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The theory for nonlinear surface acoustic waves in crystals developed using Hamiltonian mechanics [Hamilton, Il'inskii, and Zabolotskaya, J. Acoust. Soc. Am. 105, 639 (1999)] is modified to account for piezoelectric material properties. The derived spectral evolution equations permit analysis of nonlinear surface wave propagation along a cut surface of any orientation with respect to the crystallographic axes and for piezoelectric crystals with any symmetry. Numerical simulations of waveform distortion in the particle velocity and electric field components are presented for surface wave propagation in Y-cut lithium niobate along the X- and Z-crystallographic axes. The influence of piezoelectricity is illustrated by comparing the nonlinear evolution of waveforms along a surface bounded by a vacuum (free space) and an ideal conductor (short circuit). Contributions to the nonlinearity from elasticity, piezoelectricity, electrostriction, and dielectricity are quantified separately for the two boundary conditions.

The acoustic radiation force on a compressible spheroid is calculated using expansions of the scattered field in terms of both spherical and spheroidal wave functions that are matched analytically in the far field. There is no restriction on the size or impedance of the spheroid, the structure of the incident field, or the orientation of the spheroid with respect to the incident field. The form of the solution is the same as that developed previously for the radiation force on an elastic sphere, which is a summation of terms involving products of the coefficients in spherical wave expansions of the incident and scattered fields. Spheroidal wave expansions are employed to satisfy the boundary conditions and obtain the scattering coefficients. While the scattering coefficients must be obtained numerically for compressible spheroids, explicit expressions in terms of radial wave functions are available for spheroids with rigid or free surfaces. Results are compared with available analytical expressions for various limiting cases. The theoretical framework may be extended to objects of arbitrary shape.

The acoustic radiation force on a compressible spheroid is calculated using expansions of the scattered field in terms of both spherical and spheroidal wave functions that are matched analytically in the far field. There is no restriction on the size or impedance of the spheroid, the structure of the incident field, or the orientation of the spheroid with respect to the incident field. The form of the solution is the same as that developed previously for the radiation force on an elastic sphere, which is a summation of terms involving products of the coefficients in spherical wave expansions of the incident and scattered fields. Spheroidal wave expansions are employed to satisfy the boundary conditions and obtain the scattering coefficients. While the scattering coefficients must be obtained numerically for compressible spheroids, explicit expressions in terms of radial wave functions are available for spheroids with rigid or free surfaces. Results are compared with available analytical expressions for various limiting cases. The theoretical framework may be extended to objects of arbitrary shape.

When the density and compressibility of an object are similar to the corresponding properties of the surrounding fluid and the incident sound field is a standing wave, the Born approximation may be used to calculate the acoustic radiation force and torque on an object of arbitrary shape. The approximation consists of integration over the monopole and dipole contributions to the force acting at each point within the region occupied by the object. The method is applied to axisymmetric objects, for which the force and torque may be expressed as a single integral along the axis of symmetry. The integral is evaluated analytically for spheres and cylinders. The accuracy of the Born approximation is assessed by comparison with complete solutions for compressible spheres and prolate spheroids that are based on expansions of the incident, scattered, and transmitted fields in terms of eigenfunctions of the corresponding separable coordinate system. Results are presented for objects with various densities and compressibilities relative to the surrounding fluid, as well as different shapes, sizes, and orientations of the object with respect to the standing wave field. The method also accommodates spatial variations of the density and compressibility within the object.

A theoretical framework in Lagrangian coordinates is developed for calculating the acoustic radiation force on an elastic sphere in a soft elastic medium. Advantages of using Lagrangian coordinates are that the surface of the sphere is fixed in the reference frame, and nonlinearity appears only in the stress tensor. The incident field is a time-harmonic compressional wave with arbitrary spatial structure, and there is no restriction on the size of the sphere. Bulk and shear viscosities are taken into account with complex wavenumbers. A solution is presented for the radiation force due to the scattered compressional wave. For an ideal liquid surrounding the sphere, there is no scattered shear wave contributing to the radiation force and the solution is complete. The theory reproduces established results obtained in Eulerian coordinates for an elastic sphere in a fluid.

Carefully timed tandem microbubbles have been shown to produce directional and targeted membrane poration of individual cells in microfluidic systems, which could be of use in ultrasound-mediated drug and gene delivery. This study aims at contributing to the understanding of the mechanisms at play in such an interaction. The dynamics of single and tandem microbubbles between two parallel plates is studied numerically and analytically. Comparisons are then made between the numerical results and the available experimental results. Numerically, assuming a potential flow, a three-dimensional boundary element method (BEM) is used to describe complex bubble deformations, jet formation, and bubble splitting. Analytically, compressibility and viscous boundary layer effects along the channel walls, neglected in the BEM model, are considered while shape of the bubble is not considered. Comparisons show that energy losses modify the bubble dynamics when the two approaches use identical initial conditions. The initial conditions in the boundary element method can be adjusted to recover the bubble period and maximum bubble volume when in an infinite medium. Using the same conditions enables the method to recover the full dynamics of single and tandem bubbles, including large deformations and fast re-entering jet formation. This method can be used as a design tool for future tandem-bubble sonoporation experiments.

