A previous presentation compared acoustical and optical resonances of a Mie regime double sphere system that focused on a side scattering phenomenon that roughly mimicked a mirror [C. E. Dean and R. M. Hodges, JASA, 143, 1844 (2018)]. If one thinks of these scatterers as lenses the presence of a photonic or phononic “jet” suggests a caustic region with a concentration of energy near the tip of the jet, a point analogous to the focus of a lens. Since both light and sound are reversible, there are two foci on either side of such a scatterer, arrranged symmetrically about each scatterer on the axis of the line between the centers of the two sphere system. The current research examines the case when two variable sized Mie regime scatterers are arranged so as to have the backward focus of a second scatterer on or near the forward focus of the first scatterer. This is effectively a Mie regime double sphere “telescope.” Changes to far field scattering in and around the forward scattering direction are examined. This talk attempts to answer these and other questions through the use of theoretical computational acoustics models.

Two sphere systems show strong morphology dependent resonances even for two spheres of identical composition [G. W. Kattawar and C. E. Dean, Opt. Letts. 8, 48–50 (1983).]. How do analogous fluid spheres respond in the analogous situations? Do they show similar or different resonances? Preliminary investigations suggested that the response of fluid bispheres do not show as rich a panoply of resonances [C. E. Dean and J. P. Braselton, JASA, 141, 3735 (2017).]. How much do these systems differ? Would alternate boundary conditions restore some lost resonances? This talk attempts to answer these and other questions through the use of theoretical computational acoustics models.

We show that it is possible to design an invisible wavelength-sized metal-dielectric metamaterial object without evoking cloaking. Our approach is an extension of the neutral inclusion concept by Zhou and Hu [Phys.Rev.E 74, 026607 (2006)] to Mie scatterers. We demonstrate that an increase of metal fraction in the metamaterial leads to a transition from dielectric-like to metal-like scattering, which proceeds through invisibility or optical neutrality of the scatterer. Formally this is due to cancellation of multiple scattering orders, similarly to plasmonic cloaking introduced by Alu and Engheta [Phys.Rev.E 72, 016623 (2005)], but without introduction of the separation of the scatterer into cloak and hidden regions.

A comparison of results for an analogous scattering problem is made for the interactive scattering of a double sphere in both acoustics and electromagnetic scattering. For initial testing purposes, the problem is simplified to that of two identical spheres with the same size and physical properties. The acoustical scatterer is modeled to match optical characteristics of the already studied optical scattering problem [G. W. Kattawar and C. E. Dean, Opt. Lett. 8, 48-50 (1983).]. Particular study is made of situations that lead to large side scattering responses due to resonance phenomena. The model for the acoustical scattering problem was discussed in a previous Acoustical Society of America meeting [C. E. Dean and J. P. Braselton, J. Acoust. Soc. Am. 139, 1121 (2016)].

Progress in the implementation and testing of routines to compute the energy flux [J. A. Mann III, et al., J. Acoust. Soc. Am. 82, 17–30 (1987)] and energy flux streamlines [D. M. F. Chapman, J. Acoust. Soc. 124, 48–56 (2008)] for the problem of an ensonified fluid sphere near a rigid boundary are presented. The problem is isomorphic via the method of images to the problem of interactive scattering between two identical fluid spheres. Trinks provided an analytical solution to this problem in 1935 [W. Trinks, Ann. Phys. 414, 561–590 (1935)]. However, his solution is mathematically intractable for larger spheres due to the complexity of his translational addition theorem for spherical harmonics. Later, researchers Cruzan [Quart. Appl. Math. 20, 33–40 (1962)] and Liang and Lo [Radio Science 2, 1481 (1967)] simplified this translational addition theorem, making it feasible to solve the problem with larger spheres. This work follows Bruning and Lo [Tech. Rep. 69-5 (Antenna Laboratory, Univ. of Illinois, Urbana, IL 1969)], correcting typos in that work to implement efficient algorithms for acoustical scattering.

