Fluid Half-spaces Research Articles

Simulation of ultrasonic wave propagation in anisotropic poroelastic bone plate using hybrid spectral/finite element method.

This paper deals with the modeling of guided waves propagation in in vivo cortical long bone, which is known to be anisotropic medium with functionally graded porosity. The bone is modeled as an anisotropic poroelastic material by using Biot's theory formulated in high frequency domain. A hybrid spectral/finite element formulation has been developed to find the time-domain solution of ultrasonic waves propagating in a poroelastic plate immersed in two fluid halfspaces. The numerical technique is based on a combined Laplace-Fourier transform, which allows to obtain a reduced dimension problem in the frequency-wavenumber domain. In the spectral domain, as radiation conditions representing infinite fluid halfspaces may be exactly introduced, only the heterogeneous solid layer needs to be analyzed by using finite element method. Several numerical tests are presented showing very good performance of the proposed procedure. A preliminary study on the first arrived signal velocities computed by using equivalent elastic and poroelastic models will be presented.

Simulation of transient ultrasonic wave propagation in fluid-loaded heterogeneous cortical bone

This work deals with the ultrasonic wave propagation in the cortical layer of long bones which is known as being a functionally-graded anisotropic material coupled with fluids. The motivation arises from mechanical modeling of the ultrasound axial transmission technique in vivo for cortical long bone which is known as being a functionally-graded anisotropic material. The proposed method is based on a combined Laplace-Fourier transform which substitutes a problem defined by partial differential equations into a system of differential equations established in the frequency-wavenumber domain. In the spectral domain, as radiation conditions may be exactly introduced in the infinite fluid halfspaces, only the heterogeneous solid layer needs to be analyzed using finite element method. Several numerical tests are presented showing very good performance of the proposed approach. Keywords: Spectral finite element, transient wave, ultrasound, anisotropic, vibroacoustic, cortical bone, axial transmission.

Ultrasonic wave propagation in viscoelastic cortical bone plate coupled with fluids: a spectral finite element study

This work deals with the ultrasonic wave propagation in the cortical layer of long bones which is known as being a functionally graded anisotropic material coupled with fluids. The viscous effects are taken into account. The geometrical configuration mimics the one of axial transmission technique used for evaluating the bone quality. We present a numerical procedure adapted for this purpose which is based on the spectral finite element method (FEM). By using a combined Laplace–Fourier transform, the vibroacoustic problem may be transformed into the frequency–wavenumber domain in which, as radiation conditions may be exactly introduced in the infinite fluid halfspaces, only the heterogeneous solid layer needs to be analysed using FEM. Several numerical tests are presented showing very good performance of the proposed approach. We present some results to study the influence of the frequency on the first arriving signal velocity in (visco)elastic bone plate.

Time harmonic point load and dynamic contact problem of contacting fluid and poroelastic half-spaces

The dynamic response of contacting fluid and fluid-saturated poroelastic half- spaces to a time-harmonic vertical point force or a point pore pressure is investigated. The solutions are formulated using the boundary conditions at the fluid-porous medium interface. The point load solutions are then used to solve the dynamic problem of the vertical vibration of a rigid disc (both permeable and impermeable discs are included) on the surface of the poroelastic half-space. The contact problems are solved by integrating the point force and point pore pressure solutions over the contact area with unknown discontinuous force and pore pressure distributions, which are determined from the boundary conditions. The solutions are expressed in terms of dual integral equations, which are converted to Fredholm integral equations of the second kind and solved numerically. Selected numerical results for the vertical dynamic compliance coefficient for the cases with or without fluid overlying the poroelastic half-space are presented to show the effects of the fluid. The influence of the permeability condition of the disc on the compliance of the poroelastic half-space is investigated. The displacement, vertical stress, pore pressure in the poroelastic half-space and water pressure in the fluid half-space are also examined for different poroelastic materials and frequencies of excitation. The present results are helpful in the study of the dynamic response of foundations on the seabed under seawater.