Three models are considered for describing the dynamics of a pulsating cylindrical bubble. A linear solution is derived for a cylindrical bubble in an infinite compressible liquid. The solution accounts for losses due to viscosity, heat conduction, and acoustic radiation. It reveals that radiation is the dominant loss mechanism, and that it is 22 times greater than for a spherical bubble of the same radius. The predicted resonance frequency provides a basis of comparison for limiting forms of other models. The second model considered is a commonly used equation in Rayleigh-Plesset form that requires an incompressible liquid to be finite in extent in order for bubble pulsation to occur. The radial extent of the liquid becomes a fitting parameter, and it is found that considerably different values of the parameter are required for modeling inertial motion versus acoustical oscillations. The third model was developed by V. K. Kedrinskii [Hydrodynamics of Explosion (Springer, New York, 2005), pp. 23–26] in the form of the Gilmore equation for compressible liquids of infinite extent. While the correct resonance frequency and loss factor are not recovered from this model in the linear approximation, it provides reasonable agreement with observations of inertial motion.

A model is presented for a pulsating spherical bubble positioned at a fixed location in a viscous, compressible liquid between parallel viscoelastic layers of finite thickness. The Green’s function for particle displacement is found and utilized to derive an expression for the radiation load imposed on the bubble by the layers. Although the radiation load is derived for linear harmonic motion it may be incorporated into an equation for the nonlinear radial dynamics of the bubble. This expression is valid if the strain magnitudes in the viscoelastic layer remain small. Dependence of bubble pulsation on the viscoelastic and geometric parameters of the layers is demonstrated through numerical simulations.

M. P. Mortell and B. R. Seymour [M. P. Mortell and B. R. Seymour [J. Acoust. Soc. Am. 124, 3381–3385 (2008)] offer commentary on the analysis and the conclusions reached in a paper by Hamilton et al. [M. F. Hamilton, Yu. A. Ilinskii, and E. A. Zabolotskaya, J. Acoust. Soc. Am. 110, 109–119 (2001)] on linear and nonlinear frequency shifts in acoustical resonators that are close to cylindrical in shape. The present reply demonstrates that the criticisms made in the Comment are unwarranted when placed in context of the stated restrictions on the theory and the applications for which the theory is intended. It is also shown that the strongest criticisms made in the Comment stem from a mathematical error they introduced when attempting to reproduce the results of Hamilton et al.

For high gas supersaturation levels in liquids, on the order of 300% as predicted in capillaries of marine mammals following a series of dives [D. S. Houser, R. Howard, and S. Ridgway, J. Theor. Biol. 213, 183-195 (2001)], standard mathematical models of both static and rectified diffusion are found to underestimate the rate of bubble growth by 10%-20%. The discrepancy is demonstrated by comparing predictions based on existing mathematical models with direct numerical solutions of the differential equations for gas diffusion in the liquid and thermal conditions in the bubble. Underestimation of bubble growth by existing mathematical models is due to the underlying assumption that the gas concentration in the liquid is given by its value for a bubble of constant equilibrium radius. This assumption is violated when high supersaturation causes the bubble to grow too fast in relation to the time scale associated with diffusion. Rapid bubble growth results in an increased gas concentration gradient at the bubble wall and therefore a growth rate in excess of predictions based on constant equilibrium bubble radius.

A coupled pair of nonlinear parabolic equations is derived for the two components of the particle motion perpendicular to the axis of a shear wave beam in an isotropic elastic medium. The equations account for both quadratic and cubic nonlinearity. The present paper investigates, analytically and numerically, effects of cubic nonlinearity in shear wave beams for several polarizations: linear, elliptical, circular, and azimuthal. Comparisons are made with effects of quadratic nonlinearity in compressional wave beams.

Nonlinear surface waves in soft, weakly compressible elastic media are investigated theoretically, with a focus on propagation in tissue-like media. The model is obtained as a limiting case of the theory developed by Zabolotskaya [J. Acoust. Soc. Am. 91, 2569-2575 (1992)] for nonlinear surface waves in arbitrary isotropic elastic media, and it is consistent with the results obtained by Fu and Devenish [Q. J. Mech. Appl. Math. 49, 65-80 (1996)] for incompressible isotropic elastic media. In particular, the quadratic nonlinearity is found to be independent of the third-order elastic constants of the medium, and it is inversely proportional to the shear modulus. The Gol'dberg number characterizing the degree of waveform distortion due to quadratic nonlinearity is proportional to the square root of the shear modulus and inversely proportional to the shear viscosity. Simulations are presented for propagation in tissue-like media.