As an extension of recently published experimental work [Cleon E. Dean and Kendez Parker, “A ray model of sound focusing with a balloon lens: An experiment for high school students,” J. Acoust. Soc. Am. 131, 2459–2462 (2012)], preliminary results comparing energy flux streamlines [David M. F. Chapman, “Using streamlines to visualize acoustic energy flow across boundaries,” J. Acoust. Soc. Am. 124, 48–56 (2008)] versus acoustic rays for visualizing the energy flow inside and in the focal region of an acoustic lens in the form of a carbon dioxide filled balloon in air were presented. The sound field was expanded in the usual Legendre polynomials and spherical Bessel functions, and the energy flux vectors at points throughout the regions of interest were calculated [J. Adin Mann III, et al., “Instantaneous and time-averaged energy transfer in acoustic fields,” J. Acoust. Soc. Am. 82, 17–30 (1987)]. Deficiencies in the streamline plotting routines used in this earlier version of Mathematica and subtle differences between acoustic rays and acoustic energy flux streamlines lent itself to an inaccurate perception of the results. This talk uses Mathematica 10 routines to revisit these results and concentrates on testing and verification of the conclusions from the previous work.

As an extension of recently published experimental work [Dean and Parker, “A ray model of sound focusing with a balloon lens: An experiment for high school students,” J. Acoust. Soc. Am. 131, 2459–2462 (2012)], a comparison of ray acoustics and wave analysis via energy flux streamlines [Chapman, “Using streamlines to visualize acoustic energy flow across boundaries,” J. Acoust. Soc. Am. 124, 48–56 (2008)] as a means of visualizing the sound field for a positive acoustic lens in the form of a carbon dioxide filled spherical balloon in air is made. The sound field is expanded in the usual Legendre polynomials and spherical Bessel functions [Anderson, “Sound scattering from a fluid sphere,” J. Acoust. Soc. Am. 22, 426–431 (1950)], and the energy flux vectors at points throughout the regions of interest are calculated [Adin Mann III, et al., “Instantaneous and time-averaged energy transfer in acoustic fields,” J. Acoust. Soc. Am. 82, 17–30 (1987)]. Then, energy flux streamlines are plotted using Mathematica routines for comparison with conventional acoustical rays, both inside and outside the balloon. Particular attention is paid to the focal region.

Sound emitted by a circular loudspeaker can be treated as equivalent to a plane wave diffracted by a circular aperture in a rigid, sound absorbing screen. Axial symmetry leads one to expect constructive interference along the symmetry axis in the near field (the Poisson-Arago spot). An energy flux streamline model was developed to help visualize this and other features of the near sound field. The model is used to draw out similarities and differences between energy flux streamlines and acoustic rays, particularly in the transition to the far field.

An energy flux streamline model was developed in support of a simple Lloyd's mirror experiment originally intended for use by high school students wherein 10 000 Hz harmonic sound, emitted from a roughly 10 cm diameter baffled loudspeaker, was reflected off a floor, treated as a rigid boundary. The model is used to draw out similarities and differences between energy flux streamlines and acoustic rays. Particular attention is paid to conditions and angles of reflection that hold for acoustic rays reflected from a rigid boundary versus the conditions that hold for the equivalent reflection and reflection angles of energy flux streamlines.

A weather balloon filled with carbon dioxide gas is used as a positive spherical acoustic lens. High frequency but audible sound from a circular loudspeaker ensonifies the balloon and produces increased sound pressure levels in a region along the principal axis according to a ray acoustics model. This enhancement was measured experimentally and was found to agree with theory. The possibility that interference from reflected sound off walls or the floor could mask or mimic the expected focusing was countered by calculating and measuring within a "shadow zone" in which only direct rays or rays refracted by the balloon exist by the method of Fresnel volumes. The experiment described in this paper would be a suitable learning experience for junior high and high school students showing how rays and Snell's law apply to sound as well as light and giving them a measurable predicted focal region for enhanced sound pressure levels.

Energy flux streamlines yield certain advantages for the visualization of acoustical scattering processes. They give information about scattering angles, interaction with surfaces and scatterers, and can even show the relative intensity of the sound at a given location. However, the use of energy flux streamlines presents certain difficulties as well. Beyond the difficulty in calculating them, the use of the energy flux field presumes a complete solution of the sound scattering problem. Thus energy flux streamlines are usually descriptive rather than predictive. And while the set of streamlines can be chosen so as to show the relative intensities of the sound field, what works in one geometry will not necessarily work in another. Examples will be adduced illustrating these points and more.

