Transmission, reflection and absorption in Sonic and Phononic Crystals

Tesis por compendio[EN] Phononic crystals are artificial materials formed by a periodic arrangement of inclusions embedded into a host medium, where each of them can be solid or fluid. By controlling the geometry and the impedance contrast of its constituent materials, one can control the dispersive properties of waves, giving rise to a huge variety of interesting and fundamental phenomena in the context of wave propagation. When a propagating wave encounters a medium with different physical properties it can be transmitted and reflected in lossless media, but also absorbed if dissipation is taken into account. These fundamental phenomena have been classically explained in the context of homogeneous media, but it has been a subject of increasing interest in the context of periodic structures in recent years as well. This thesis is devoted to the study of different effects found in sonic and phononic crystals associated with transmission, reflection and absorption of waves, as well as the development of a technique for the characterization of its dispersive properties, described by the band structure. We start discussing the control of wave propagation in transmission in conservative systems. Specifically, our interest is to show how sonic crystals can modify the spatial dispersion of propagating waves leading to control the diffractive broadening of sound beams. Making use of the spatial dispersion curves extracted from the analysis of the band structure, we first predict zero and negative diffraction of waves at frequencies close to the band-edge, resulting in collimation and focusing of sound beams in and behind a 3D sonic crystal, and later demonstrate it through experimental measurements. The focusing efficiency of a 3D sonic crystal is limited due to the strong scattering inside the crystal, characteristic of the diffraction regime. To overcome this limitation we consider axisymmetric structures working in the long wavelength regime, as a gradient index lens. In this regime, the scattering is strongly reduced and, in an axisymmetric configuration, the symmetry matching with acoustic sources radiating sound beams increase its efficiency dramatically. Moreover, the homogenization theory can be used to model the structure as an effective medium with effective physical properties, allowing the study of the wave front profile in terms of refraction. We will show the model, design and characterization of an efficient focusing device based on these concepts. Consider now a periodic structure in which one of the parameters of the lattice, such as the lattice constant or the filling fraction, gradually changes along the propagation direction. Chirped crystals represent this concept and are used here to demonstrate a novel mechanism of sound wave enhancement based on a phenomenon known as "soft" reflection. The enhancement is related to a progressive slowing down of the wave as it propagates along the material, which is associated with the group velocity of the local dispersion relation at the planes of the crystal. A model based on the coupled mode theory is proposed to predict and interpret this effect. Two different phenomena are observed here when dealing with dissipation in periodic structures. On one hand, when considering the propagation of in-plane sound waves in a periodic array of absorbing layers, an anomalous decrease in the absorption, combined with a simultaneous increase of reflection and transmission at Bragg frequencies is observed, in contrast to the usual decrease of transmission, characteristic in conservative periodic systems at these frequencies. For a similar layered media, backed now by a rigid reflector, out-of-plane waves impinging the structure from a homogeneous medium will increase dramatically the interaction strength. In other words, the time delay of sound waves inside the periodic system will be considerably increased resulting in an enhanced absorption, for a broadband spectral range.[ES] Los cristales fonónicos son materiales artificiales formados por una disposición periódica de inclusiones en un medio, pudiendo ambos ser de carácter sólido o fluido. Controlando la geometría y el contraste de impedancias entre los materiales constituyentes se pueden controlar las propiedades dispersivas de las ondas. Cuando una onda propagante se encuentra un medio con diferentes propiedades físicas puede ser transmitida y reflejada, en medios sin pérdidas, pero también absorbida, si la disipación es tenida en cuenta. La presente tesis está dedicada al estudio de diferentes efectos presentes en cristales sónicos y fonónicos relacionados con la transmisión, reflexión y absorción de ondas, así como el desarrollo de una técnica para la caracterización de sus propiedades dispersivas, descritas por la estructura de bandas. En primer lugar, se estudia el control de la propagación de ondas en transmisión en sistemas conservativos. Específicamente, nuestro interés se centra en mostrar cómo los cristales sónicos son capaces de modificar la dispersión espacial de las ondas propagantes, dando lugar al control del ensanchamiento de haces de sonido. Haciendo uso de las curvas de dispersión espacial extraídas del análisis de la estructura de bandas, se predice primero la difracción nula y negativa de ondas a frecuencias cercanas al borde de la banda, resultando en la colimación y focalización de haces acústicos en el interior y detrás de un cristal sónico 3D, y posteriormente se demuestra mediante medidas experimentales. La eficiencia de focalización de un cristal sónico 3D está limitada debido a las múltiples reflexiones existentes en el interior del cristal. Para superar esta limitación se consideran estructuras axisimétricas trabajando en el régimen de longitud de onda larga, como lentes de gradiente de índice. En este régimen, las reflexiones internas se reducen fuertemente y, en configuración axisimétrica, la adaptación de simetría con fuentes acústicas radiando haces