Paraxial and ray approximations of acoustic vortex beams.

A compact analytical solution obtained in the paraxial approximation is used to investigate focused and unfocused vortex beams radiated by a source with a Gaussian amplitude distribution. Comparisons with solutions of the Helmholtz equation are conducted to determine bounds on the parameter space in which the paraxial approximation is accurate. A linear relation is obtained for the dependence of the vortex ring radius on the topological charge, characterized by its orbital number, in the far field of an unfocused beam and in the focal plane of a focused beam. For a focused beam, it is shown that as the orbital number increases, the vortex ring not only increases in radius but also moves out of the focal plane in the direction of the source. For certain parameters, it is demonstrated that with increasing orbital number, the maximum amplitude in a focused beam becomes localized along a spheroidal surface enclosing a shadow zone in the prefocal region. This field structure is described analytically by ray theory developed in the present work, showing that the spheroidal surface in the prefocal region coincides with a simple expression for the coordinates of the caustic surface formed in a focused vortex beam.

Data-driven Identification of Parametric Governing Equations of Dynamical Systems Using the Signed Cumulative Distribution Transform.

This paper presents a novel data-driven approach to identify partial differential equation (PDE) parameters of a dynamical system. Specifically, we adopt a mathematical "transport" model for the solution of the dynamical system at specific spatial locations that allows us to accurately estimate the model parameters, including those associated with structural damage. This is accomplished by means of a newly-developed mathematical transform, the signed cumulative distribution transform (SCDT), which is shown to convert the general nonlinear parameter estimation problem into a simple linear regression. This approach has the additional practical advantage of requiring no a priori knowledge of the source of the excitation (or, alternatively, the initial conditions). By using training data, we devise a coarse regression procedure to recover different PDE parameters from the PDE solution measured at a single location. Numerical experiments show that the proposed regression procedure is capable of detecting and estimating PDE parameters with superior accuracy compared to a number of recently developed machine learning methods. Furthermore, a damage identification experiment conducted on a publicly available dataset provides strong evidence of the proposed method's effectiveness in structural health monitoring (SHM) applications. The Python implementation of the proposed system identification technique is integrated as a part of the software package PyTransKit [1].

Exact radiation boundary conditions to determine the complex wavenumber of an underwater acoustic leaky wave antenna.

Underwater elastic leaky wave antennas (LWAs) steer acoustic energy as a function of frequency by exploiting fluid-solid coupling. LWAs present a modeling challenge due to complex radiation impedance on the waveguide surface that leads to changes in dynamic response. This work presents an approach to model underwater LWAs that considers an elastic unit cell surrounded by a fluid domain and includes a radiation boundary condition to simulate an open boundary. The model solves an eigenvalue problem for the complex-valued wavenumber given a specified frequency, forming an accurate representation for the free response of an elastic LWA in an underwater environment.

Controlling Displacement Fields in Polar Willis Solids via Gauge Transformations.

Rigid-body displacement and deformation constitute the total displacement field of a solid. Harnessing the former calls for well-organized kinematic elements, and controlling the latter allows for creation of shape-morphing materials. A solid capable of simultaneously controlling both rigid-body displacement and deformation remains unknown. Here, we exploit gauge transformations to show how the total displacement field in elastostatic polar Willis solids can be harnessed at will and how those solids can be realized in the form of lattice metamaterials. The transformation method we develop leverages a displacement gauge in linear transformation elasticity, giving rise to polarity and Willis coupling such that the resulting solids not only break minor symmetries of the stiffness tensor, but display cross coupling between stress and displacement. We realize those solids using a combination of tailored geometries, grounded springs, and a set of coupled gears and numerically demonstrate a range of satisfactory, and peculiar, displacement control functions. Our work provides an analytical framework for the inverse design of grounded polar Willis metamaterials to achieve arbitrary displacement control functions by design.

Design and characterization of a three-dimensional anisotropic additively manufactured pentamode material.

A metamaterial of particular interest for underwater applications is the three-dimensional (3D) anisotropic pentamode (PM), i.e., a structure designed to support a single longitudinal wave with a sound speed that depends on the propagation direction. The present work attempts to experimentally verify anisotropic sound speeds predicted by finite element simulations using additively manufactured anisotropic 3D PM samples made of titanium. The samples were suspended in front of a plane wave source emitting a broadband chirp in a water tank to measure time of flight for wavefronts with and without the PM present. The measurement utilizes a deconvolution method that extracts the band limited impulse response of data gathered by a scanning hydrophone in a plane of constant depth behind the samples. Supporting material takes the form of finite element simulations developed to model the response of a semi-infinite PM medium to an incident normal plane wave. A technique to extract the longitudinal PM wave speed for frequency domain simulations based on Fourier series expansions is given.

