J. Sánchez-pérez - Academia.edu (original) (raw)

Papers by J. Sánchez-pérez

The Journal of the Acoustical Society of America, 1999

The Journal of the Acoustical Society of America, 2000

Journal of Lightwave Technology, 1999

Theoretical and experimental determination of sonic band structures of two-dimensional (2-D) arra... more Theoretical and experimental determination of sonic band structures of two-dimensional (2-D) arrays of rigid cylinders in air is reported. We present measurements for square and triangular lattices. A variational method is employed to calculate the acoustic dispersion relation. Experimentally, a transmission technique and the analysis of the phase delay between the incident and scattered waves by the structure are used to construct the acoustic bands. The comparison between theory and experiments allows to fully characterize the band gaps and it has also demonstrated the existence of deaf bands; i.e., bands which cannot be excited due to symmetry reasons. For the case of square lattice we show that the structure with a filling fraction of 0.41 has a full acoustic gap.

The Journal of the Acoustical Society of America, 1998

We show that absolute sonic band gaps produced by two-dimensional square and triangular lattices ... more We show that absolute sonic band gaps produced by two-dimensional square and triangular lattices of rigid cylinders in air can be increased by reducing the structure symmetry. In the case of square lattices, symmetry reduction is achieved by a smaller diameter cylinder placed at the center of each unit cell. For triangular lattices the reduction is achieved by decreasing the diameter of the cylinder at the center of the hexagons in the lattice. Theoretical predictions are also demonstrated experimentally: starting from a honeycomb lattice (using cylinders of 4 cm of diameter size and 6.35 cm nearest-neighbor distance) we have studied the transition to a triangular symmetry by putting rods with increasing diameter (in the range 0.6-4 cm) at the center. The greatest enhancement of the attenuation strength observed in transmission experiments has been obtained in the high frequency region for diameter ratios in the range 0.1-0.3.

The Journal of the Acoustical Society of America, 2002

We show that a sonic crystal made of periodic distributions of rigid cylinders in air acts as a n... more We show that a sonic crystal made of periodic distributions of rigid cylinders in air acts as a new material which allows the construction of refractive acoustic devices for airborne sound. It is demonstrated that, in the long-wave regime, the crystal has low impedance and the sound is transmitted at subsonic velocities. Here, the fabrication and characterization of a convergent lens are presented. Also, an example of a Fabry-Perot interferometer based on this crystal is analyzed. It is concluded that refractive devices based on sonic crystals behave in a manner similar to that of optical systems.

The elastic and geometrical properties of low density polyethylene (LDPE) foam are used in this p... more The elastic and geometrical properties of low density polyethylene (LDPE) foam are used in this paper to improve the attenuation properties of periodic arrangements of acoustic scatterers known as Sonic Crystals (SCs). A specific recycled profile of LDPE foam is used as elastic-acoustic scatterer. The acoustic spectrum of the single scatterer shows two attenuation peaks in the low frequency range. If the elastic scatterers are arranged as a SC, the attenuation spectrum shows the attenuation peaks of the single scatterers at the same position, as well as the attenuation peaks related to the periodicity of the structure. COMSOL Multiphysics is used to reproduce numerically the frequency response measured experimentally for both the single scatterers and the periodic arrangement. The results provided by COMSOL are in good agreement with the experimental data.

The acoustical performances of regular arrays of cylindrical elements with their axes aligned and... more The acoustical performances of regular arrays of cylindrical elements with their axes aligned and parallel to a ground plane have been investigated through predictions and laboratory experiments. Semianalytical predictions based on multiple scattering theory and numerical simulations based on a Boundary Element formulation have been made. In an anechoic chamber, arrays of (a) cylindrical acousticallyrigid scatterers (PVC pipes) and (b) thin elastic shells have been installed with their axis parallel to ground planes consisting either of Medium Density Fibreboard (MDF) plate or a sheet of partially reticulated polyurethane foam. Measurements of Insertion Loss (IL) spectra due to the arrays have been made without and with ground planes for several receiver heights. The data have been compared with predictions and numerical simulations. The minima in the excess attenuation spectrum due to the ground alone resulting from destructive interference between direct and ground-reflected sound waves, tend to have an adverse influence on the band gaps related to a periodic array in the free field when these two effects coincide. On the other hand, the presence of rigid ground may result in an IL for an array near the ground similar to or, in the case of the first Bragg band gap, greater than that resulting from a double array, equivalent to the original array plus its ground plane mirror image, in the free field.