de sonido incrementa la eficiencia drásticamente. Además, la teoría de homogenización puede ser empleada para modelar la estructura como un medio efectivo con propiedades físicas efectivas, permitiendo el estudio del frente de ondas en términos refractivos. Se mostrará el modelado, diseño y caracterización de un dispositivo de focalización eficiente basado en los conceptos anteriores. Considérese ahora una estructura periódica en la que uno de los parámetros de la red, sea el paso de red o el factor de llenado, cambia gradualmente a lo largo de la dirección de propagación. Los cristales chirp representan este concepto y son empleados aquí para demostrar un mecanismo novedoso de incremento de la intensidad de la onda sonora basado en un fenómeno conocido como reflexión "suave". Este incremento está relacionado con una ralentización progresiva de la onda conforme se propaga a través del material, asociado con la velocidad de grupo de la relación de dispersión local en los planos del cristal. Un modelo basado en la teoría de modos acoplados es propuesto para predecir e interpretar este efecto. Se observan dos fenómenos diferentes al considerar pérdidas en estructuras periódicas. Por un lado, si se considera la propagación de ondas sonoras en un array periódico de capas absorbentes, cuyo frente de ondas es paralelo a los planos del cristal, se produce una reducción anómala en la absorción combinada con un incremento simultáneo de la reflexión y transmisión a las frecuencias de Bragg, de forma contraria a la habitual reducción de la transmisión, característica de sistemas periódicos conservativos a estas frecuencias. En el caso de la misma estructura laminada en la que se cubre uno de sus lados mediante un reflector rígido, la incidencia de ondas sonoras desde un medio homogéneo, cuyo frente de ondas es perpendicular a los planos del cristal, produce un gran incremento de la fuerza de[CA] Els cristalls fonònics són materials artificials formats per una disposició d'inclusions en un medi, ambdós poden ser sòlids o fluids. Controlant la geometría i el contrast d'impedàncies dels seus materials constituents, és poden controlar les propietats dispersives de les ondes, permetent una gran varietatde fenòmens fonamentals interessants en el context de la propagació d'ones. Quan una ona propagant troba un medi amb pèrdues amb propietats físiques diferents es pot transmetre i reflectir, però també absorbida si la dissipació es té en compte. Aquests fenòmens fonamentals s'han explicat clàssicament en el context de medis homogenis, però també ha sigut un tema de creixent interés en el context d'estructures periòdiques en els últims anys. Aquesta tesi doctoral tracta de l'estudi de diferents efectes en cristalls fonònics i sònics lligats a la transmissió, reflexió i absorció d'ones, així com del desenvolupament d'una tècnica de caracterització de les propietats dispersives, descrites mitjançant la estructura de bandes. En primer lloc, s'estudia el control de la propagació ondulatori en transmissió en sistemes conservatius. Més específicament, el nostre interés és mostrar com els cristalls sonors poden modificar la dispersió espacial d'ones propagants donant lloc al control de l'amplària per difracció dels feixos sonors. Mitjançant les corbes dispersió espacial obtingudes de l'anàlisi de l'estructura de bandes, es prediu, en primer lloc, la difracció d'ones zero i negativa a freqüències próximes al final de banda. El resultat és la collimació i focalització de feixos sonors dins i darrere de cristalls de so. Després es mostra amb mesures experimentals. L'eficiència de focalització d'un cristall de so 3D està limitada per la gran dispersió d'ones dins del cristall, que és característic del règim difractiu. Per a superar aquesta limitació, estructures axisimètriques que treballen en el règim de llargues longituds d'ona, i es comporten com a lents de gradient d'índex. En aquest règim, la dispersió es redueix enormement i, en una configuració axisimètrica, a causa de l'acoblament de la simetría amb les fonts acústiques que radien feixos sonors, l'eficiència de radiació s'incrementa significativament. D'altra banda, la teoria d'homogeneïtzació es pot utilitzar per a modelar, dissenyar i caracteritzar un dispositiu eficient de focalització basat en aquests conceptes. Considerem ara una estructura periòdica en la qual un dels seus paràmetres de xarxa, com ara la constant de xarxa o el factor d'ompliment canvia gradualment al llarg de la direcció de propagació. Els cristalls chirped representen aquest concepte i s'utilitzen ací per a demostrar un mecanisme nou d'intensificació d'ones sonores basat en el fenòmen conegut com a reflexió "suau". La intensificació està relacionada amb la alentiment progressiva de l'ona conforme propaga al llarg del material, que està associada amb la velocitat de grup de la relació de dispersió local en els diferents plànols del cristall. Es proposa un model basat en la teoria de modes acoblats per a predir i interpretar este efecte. Dos fenòmens diferents cal destacar quan es tracta d'estructures periòdiques amb dissipació. Per un costat, al considerar la propagació d'ones sonores en el plànol en un array periòdic de capes absorbents, s'observa una disminució anòmala de l'absorció i es combina amb un augment simultani de reflexió i transmissió en les freqüències de Bragg que contrasta amb la usual disminució de transmissió, característica dels sistemes conservatius a eixes freqüències. Per a un medi similar de capes, amb un reflector rígid darrere, les ones fora del pla incidint l'estructura des de un medi homogeni, augmentaran considerablement la interacció. En altres paraules, el retràs temporal de les ones sonores dins del sistema periòdic augmentarà significativament produint un augmenCebrecos Ruiz, A. (2015). Transmission, reflection and absorption in Sonic and Phononic Crystals [Tesis doctoral]. Universitat Politècnica de València. https://doi.org/10.4995/Thesis/10251/56463TESISPremios Extraordinarios de tesis doctoralesCompendi