Two models of interacting bubble dynamics are presented, a coupled system of second-order differential equations based on Lagrangian mechanics, and a first-order system based on Hamiltonian mechanics. Both account for pulsation and translation of an arbitrary number of spherical bubbles. For large numbers of interacting bubbles, numerical solution of the Hamiltonian equations provides greater stability. The presence of external acoustic sources is taken into account explicitly in the derivation of both sets of equations. In addition to the acoustic pressure and its gradient, it is found that the particle velocity associated with external sources appears in the dynamical equations.

Ultrasonic standing waves are widely used for separation applications. In MEMS applications, a half wavelength standing wave field is generated perpendicular to a laminar flow. The acoustic radiation force exerted on the particle drives the particle to the center of the MEMS channel, where concentrated particles are harvested. In macro-scale applications, the ultrasonic standing wave spans multiple wavelengths. Examples of such applications are oil/water emulsion splitting [1], and blood/lipid separation [2]. In macro-scale applications, particles are typically trapped in the standing wave, resulting in clumping or coalescence of particles/droplets. Subsequent gravitational settling results in separation of the secondary phase. An often used expression for the radiation force on a particle is that derived by Gorkov [3]. The assumptions are that the particle size is small relative to the wavelength, and therefore, only monopole and dipole scattering contributions are used to calculate the radiation force. This framework seems satisfactory for MEMS scale applications where each particle is treated separately by the standing wave, and concentrations are typically low. In macro-scale applications, particle concentration is high, and particle clumping or droplet coalescence results in particle sizes not necessarily small relative to the wavelength. Ilinskii et al. developed a framework for calculation of the acoustic radiation force valid for any size particle [4]. However, this model does not take into account particle to particle effects, which can become important as particle concentration increases. It is known that an acoustic radiation force on a particle or a droplet is determined by the local field. An acoustic radiation force expression is developed that includes the effect of particle to particle interaction. The case of two neighboring particles is considered. The approach is based on sound scattering by the particles. The acoustic field at the location of one particle then consists of two components, the incident sound wave and the scattered field generated by the neighboring particle. The radiation force calculation then includes the contributions of these two fields and incorporates the mutual particle influence. In this investigation the droplet/particle influence on each other has been analyzed theoretically by using the method developed by Gorkov and modified by Ilinskii et al.

Acoustic radiation force is of interest in a wide variety of biomedical applications ranging from tissue characterization (e.g. elastography) to tissue treatment (e.g. high intensity focused ultrasound, kidney stone fragment removal). As tissue mechanical properties are reliable indicators of tissue health, the former is the focus of the present contribution. This is accomplished through an investigation of the acoustic radiation force on a spherical scatterer embedded in tissue. Properties of both the scatterer and the surrounding tissue are important in determining the magnitude and the direction of the force. As these properties vary, the force computation shows changes in magnitude and direction, which may enable more accurate noninvasive determination of tissue properties.

A coupled pair of nonlinear parabolic equations was derived by Zabolotskaya [1] that model the transverse components of the particle motion in a collimated shear wave beam propagating in an isotropic elastic solid. Like the KZK equation, the parabolic equation for shear wave beams accounts consistently for the leading order effects of diffraction, viscosity and nonlinearity. The nonlinearity includes a cubic nonlinear term that is equivalent to that present in plane shear waves, as well as a quadratic nonlinear term that is unique to diffracting beams. The work by Wochner et al. [2] considered shear wave beams with translational polarizations (linear, circular and elliptical), wherein second-order nonlinear effects vanish and the leading order nonlinear effect is third-harmonic generation by the cubic nonlinearity. The purpose of the current work is to investigate the quadratic nonlinear term present in the parabolic equation for shear wave beams by considering second-harmonic generation in Gaussian beams as a second-order nonlinear effect using standard perturbation theory. In order for second-order nonlinear effects to be present, a broader class of source polarizations must be considered that includes not only the familiar translational polarizations, but also polarizations accounting for stretching, shearing and rotation of the source plane. It is found that the polarization of the second harmonic generated by the quadratic nonlinearity is not necessarily the same as the polarization of the source-frequency beam, and we are able to derive a general analytic solution for second-harmonic generation from a Gaussian source condition that gives explicitly the relationship between the polarization of the source-frequency beam and the polarization of the second harmonic.