A baffled circular loudspeaker is driven at 10 000 Hz to provide a sound source for a high school level Lloyd’s mirror experiment. The experiment was to be performed in an ordinary building hallway. The experiment was also designed to produce interference using reflected sound from a hard floor while avoiding reflected sound from other surfaces. Modeling the sound as a Fresnel volume proved useful, but certain assumptions about the effectiveness of sound absorbing tile and flat foam rubber mats were less so. Substitution of more effective sound absorbing materials gave improved results when reflected sound was to be eliminated or the interference effect annulled.

Energy flux streamlines are compared with alternative presentations of the energy flux vector field to visualize energy coupling inside and outside the surface of an insonified spherical acoustic lens. The emphasis is on the energy flux streamline as a natural bridge between necessarily approximate ray solutions and full-fledged wave solutions. The present work is the wave solution for the experimental work performed by Kendez C. Parker and Cleon E. Dean also being presented at this meeting that primarily used ray methods in its theoretical analysis; thus a comparison between energy flux representations and ray analysis results is made. Difficulties in the interpretation of the energy flux streamline representation due to the spherical geometry of the scatterer are noted.

A simple classroom demonstration consists of a weather balloon filled with carbon dioxide, a sound source, and a microphone. Since the speed of sound is slower in carbon dioxide than in air at room temperature and pressure, the balloon acts as a positive spherical acoustic lens. Preliminary experimental results have been presented previously [C. E. Dean and J. P. Braselton, “The energy flow for a spherical acoustic lens: ray and wave methods vs. experiment.,” J. Acoust. Soc. Am. 125, 2627 (2009)]. The possibility of interference effects from the reflection of sound off surfaces was brought up in the ensuing discussion. The current results have been measured in a way that minimizes the effect of interference due to reflections off walls, floor, or other surfaces.

Energy flux streamlines are compared with alternative presentations of the energy flux vector field to visualize energy coupling at the surface and inside an ensonified fluid loaded elastic cylindrical shell with vacuum interior for both forward and retrograde guided propagating waves. The present work uses a method adapted from a simpler technique due to Kaduchak and Marston [Gregory Kaduchak and Philip L. Marston, “Traveling-wave decomposition of surface displacements associated with scattering by a cylindrical shell: Numerical evaluation displaying guided forward and backward wave properties,” J. Acoust. Soc. Am. 98, 3501–3507 (1995)] to isolate unidirectional energy flows.

A simple classroom demonstration consists of a weather balloon filled with carbon dioxide, a sound source, and a microphone. Since the speed of sound is slower in carbon dioxide than in air at room temperature and pressure, the balloon acts as a positive spherical acoustic lens. The accuracy of ray methods in locating the acoustic focus versus a full blown wave solution approach is probed. This problem presents particular difficulties if the sound source lies in the near field region. The sound emitter is treated as a dipole source equivalent to a rigid oscillating sphere of small size and amplitude of motion relative to the scatterer. The energy flux around the balloon is visualized by both ray methods and by the acoustic Poynting vector field. The geometrical ray results and the acoustic Poynting vector field resulting from the wave solution are compared.

A variety of ways of visualizing the energy flux or elastodynamic Poynting vector field are demonstrated including gridded vector field, color coding, and other methods for the simple example of a Rayleigh wave. In particular an improved version of a technique shown at the 8th International Conference on Theoretical and Computational Acoustics, ICTCA 2007 is demonstrated [David M. F. Chapman, "Visualizing acoustic energy flow into the seabed using energy streamlines," Eighth International Conference on Theoretical and Computational Acoustics, ICTCA 2007, Heraklion, Crete, GREECE, 2-5 July 2007.]. The improved streamline method shows both the direction and the intensity of the energy flow.

The Poynting vector field is used to show preliminary results for the energy flow both at the surface and inside an ensonified fluid loaded elastic cylindrical shell for both forward and retrograde propagating waves. The present work uses a method adapted from a simpler technique due to Kaduchak and Marston [G. Kaduchak and P. L. Marston, ‘‘Traveling-wave decomposition of surface displacements associated with scattering by a cylindrical shell: Numerical evaluation displaying guided forward and backward wave properties,’’ J. Acoust. Soc. Am. 98, 3501–3507 (1995)] to isolate unidirectional energy flows.