The Journal of the Acoustical Society of America, 2011

The acoustical performances of regular arrays of cylindrical elements with their axes aligned and... more The acoustical performances of regular arrays of cylindrical elements with their axes aligned and parallel to a ground plane have been investigated through predictions and laboratory experiments. Semi-analytical predictions based on multiple scattering theory and numerical simulations based on Boundary Elements formulation have been made. In an anechoic chamber, 7 × 3 arrays of solid scatterers (hollow PVC cylinders of 55 mm diameter) have been located near to ground planes consisting of Medium Density Fibreboard (MDF) plate or sheets of porous materials. Measurements of Insertion Loss (IL) spectra due to the arrays have been made without and with ground planes for several source and receiver heights. The resulting data have been compared with predictions and numerical simulations. The introduction of a ground plane is shown to have noticeable influences on free field IL spectra. The effects are most adverse when the frequencies of the band gaps coincide with the minima in the excess attenuation spectrum due to the ground alone. On the other hand if the ground effect excess attenuation minimum coincides with an array pass band then it is retained in the insertion loss. Cases are identified where it is possible to obtain an insertion loss with the array in the presence of ground that is greater than that of the same array in the free field.

The use of sonic crystals (SCs) as environmental noise barriers has certain advantages from both ... more The use of sonic crystals (SCs) as environmental noise barriers has certain advantages from both the acoustical and the constructive points of view with regard to conventional ones. However, the interaction between the SCs and the ground has not been studied yet. In this work we are reporting a semi-analytical model, based on the multiple scattering theory and on the method of images, to study this interaction considering the ground as a finite impedance surface. The results obtained here show that this model could be used to design more effective noise barriers based on SCs because the excess attenuation of the ground could be modelled in order to improve the attenuation properties of the array of scatterers. The results are compared with experimental data and numerical predictions thus finding good agreement between them. (Some figures in this article are in colour only in the electronic version)

The Journal of the Acoustical Society of America, 2010

ABSTRACT Sonic crystals (SCs) are periodic arrangement of scatterers embedded in a fluid medium w... more ABSTRACT Sonic crystals (SCs) are periodic arrangement of scatterers embedded in a fluid medium with different elastic properties. One of the main acoustical properties of these materials is the existence of ranges of frequencies where the transmission is forbidden according to Bragg's law of diffraction. This property enables the use of these devices in the design of acoustic screens for environmental noise. One factor that can affect the attenuation of these systems is the presence of the ground. Experimental, numerical, and analytical results are shown in this work, related to the influence of the hard ground on the attenuation properties of 2-D SC consisting of rigid cylinder arrays with the cylinder axes parallel to the ground.

The acoustical performances of regular arrays of cylindrical elements with their axes aligned and... more The acoustical performances of regular arrays of cylindrical elements with their axes aligned and parallel to a ground plane have been investigated through predictions and laboratory experiments. Semianalytical predictions based on multiple scattering theory and numerical simulations based on a Boundary Element formulation have been made. In an anechoic chamber, arrays of (a) cylindrical acousticallyrigid scatterers (PVC pipes) and (b) thin elastic shells have been installed with their axis parallel to ground planes consisting either of Medium Density Fibreboard (MDF) plate or a sheet of partially reticulated polyurethane foam. Measurements of Insertion Loss (IL) spectra due to the arrays have been made without and with ground planes for several receiver heights. The data have been compared with predictions and numerical simulations. The minima in the excess attenuation spectrum due to the ground alone resulting from destructive interference between direct and ground-reflected sound waves, tend to have an adverse influence on the band gaps related to a periodic array in the free field when these two effects coincide. On the other hand, the presence of rigid ground may result in an IL for an array near the ground similar to or, in the case of the first Bragg band gap, greater than that resulting from a double array, equivalent to the original array plus its ground plane mirror image, in the free field.