Estimación indirecta del target strength (TS) con ecosondas científicas de haz simple

El objetivo de este trabajo es desarrollar y comprobar la fiabilidad de una aplicación de procesado pensada para la medida indirecta de la potencia de blanco (TS) de peces, con vistas a la integración de ésta en una ecosonda científica de haz simple. Se emplean mediciones dorsales y ventrales de 5 clases de doradas (Sparus Aurata) clasificadas según su tamaño, realizadas con la ecosonda de haz dividido EK60. The aim of this work is to develop and verify the reliability of a processing application thought for the indirect measurement of target strength (TS), with a view to the integration into a single-beam system. Dorsal and ventral measurements are carried out for Gilt-head sea-bream (Sparus Aurata) for 5 size classes using a split-beam EK60 echosounder and single-beam data obtained from one of the channels from the echosounder.Cebrecos Ruiz, A. (2010). Estimación indirecta del target strength (TS) con ecosondas científicas de haz simple. Universitat Politècnica de València. http://hdl.handle.net/10251/10339Archivo delegad

Complex Dispersion Relation Recovery from 2D Periodic Resonant Systems of Finite Size

[EN] The complex dispersion relations along the main symmetry directions of two-dimensional finite size periodic arrangements of resonant or non-resonant scatterers are recovered by using an extension of the SLaTCoW (Spatial LAplace Transform for COmplex Wavenumber) method. This method relies on the analysis of the spatial Laplace transform instead of the usual spatial Fourier transform of the measured wavefield in the frequency domain. We apply this method to finite dimension square periodic arrangements of both rigid and resonant scatterers embedded in air, i.e., to finite size sonic crystals and finite size acoustic metamaterials, respectively. The main hypothesis considered in this work is the mirror symmetry of the finite structure with respect to its median axis along the analyzed direction. However, we show that the method is robust enough to provide excellent results even if this hypothesis is not fully satisfied. Effectively, a minor asymmetry could be considered as a side effect when the structure is large enough because Laplace transforming the field along the main symmetry directions also implies averaging the field in the perpendicular one. The calculated complex dispersion relations are in excellent agreement with those obtained by an already validated technique, like the Extended Plane Wave Expansion (EPWE). The methodology employed in this work is intended to be directly used for the experimental characterization of real 2D periodic and resonant systems.This article is based on work from COST Action DENORMS CA15125, supported by COST (European Cooperation in Science and Technology).Cebrecos, A.; Romero-García, V.; Groby, JP. (2019). Complex Dispersion Relation Recovery from 2D Periodic Resonant Systems of Finite Size. Applied Sciences. 9(3). https://doi.org/10.3390/app90304789

Magnetic force induced vibration on a ferromagnetic sphere for viscoelastic media characterization

[EN] A new method that combines transient magnetic forces with ultrasonic imaging and allows the local experimental characterization of the complex shear modulus of a viscoelastic medium is presented. By measuring the dynamics of a ferromagnetic inclusion under the application of a magnetic force, the viscoelastic properties of the medium are extracted. The system is composed of a coil, which creates a magnetic field that induces displacement on a ferromagnetic particle located inside a test phantom, and an ultrasound transducer operating in pulsed-echo mode, utilized to track the displacement of the particle with spatial resolution of several um. Experiments were conducted embedding a ferromagnetic sphere on test phantoms with different compositions and at different temperatures. The obtained results are in good agreement with the theoretical estimation of the dynamical response of a sphere and show robustness on the estimation of the viscoelastic parameters. Moreover, temperature dependent results show asymptotic elasticity values which are physically consistent for soft-solid media.This work is funded by the Spanish Ministerio de Economía e lnnovación (MlNECO) Generalitat Valenciana (GVA) through the projects TEC2016-80976-R and AlCO2016-108. N.J. and A. C acknowledge the support of GVA through the contracts APOSTD/2017/042 and APOSTD/2018/A/229.Cebrecos, A.; Company, M.; Jimenez, N.; Benlloch Baviera, JM.; Camarena Femenia, F. (2019). Magnetic force induced vibration on a ferromagnetic sphere for viscoelastic media characterization. Acoustical Society of America. 1-5. https://doi.org/10.1121/2.0001200S1