A model for the interaction of two bubbles in a sound field [E. A. Zabolotskaya, Sov. Phys. Acoust. 30, 365–368 (1984)] is extended to account for an arbitrary number of bubbles interacting in a cluster. Compressibility of the liquid is taken into account through radiation damping and time delays due to the finite sound speed. Bubble coalescence is also included. A numerical method for implementing the model is described, and simulations of the growth and collapse of a bubble cluster near a rigid surface are presented. The relative effects of compressibility and coalescence are discussed.

An evolution equation for nonlinear shear waves in soft isotropic solids is derived using an expansion of the strain energy density that permits separation of compressibility and shear deformation. The advantage of this approach is that the coefficient of nonlinearity for shear waves depends on only three elastic constants, one each at second, third, and fourth order, and these coefficients have comparable numerical values. In contrast, previous formulations yield coefficients of nonlinearity that depend on elastic constants whose values may differ by many orders of magnitude because they account for effects of compressibility as well as shear. It is proposed that the present formulation is a more natural description of nonlinear shear waves in soft solids, and therefore it is especially applicable to biomaterials like soft tissues. Calculations are presented for harmonic generation and shock formation in both linearly and elliptically polarized shear waves.

A formulation of the elastic energy density for an isotropic medium is presented that permits separation of effects due to compressibility and shear deformation. The motivation is to obtain an expansion of the energy density for soft elastic media in which the elastic constants accounting for shear effects are of comparable order. The expansion is carried out to fourth order to ensure that nonlinear effects in shear waves are taken into account. The result is E≃E0(ρ)+μI2+13AI3+DI22, where ρ is density, I2 and I3 are the second- and third-order Lagrangian strain invariants used by Landau and Lifshitz, μ is the shear modulus, A is one of the third-order elastic constants introduced by Landau and Lifshitz, and D is a new fourth-order elastic constant. For processes involving mainly compressibility E≃E0(ρ), and for processes involving mainly shear deformation E≃μI2+13AI3+DI22.

Acoustic streaming generated by standing waves in channels of arbitrary width is investigated analytically. In a previous paper by the authors [J. Acoust. Soc. Am. 113, 153-160 (2003)], a purely viscous fluid in a two-dimensional channel was considered. That analysis is extended here to a gas in which heat conduction and dependence of the viscosity on temperature are taken into account. Calculations are presented for typical working gases used in thermoacoustic engines at standard temperature and pressure. In channels that are very wide in comparison with the viscous penetration depth, which is the Rayleigh streaming regime, the influence of the two thermal effects is comparable but small. The same is true in very narrow channels, having widths on the order of the viscous penetration depth. In channels having intermediate widths, 10-20 times the viscous penetration depth, the effect of heat conduction can be substantial. The analysis is performed for cylindrical tubes as well as two-dimensional channels, and the results are found to be qualitatively the same.

An analytic solution is derived for acoustic streaming generated by a standing wave in a viscous fluid that occupies a two-dimensional channel of arbitrary width. The main restriction is that the boundary layer thickness is a small fraction of the acoustic wavelength. Both the outer, Rayleigh streaming vortices and the inner, boundary layer vortices are accurately described. For wide channels and outside the boundary layer, the solution is in agreement with results obtained by others for Rayleigh streaming. As channel width is reduced, the inner vortices increase in size relative to the Rayleigh vortices. For channel widths less than about 10 times the boundary layer thickness, the Rayleigh vortices disappear and only the inner vortices exist. The obtained solution is compared with those derived by Rayleigh, Westervelt, Nyborg, and Zarembo.

Previously, a study of the acoustic radiation force acting on a spheroidal object in liquid showed that the radiation force depends on the angle between the incident acoustic wave and the main axis of a spheroid. This investigation demonstrated that there is a radiation force moment which acts on the spheroidal object and depends on particle orientation. This contribution is a continuation of the previously reported work. Here, the acoustic radiation force moment on prolate objects in an acoustic field in liquid is investigated analytically, i.e., the equations to describe the moment are derived. The incident acoustic and scattered field are expanded with respect to spherical waves. Analytically, scattering amplitudes are calculated from boundary conditions for spheroidal functions that are solutions of a wave equation in spheroidal coordinates ζ, η, φ. The radiation force moment is analyzed numerically. Randomly oriented spheroidal particles distributed in liquid align in an acoustic field as scattering ...

Macro-scale acousto-fluidics involves the interaction between acoustic radiation force exerted on a particle by bulk acoustic standing waves spanning many wavelengths, fluid drag force, and the gravitational force of the particle. Parameters are particle size, and ratio of particle to fluid density and particle to fluid compressibility. Different acousto-fluidics configurations can be used to manipulate particles in multiple ways. In cell clarification, the configuration is that of a depth flow filter with the added benefit of separating the cells out of the acoustic field, thereby eliminating any issues with filter clogging or fouling. In perfusion of stirred bioreactors, the configuration resembles that of a tangential flow filter. In a third configuration, the bulk acoustic standing wave is angled relative to the fluid velocity resulting in a label-free fractionation tool. Several underlying theoretical and numerical results of acoustic radiation force and particle trajectory calculations will be presented. A theoretical framework to calculate the acoustic radiation force on spherical, spheroidal, and cylindrical particles has been developed for any particle size relative to wavelength. An analytical solution for particle deflection angle in a planar angled standing wave and uniform flow has been developed. Experimental results will be shown to support the theory.