Let λ be a countably additive vector measure with values in a separable real Hilbert space H. We ... more Let λ be a countably additive vector measure with values in a separable real Hilbert space H. We define and study a pseudo metric on a Banach lattice of integrable functions related to λ that we call a λ-weighted distance. We compute the best approximation with respect to this distance to elements of the function space by the use of sequences with special geometric properties. The requirements on the sequence of functions are given in terms of a commutation relation between these functions that involves integration with respect to λ. We also compare the approximation that is obtained in this way with the corresponding projection on a particular Hilbert space.

SUMMARY In this article we present a program of interdisciplinary practices for a first engineeri... more SUMMARY In this article we present a program of interdisciplinary practices for a first engineering course. We followed this program during the course 1996-1997, and we give several examples of the kind of exercises that we used. We also present some remarks and conclusions that we have achieved in our experience. Our purpose is to introduce a teaching strategy for Mathematics and Physics at the University level.

Sonic crystals are periodic structures that have acoustic band gaps centred at frequencies depend... more Sonic crystals are periodic structures that have acoustic band gaps centred at frequencies depending on the lattice constant of the array and on the direction of the incident acoustic wave. To eliminate this dependence, this work presents designed mixed structures constructed with rigid scatterers and resonators embedded in air. Specifically, balloons filled with a blend of air and helium were used as resonators, showing experimental evidence about the resonant behavior of an array formed with these balloons. As a result, the authors obtain full band gaps in a predetermined range of frequencies desired.

Sonic crystals are defined as materials consisting of periodic distributions of acoustic scattere... more Sonic crystals are defined as materials consisting of periodic distributions of acoustic scatterers in another medium with different elastic properties. These materials present two main acoustical features: they present ranges of frequencies where the acoustic wave propagation is forbidden and, moreover, they can be used to construct acoustical refractive devices. We present here an interferometric method of characterizing directly the refraction index of these materials. We focused our research on sonic crystals constructed with a periodic distribution of rigid cylinders in air. The interferometric method used is based on the acoustical analog of an optical device called Fresnel's biprism. Our numerical predictions are in good agreement with the experimental results obtained and allow us to estimate the refraction index for very low frequencies. Moreover, in this range we show the nonlinear behavior in the dispersion relation refraction index versus frequency. Finally, we analyze the relationship between the refraction index and some typical sonic crystal parameters. In view of the results we can conclude that this method offers certain advantages with respect to the classical method used to obtain the refraction index in sonic crystals with airborne propagation: the phase delay method.

An improvement in the attenuation capabilities of acoustic metamaterials by means of the creation... more An improvement in the attenuation capabilities of acoustic metamaterials by means of the creation of defects is considered here as a multiobjective optimization problem. From this point of view, it is possible to define the optimum strategy in the creation of defects to achieve an important increase in acoustic attenuation in a predetermined range of frequencies. A powerful multiobjective optimization algorithm called evMOGA has been used to solve this problem. The study has been restricted to the case of a two-dimensional sonic crystal formed by rigid cylinders in air, the defects being vacancies in the initial structure.

This paper shows a promising method for acoustic barrier design using a new acoustic material cal... more This paper shows a promising method for acoustic barrier design using a new acoustic material called Sonic Crystals (SCs). The configuration of these SCs is set as a multiobjective optimization problem which is very difficult to solve with conventional optimization techniques. The paper presents a new parallel implementation of a Multiobjective Evolutionary Algorithm called ev-MOGA (also known as-MOGA) and its application in a complex design problem. ev-MOGA algorithm has been designed to converge towards a reduced, but well distributed, representation of the Pareto Front (solution of the multiobjective optimization problem). The algorithm is presented in detail and its most important properties are discussed. To reduce the ev-MOGA computational cost when objective functions are substantial, a basic paralleliza-tion has been implemented on a distributed platform.

An exhaustive study has been made into the potential improvement in attenuation and focusing of p... more An exhaustive study has been made into the potential improvement in attenuation and focusing of phononic crystal arrays resulting from the deliberate creation of vacancies. Use is made of a stochastic search algorithm based on evolutionary algorithms called the epsilon variable multi-objective genetic algorithm which, in conjunction with the application of multiple scattering theory, enables the design of devices for effectively controlling sound waves. Several parameters are analyzed, including the symmetries used in the distribution of holes and the optimum number of holes. The validity and utility of the general rules obtained have been confirmed experimentally.

