In resonant ultrasound spectroscopy (RUS) one measures a set of resonance frequencies for a sample of material and then adjusts a set of parameters in a model to least squares fit those frequencies. Some problems may occur in the least squares fitting procedure. For example, there may be a parameter which has only a weak influence on the fit, and the fitting procedure may give this parameter an unphysical value; this may cause other parameters to take on reasonable but inaccurate values. Another problem is that the space spanned by the adjustable parameters may contain local minima in addition to the desired global minimum, so that a sufficiently fine mesh global search over the parameter space may be necessary. If there were six adjustable parameters (or more, e.g., for textured or piezoelectric samples), each with 10 possible initial values, then one would have to run a RUS computer program a million times (12 days per second of RUS program runtime), which is problematic. In this talk, solutions to these problems will be presented.

Resonant ultrasound spectroscopy (RUS) is a method in which the least-squares fitting of a sample's measured resonance frequencies is used to determine the sample material's elastic constants. Difficulties arise when RUS is applied to textured materials, which are composed of crystallites that are neither completely aligned nor randomly oriented; such materials would have effective elastic tensors with 21 independent elastic constants, and this would overburden the conventional RUS analysis process. In any case, the elastic nature of some textured materials may be represented as the weighted average of the elastic tensors of the constituent crystalline material rotated in the directions of the individual crystallites. The collection of weights is the orientation distribution function (ODF). Now the least-squares fitting of a sample's measured resonance frequencies may be used to determine not the sample's elastic constants but instead its ODF. This is not trivial since the weights must be positive and sum to one, but it may be done with a simple modification of the conventional RUS software.

Resonant ultrasound spectroscopy (RUS) is a method in which knowledge of the shape, mass, and some natural frequencies of a sample of solid material may be used to determine the elastic properties of the material. Customarily, the natural frequencies are measured by driving the sample with a sine wave in one transducer and monitoring the sample's response with a second transducer; when the frequency of the drive is swept through a natural frequency, a tuning curve is measured, from which the natural frequency and a quality factor may be determined. Usually, a Rayleigh-Ritz method involving rectangular parallelepiped or cylindrical sample shapes and polynomial basis functions is used to analyze the data to determine the elastic properties. This talk will discuss the customary method and some extensions, including: 1) overcoming a limitation that for some problematic materials (e.g., elastomers), the method can measure only one elastic constant accurately; 2) modifying the method to include piezoelectric materials; 3) overcoming the inaccuracy of the method for samples having sharp interfaces between different materials.

A powerful method for measuring the elastic properties of solid materials is resonance spectroscopy. An advantage of this method is that the experimental measurement is relatively easy, and the difficult analysis is accomplished with a computer program. A particular computer algorithm has become standard because it is fast and may be applied to a wide range of shapes of solid samples. However, the algorithm does not work well for solids with layers of different materials. Fortunately, a relatively easy modification of the customary computer algorithm enables it to work for layered solids.

When a system, composed of some substance, is displaced from equilibrium, the forces which were maintaining equilibrium will bring the system back toward equilibrium. Because of the mass of the substance, the system will overshoot equilibrium, then have to return, and subsequently the system will oscillate about equilibrium. This is the essence of acoustics, and it applies to virtually everything, from molecules to galaxies. In acoustic experiments, measurements of acoustic motion and damping, over a wide range of conditions, can be made to high precision and accuracy, and significant information may be obtained about the thermodynamics and interatomic forces involved in condensed matter physics.

The determination of the elastic constants for a solid material may be conveniently accomplished by measuring resonances corresponding to the elastic normal modes of a sample. Isotropic materials should be particularly straightforward, since they involve only two independent elastic constants, the shear modulus and the longitudinal modulus. In practice one typically measures the shear modulus and Young's modulus, but for some problematic materials these are not truly independent. Probing the microscopic processes of solid mechanical behavior requires knowledge of independent moduli. In this paper, a resonance method for directly measuring the independent moduli for problematic materials is described.

