Trapping in acoustic standing waves: Effect of liquid drop compressibility.

The Reflections series takes a look back on historical articles from The Journal of the Acoustical Society of America that have had a significant impact on the science and practice of acoustics.

Radiation forces on highly reflecting circular cylinders in two slanted plane waves: Specular-reflection contributions.

Situations arise where it is desirable to understand and estimate the radiation force on large smooth highly reflecting objects in water illuminated by beams of ultrasound. The approach examined here is to extend a formulation experimentally confirmed by Herrey [J. Acoust. Soc. Am. 27, 891-896 (1955)] for tilted reflecting surfaces in fluids that are modeled as being inviscid. The formulation applies Brillouin's analysis of the Langevin-like radiation force on objects in open containers. The specular reflection contributions to the radiation force of two slanted plane waves incident on a rigid cylinder is approximated and compared with a full partial wave series (PWS) solution for an infinitely long cylinder in an inviscid fluid. The availability of the PWS solution gives support to approximations introduced in the geometric analysis, provided ka (the wave number-cylinder-radius product) is sufficiently large. The normalized force projection is plotted as a function of the wave slant angle relative to the symmetry axis. Deviations between the specular and PWS analysis for ka of 7.5 are diminished for ka of 15 and 25. A region of enhanced force associated with constructive interference narrows with increasing ka.

Specular reflection contributions to dynamic radiation forces on highly reflecting spheres (L).

Specular reflection contributions to dynamic radiation forces were recently mentioned for highly reflecting spheres to facilitate comparison with forces on cylinders [Marston, Daniel, Fortuner, Kirsteins, and Abawi, J. Acoust. Soc. Am. 149, 3042-3051 (2021)]. Both shapes of reflectors were taken to be illuminated by short-wavelength plane wave double-sideband suppressed-carrier ultrasound. Here, the geometric method of evaluating dynamic forces on spheres is illustrated along with an analysis of the phase of the modulated radiation force. Comparison with partial wave series solutions supports the relevance of the specular reflection analysis for insight into forces on highly reflecting objects in water.

Specular-reflection contributions to static and dynamic radiation forces on circular cylinders.

Interest in the response of highly reflecting objects in water to modulated acoustical radiation forces makes it appropriate to consider contributions to such forces from perfectly reflecting objects to provide insight into radiation forces. The acoustic illumination can have wavelengths much smaller than the object's size, and objects of interest may have complicated shapes. Here, the specular contribution to the oscillating radiation force on an infinite circular cylinder at normal incidence is considered for double-sideband-suppressed carrier-modulated acoustic illumination. The oscillatory magnitude of the specular force decreases monotonically with increasing modulation frequency, and the phase of the oscillating force depends on the relative phase of the sidebands. The phase dependence on the modulation frequency can be reduced with the appropriate selection of a sideband relative-phase parameter. That is a consequence of the significance of rays that are incident on the cylinder having small impact parameters that are nearly backscattered. For one choice of a relative sideband phase, a prior partial wave series (PWS) solution is available, which supports the specular analysis when the PWS is evaluated for a rigid cylinder. The importance of specular contributions for aluminum cylinders in water is noted. A specular analysis for an analogous spherical reflector is also summarized.

Phase and amplitude evolution of backscattering by a sphere scanned through an acoustic vortex beam: Measured helicity projections.

While acoustic vortex beams have many potential applications, the full implication of the phase information available in scattering experiments has not been developed. The present paper concerns observables in measured near-backward scattering from a sphere in water raster scanned through a first-order acoustic vortex beam. Symmetrically placed transducer elements were operated in a transmit-receive mode. Helicity-dependent projections of the spatial evolution of the scattering were used to display magnitude and phase information. The resulting phase swirl patterns were projection dependent and especially sensitive to the transverse position of the sphere. The magnitude also depended on the sphere's position relative to the beam's axial null.