Simulation of transient ultrasonic wave propagation in fluid-loaded heterogeneous cortical bone

This work deals with the ultrasonic wave propagation in the cortical layer of long bones which is known as being a functionally-graded anisotropic material coupled with fluids. The motivation arises from mechanical modeling of the ultrasound axial transmission technique in vivo for cortical long bone which is known as being a functionally-graded anisotropic material. The proposed method is based on a combined Laplace-Fourier transform which substitutes a problem defined by partial differential equations into a system of differential equations established in the frequency-wavenumber domain. In the spectral domain, as radiation conditions may be exactly introduced in the infinite fluid half-spaces, only the heterogeneous solid layer needs to be analyzed using finite element method. Several numerical tests are presented showing very good performance of the proposed approach.

A Rayleigh-type wave at the plane interface of two homogeneous fluid half-spaces

It is shown that at the plane interface of two homogeneous fluid semi-infinite media, of which one is an ordinary fluid and the other a double negative—i.e., a fluid with negative density and negative compressibility—there can exist a Rayleigh-type wave propagating without attenuation, whose amplitude exponentially decreases with distance on both sides of the interface. In the parameter space of the media, the range of its existence is determined and the main properties are studied. A unique feature of this wave is the arbitrarily small propagation velocity, which does not depend on frequency, and the energy concentration in an arbitrarily thin layer close to the interface.

Resonance frequencies of piezoelectric plates surrounded by solid and fluid half-spaces

The design of ultrasound transducers, resonators and other piezoelectric devices usually requires the calculation of the resonance frequencies of piezoelectric plates. Recent studies have shown that the resonance frequencies for plates in vacuum correspond to frequencies where the waveguide group velocity vanishes (zero-group-velocity points). However, those studies are limited to vacuum boundary conditions. The objective of the present study is to analyze the resonance frequencies of layered piezoelectric plates in contact with solid and fluid half-spaces and their relation to the dispersion behavior of the elastic guided wave propagation. Theoretical analysis using partial-wave approach of leaky Lamb waves is performed to study wave propagation in, and resonance behavior of, multilayered plates in contact with solid and fluid half-spaces. A novel observation resulted from this analysis is that, for plates in contact with solid and fluid half-spaces, the resonance frequencies occur at points where the magnitude of the wavenumber reaches a minimum. This frequency is named as a ‘ transition frequency’. Such observations are important because they allow an easy identification of resonance frequencies with high amplitude response directly from the dispersion curves. This study will be helpful for the design of piezoelectric components used for resonators and sensors.

Viscosity and Diffusion Effects at the Boundary Surface of Viscous Fluid and Thermoelastic Diffusive Solid Medium

AbstractThis paper concentrates on the wave motion at the interface of viscous compressible fluid half-space and homogeneous isotropic, generalized thermoelastic diffusive half-space. The wave solutions in both the fluid and thermoelastic diffusive half-spaces have been investigated; and the complex dispersion equation of leaky Rayleigh wave motion have been derived. The phase velocity and attenuation coefficient of leaky Rayleigh waves have been computed from the complex dispersion equation by using the Muller’s method. The amplitudes of displacements, temperature change and concentration have been obtained. The effects of viscosity and diffusion on phase velocity and attenuation coefficient of leaky Rayleigh waves motion for different theories of thermoelastic diffusion have been depicted graphically. The magnitude of heat and mass diffusion flux vectors for different theories of thermoelastic diffusion have also been computed and represented graphically.

Monochromatic surface waves at the interface between poroelastic and fluid halfspaces

The topic of the previous work of Albers and Wilmanski was the study of monochromatic surface waves at the boundary between a porous medium and a vacuum. This article is an extension of this research to the propagation of surface waves on the interface between a porous halfspace and a fluid halfspace. Results for phase and group velocities and attenuations are shown in dependence on both the frequency and the surface permeability. In contrast to classical papers on surface waves where only the limits of the frequencyω→0,ω→∞ and the limits of the surface permeability (fully sealed and fully open boundary) were studied, we investigate the problem in the full range of both parameters. For the analysis we use the ‘simple mixture model’ which is a simplification of the classical Biot model for poroelastic media. The construction of a solution is shown and the dispersion relation solved numerically. There exist three surface waves for this boundary: a leaky Rayleigh wave and both a true and a leaky Stoneley wave. The true Stoneley wave exists only in a limited range of the surface permeability.