Acoustic radiation force on a sphere in soft tissue can be written as the sum of four distinct contributions. Two arise from incident and scattered compressional waves only, one from direct integration of the time-averaged Piola-Kirchhoff stress tensor over the surface of the sphere, and one from the irrotational component of the body force producing deformation of the surrounding medium. The other two contributions also incorporate scattered shear waves, and they are found by the same procedures. Three of these terms are known analytically [Ilinskii et al., Proc. Meet. Acoust. 19, 045004 (2013)], but the contribution relating to the shear field body force must be found numerically. Preliminary results for this term were obtained through simplifying approximations and presented at the fall 2016 ASA meeting; the present submission extends this work to cases where these approximations do not hold. Helmholtz decomposition of the shear field body force is performed using 3D Fourier transforms, then the irrotational potential is integrated over the surface of the sphere. Various sphere materials are considered, and comparisons are made with known results for a sphere in an ideal fluid. [Work supported by the ARL:UT McKinney Fellowship in Acoustics.]

A three-dimensional model is developed to describe an acoustic field excited by a piezoelectric plate of finite size in a fluid filled resonator. First, the eigenfunctions (modes) of a bare plate are derived using general piezoelectric equations considering the elastic and electric properties of the plate. Then, the piezoelectric plate is placed into a fluid media such that only one plate side is in fluid and an acoustic field generated by the plate in the fluid is estimated. Finally, a reflector is placed to be parallel to the piezoelectric plate and acoustic field in a resonator is evaluated. The solution for a piezoelectric plate of finite size is obtained using Singular Value Decomposition (SVD) method. Equations for acoustic and electric variables are presented. Radiation force on spherical particles in the standing wave field is derived and discussed. Numerical results are presented to show the three-dimensional modal displacement and electrical characteristics of the plate at various frequencies an...

The frequency response of a nonlinear acoustical resonator is investigated analytically and numerically. The cross-sectional area is assumed to vary slowly but otherwise arbitrarily along the axis of the resonator, such that the Webster horn equation provides a reasonable one-dimensional model in the linear approximation. First, perturbation theory is used to derive an asymptotic formula for the natural frequencies as a function of resonator shape. The solution shows that each natural frequency can be shifted independently via appropriate spatial modulation of the resonator wall. Numerical calculations for resonators of different shapes establish the limits of the asymptotic formula. Second, the nonlinear interactions of modes in the resonator are investigated with Lagrangian mechanics. An analytical result is obtained for the amplitude-frequency response curve and nonlinear resonance frequency shift for the fundamental mode. For a resonator driven at its lowest natural frequency, it is found that whether hardening or softening behavior occurs depends primarily on whether the nonlinearly generated second-harmonic frequency is greater or less than the second natural frequency of the resonator. A fully nonlinear one-dimensional numerical code is used to verify the analytical result.

Spectral evolution equations are derived for plane, progressive, finite-amplitude Stoneley and Scholte waves that propagate along plane interfaces formed by two semi-infinite, isotropic media in contact. The evolution equations have mathematical forms identical to those obtained previously for Rayleigh waves [E. A. Zabolotskaya, J. Acoust. Soc. Am. 91, 2569–2575 (1992)], and they are expressed explicitly in terms of the second- and third-order elastic constants of the media. Calculations were performed to simulate nonlinear surface wave propagation in several pairs of real media. Harmonic generation and shock formation associated with the Stoneley and Scholte modes are compared with the corresponding processes in Rayleigh waves. Waveform distortion is shown to be very similar for the three types of surface waves when the propagation distance is normalized by an appropriate shock formation distance.

A theory for nonlinear surface acoustic waves in isotropic solids [E. A. Zabolotskaya, J. Acoust. Soc. Am. 91, 2569–2575 (1992)] is generalized to include the anisotropy of crystals. There is no restriction to material symmetry, orientation of the free surface with respect to crystal axes, or propagation direction in the plane of the free surface. Numerical simulations of waveform distortion, shock formation, and harmonic propagation curves are presented for two different cuts of potassium chloride.