Robust design of an asymmetrically absorbing Willis acoustic metasurface subject to manufacturing-induced dimensional variations.

Advancements in additive manufacturing (AM) technology are promising for the creation of acoustic materials. Acoustic metamaterials and metasurfaces are of particular interest for the application of AM technologies as theoretical predictions suggest the need for precise arrangements of dissimilar materials within specified regions of space to reflect, transmit, guide, or absorb acoustic waves in ways that exceed the capabilities of currently available acoustic materials. This work presents the design of an acoustic metasurface (AMS) with Willis constitutive behavior, which is created from an array of multi-material inclusions embedded in an elastomeric matrix, which displays the asymmetric acoustic absorption. The finite element models of the AMS show that the asymmetric absorption is dependent on asymmetry in the distribution of materials within the inclusion and highly sensitive to small changes in the inclusion geometry. It is shown that the performance variability can be used to place constraints on the manufacturing-induced variability to ensure that an as-built AMS will perform using the as-designed parameters. The evaluation of the AMS performance is computationally expensive, thus, the design is performed with a classifier-based metamodel to support more efficient Monte Carlo simulations and quantify the sensitivity of the candidate design performance to the manufacturing variability. This work explores combinations of material choices and dimensional accuracies to demonstrate how a robust design approach can be used to help select AM fabrication methods or guide process development toward an AM process that is capable of fabricating acoustic material structures.

Introduction to the special issue on Additive Manufacturing and Acoustics.

Additive manufacturing (AM) has expanded to a wide range of applications over the last few years, and acoustic applications are no exception. This article is an introduction to the special issue of the Journal of the Acoustical Society of America on AM and acoustics. To provide background to the reader, a brief introduction to the manufacturing approach of AM is included. The ways in which the articles in this special issue advance the field of acoustics are described for a range of applications.

Characterization of an underwater metamaterial made of aluminum honeycomb panels at low frequencies.

This paper presents a method to characterize the effective properties of inertial acoustic metamaterial unit cells for underwater operation. The method is manifested by a fast and reliable parameter retrieval procedure utilizing both numerical simulations and measurements. The effectiveness of the method was proved to be self-consistent by a metamaterial unit cell composed of aluminum honeycomb panels with soft rubber spacers. Simulated results agree well with the measured responses of this metamaterial in a water-filled resonator tube. A sub-unity density ratio and an anisotropic mass density are simultaneously achieved by the metamaterial unit cell, making it useful in implementations of transformation acoustics. The metamaterial, together with the approach for its characterization, are expected to be useful for underwater acoustic devices.

Numerical study of acoustic focusing using a bianisotropic acoustic lens.

Materials with sub-wavelength asymmetry and long-range order have recently been shown to demonstrate acoustical properties analogous to electromagnetic bianisotropy. One characteristic of bianisotropic acoustic media is the existence of direction-dependent acoustic impedance. Therefore, the magnitude and phase of the acoustic fields transmitted through bianisotropic acoustic media are dependent on the direction of bianisotropic polarization. These materials can therefore be used as acoustic metasurfaces to control acoustic fields. To demonstrate this behavior, a numerical model of bianisotropic acoustic waveguides is utilized to design a lens that focuses an incident plane wave by only manipulating the orientation of the bianisotropic coupling vector.

An active mechanical Willis meta-layer with asymmetric polarizabilities.

Willis materials exhibit macroscopic cross-coupling between particle velocity and stress as well as momentum and strain. However, Willis coupling coefficients designed so far are intrinsically coupled, which inhibits their full implementation in structural dynamic applications. This work presents a means to eliminate these limitations by introducing an active scatterer in a mechanical meta-layer that exploits piezoelectric sensor-actuator pairs controlled by digital circuits. We experimentally demonstrate abilities of the Willis meta-layer, in beams and plates, for independently engineering transmission and reflection coefficients of flexural waves in both amplitude and phase and nonreciprocal wave propagations. The meta-layer is described by a flexural wave polarizability tensor, which captures independent higher-order symmetric-to-symmetric and symmetric-to-antisymmetric couplings. The active meta-layer is adaptive in real time for reconfigurable broadband operation thanks to its programmability. This work sheds a new light on unsurpassed control of elastic waves, ranging from vibration protections to ultrasonic sensing and evaluation of engineering structures.

Acoustic response for nonlinear, coupled multiscale model containing subwavelength designed microstructure instabilities.