Maxwell stress excitation of wire vibrations at difference and sum frequencies of electric currents in separate circuits.

Oscillating electric currents through a wire under tension can excite transverse vibrational modes of the wire when a perpendicular static magnetic field is present and the frequency of the current is close to the natural frequency of the mode of interest. The excitation of the mode is associated with temporally oscillating Maxwell stresses on the wire, often also known as oscillating Lorentz forces. That excitation process is sometimes demonstrated in educational contexts. The investigation here concerns situations where a temporally oscillating magnetic field generated by oscillating electric currents in a cylindrical coil replaces the imposed perpendicular static magnetic field. The frequencies of the currents in the wire and in the coil are related to the frequency of the oscillating stress. In this experiment, this effect is documented for sum-frequency excitation (with input frequencies in the range of half that of the excited lowest vibrational mode of the wire) and the difference-frequency excitation (with input frequencies an order-of-magnitude larger than the mode frequency). This coupling may be useful when it is desirable to use only high-frequency currents. The experiment uses tone-burst stress excitation and a differential photodiode for detecting transverse low-amplitude wire oscillations. Signal envelopes decayed exponentially after the tone-burst.

Scattering and radiation force dependence on properties of empty elastic spherical shells: Low-frequency phase-shift derivation.

It is helpful to evaluate scattering and acoustic radiation forces on spheres for idealized cases in which the effects of energy dissipation are ignorable. Let x denote the product of the acoustic wave number and the sphere's radius. Previously expansions were obtained for fluid and solid spheres involving powers of x and algebraic expressions containing material properties. The present analysis concerns the case of empty elastic shells and reveals how expansion coefficients also depend on shell thickness. Incident waves considered are plane traveling and standing waves, though relevance to Bessel wave-fields is also noted. The expansions give leading-order corrections to the usual Rayleigh scattering approximation.

Phase-shift derivation of expansions for material and frequency dependence of progressive-wave radiation forces and backscattering by spheres.

When considering the scattering of sound and radiation forces for spheres, it has historically been helpful to understand situations lacking dissipation. In that case the scattering is characterized by real partial-wave phase shifts. At low frequencies expansions show the dependence of each phase shift on material properties and on frequency. Those expansions are used here to describe the frequency and material dependence of scattering and radiation forces beyond the usual Rayleigh-scattering approximation. Results for radiation forces on spheres in standing waves are extended to plane progressive waves. The expansion coefficients use algebraic functions. Results for movable and fixed rigid spheres are shown.

Bessel beam expansion of linear focused ultrasound.

Previous work on scattering by Bessel beams shows that expansion of incident sound fields in term of these beams has application to scattering [P. L. Marston, J. Acoust. Soc. Am. 122, 247-252 (2007)]. In this work, an expression for the expansion coefficients of propagating, axisymmetric, sound fields are derived. In this paper, this expression is applied to a linear focused axisymmetric sound field and is expanded in terms of Bessel beam components. This is done for focused beams radiated from a spherical cap source. A physical optics model is applied to sound propagation close to the source to facilitate the calculation of the Bessel beam expansion coefficients. This type of model is useful for focused scattering [P. L. Marston and D. S. Langley, J. Acoust. Soc. Am. 73, 1464-1475 (1983)]. Once the expansion coefficients are found, the sound field can be evaluated by superposition. The model agrees approximately with Chen, Schwarz, and Parker [J. Acoust. Soc. Am. 94, 2979-2991 (1993)] and O'Neil [J. Acoust. Soc. Am. 21, 516-526 (1949)] on axis and with direct integration of a Kirchhoff integral both on and off axis. This type of expansion will have applications to scattering problems.

Observation and modeling of acoustic scattering from a rubber spherical shell.