As a Scientific Officer for the Office of Naval Research, Logan Hargrove would support research which addressed Navy problems, such as making ships quieter. Logan recognized that this problem could be tackled with nearfield acoustical holography (NAH), which, unlike conventional holography, could image sources with a resolution not limited by the wavelength of the radiation. NAH was cited as one the Navy's 35 big achievements in 75 years. A related problem was sound radiation from a rib-stiffened plate; in this case it was conjectured that a condensed matter physics concept, Anderson localization (cited in a Nobel prize), could be used to reduce noise radiation from a plate. Logan supported this research, and extended it to include research with other condensed matter concepts merged with acoustics. This led to an acoustic model of a quasicrystal formed with 150 musical tuning forks coupled in a Penrose tile pattern. While the Navy was indeed interested in novel materials such as alloy quasicrystals, Logan was pleased that part of his supported research, the tuning fork quasicrystal, was featured in the New York Times (September 5, 1989, Science Section).

Some common exercises presented in introductory acoustics courses and texts illustrate solutions involving eigenvalues and eigenfunctions. Challenging extensions of these, even for one-dimensional (1D) systems, might involve a mass or spring loading the acoustic medium at an end point or at an interior point. These problems might be extended further by requiring that some given function be expanded in a series of the eigenfunctions, but such extended problems may lead to unexpected complications in regard to eigenfunction orthogonality. In this paper, Sturm-Liouville theory is used to develop a systematic method for predetermining eigenfunction orthogonality for 1D systems loaded at end points or interior points or having properties that change with jump discontinuities.

A significant success of modern theoretical physics has been in the study of critical phenomena, where a system displays singular properties while undergoing a transition between different phases of matter. The modern theory (recognized by a Wolf Prize and a Nobel Prize) involves the notions of length scale invariance, power law singularities, the renormalization group etc. A significant experimental test of the theory has been the study of the transition between normal liquid and superfluid behavior of liquid helium-4 (recognized with a London Prize). The singular properties which characterize a critical transition typically involve a divergence in a susceptibility, such as a heat capacity, compressibility, or other property which may couple to a local configuration of atoms or molecules. Since these latter properties may be probed with sound waves, acoustics can be a useful probe of critical behavior. In the case of superfluid helium there are five fundamentally different modes of sound propagation, and this acoustic abundance contributed to the success of the critical phenomenon theory test. This talk will review the theory of critical phenomena, describe the different modes of sound propagation in superfluid helium, and summarize the results of the key test of the critical phenomenon theory.

Piezoelectrics are often used at low temperatures, but among the large number of piezoelectric materials, only one (quartz) has had its properties measured at low temperatures. Because measuring piezoelectric constants with the traditional electrical impedance method has shortcomings, particularly at liquid helium temperatures, it would be advantageous to make measurements with resonant ultrasound spectroscopy (RUS). EerNisse and Holland (1967) established a theoretical basis and Ogi et al. (2002) demonstrated an experimental RUS method. A problem with RUS for piezoelectrics is that resonance frequencies are much more sensitive to elastic behavior than to piezoelectric behavior, so that extraordinary precision is required. However, one may make a RUS measurement on a sample twice, once with an electrically conducting coating on sample faces and once without, and analyze the differences in the frequency spectra. Because the effect of the conducting coating depends more on the piezoelectric behavior than on the elastic behavior, analyzing the frequency differences suppresses the dependence on the elastic constants and enhances the measurement of the piezoelectric constants. This paper will present theoretical and experimental results for this RUS method.

An historic transducer to which one should pay attention is the siren. While its early application was as a source for a musical instrument, the siren soon became the transducer of choice for long-range audible warning because of its high intensity and recognizable tone. The components defining the siren include a solid stator and rotor, each with periodic apertures, and a compressed fluid (usually air but could be other fluids). With the rotor rotating in close proximity to the stator, and the resulting opening and closing of passageways through the apertures for the compressed fluid results in periodic sound waves in the surrounding fluid; usually a horn is used to enhance the radiation efficiency. The high potential energy of the compressed fluid permits high intensity sound. Some sirens which received scientific study include that of R. Clark Jones (1946), a 50 horsepower siren with an efficiency of about 70%, and that of C. H. Allen and I. Rudnick (1947), capable of ultrasonic frequencies and described as a “supersonic death ray” in the news media. Some design considerations, performance results, and applications for these sirens will be presented.

