kronig relationships.

Negative dispersion in bone: The role of interference in measurements of the apparent phase velocity of two temporally overlapping signals

In this study the attenuation coefficient and dispersion (frequency dependence of phase velocity) are measured using a phase sensitive (piezoelectric) receiver in a phantom in which two temporally overlapping signals are detected, analogous to the fast and slow waves typically found in measurements of cancellous bone. The phantom consisted of a flat and parallel Plexiglas plate into which a step discontinuity was milled. The phase velocity and attenuation coefficient of the plate were measured using both broadband and narrowband data and were calculated using standard magnitude and phase spectroscopy techniques. The observed frequency dependence of the phase velocity and attenuation coefficient exhibit significant changes in their frequency dependences as the interrogating ultrasonic field is translated across the step discontinuity of the plate. Negative dispersion is observed at specific spatial locations of the plate at which the attenuation coefficient rises linearly with frequency, a behavior analogous to that of bone measurements reported in the literature. For all sites investigated, broadband and narrowband data (3-7 MHz) demonstrate excellent consistency. Evidence suggests that the interference between the two signals simultaneously reaching the phase sensitive piezoelectric receiver is responsible for this negative dispersion.

Explicit causal relations between material damping ratio and phase velocity from exact solutions of the dispersion equations of linear viscoelasticity

SUMMARY The theory of linear viscoelasticity is the simplest constitutive model that can be adopted to accurately predict the small-strain mechanical response of materials exhibiting the ability to both store and dissipate strain energy. An important result implied by this theory is the relationship existing between material attenuation and the velocity of propagation of a mechanical disturbance. The functional dependence of these important parameters is represented by the Kramers‐Kronig (KK) equations, also known as dispersion equations, which are nothing but a statement of the necessary and sufficient conditions to satisfy physical causality. This paper illustrates the derivation of exact solutions of the KK equations to provide explicit relations between frequency-dependent phase velocity and material damping ratio (or equivalently, quality factor). The assumptions that form the basis of the derivation are not beyond those established by the standard theory of viscoelasticity for a viscoelastic solid. The explicit expression for phase velocity as a function of damping ratio was derived by means of the theory of linear singular integral equations, and in particular by the solution of the associated Homogeneous Riemann Boundary Value Problem. It is shown that the same solution may be obtained also by using the implications of physical causality on the Fourier Transform. On the other hand, the explicit solution for damping ratio as a function of phase velocity was found through the components of the complex wavenumber. The exact solutions make it possible to obtain frequency-dependent material damping ratio solely from phase velocity measurements, and conversely. Hence, these relations provide an innovative and inexpensive tool to determine the small-strain dynamic properties of geomaterials. It is shown that the obtained rigorous solutions are in good agreement with well-known solutions based on simplifying assumptions that have been developed in the fields of seismology and geotechnical engineering. In addition, the theoretical results have been validated using recently published experimental material functions for fine-grained soils and for polymethyl methacrylate.

Phase velocity and normalized broadband ultrasonic attenuation in Polyacetal cuboid bone-mimicking phantoms

The variations of phase velocity and normalized broadband ultrasonic attenuation (nBUA) with porosity were investigated in Polyacetal cuboid bone-mimicking phantoms with circular cylindrical pores running normal to the surface along the three orthogonal axes. The frequency-dependent phase velocity and attenuation coefficient in the phantoms with porosities from 0% to 65.9% were measured from 0.65 to 1.10 MHz. The results showed that the phase velocity at 880 kHz decreased linearly with porosity, whereas the nBUA increased linearly with porosity. This study provides a useful insight into the relationships between ultrasonic properties and porosity in bone at porosities lower than 70%.

Frequency dependencies of phase velocity and attenuation coefficient in a water-saturated sandy sediment from 0.3to1.0MHz

The frequency-dependent phase velocity and attenuation coefficient for the fast longitudinal wave in a water-saturated sandy sediment were measured over the frequency range from 0.3 to 1.0 MHz. The experimental data of phase velocity exhibited the significant negative dispersion, with the mean rate of decline of 120 +/- 20 m/s/MHz. The Biot model predicted the approximately nondispersive phase velocity and the grain-shearing (GS) model exhibited the slightly positive dispersion. In contrast, the predictions of the multiple scattering models for the negative dispersion in the glass-grain composite were in general agreement with the experimental data for the water-saturated sandy sediment measured here. The experimental data of attenuation coefficient was found to increase nonlinearly with frequency from 0.3 to 1.0 MHz. However, both the Biot and the GS models yielded the attenuation coefficient increasing almost linearly with frequency. The total attenuation coefficient given by the algebraic sum of absorption and scattering components showed a reasonable agreement with the experimental data for overall frequencies. This study suggests that the scattering is the principal mechanism responsible for the variations of phase velocity and attenuation coefficient with frequency in water-saturated sandy sediments at high frequencies.