Acoustic backscattering from a rubber spherical shell in water is observed to contain a delayed enhancement, demonstrated to be associated with a waveguide path along the shell. This path is somewhat analogous to that of the Lamb wave observed on metallic shells. Rubber is a unique material because of its subsonic sound speed relative to water, and because shear coupling is often small enough to be neglected in typical models, making it fluid-like. This makes rubber a material of interest for coating and cloaking underwater devices and vehicles. Both fluid and elastic rubber partial wave series models are tested, using experimentally measured longitudinal and shear speeds, attenuation, and rubber density. A finite element model for the shell is also developed. Comparison of the models and experiments highlights the importance of the waveguide path to the overall scattering. Estimates for the group and phase velocities of the lowest order propagating mode in the shell are determined through waveguide normal mode analysis and Sommerfeld-Watson theory, and are shown to give good agreement with experiments in predicting the time of arrival of the waveguide path.

Finite-size radiation force correction for inviscid spheres in standing waves.

Yosioka and Kawasima gave a widely used approximation for the acoustic radiation force on small liquid spheres surrounded by an immiscible liquid in 1955. Considering the liquids to be inviscid with negligible thermal dissipation, in their approximation the force on the sphere is proportional to the sphere's volume and the levitation position in a vertical standing wave becomes independent of the size. The analysis given here introduces a small correction term proportional to the square of the sphere's radius relative to the aforementioned small-sphere force. The significance of this term also depends on the relative density and sound velocity of the sphere. The improved approximation is supported by comparison with the exact partial-wave-series based radiation force for ideal fluid spheres in ideal fluids.

Phase-shift expansions for approximate radiation forces on solid spheres in inviscid-acoustic standing waves.

Previously acoustic radiation forces on spheres have been expressed using scattering phase shifts associated with the corresponding traveling wave scattering situation. That approach is applied here to spheres in inviscid standing waves that are solid, fixed-rigid, or movable-rigid of finite density. Low frequency truncated expansions of the phase shifts result in expressions for radiation forces that have simple forms. The expansion expresses the leading finite-size correction to the common low-frequency approximation associated with Rayleigh scattering in which the radiation force is proportional to the solid sphere's volume.

Relationship of scattering phase shifts to special radiation force conditions for spheres in axisymmetric wave-fields.

When investigating the radiation forces on spheres in complicated wave-fields, the interpretation of analytical results can be simplified by retaining the s-function notation and associated phase shifts imported into acoustics from quantum scattering theory. For situations in which dissipation is negligible, as taken to be the case in the present investigation, there is an additional simplification in that partial-wave phase shifts become real numbers that vanish when the partial-wave index becomes large and when the wave-number-sphere-radius product vanishes. By restricting attention to monopole and dipole phase shifts, transitions in the axial radiation force for axisymmetric wave-fields are found to be related to wave-field parameters for traveling and standing Bessel wave-fields by considering the ratio of the phase shifts. For traveling waves, the special force conditions concern negative forces while for standing waves, the special force conditions concern vanishing radiation forces. An intermediate step involves considering the functional dependence on phase shifts. An appendix gives an approximation for zero-force plane standing wave conditions. Connections with early investigations of acoustic levitation are mentioned and some complications associated with viscosity are briefly noted.

Spectral analysis of bistatic scattering from underwater elastic cylinders and spheres.

Far field sound scattering from underwater elastic spheres and finite cylinders is considered over the full range of scattering angles. Three models for the frequency response of the scattered field are evaluated: a hybrid finite element/propagation simulation for a finite cylinder with broadside illumination, an approximate solution for the finite cylinder, and the exact solution for a sphere. The cylinder models are shown to give comparable results, attesting to the strength of the finite cylinder approximate solution. Interference and resonance structure present in the frequency response of the targets is identified and discussed, and the bistatic spectra for a variety of elastic sphere materials are presented. A thorough understanding of the complicated angle and frequency dependence of the scattering from simple elastic targets is helpful for interpretation of backscattering data from targets at or near an interface, or for scattering data taken by moving automated underwater vehicles, acoustic arrays, or other forms of data collection involving bistatic scattering.