A laboratory underwater acoustic measurement technique, Supersonic Intensity in Reverberant Environments (SIRE), is developed analytically and validated experimentally and numerically. Unlike standard free or diffuse field techniques, SIRE enables the measurement of narrowband sound power and directivity in an environment with inexact field conditions. The technique takes advantage of underwater vector sensors, measuring only acoustic pressure and the normal component of particle velocity/acceleration, and supersonic wavenumber filtering in the near field of a source. The result is outward-propagating acoustic waves separated from interfering incoming and/or evanescent waves. The SIRE technique was experimentally applied to monopole and dipole sources and the results are compared with theory and standard methods. SIRE is shown to accurately measure radiated sound power to within the limits of ANSI S12.51 and to accurately measure the directivity indices of simple sources to within ±3dB. A coupled finite element/boundary element model of a point-driven, thin-walled cylinder is also developed to establish the limitations of the SIRE technique. The model results show that the measurement standoff distance should be less than the reciprocal of the largest wavenumber in the frequency band of interest. Furthermore, the maximum measurement grid spacing must be less than twice the standoff distance.

One of the most important undertakings for materials is the measurement of the elastic behavior. As derivatives of the free energy with respect to atomic displacements, the elastic properties are closely connected to the thermodynamic properties of the material. Elastic behavior is a sensitive probe of the lattice environment in which all solid state phenomena occur, particularly in the vicinity of a phase transition. A useful method for measuring elastic properties is resonant ultrasound spectroscopy (RUS). Some novel materials to which RUS might be applied are often fragile or chemically reactive so that they cannot be polished into the shapes required by conventional RUS; for such cases a finite element method may be used. In this paper a discussion and test of a finite element method for RUS with arbitrarily shaped samples is provided.

When new materials are developed, among the first properties which should be measured are the elastic constants, and resonant ultrasound spectroscopy (RUS) has been found to be an appropriate method for accomplishing such measurements. However, novel materials to which RUS might be applied are often available only as small samples, only a few hundred micrometers in size, or as thin films deposited on a substrate, with film thicknesses of only a few hundred nanometers. On some occasions samples are fragile or chemically reactive so that they cannot be polished into the shapes required by conventional RUS; for such cases a finite element method is required. In this talk the development of RUS for small or arbitrarily shaped samples and thin films will be presented. [Research supported by NSF DMR 0804105.]

The appearance of mathematical regularities in the disposition of leaves on a stem, scales on a pine-cone, and spines on a cactus has puzzled scholars for millennia; similar so-called phyllotactic patterns are seen in self-organized growth, polypeptides, convection, magnetic flux lattices and ion beams. Levitov showed that a cylindrical lattice of repulsive particles can reproduce phyllotaxis under the (unproved) assumption that minimum of energy would be achieved by two-dimensional Bravais lattices. Here we provide experimental and numerical evidence that the Phyllotactic lattice is actually a ground state. When mechanically annealed, our experimental "magnetic cactus" precisely reproduces botanical phyllotaxis, along with domain boundaries (called transitions in Botany) between different phyllotactic patterns. We employ a structural genetic algorithm to explore the more general axially unconstrained case, which reveals multijugate (multiple spirals) as well as monojugate (single-spiral) phyllotaxis.

The technique of resonant ultrasound spectroscopy was used to measure the elastic properties of a polycrystalline cubic silicon carbide (3C-SiC) thin film. The film, grown on a silicon (100) substrate, was 1.69 microns thick with columnar crystalline grains and a (111) texture. The substrate with the film was placed between two transducers and the resonant frequencies were measured; measurements were repeated after selective, timed dry etching of the film, allowing a determination of the elastic constants of the film alone. The film elastic constants, c(11)=371 and c(12)=146 GPa, were within a few percent of the literature values (c(11)=386, c(12)=136 GPa) of crystalline 3C-SiC. However, the film elastic constant c(44), 111 GPa, was significantly smaller than the bulk literature value, 254 GPa. For the film, c44 approximately (c(11)-c(12))/2, indicating that, quite unlike a bulk 3C-SiC crystal, the thin film is elastically isotropic.