Ultrasonic Field Modeling in Multilayered Fluid Structures Using the Distributed Point Source Method Technique

In the field of nondestructive evaluation (NDE), the newly developed distributed point source method (DPSM) is gradually gaining popularity. DPSM is a semi-analytical technique used to calculate the ultrasonic field (pressure and velocity fields) generated by ultrasonic transducers. This technique is extended in this paper to model the ultrasonic field generated in multilayered nonhomogeneous fluid systems when the ultrasonic transducers are placed on both sides of the layered fluid structure. Two different cases have been analyzed. In the first case, three layers of nonhomogeneous fluids constitute the problem geometry; the higher density fluid is sandwiched between two identical fluid half-spaces. In the second case, four layers of nonhomogeneous fluids have been considered with the fluid density monotonically increasing from the bottom to the top layer. In both cases, analyses have been carried out for two different frequencies of excitation with various orientations of the transducers. As expected, the results show that the ultrasonic field is very sensitive to the fluid properties, the orientation of the fluid layers, and the frequency of excitation. The interaction effect between the transducers is also visible in the computed results. In the pictorial view of the resulting ultrasonic field, the interface between two fluid layers can easily be seen.

Analysis of the dispersion spectrum of fluid-loaded anisotropic plates: flexural-type branches and real-valued loops

The subsonic spectrum of complex velocity versus real frequency in immersed anisotropic plates is considered for the various options of presupposed choice of modes in the loading fluid half-spaces. Two principal implications of the latter prerequisite are investigated. The first is related to the family of flexural-type branches (with origin at v = 0 , k = 0 ), which evolve from the so-called A 0 and A 1 free-plate branches in a way that depends on the choice of fluid modes. By inspecting the low-frequency solutions of the dispersion equations corresponding to the different choices of fluid modes, a complete set of the flexural-type branches in an anisotropic fluid-loaded plate is identified. The result generalizes and also rectifies the interpretation commonly adopted for the isotropic case. The second issue is concerned with the real-valued loops on otherwise complex subsonic branches, involving the fluid mode increasing away from the plate. This phenomenon has been broadly discussed in numerical and experimental works on isotropic plates. The topological origin and shape of the real loops for an arbitrary plate can be readily viewed by way of the graphical layout of the sextic plate formalism. To this end, three possibilities are considered, namely, the sound velocity in the fluid, c f , being (i) less than the Rayleigh velocity, (ii) greater than that but less than the bulk-wave threshold in the plate, and (iii) greater than the bulk-wave threshold. These lead to three basic configurations. In case (i), a closed real loop exists provided than the fluid-to-solid density ratio is smaller than a certain critical value, estimated here in a simple explicit form. Case (ii) is typically characterized by the presence of two open real-valued arches with common high-frequency limits. Case (iii) produces an infinite sequence of progressively narrowing real-valued arches, whose upper arms rise up to c f and then transform into descending curves of solutions associated with the decreasing fluid modes (akin to the Sezawa continuum). A general overview of the locus of looping branches is supplied by analytical estimates and numerical examples.

Directional scholte wave generation and detection using interdigital capacitive micromachined ultrasonic transducers

Directional generation and detection of Scholte waves and other guided modes in liquids and microfluidic channels by capacitive micromachined ultrasonic transducers (cMUTs) is reported. An interdigital transducer structure along with a phased-excitation scheme is used to enhance the directionality of Scholte interface waves in microfluidic environments. Finite element models are developed to predict the performance of the devices in both fluid half-spaces and microchannels. Experiments on the interdigital cMUTs show that a five-finger-pair device in a water half-space has 12 dB of directionality in generating Scholte waves at the design frequency of 10 MHz. A 10-finger device operating at 10 MHz in a water-filled microchannel has 13.4 dB of directionality. These directionality figures agree well with the modeling results. Using the results of the finite element model of a cMUT in a fluid half-space, it was determined that 41% of the acoustic power radiated into the fluid is contained in the Scholte wave propagating in the desired lateral direction. Transducers are demonstrated to perform bidirectional pumping in fluid channels with input power levels in the milliwatt range. Interdigital cMUTs fabricated using low temperature processes can be used as compact ultrasonic transducers with integrated electronics for sensing and actuation in fluidic environments.