The finite-element time-domain method for elastic band-structure calculations

[EN] The finite-element time-domain method for elastic band-structure calculations is presented in this paper. The method is based on discretizing the appropriate equations of motion by finite elements, applying Bloch boundary conditions to reduce the analysis to a single unit cell, and conducting a simulation using a standard time-integration scheme. The unit cell is excited by a wide-band frequency signal designed to enable a large number of modes to be identified from the time-history response. By spanning the desired wave-vector space within the Brillouin zone, the band structure is then robustly generated. Bloch mode shapes are computed using the well-known concept of modal analysis, especially as implemented in an experimental setting. The performance of the method is analyzed in terms of accuracy, convergence, and computation time, and is compared to the finite-difference time-domain method as well as to a direct finite-element (FE) solution of the corresponding eigenvalue problem. The proposed method is advantageous over FD-based methods for unit cells with complex geometries, and over direct FE in situations where the formulation of an eigenvalue problem is not straightforward. For example, the new method makes it possible to accurately solve a time-dependent Bloch problem, such as the case of a complex unit cell model of a topological insulator where an internal fluid flow or other externally controlled physical fields are present. (C) 2018 Elsevier B.V. All rights reserved.A.C. is grateful for the support of Programa de Ayudas de Investigacion y Desarrollo (PAID) and Programa de Movilidad e Internacionalizacion Academica (PMIA-2013) of the UPV. This research was partially funded by the funded by the Ministerio de Economia e Innovacion (MINECO), Spain through project FIS2015-65998-C2-2-P, and partially funded by the National Science Foundation (NSF), USA through grant number 1538596. The authors acknowledge Dr. Noe Jimenez for fruitful discussions.Cebrecos, A.; Krattiger, D.; Sánchez Morcillo, VJ.; Romero García, V.; Hussein, MI. (2019). The finite-element time-domain method for elastic band-structure calculations. Computer Physics Communications. 238:77-87. https://doi.org/10.1016/j.cpc.2018.12.016S778723

Angular bandgaps in sonic crystals: evanescent waves and spatial complex dispersion relation