Multipole expansion of acoustical Bessel beams with arbitrary order and location.

An exact solution of expansion coefficients for a T-matrix method interacting with acoustic scattering of arbitrary order Bessel beams from an obstacle of arbitrary location is derived analytically. Because of the failure of the addition theorem for spherical harmonics for expansion coefficients of helicoidal Bessel beams, an addition theorem for cylindrical Bessel functions is introduced. Meanwhile, an analytical expression for the integral of products including Bessel and associated Legendre functions is applied to eliminate the integration over the polar angle. Note that this multipole expansion may also benefit other scattering methods and expansions of incident waves, for instance, partial-wave series solutions.

Unphysical consequences of negative absorbed power in linear passive scattering: Implications for radiation force and torque.

Contrary to some claims, the absorbed power associated with linear scattering of sound by passive objects in ideal fluids must be non-negative. Such unphysical claims suggest analytical or computational error, or use of an unphysical constitutive relation for material properties. The close connection with the evaluation of acoustic radiation force on targets according to Westervelt's formulation [J. Acoust. Soc. Am. 29, 26-29 (1957)], recently generalized to certain acoustic beams, is briefly reviewed along with the theory of acoustic radiation torque on axisymmetric targets with power absorption. Applications to viscous dissipation and to issues pertaining to active targets are also examined.

Kirchhoff approximation for backscattering from a partially exposed rigid sphere at a flat interface.

The Kirchhoff approximation (KA) is used to model backscatter of sound from a partially exposed, rigid sphere at a flat free interface of two homogenous media. Scattered wavefields are calculated through numerical integration on the sphere of the Kirchhoff integral, requiring detailed knowledge of the illuminated region for each scattering path. This approach avoids amplitude discontinuities resulting from geometric transitions in the number of reflected rays. Reflections from the interface are modeled through use of an image source, positioned symmetrically relative to the real source. Results are compared to experimentally obtained backscattering records from elastic spheres at an air-water interface, as well as to an exact partial wave series for a half exposed sphere. These comparisons highlight the omission of Franz-type reflections from consideration within the KA, and the consequences of this omission are discussed. The results can be extended to boundary conditions beyond the ideal free surface limit, and are applicable to the problem of scattering by underwater objects partially buried in sand.

Acoustic radiation force expressed using complex phase shifts and momentum-transfer cross sections.

Acoustic radiation force is expressed using complex phase shifts of partial wave scattering functions and the momentum-transfer cross section, herein incorporated into acoustics from quantum mechanisms. Imaginary parts of the phase shifts represent dissipation in the object and/or in the boundary layer adjacent to the object. The formula simplifies the force as summation of functions of complex phase shifts of adjacent partial waves involving differences of real parts and sums of imaginary parts, providing an efficient way of exploring the force parameter-space. The formula for the force is proportional to a generalized momentum-transfer cross section for plane waves and no dissipation.

High frequency imaging and elastic effects for a solid cylinder with axis oblique relative to a nearby horizontal surface.

The calibrated acoustic backscattering spectrum versus aspect angle, also called the "acoustic color" or "acoustic template," of solid cylinders located near a flat interface was previously studied for the case where the cylinder axis was vertically oblique relative to the interface and was insonified by a beam at a non-zero grazing angle. The presence of the interface allows for multiple paths by which sound is backscattered. These multipaths are highly dependent on the relative orientations of the target, the interface, and the source/receiver. In this work, the effects of vertical obliquity on the reconstructed synthetic aperture acoustic images is presented. Several robust orientation dependent features are considered and the physical mechanisms responsible identified through geometric arguments. Information about a target's three-dimensional orientation and size may be gleaned from these images, aiding the interpretation of features within the target's acoustic template. Specific features observed in the reconstructed image are associated with features in the acoustic color. The coupling conditions for surface elastic waves are also considered.