Best Constants in the Miranda-Agmon Inequalities for Solutions of Elliptic Systems and the Classical Maximum Modulus Principle for Fluid and Elastic Half-spaces

The Dirichlet problem for elliptic systems of the second order with constant real and complex coefficients in the half-space 𝔑 k + = {x = (x 1,…,xk ): xk > 0} is considered. It is assumed that the boundary values of a solution u = (u 1,…,u m) have the form ψ 1 ξ 1 + · · · + ψ n ξ n, 1 ≤ n ≤ m, where ξ 1,· · ·,ξ n is an orthogonal system of m-component normed vectors and ψ 1,· · ·,ψ n are continuous and bounded functions on ∂𝔑 k +. We study the mappings [C(∂𝔑 k +)] n ∋ (ψ 1,…,ψ n) → u(x) ∋ 𝔑 m and [C(∂𝔑 k +)] n ∋ (ψ 1,…,ψ n) → u(x) ∋ 𝔑 m generated by real and complex vector valued double layer potentials. We obtain representations for the sharp constants in inequalities between |u(x)| or |(z, u(x))| and ∥u| xk =0∥, where z is a fixed unit m-component vector, | · | is the length of a vector in a finite-dimensional unitary space or in Euclidean space, and (·,·) is the inner product in the same space. Explicit representations of these sharp constants for the Stokes and Lamé systems are given. We show, in particular, that if the velocity vector (the elastic displacement vector) is parallel to a constant vector at the boundary of a half-space and if the modulus of the boundary data does not exceed 1, then the velocity vector (the elastic displacement vector) is majorised by 1 at an arbitrary point of the half-space. An analogous classical maximum modulus principle is obtained for two components of the stress tensor of the planar deformed state as well as for the gradient of a biharmonic function in a half-plane.

Theory of Thin Layer Photoacoustic Cells for Determination of Volume Changes in Solution

The wave equation for the photoacoustic effect generated by heat deposition and molecular volume change is solved for a one-dimensional geometry for the case of excitation of a single molecular electronic state that decays to the ground state exponentially in time. The solution for a uniformly illuminated, infinite layer of fluid in contact with two infinite fluid or solid half-spaces, corresponding to a recently developed experimental geometry for generation of the photoacoustic effect is given. The solutions show the effects of finite decay of the excited state and of reflections at the interfaces.

Comments on “Flexural wave reduction using a compliant tube baffle” [J. Acoust. Soc. Am. 99(2), 691–699 (1996)] and “Signal pressure received by a hydrophone placed on a plate backed by a compliant baffle” [J. Acoust. Soc. Am. 89(2), 559–564 (1991)

The cited formulations for studying the baffling of flexural waves by a compliant tube grating [S. H. Ko, J. Acoust. Soc. Am. 99, 691–699 (1996)] and for determining the acoustic receive sensitivity of a hydrophone placed near a plate backed by a compliant tube grating [S. H. Ko and H. H. Schloemer, J. Acoust. Soc. Am. 89, 559–564 (1991)] are dependent on the validity of the lumped-fluid parameter approximations developed by Junger [M. C. Junger, J. Acoust. Soc. Am. 78, 1010–1012 (1985)] for a grating of closely spaced compliant tubes between two semi-infinite fluid half-spaces. If the fluid between the plate and the grating is great enough to minimize their interaction, the incorporation of the effective layer parameters developed by Junger should be reasonably valid. Unfortunately, the sample calculations in Ko (Ref. 4) and Ko and Schloemer (Ref. 5) have been made for the limiting case of the compliant tube grating in direct contact with the plates where the use of Junger’s approximations is inappropriate.