We have studied the thermal history of the resonant frequency of a torsional oscillator containing solid $^{4}\text{H}\text{e}$. We find that the magnitude of the frequency shift that occurs below $\ensuremath{\sim}100\text{ }\text{mK}$ is multivalued in the low-temperature limit, depending strongly on how the state is prepared. This result can be qualitatively explained in terms of the motion and pinning of quantized vortices within the sample. Several aspects of the data are also consistent with the response of dislocation lines to oscillating stress fields imposed on the solid. However, studies of solid helium in porous media, several control experiments, and the magnitude of the frequency shift found in this and other experiments all indicate that the most appropriate interpretation of the torsional oscillator results is the existence of a supersolid $^{4}\text{H}\text{e}$ phase.

Periodic array of scatterers The importance of understanding wave propagation in a periodic array of scatterers is illustrated in Fig. 3. Consider a plate (e.g., a floorboard) with a source of vibration at one end and some listeners at the far end. The source generates transverse (flexural) waves, and these waves propagate to the far end of the plate, radiate sound and create an annoyance for the listeners (Fig. 3a). Usually, for structural reasons, a plate will have a rib on it, and that rib reflects the vibration (as shown in Fig. 3b) so there is less vibration transmitted, and less noise at the other end of the plate. If one rib reflects the vibration and reduces the annoyance, why not a series of many ribs? For ease of manufacture, the ribs may be identical and arranged periodically, as illustrated in Fig. 3c. One might expect that the array of ribs would greatly decrease the vibration, with transmission proportional to some large power of the single-rib transmission coefficient. However, this is not at all what happens. The remarkable result for the periodic array is that the first rib reflects the vibration, but all the rest transmit without any further reduction, at least within ranges of frequencies, called “pass bands.” This remarkable property of periodic arrays of scatterers is well known in solid state physics, where a typical application would be the understanding of the electrical conductivity of a metallic crystal. In a crystalline wire consisting of moving electrons and fixed positive ions, as illustrated in Fig. 4, a classical electron would be very strongly scattered by the ions as it tried to move down the wire, and one would have to conclude that a metal should be a very poor conductor of electricity. The fact that metals are actually good conductors When sound waves, propagating in a uniform medium, encounter a foreign object, the sound is scattered in all directions. When the size of the object is about the same size as the wavelength of the sound, the scattering may be called “strong” and its pattern may be complicated. If the geometry of the object is relatively simple, it would be possible to calculate the scattered sound field—such a situation is represented by Fig. 1a. However, if there is a second object, located a few wavelengths away from the first object, then the scattered sound may reflect back and forth between the objects, undergoing “multiple scattering,” as shown in Fig. 1b. The multiple scattering makes any calculation far more difficult, and the problem would rapidly grow in difficulty if several more scattering objects were added. What then if a medium were filled with such scatterers, as illustrated in Fig. 2? One might imagine that such a situation would be hopeless. However, if the scatterers are arranged periodically, like the atoms in a crystal, then a sound field may be readily calculated to high accuracy. Furthermore, if the scatterers were in a disordered configuration, one could make significant qualitative and quantitative predictions as to how sound would behave if it encountered such a system. There is also a possible configuration of scatterers intermediate between periodic and disordered, referred to as quasicrystalline, for which an accurate understanding of the behavior of the sound is again possible. This article is an introduction to understanding sound propagation in periodic arrays of scatterers. Later articles will treat the cases of disordered and quasicrystalline arrays of scatterers. “If scatterers are arranged

The complete elastic tensor of single crystal GdScO3 was determined using resonant ultrasound spectroscopy (RUS) in combination with ab initio calculations. The experimental determination of all nine elastic constants also provides a method for probing the dynamic lattice properties for this recently developed orthorhombic material. The experimentally determined elastic constants differed from theoretical values on average by 10%, and all but three of the nine elastic constants varied by less than 10%. These results indicate that ab initio calculations are now sufficiently accurate for the precise determination of the elastic tensor using RUS as the sole experimental source.