A one-dimensional model is developed to analyze nonlinear standing waves in an acoustical resonator. The time domain model equation is derived from the fundamental gasdynamics equations for an ideal gas. Attenuation associated with viscosity is included. The resonator is assumed to be of an axisymmetric, but otherwise arbitrary shape. In the model the entire resonator is driven harmonically with an acceleration of constant amplitude. The nonlinear spectral equations are integrated numerically. Results are presented for three geometries: a cylinder, a cone, and a bulb. Theoretical predictions describe the amplitude related resonance frequency shift, hysteresis effects, and waveform distortion. Both resonance hardening and softening behavior are observed and reveal dependence on resonator geometry. Waveform distortion depends on the amplitude of oscillation and the resonator shape. A comparison of measured and calculated wave shapes shows good agreement.

Rayleigh wave nonlinearity is investigated theoretically. First, spectral forms of the evolution equations for Rayleigh waves and sound waves are used to compare the relative efficiencies of sum and difference frequency generation. Next, time domain forms of the same evolution equations are used to compare the relative importance of local and nonlocal nonlinearity on the distortion of Rayleigh waves in the preshock region. Finally, new analytical approximations are derived for the shock formation distance and the corresponding coefficient of nonlinearity for Rayleigh waves.

A model is developed for a pulsating and translating gas bubble immersed in liquid in a channel formed by two soft, thin elastic parallel layers having densities equal to that of the surrounding liquid and small, but finite, shear moduli. The bubble is nominally spherical but free to undergo small shape deformations. Shear strain in the elastic layers is estimated in a way which is valid for short, transient excitations of the system. Coupled nonlinear second-order differential equations are obtained for the shape and position of the bubble, and numerical integration of an expression for the liquid velocity at the layer interfaces yields an estimate of the elastic layer displacement. Numerical integration of the dynamical equations reveals behavior consistent with laboratory observations of acoustically excited bubbles in ex vivo vessels reported by Chen et al. [Phys. Rev. Lett. 106, 034301 (2011) and Ultrasound Med. Biol. 37, 2139–2148 (2011)].

The analysis presented at the previous ASA meeting related to investigation of the acoustic radiation force on a sphere embedded in a soft elastic medium with shear modulus that is several orders of magnitude smaller than its bulk modulus. The acoustic field was assumed to be axisymmetric and the spherical scatterer to be located on the axis of the acoustic beam. When one of these conditions is violated, the problem loses its symmetry. In this talk, the acoustic radiation force is considered in the more general case of nonaxisymmetric fields. The calculation is performed in Lagrangian coordinates. All acoustic fields, incident as well as scattered, depend on all three spherical coordinates. The incident and scattered waves, which include both potential and solenoidal parts, are expanded with respect to spherical harmonics. An analytical expression for the acoustic radiation force derived in this investigation may contain as many spherical harmonics as needed. In limiting cases when the scatterer is in liquid and only two modes, monopole and dipole, remain in the scattered fields, the solution for the acoustic radiation force recovers the results reported by Gor’kov [Sov. Phys. Doklady 6, 773 (1962)]. [Work supported by NIH DK070618 and EB011603.]

Previous models and experiments have shown that direction of bubble translation near a viscoelastic layer depends on both the standoff distance of the bubble and the elastic properties of the layer. Here the individual forces due to the incident sound field and the field reflected from the viscoelastic layer are shown to compete with one another and ultimately determine the direction of bubble translation. In addition, many other factors pertinent to the direction of bubble translation such as the incident acoustic waveform, the phase and propagation direction of the incident field, and the radial bubble dynamics are considered. The force due to the viscoelastic layer is calculated using a Green's function, which takes into account elastic waves and viscosity in the layer and the viscous boundary layer at the solid-liquid interface. [Work supported by the ARL:UT McKinney Fellowship in Acoustics and NIH DK070618.]

A theory for acoustic radiation force on a sphere in soft tissue was developed for arbitrary incident compressional wave fields [Ilinskii et al., Proc. Meet. Acoust. 19, 045004 (2013)]. This theory includes two contributions to the radiation force. The first depends only on the incident and scattered compressional waves, whereas the second depends on the scattered shear waves as well. Each contribution in turn has two parts, one due to direct integration of the time-averaged Piola-Kirchhoff stress tensor over the surface of the sphere, and the other due to the irrotational component of the body force on the sphere. While both parts are known analytically for the compressional waves, only the first part has been obtained analytically for the contribution involving shear waves. The irrotational portion associated with shear waves and its effect on the total radiation force is the subject of this presentation. The analysis is conducted via Helmholtz decomposition of the body force associated with shear waves...