Phononic crystals are artificial materials made of a periodic distribution of solid scatterers embedded into a solid host medium with different physical properties. An interesting case of phononic crystals, known as sonic crystals (SCs), appears when the solid scatterers are periodically embedded in a fluid medium. In SCs only longitudinal modes are allowed to propagate and both the theoretical and the experimental studies of the properties of the system are simplified without loss of generality. The most celebrated property of these systems is perhaps the existence of spectral band gaps. However, the periodicity of the system can also affect to the spatial dispersion, making possible the control of the diffraction inside these structures. In this work we study the main features of the spatial dispersion in SCs from a novel point of view taking into account the evanescent properties of the system, i.e., studying the complex spatial dispersion relations. The evanescent behavior of the propagation of waves in the angular band gaps are theoretically and experimentally observed in this work. Both the numerical predictions and the experimental results show the presence of angular band gaps in good agreement with the complex spatial dispersion relation. The results shown in this work are independent of the spatial scale of the structure, and in principle the fundamental role of the evanescent waves could be also expected in micro- or nanoscale phononic crystals.This work was supported by MCI Secretaria de Estado de Investigacion (Spanish government) and the FEDER funds, under Grant Nos. MAT2009-09438, FIS2011-29734-C02-02, and from Generalitat Valencia through Project No. GV/2011/055. V.R.G. is grateful for the support of "Programa de Contratos Post-Doctorales con Movilidad UPV (CEI-01-11)." We acknowledge the Centro de Tecnologias Fisicas: Acustica, Materiales y Astrofisica and the Sonic Crystal Technologies Research Group of the Universitat Politecnica de Valencia for the use of the anechoic chamber and the 3DReAMS respectively.Romero García, V.; Picó Vila, R.; Cebrecos Ruiz, A.; Staliünas, K.; Sánchez Morcillo, VJ. (2013). Angular bandgaps in sonic crystals: evanescent waves and spatial complex dispersion relation. Journal of Vibration and Acoustics. 135(4):410121-410126. https://doi.org/10.1115/1.4023832S4101214101261354Yablonovitch, E. (1987). Inhibited Spontaneous Emission in Solid-State Physics and Electronics. Physical Review Letters, 58(20), 2059-2062. doi:10.1103/physrevlett.58.2059John, S. (1987). Strong localization of photons in certain disordered dielectric superlattices. Physical Review Letters, 58(23), 2486-2489. doi:10.1103/physrevlett.58.2486Ruffa, A. A. (1992). Acoustic wave propagation through periodic bubbly liquids. 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G., Joannopoulos, J. D., & Pendry, J. B. (2003). Subwavelength imaging in photonic crystals. Physical Review B, 68(4). doi:10.1103/physrevb.68.045115Yang, S., Page, J. H., Liu, Z., Cowan, M. L., Chan, C. T., & Sheng, P. (2004). Focusing of Sound in a 3D Phononic Crystal. Physical Review Letters, 93(2). doi:10.1103/physrevlett.93.024301Ke, M., Liu, Z., Qiu, C., Wang, W., Shi, J., Wen, W., & Sheng, P. (2005). Negative-refraction imaging with two-dimensional phononic crystals. Physical Review B, 72(6). doi:10.1103/physrevb.72.064306Feng, L., Liu, X.-P., Chen, Y.-B., Huang, Z.-P., Mao, Y.-W., Chen, Y.-F., … Zhu, Y.-Y. (2005). Negative refraction of acoustic waves in two-dimensional sonic crystals. Physical Review B, 72(3). doi:10.1103/physrevb.72.033108Romero-García, V., Sánchez-Pérez, J. V., & Garcia-Raffi, L. M. (2010). Evanescent modes in sonic crystals: Complex dispersion relation and supercell approximation. Journal of Applied Physics, 108(4), 044907. doi:10.1063/1.3466988Laude, V., Achaoui, Y., Benchabane, S., & Khelif, A. (2009). Evanescent Bloch waves and the complex band structure of phononic crystals. Physical Review B, 80(9). doi:10.1103/physrevb.80.092301Romero-García, V., Sánchez-Pérez, J. V., & Garcia-Raffi, L. M. (2010). Propagating and evanescent properties of double-point defects in sonic crystals. New Journal of Physics, 12(8), 083024. doi:10.1088/1367-2630/12/8/083024Romero-García, V., Sánchez-Pérez, J. V., Castiñeira-Ibáñez, S., & Garcia-Raffi, L. M. (2010). Evidences of evanescent Bloch waves in phononic crystals. Applied Physics Letters, 96(12), 124102. doi:10.1063/1.3367739Romero-García, V., Garcia-Raffi, L. M., & Sánchez-Pérez, J. V. (2011). Evanescent waves and deaf bands in sonic crystals. AIP Advances, 1(4), 041601. doi:10.1063/1.3675801Li, J., & Chan, C. T. (2004). Double-negative acoustic metamaterial. 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Dynamic beamforming for large area scan in array-based photoacoustic microscopy

[EN] We explore the use of a beamforming method intended for large-area scanning in optical-resolution photoacoustic microscopy. It has been evaluated in a experimental setup that comprises a low-cost laser diode and a phase array with a 128-elements linear probe. Three different beamforming strategies are discussed: no-beamforming, static beamforming and dynamic beamforming. The method has been tested in gelatine-based phantoms as well as ex-vivo organs. Results show that, compared with the other two, dynamic beamforming increases up to 15dB and homogenizes signal-to-noise ratio (SNR) along images of roughly 1 cm2. The method and system presented here could be the baseline for more advanced array-based systems that leverage the low-cost laser sources for clinical applications.This research has been supported by the Spanish Ministry of Science, Innovation and Universities through grant Juan de la Cierva - Incorporacion (IJC2018-037897-I), and program Proyectos I+D+i 2019 (PID2019-111436RB-C22). Action co-financed by the European Union through the Programa Operativo del Fondo Europeo de Desarrollo Regional (FEDER) of the Comunitat Valenciana 2014-2020 (IDIFEDER/2018/022). A.C. received financial support from Generalitat Valenciana and Universitat Politecnica de Val ` encia through the grants APOSTD/2018/229 and program PAID-10-19, respectively.Cebrecos, A.; García-Garrigós, JJ.; Descals, A.; Jimenez, N.; Benlloch Baviera, JM.; Camarena Femenia, F. (2020). Dynamic beamforming for large area scan in array-based photoacoustic microscopy. IEEE. 1-4. https://doi.org/10.1109/IUS46767.2020.9251519S1