The transmission of acoustic waves in a fractal distribution of rigid scatterers embedded in air ... more The transmission of acoustic waves in a fractal distribution of rigid scatterers embedded in air is reported in this work. The Sierpinsky fractal is used to produce a compact small device containing several periodicities, therefore the fractal distribution contains several finite sonic crystals. The attenuation band produced by the fractal distribution results from the sum of the Bragg peaks of each periodicity. On the other hand, bandgaps of sonic crystal depend on the well-known filling fraction, thus the radii of the scatterers in the fractal distribution has been optimized using genetic algorithm in order to overlap the bandgaps of each periodicity obtaining a wide and full attenuation band.

We show both experimentally and theoretically the evanescent behavior of modes in the band gap of... more We show both experimentally and theoretically the evanescent behavior of modes in the band gap of finite phononic crystal PC. Based on experimental and numerical data we obtain the imaginary part of the wave vector in good agreement with the complex band structures obtained by the extended plane wave expansion. The calculated and measured acoustic field of a localized mode out of the point defect inside the PC presents also evanescent behavior. The correct understanding of evanescent modes is fundamental for designing narrow filters and waveguides based on PCs with defects. During the past few years, there has been a great deal of interest in studying propagation of waves inside periodic structures. These systems are composites made of inhomoge-neous distribution of some material periodically embedded in other with different physical properties. Phononic crystals PC 1,2 are one of the examples of these systems. PCs are the extension of the so-called photonic crystals 3 when elastic and acoustic waves propagate in periodic structures made of materials with different elastic properties. When one of these elastic materials is a fluid medium, then PC are called sonic crystals SC. 4,5 For these artificial materials, both theoretical and experimental results have shown several interesting physical properties. 6 In the homogenization limit, 7 it is possible to design acoustic metamaterials that can be used to build re-fractive devices. 8 In the range of wavelengths similar to the periodicity of the PC a, multiple scattering process inside the PC leads to the phenomenon of so called band gaps BGs, which are required for filtering sound, 5 trapping sound in defects, 9,10 and for acoustic wave guiding. 11 Propagating waves inside a periodic media are a set of solutions of the wave equations satisfying the translational symmetry property. However, periodic media with point defects where the translational symmetry is broken, or finite periodic media, can support evanescent modes as well. Recently Laude et al. 13 have analyzed the evanescent Bloch waves and the complex band structure of PC. Complex band structures show bands that are simply not revealed by the traditional k method. By means of the complex band structures, BG can be defined as ranges of frequencies where all Bloch waves must be evanescent. The goal of the paper is to characterize the evanescent behavior of waves with frequencies in the BG inside of PC. Analytical, numerical, and experimental data show the evi-dences for the exponential-like decay of these modes. En-gelen et al. 12 shown that modes inside BG in photonic crystals decay multiexponentially. Supercell approximation in extended plane wave expansion EPWE 13–16 has been used in the present work to determine the imaginary part of the wave vector of evanescent modes. Specifically, we have deduced both analytically and experimentally the imaginary part of the wave vector observing that only the first harmonic contributes substantially to the decay of the acoustic field inside complete SC. In all cases we have obtained a very good agreement between theoretical and experimental results. We have performed experiments in an echo-free chamber of dimensions 8 6 3 m 3. The finite two-dimensional SC used in this paper forms a square array with lattice constant a = 22 cm. The size of the SC is 5a 5a and the radius of the cylinders is r = 10 cm. A prepolarized free-field 1 / 2 microphone Type 4189 B&K has been used throughout the experiments. The diameter of the microphone is 1.32 cm, which is approximately 0.06a. Our system 3DReAMS three-dimensional Robotized e-Acoustic Measurement System is capable of sweeping the microphone through a 3D grid of measuring points located at any trajectory inside the echo-free chamber. Motion of the robot is controlled by NI-PCI 7334. Figure 1 shows the complex and real band structures for the SC with a point defect. The complex band structures and the value of the k number for the modes inside the BG can be obtained by EPWE and it becomes in a purely real value for the localized mode. We can observe the localized mode at 920 Hz continuous line. That value exactly coincides with the value obtained by PWE with supercell approximation. We have compared these results with experimental data measuring the insertion loss IL behind the SC with and without the point defect. In Fig. 1, we can observe that the experimental IL for the localized mode at frequency 920 Hz continuous line is lower than the case of the complete SC dashed line, i.e., it can be concluded that there is a passing mode. This results because the localized mode is not killed completely by the SC around the point defect see also Ref. 10. In fact, although the localized mode has an evanescent behavior, as we will see later, in this case there is not enough number of rows around the point defect to kill it. For frequencies in the BG, the borders of the point defect act as perfect mirrors producing the localization in this a Electronic mail: virogar1@mat.upv.es.