Nonperiodic arrangements of inclusions with incremental linear negative stiffness embedded within a host material offer the ability to achieve unique and useful material properties on the macroscale. In an effort to study such types of inclusions, the present paper develops a time-domain model to capture the nonlinear dynamic response of a heterogeneous medium containing a dilute concentration of subwavelength nonlinear inclusions embedded in a lossy, nearly incompressible medium. Each length scale is modeled via a modified Rayleigh-Plesset equation, which differs from the standard form used in bubble dynamics by accounting for inertial and viscoelastic effects of the oscillating spherical element and includes constitutive equations formulated with incremental deformations. The two length scales are coupled through the constitutive relations and viscoelastic loss for the effective medium, both dependent on the inclusion and matrix properties. The model is then applied to an example nonlinear inclusion with incremental negative linear stiffness stemming from microscale elastic instabilities embedded in a lossy, nearly incompressible host medium. The macroscopic damping performance is shown to be tunable via an externally applied hydrostatic pressure with the example system displaying over two orders of magnitude change in energy dissipation due to changes in prestrain. The numerical results for radial oscillations versus time, frequency spectra, and energy dissipation obtained from the coupled dynamic model captures the expected response for quasistatic and dynamic regimes for an example buckling inclusion for both constrained and unconstrained negative stiffness inclusions.

Physical Observation of a Robust Acoustic Pumping in Waveguides with Dynamic Boundary.

Research on breaking time-reversal symmetry to realize one-way wave propagation is a growing area in photonic and phononic crystals and metamaterials. In this Letter, we present physical realization of an acoustic waveguide with spatiotemporally modulated boundary conditions to realize nonreciprocal transport and acoustic topological pumping. The modulated waveguide inspired by a water wheel consists of a helical tube rotating around a slotted tube at a controllable speed. The rotation of the helical tube creates moving boundary conditions for the exposed waveguide sections at a constant speed. We experimentally demonstrate acoustic nonreciprocity and topologically robust bulk-edge correspondences for this system, which is in good agreement with analytical and numerical predictions. The nonreciprocal waveguide is a one-dimensional analog to the two-dimensional quantum Hall effect for acoustic circulators and is characterized by a robust integer-valued Chern number. These findings provide insight into practical implications of topological modes in acoustics and the implementation of higher-dimensional topological acoustics where time serves as a synthetic dimension.

Non-reciprocal wave propagation in mechanically-modulated continuous elastic metamaterials.

Acoustic and elastic metamaterials with time- and space-dependent effective material properties have recently received significant attention as a means to induce non-reciprocal wave propagation. Recent analytical models of spring-mass chains have shown that external application of a nonlinear mechanical deformation, when applied on time scales that are slow compared to the characteristic times of propagating linear elastic waves, may induce non-reciprocity via changes in the apparent elastic modulus for perturbations around that deformation. Unfortunately, it is rarely possible to derive analogous analytical models for continuous elastic metamaterials due to complex unit cell geometry. The present work derives and implements a finite element approach to simulate elastic wave propagation in a mechanically-modulated metamaterial. This approach is implemented on a metamaterial supercell to account for the modulation wavelength. The small-on-large approximation is utilized to separate the nonlinear mechanical deformation (the "large" wave) from superimposed linear elastic waves (the "small" waves), which are then analyzed via Bloch wave analysis with a Fourier expansion in the harmonics of the modulation frequency. Results on non-reciprocal wave propagation in a negative stiffness chain, a structure exhibiting large stiffness modulations due to the presence of mechanical instabilities, are then shown as a case example.

Frequency-dependent behavior of media containing pre-strained nonlinear inclusions: Application to nonlinear acoustic metamaterials.

One emerging research area within the fields of acoustic and elastic metamaterials involves designing subwavelength structures that display elastic instabilities in order to generate an effective medium response that is strongly nonlinear. To capture the overall frequency-dependent and dispersive macroscopic response of such heterogeneous media with subwavelength heterogeneities, a theoretical framework is developed that accounts for higher-order stiffnesses of a resonant, nonlinear inclusion that varies with a macroscopic pre-strain, and the inherent inertia associated with an inclusion embedded in a nearly incompressible elastic matrix material. Such a model can be used to study varying macroscopic material properties as a function of both frequency and pre-strain and the activation of such microscale instabilities due to an external, macroscopic loading, as demonstrated with a buckling metamaterial inclusion that is of interest due to its tunable and tailorable nature. The dynamic results obtained are consistent with similar static behavior reported in the literature for structures with elastic instabilities.

Acoustic scattering from a fluid cylinder with Willis constitutive properties.