Sound absorption and diffusion by 2D arrays of Helmholtz resonators

[EN] We report a theoretical and experimental study of an array of Helmholtz resonators optimized to achieve both efficient sound absorption and diffusion. The analysis starts with a simplified 1D model where the plane wave approximation is used to design an array of resonators showing perfect absorption for a targeted range of frequencies. The absorption is optimized by tuning the geometry of the resonators, i.e., by tuning the viscothermal losses of each element. Experiments with the 1D array were performed in an impedance tube. The designed system is extended to 2D by periodically replicating the 1D array. The 2D system has been numerically modeled and experimentally tested in an anechoic chamber. It preserves the absorption properties of the 1D system and introduces efficient diffusion at higher frequencies due to the joint effect of resonances and multiple scattering inside the discrete 2D structure. The combined effect of sound absorption at low frequencies and sound diffusion at higher frequencies, may play a relevant role in the design of noise reduction systems for different applications.This research was funded by the European Space Agency under the Networking/Partnering Initiative (NPI) contract number 441-2015. In memoriam to Julián Santiago-Prowald, Senior Advisor for the Structures, Mechanisms and Materials Division of ESA, a great man that always gave us his tireless support. AC acknowledges financial support from Generalitat Valenciana through the grant APOSTD/2018/229. VRG acknowledges the financial support from RFI Le Mans Acoustique (Région Pays de la Loire) in the framework of the project HYPERMETA funded under the program Étoiles Montantes of the Région Pays de la Loire. Authors acknowledge the support of the European Space Agency under contract 441-2015 Co- Sponsored PhD ¿Acoustic Reduction Methods for the Launch Pad¿ and project TRP ESA AO/1-9479/18/NL/LvH ¿Launch Sound Level Reduction¿. This article is based upon work from COST Action DENORMS CA15125, supported by COST (European Cooperation in Science and Technology).Herrero-Durá, I.; Cebrecos, A.; Picó Vila, R.; Romero-García, V.; García-Raffi, LM.; Sánchez Morcillo, VJ. (2020). Sound absorption and diffusion by 2D arrays of Helmholtz resonators. Applied Sciences. 10(5):1-15. https://doi.org/10.3390/app10051690S115105Sigalas, M. M., & Economou, E. N. (1992). Elastic and acoustic wave band structure. Journal of Sound and Vibration, 158(2), 377-382. doi:10.1016/0022-460x(92)90059-7Matlack, K. H., Bauhofer, A., Krödel, S., Palermo, A., & Daraio, C. (2016). Composite 3D-printed metastructures for low-frequency and broadband vibration absorption. Proceedings of the National Academy of Sciences, 113(30), 8386-8390. doi:10.1073/pnas.1600171113Wormser, M., Wein, F., Stingl, M., & Körner, C. (2017). Design and Additive Manufacturing of 3D Phononic Band Gap Structures Based on Gradient Based Optimization. Materials, 10(10), 1125. doi:10.3390/ma10101125Lucklum, F., & Vellekoop, M. J. (2018). Bandgap engineering of three-dimensional phononic crystals in a simple cubic lattice. Applied Physics Letters, 113(20), 201902. doi:10.1063/1.5049663D’Alessandro, L., Ardito, R., Braghin, F., & Corigliano, A. (2019). Low frequency 3D ultra-wide vibration attenuation via elastic metamaterial. Scientific Reports, 9(1). doi:10.1038/s41598-019-44507-6Martínez-Sala, R., Sancho, J., Sánchez, J. V., Gómez, V., Llinares, J., & Meseguer, F. (1995). Sound attenuation by sculpture. Nature, 378(6554), 241-241. doi:10.1038/378241a0Cebrecos, A., Krattiger, D., Sánchez-Morcillo, V. J., Romero-García, V., & Hussein, M. I. (2019). The finite-element time-domain method for elastic band-structure calculations. Computer Physics Communications, 238, 77-87. doi:10.1016/j.cpc.2018.12.016Cebrecos, A., Romero-García, V., & Groby, J. (2019). Complex Dispersion Relation Recovery from 2D Periodic Resonant Systems of Finite Size. Applied Sciences, 9(3), 478. doi:10.3390/app9030478Hussein, M. I., Leamy, M. J., & Ruzzene, M. (2014). Dynamics of Phononic Materials and Structures: Historical Origins, Recent Progress, and Future Outlook. Applied Mechanics Reviews, 