On formulation of the high-order boundary conditions for stratified media

Boundary conditions containing spatial derivatives of high orders, Â(∂x,∂y)(∂U/∂z)+ikB̂(∂x, ∂y)U‖z=0=0, where z is the coordinate orthogonal to a plane surface of a layered medium, Â, B̂ are the differential operators with respect to the tangent coordinates, U is a scalar function representing acoustic pressure in the region above the surface (z≥0), k is the wave number, are widely used to describe the complicated process of interaction of sound waves with thin elastic plates or stratified slabs. The high-order boundary conditions are usually obtained from the equations of motion of the elastic structures by expanding the wave fields into power series according to the slab thickness. This paper describes a new approach to formulation of the high-order boundary conditions for isotropic stratified media. Via Fourier transform the problem is rigorously reduced the approximation of certain entire functions by polynomials in the complex plane. Choosing an appropriate method of approximation such boundary conditions are constructed that are specially adjusted to model the structure of interest. The technique proposed is regular; it does not require the separation of symmetrical and antisymmetrical processes, and it includes the slim layer theories as a special case. For structures placed between two fluid half-spaces, the scalar boundary conditions must be replaced by matrix ones combining the values of U and its derivatives on the opposite sides of the slab.

Scattering by objects buried in underwater sediments: Theory and experiment

The scattering of sound by objects buried in underwater sediments is studied in the context of an exactly soluble model. The model consists of two fluid half-spaces separated by a planar, fluid, transition layer of arbitrary thickness. Attenuation is included in any of these regions by using complex wave numbers. A directional source field, generated in the upper half-space by a continuous line array, insonifies an object placed in the lower half-space. The scattered field detected by another line array placed anywhere in the system may be calculated. The solution is determined from the T matrix for the bounded scattering system and is exact (in linear acoustics) to all orders of multiple scattering among the interfaces and object. Numerical results are presented to investigate the effect of the local acoustic environment on the free-field, in-water scattering resonances of thin spherical shells. The field scattered by a shallowly buried object is discussed with emphasis on the importance of evanescent wave scattering in detection from above the sediment over an extended range. An initial set of experiments meant to verify the model are described. Results are presented and discussed for the measured scattering response of buried, spherical, evacuated, steel shells, that are 2.25% and 11% of the outer radius in thickness.

Elastodynamic response of contacting fluid and solid half-spaces to a three-dimensional point load

The dynamic response of contacting fluid and solid elastic half-spaces to a point load is addressed. The problem considered is the three-dimensional one in which the point load acts in the fluid or in the solid in an arbitrary direction. Integral expressions of displacements and stresses are obtained by using integral transform techniques, and techniques for numerical evaluation are discussed. Results are presented for time harmonic and impulsive loads acting in both the solid and the fluid. Surface and subsurface displacements are compared for fluid half-spaces of water and air. This comparison shows the effect of wavefield coupling at the interface.

Acoustic emission from the Arctic ice sheet

A theoretical analysis of the sound field in the Arctic Ocean due to a crack occurring in the ice sheet is given. Particular attention is paid to the contribution of that field to Arctic Ocean ambient noise. The crack is modeled by an approximate ‘‘equivalent source,’’ and the example chosen represents a small failure of the ice sheet under tension. It is shown that if the ice sheet were completely continuous, only a very small fraction of the energy released by the crack could be trapped as sound in the surface half-channel of the ocean, where it would contribute to the ambient noise. This suggests that scattering of energetic elastic waves in the ice by natural inhomogeneities may be an important additional mechanism by which acoustic energy can leak into the surface channel. The analysis is relevant to acoustic emission from any unbounded, elastic plate in contact with two, possibly different, fluid half-spaces.

Ultrasonic leaky waves in a solid plate separating a fluid and vacuum

The reflection of a finite acoustic beam from a solid plate separating fluid and vacuum half-spaces has been measured and analyzed. Material viscous damping has been included in the reflection model and the finite beam has been approximated by an integral of the plane wave reflection coefficient times the spatial beam transform over the incident wave vector. The effects of viscosity and the finite beam on the reflected field are separately demonstrated, and generally good agreement is found between the model calculation and experiments on an aluminum plate with water chosen for the fluid. It is shown that the finite beam alone modified the plane-wave reflection coefficient substantially, whereas the introduction of viscosity exerts a smaller, but still important, influence.