Macro-scale acoustophoretic separation has applications in oil/water separation and biopharmaceutical cell clarification. Interesting phenomena include deformation of oil droplets and clustering of cells into cylindrical shapes. Particle and cluster size are not necessarily small relative to the wavelength. A general computational framework is needed to calculate the acoustic radiation force on particles, clusters, and cells of various shapes. Ilinskii et al. presented a method to evaluate an acoustic radiation force on spherical objects, based on expansion of incident and scattered field in spherical waves. The acoustic force is given by the incident wave amplitudes and dimensionless scattering coefficients. This method is generalized for particle shapes which are not spherical. The incident acoustic and scattered field are expanded with respect to spherical waves. An evaluation of the scattering amplitudes for non-spherical objects can be done numerically. Analytically, scattering amplitudes are calcula...

Green’s functions are derived for elastic waves generated by a volume source in a homogeneous isotropic half-space. The context is sources at shallow burial depths, for which surface (Rayleigh) and bulk waves, both longitudinal and transverse, can be generated with comparable magnitudes. Two approaches are followed. First, the Green’s function is expanded with respect to eigenmodes that correspond to Rayleigh waves. While bulk waves are thus ignored, this approximation is valid on the surface far from the source, where the Rayleigh wave modes dominate. The second approach employs an angular spectrum that accounts for the bulk waves and yields a solution that may be separated into two terms. One is associated with bulk waves, the other with Rayleigh waves. The latter is proved to be identical to the Green’s function obtained following the first approach. The Green’s function obtained via angular spectrum decomposition is analyzed numerically in the time domain for different burial depths and distances to the receiver, and for parameters relevant to seismo-acoustic detection of land mines and other buried objects.

A theory is presented for the acoustic radiation force on a sphere embedded in a soft elastic medium that possesses a shear modulus μ several orders of magnitude smaller than its bulk modulus. Scattering of both compressional and shear waves is taken into account. There is no restriction on the size of the sphere or, apart from axisymmetry, the form of the incident compressional wave. The analysis employs the Piola-Kirchhoff pseudostress tensor and Lagrangian coordinates. In the linear approximation an analytical solution is obtained for the scattered waves. The nonlinear stress and full radiation force are calculated at the next order of approximation. For a small sphere and μ≈0 the classical result for a particle in liquid is recovered. For small but finite shear modulus the radiation force is evaluated for a gas bubble driven at a frequency below resonance. The predicted magnitude of the radiation force on the bubble is found to be less than that in liquid by the factor [1+(4/3)μ/γ P 0]-1, where P 0 is...

A model is presented for a pulsating and translating gas bubble in a channel formed by two soft elastic parallel layers. The bubble is free to undergo small shape deformations. Coupled nonlinear second-order differential equations are obtained for the shape and position of the bubble, and numerical integration of an expression for the liquid velocity at the layer interfaces yields an estimate of their displacement. Simulations reveal behavior consistent with laboratory observations.

Interaction between acoustically driven or laser-generated bubbles causes the bubble surfaces to deform. Dynamical equations describing the motion of two translating, nominally spherical bubbles undergoing small shape oscillations in a viscous liquid are derived using Lagrangian mechanics. Deformation of the bubble surfaces is taken into account by including quadrupole and octupole perturbations in the spherical-harmonic expansion of the boundary conditions on the bubbles. Quadratic terms in the quadrupole and octupole amplitudes are retained, and surface tension and shear viscosity are included in a consistent manner. A set of eight coupled second-order ordinary differential equations is obtained. Simulation results, obtained by numerical integration of the model equations, exhibit qualitative agreement with experimental observations by predicting the formation of liquid jets. Simulations also suggest that bubble-bubble interactions act to enhance surface mode instability.

The theory for acoustic radiation force on a viscoelastic sphere of arbitrary size in tissue was extended recently to account for nonaxisymmetric incident fields [Ilinskii et al., POMA 19, 045004 (2013)]. A spherical harmonic expansion was used to describe the incident field. This work was specialized at the spring 2014 ASA meeting to focused axisymmetric sound beams with various focal spot sizes and a scatterer located at the focus. The emphasis of the present contribution is nonaxisymmetric fields, either through moving the scatterer off the axis of an axisymmetric beam or through explicitly defining a nonaxisymmetric beam. This is accomplished via angular spectrum decomposition of the incident field, spherical wave expansions of the resulting plane waves about the center of the scatterer, Wigner D-matrix transformations to express these spherical waves in a coordinate system with the polar axis aligned with the desired radiation force component, and finally integration over solid angle to obtain spherical wave amplitudes as required in the theory. Various scatterer sizes and positions relative to the focus are considered, and the effects of changing properties of both the scatterer and the surrounding tissue are examined. [Work supported by the ARL:UT McKinney Fellowship in Acoustics.]