66(4). doi:10.1115/1.4026911Sanchez-Perez, J. V., Rubio, C., Martinez-Sala, R., Sanchez-Grandia, R., & Gomez, V. (2002). Acoustic barriers based on periodic arrays of scatterers. Applied Physics Letters, 81(27), 5240-5242. doi:10.1063/1.1533112Martínez-Sala, R., Rubio, C., García-Raffi, L. M., Sánchez-Pérez, J. V., Sánchez-Pérez, E. A., & Llinares, J. (2006). Control of noise by trees arranged like sonic crystals. Journal of Sound and Vibration, 291(1-2), 100-106. doi:10.1016/j.jsv.2005.05.030Garcia-Raffi, L. M., Salmerón-Contreras, L. J., Herrero-Durá, I., Picó, R., Redondo, J., Sánchez-Morcillo, V. J., … Romero-García, V. (2018). Broadband reduction of the specular reflections by using sonic crystals: A proof of concept for noise mitigation in aerospace applications. Aerospace Science and Technology, 73, 300-308. doi:10.1016/j.ast.2017.11.048Sanchez-Perez, J. V., Castineira-Ibanez, S., Romero-Garcia, V., & Garcia-Raffi, L. M. (2015). PERIODIC SYSTEMS AS ROAD TRAFFIC NOISE REDUCING DEVICES: PROTOTYPE AND STANDARDIZATION. Environmental Engineering and Management Journal, 14(12), 2759-2769. doi:10.30638/eemj.2015.293Kandula, M. (2009). Broadband shock noise reduction in turbulent jets by water injection. Applied Acoustics, 70(7), 1009-1014. doi:10.1016/j.apacoust.2008.12.001Liu, Z., Zhang, X., Mao, Y., Zhu, Y. Y., Yang, Z., Chan, C. T., & Sheng, P. (2000). Locally Resonant Sonic Materials. Science, 289(5485), 1734-1736. doi:10.1126/science.289.5485.1734Fang, N., Xi, D., Xu, J., Ambati, M., Srituravanich, W., Sun, C., & Zhang, X. (2006). Ultrasonic metamaterials with negative modulus. Nature Materials, 5(6), 452-456. doi:10.1038/nmat1644Sugimoto, N., & Horioka, T. (1995). Dispersion characteristics of sound waves in a tunnel with an array of Helmholtz resonators. The Journal of the Acoustical Society of America, 97(3), 1446-1459. doi:10.1121/1.412085Theocharis, G., Richoux, O., García, V. R., Merkel, A., & Tournat, V. (2014). Limits of slow sound propagation and transparency in lossy, locally resonant periodic structures. New Journal of Physics, 16(9), 093017. doi:10.1088/1367-2630/16/9/093017Jiménez, N., Cox, T. J., Romero-García, V., & Groby, J.-P. (2017). Metadiffusers: Deep-subwavelength sound diffusers. Scientific Reports, 7(1). doi:10.1038/s41598-017-05710-5Ballestero, E., Jiménez, N., Groby, J.-P., Dance, S., Aygun, H., & Romero-García, V. (2019). Experimental validation of deep-subwavelength diffusion by acoustic metadiffusers. Applied Physics Letters, 115(8), 081901. doi:10.1063/1.5114877Romero-García, V., Sánchez-Pérez, J. V., & Garcia-Raffi, L. M. (2011). Tunable wideband bandstop acoustic filter based on two-dimensional multiphysical phenomena periodic systems. Journal of Applied Physics, 110(1), 014904. doi:10.1063/1.3599886Lagarrigue, C., Groby, J. P., & Tournat, V. (2013). Sustainable sonic crystal made of resonating bamboo rods. The Journal of the Acoustical Society of America, 133(1), 247-254. doi:10.1121/1.4769783Krynkin, A., Umnova, O., Yung Boon Chong, A., Taherzadeh, S., & Attenborough, K. (2010). Predictions and measurements of sound transmission through a periodic array of elastic shells in air. The Journal of the Acoustical Society of America, 128(6), 3496-3506. doi:10.1121/1.3506342Koussa, F., Defrance, J., Jean, P., & Blanc-Benon, P. (2013). Acoustical Efficiency of a Sonic Crystal Assisted Noise Barrier. Acta Acustica united with Acustica, 99(3), 399-409. doi:10.3813/aaa.918621Castiñeira-Ibáñez, S., Romero-García, V., Sánchez-Pérez, J. V., & Garcia-Raffi, L. M. (2010). Overlapping of acoustic bandgaps using fractal geometries. EPL (Europhysics Letters), 92(2), 24007. doi:10.1209/0295-5075/92/24007García-Chocano, V. M., Cabrera, S., & Sánchez-Dehesa, J. (2012). Broadband sound absorption by lattices of microperforated cylindrical shells. Applied Physics Letters, 101(18), 184101. doi:10.1063/1.4764560Lardeau, A., Groby, J.-P., & Romero-García, V. (2016). Broadband Transmission Loss Using the Overlap of Resonances in 3D Sonic Crystals. Crystals, 6(5), 51. doi:10.3390/cryst6050051Cavalieri, T., Cebrecos, A., Groby, J.