A material that exhibits Willis coupling has constitutive equations that couple the pressure-strain and momentum-velocity relationships. This coupling arises from subwavelength asymmetry and non-locality in heterogeneous media. This paper considers the problem of the scattering of a plane wave by a cylinder exhibiting Willis coupling using both analytical and numerical approaches. First, a perturbation method is used to describe the influence of Willis coupling on the scattered field to a first-order approximation. A higher order analysis of the scattering based on generalized impedances is then derived. Finally, a finite-element method-based numerical scheme for calculating the scattered field is presented. These three analyses are compared and show strong agreement for low to moderate levels of Willis coupling.

Experimental evidence of Willis coupling in a one-dimensional effective material element.

The primary objective of acoustic metamaterial research is to design subwavelength systems that behave as effective materials with novel acoustical properties. One such property couples the stress-strain and the momentum-velocity relations. This response is analogous to bianisotropy in electromagnetism, is absent from common materials, and is often referred to as Willis coupling after J.R., Willis, who first described it in the context of the dynamic response of heterogeneous elastic media. This work presents two principal results: first, experimental and theoretical demonstrations, illustrating that Willis properties are required to obtain physically meaningful effective material properties resulting solely from local behaviour of an asymmetric one-dimensional isolated element and, second, an experimental procedure to extract the effective material properties from a one-dimensional isolated element. The measured material properties are in very good agreement with theoretical predictions and thus provide improved understanding of the physical mechanisms leading to Willis coupling in acoustic metamaterials.

Broadband focusing of underwater sound using a transparent pentamode lens.

An inhomogeneous acoustic metamaterial lens based on spatial variation of refractive index for broadband focusing of underwater sound is reported. The index gradient follows a modified hyperbolic secant profile designed to reduce aberration and suppress side lobes. The gradient index (GRIN) lens is comprised of transversely isotropic hexagonal microstructures with tunable quasi-static bulk modulus and mass density. In addition, the unit cells are impedance-matched to water and have in-plane shear modulus negligible compared to the effective bulk modulus. The flat GRIN lens is fabricated by cutting hexagonal centimeter scale hollow microstructures in aluminum plates, which are then stacked and sealed from the exterior water. Broadband focusing effects are observed within the homogenization regime of the lattice in both finite element simulations and underwater measurements (20-40 kHz). This design approach has potential applications in medical ultrasound imaging and underwater acoustic communications.

Reciprocity, passivity and causality in Willis materials.

Materials that require coupling between the stress-strain and momentum-velocity constitutive relations were first proposed by Willis (Willis 1981 Wave Motion3, 1-11. (doi:10.1016/0165-2125(81)90008-1)) and are now known as elastic materials of the Willis type, or simply Willis materials. As coupling between these two constitutive equations is a generalization of standard elastodynamic theory, restrictions on the physically admissible material properties for Willis materials should be similarly generalized. This paper derives restrictions imposed on the material properties of Willis materials when they are assumed to be reciprocal, passive and causal. Considerations of causality and low-order dispersion suggest an alternative formulation of the standard Willis equations. The alternative formulation provides improved insight into the subwavelength physical behaviour leading to Willis material properties and is amenable to time-domain analyses. Finally, the results initially obtained for a generally elastic material are specialized to the acoustic limit.

A micromechanical approach for homogenization of elastic metamaterials with dynamic microstructure.

An approximate homogenization technique is presented for generally anisotropic elastic metamaterials consisting of an elastic host material containing randomly distributed heterogeneities displaying frequency-dependent material properties. The dynamic response may arise from relaxation processes such as viscoelasticity or from dynamic microstructure. A Green's function approach is used to model elastic inhomogeneities embedded within a uniform elastic matrix as force sources that are excited by a time-varying, spatially uniform displacement field. Assuming dynamic subwavelength inhomogeneities only interact through their volume-averaged fields implies the macroscopic stress and momentum density fields are functions of both the microscopic strain and velocity fields, and may be related to the macroscopic strain and velocity fields through localization tensors. The macroscopic and microscopic fields are combined to yield a homogenization scheme that predicts the local effective stiffness, density and coupling tensors for an effective Willis-type constitutive equation. It is shown that when internal degrees of freedom of the inhomogeneities are present, Willis-type coupling becomes necessary on the macroscale. To demonstrate the utility of the homogenization technique, the effective properties of an isotropic elastic matrix material containing isotropic and anisotropic spherical inhomogeneities, isotropic spheroidal inhomogeneities and isotropic dynamic spherical inhomogeneities are presented and discussed.