An ultrasound-based method to locally assess the shear modulus of a medium is reported. The proposed approach is based on the application of an impulse acoustic radiation force to an inhomogeneity in the medium and subsequent monitoring of the spatio-temporal response. In our experimental studies, a short pulse produced by a 1.5-MHz highly focused ultrasound transducer was used to initiate the motion of a rigid sphere embedded into an elastic medium. Another 25 MHz focused ultrasound transducer operating in pulse-echo mode was used to track the displacement of the sphere. The experiments were performed in gel phantoms with varying shear modulus to demonstrate the relationship between the displacement of the sphere and shear modulus of the surrounding medium. Because the magnitude of acoustic force applied to sphere depends on the acoustic material properties and, therefore, cannot be used to assess the absolute value of shear modulus, the temporal behavior of the displacement of the sphere was analyzed. The results of this study indicate that there is a strong correlation between the shear modulus of a medium and spatio-temporal characteristics of the motion of the rigid sphere embedded in this medium.

A nonlinear model equation in Rayleigh–Plesset form is developed for volume oscillations of a compliant object buried close to the surface in soil. The equation takes into account the stress-free boundary condition on the surface of the ground. The model is fully nonlinear given exact relations for the elastic potential energy stored in deformation of the object and the soil. Expansions of the potential energies for weak nonlinearity are provided in terms of elastic constants that can be determined experimentally. When the shear modulus is allowed to increase with depth below the surface, the natural frequency predicted by the model first decreases and thereafter increases with burial depth, in agreement with reported observations. Perturbation solutions are derived for the displacements on the surface of the ground at the second harmonic and difference frequency due to the nonlinear response of the object to acoustic excitation.

The frequency response and nonlinear resonance frequency shift of an acoustical resonator with losses and having a varying cross section were investigated previously using Lagrangian mechanics and perturbation for resonator shapes that are close to cylindrical [M. F. Hamilton, et al., J. Acoust. Soc. Am. 110, 109-119 (2001)]. The same approach is extended here to include resonators having any shape for which the Webster horn equation is a valid model in the linear approximation. Admissible shapes include cones and bulbs proposed for acoustical compressors. The approach is appropriate for approximate but rapid parameter estimations for resonators with complicated shapes, requiring far less computation time than for direct numerical solution of the one-dimensional model equation frequently used for such resonators [Ilinskii et al., J. Acoust. Soc. Am. 104, 2664-2674 (1998)]. Results for cone and bulb shaped resonators with losses are compared with results from the direct numerical solution. The direction of the resonance frequency shift is determined by the efficiency of second-harmonic generation in modes having natural frequencies below versus above the frequency of the second harmonic, and how the net effect of this coupling compares with the frequency shifts due to cubic nonlinearity and static deformation.

A model of the interaction of a spherical gas bubble and a rigid spherical particle is derived as a coupled system of second-order differential equations using Lagrangian mechanics. The model accounts for pulsation and translation of the bubble as well as translation of the particle in an infinite, incompressible liquid. The model derived here is accurate to order R(5)d(5), where R is a characteristic radius and d is the separation distance between the bubble and particle. This order is the minimum accuracy required to account for the interaction of the bubble and particle. Dependence on the size and density of the particle is demonstrated through numerical integration of the dynamical equations for both the free and forced response of the system. Numerical results are presented for models accurate to orders higher than R(5)d(5) to demonstrate the consequences of truncating the equations at order R(5)d(5).

For high gas supersaturation levels in liquids, on the order of 300% as predicted in capillaries of marine mammals following a series of dives, standard mathematical models of both static and rectified diffusion are found to underestimate the rate of bubble growth by 10%-20%. The discrepancy is demonstrated by comparing predictions based on existing mathematical models with direct numerical solutions of the differential equations for gas diffusion in the liquid and thermal conditions in the bubble. Underestimation of bubble growth by existing mathematical models is due to the underlying assumption that the gas concentration in the liquid is given by its value for a bubble of constant equilibrium radius. This assumption is violated when high supersaturation causes the bubble to grow too fast in relation to the time scale associated with diffusion. Rapid bubble growth results in an increased gas concentration gradient at the bubble wall, and therefore a growth rate in excess of predictions based on constant equilibrium bubble radius. The effect of gas supersaturation level, excitation frequency, duty cycle and sound pressure level on bubble growth were also studied.

The motion of a rigid sphere in a viscoelastic medium in response to an acoustic radiation force of short duration was investigated. Theoretical and numerical studies were carried out first. To verify the developed model, experiments were performed using rigid spheres of various diameters and densities embedded into tissue-like, gel-based phantoms of varying mechanical properties. A 1.5 MHz, single-element, focused transducer was used to apply the desired radiation force. Another single-element, focused transducer operating at 25 MHz was used to track the displacements of the sphere. The results of this study demonstrate good agreement between theoretical predictions and experimental measurements. The developed theoretical model accurately describes the displacement of the solid spheres in a viscoelastic medium in response to the acoustic radiation force.