-P., Chaufour, C., & Romero-García, V. (2019). Three-dimensional multiresonant lossy sonic crystal for broadband acoustic attenuation: Application to train noise reduction. Applied Acoustics, 146, 1-8. doi:10.1016/j.apacoust.2018.10.020Dimitrijević, S. M., García-Chocano, V. M., Cervera, F., Roth, E., & Sánchez-Dehesa, J. (2019). Sound Insulation and Reflection Properties of Sonic Crystal Barrier Based on Micro-Perforated Cylinders. Materials, 12(17), 2806. doi:10.3390/ma12172806Stinson, M. R. (1991). The propagation of plane sound waves in narrow and wide circular tubes, and generalization to uniform tubes of arbitrary cross‐sectional shape. The Journal of the Acoustical Society of America, 89(2), 550-558. doi:10.1121/1.400379Duclos, A., Lafarge, D., & Pagneux, V. (2009). Transmission of acoustic waves through 2D phononic crystal: visco-thermal and multiple scattering effects. The European Physical Journal Applied Physics, 45(1), 11302. doi:10.1051/epjap:2008203Romero-García, V., Theocharis, G., Richoux, O., & Pagneux, V. (2016). Use of complex frequency plane to design broadband and sub-wavelength absorbers. The Journal of the Acoustical Society of America, 139(6), 3395-3403. doi:10.1121/1.4950708Romero-García, V., Theocharis, G., Richoux, O., Merkel, A., Tournat, V., & Pagneux, V. (2016). Perfect and broadband acoustic absorption by critically coupled sub-wavelength resonators. Scientific Reports, 6(1). doi:10.1038/srep19519Jiménez, N., Huang, W., Romero-García, V., Pagneux, V., & Groby, J.-P. (2016). Ultra-thin metamaterial for perfect and quasi-omnidirectional sound absorption. Applied Physics Letters, 109(12), 121902. doi:10.1063/1.4962328Jiménez, N., Romero-García, V., Pagneux, V., & Groby, J.-P. (2017). Quasiperfect absorption by subwavelength acoustic panels in transmission using accumulation of resonances due to slow sound. Physical Review B, 95(1). doi:10.1103/physrevb.95.014205Jiménez, N., Romero-García, V., Pagneux, V., & Groby, J.-P. (2017). Rainbow-trapping absorbers: Broadband, perfect and asymmetric sound absorption by subwavelength panels for transmission problems. Scientific Reports, 7(1). doi:10.1038/s41598-017-13706-4Merkel, A., Theocharis, G., Richoux, O., Romero-García, V., & Pagneux, V. (2015). Control of acoustic absorption in one-dimensional scattering by resonant scatterers. Applied Physics Letters, 107(24), 244102. doi:10.1063/1.4938121Kergomard, J., & Garcia, A. (1987). Simple discontinuities in acoustic waveguides at low frequencies: Critical analysis and formulae. Journal of Sound and Vibration, 114(3), 465-479. doi:10.1016/s0022-460x(87)80017-2Sánchez-Dehesa, J., Garcia-Chocano, V. M., Torrent, D., Cervera, F., Cabrera, S., & Simon, F. (2011). Noise control by sonic crystal barriers made of recycled materials. The Journal of the Acoustical Society of America, 129(3), 1173-1183. doi:10.1121/1.3531815Christensen, J., Romero-García, V., Picó, R., Cebrecos, A., de Abajo, F. J. G., Mortensen, N. A., … Sánchez-Morcillo, V. J. (2014). Extraordinary absorption of sound in porous lamella-crystals. 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Acoustic Bessel-like beam formation by an axisymmetric grating

We report Bessel-like beam formation of acoustic waves by means of an axisymmetric grating of rigid tori. The results show that the generated beam pattern is similar to that of Bessel beams, characterized by elongated non-diffracting focal spots. A multiple foci structure is observed, due to the finite size of the lens. The dependence of the focal distance on the frequency is also discussed, on the basis of an extended grating theory. Experimental validation of acoustic Bessel-like beam formation is also reported for sound waves. The results can be generalized to wave beams of different nature, as optical or matter waves.The work was supported by the Spanish Ministry of Science and Innovation and the European Union FEDER through projects FIS2011-29731-C02-01 and -02, also MAT2009-09438, MTM2012-36740-C02-02 and UPV-PAID 2012/253. VR-G acknowledges financial support from the "Pays de la Loire" through the post-doctoral programme.Jimenez, N.; Romero García, V.; Picó Vila, R.; Cebrecos Ruiz, A.; Sánchez Morcillo, VJ.; García-Raffi, LM.; Sánchez Pérez, JV.... (2014). Acoustic Bessel-like beam formation by an axisymmetric grating. EPL. 106(2):240051-240055. doi:10.1209/0295-5075/106/24005240051240055106