The Journal of the Acoustical Society of America, 1999

The Journal of the Acoustical Society of America, 2000

Journal of Lightwave Technology, 1999

Theoretical and experimental determination of sonic band structures of two-dimensional (2-D) arra... more Theoretical and experimental determination of sonic band structures of two-dimensional (2-D) arrays of rigid cylinders in air is reported. We present measurements for square and triangular lattices. A variational method is employed to calculate the acoustic dispersion relation. Experimentally, a transmission technique and the analysis of the phase delay between the incident and scattered waves by the structure are used to construct the acoustic bands. The comparison between theory and experiments allows to fully characterize the band gaps and it has also demonstrated the existence of deaf bands; i.e., bands which cannot be excited due to symmetry reasons. For the case of square lattice we show that the structure with a filling fraction of 0.41 has a full acoustic gap.

The Journal of the Acoustical Society of America, 1998

We show that absolute sonic band gaps produced by two-dimensional square and triangular lattices ... more We show that absolute sonic band gaps produced by two-dimensional square and triangular lattices of rigid cylinders in air can be increased by reducing the structure symmetry. In the case of square lattices, symmetry reduction is achieved by a smaller diameter cylinder placed at the center of each unit cell. For triangular lattices the reduction is achieved by decreasing the diameter of the cylinder at the center of the hexagons in the lattice. Theoretical predictions are also demonstrated experimentally: starting from a honeycomb lattice (using cylinders of 4 cm of diameter size and 6.35 cm nearest-neighbor distance) we have studied the transition to a triangular symmetry by putting rods with increasing diameter (in the range 0.6-4 cm) at the center. The greatest enhancement of the attenuation strength observed in transmission experiments has been obtained in the high frequency region for diameter ratios in the range 0.1-0.3.

The Journal of the Acoustical Society of America, 2002

We show that a sonic crystal made of periodic distributions of rigid cylinders in air acts as a n... more We show that a sonic crystal made of periodic distributions of rigid cylinders in air acts as a new material which allows the construction of refractive acoustic devices for airborne sound. It is demonstrated that, in the long-wave regime, the crystal has low impedance and the sound is transmitted at subsonic velocities. Here, the fabrication and characterization of a convergent lens are presented. Also, an example of a Fabry-Perot interferometer based on this crystal is analyzed. It is concluded that refractive devices based on sonic crystals behave in a manner similar to that of optical systems.

The elastic and geometrical properties of low density polyethylene (LDPE) foam are used in this p... more The elastic and geometrical properties of low density polyethylene (LDPE) foam are used in this paper to improve the attenuation properties of periodic arrangements of acoustic scatterers known as Sonic Crystals (SCs). A specific recycled profile of LDPE foam is used as elastic-acoustic scatterer. The acoustic spectrum of the single scatterer shows two attenuation peaks in the low frequency range. If the elastic scatterers are arranged as a SC, the attenuation spectrum shows the attenuation peaks of the single scatterers at the same position, as well as the attenuation peaks related to the periodicity of the structure. COMSOL Multiphysics is used to reproduce numerically the frequency response measured experimentally for both the single scatterers and the periodic arrangement. The results provided by COMSOL are in good agreement with the experimental data.

The acoustical performances of regular arrays of cylindrical elements with their axes aligned and... more The acoustical performances of regular arrays of cylindrical elements with their axes aligned and parallel to a ground plane have been investigated through predictions and laboratory experiments. Semianalytical predictions based on multiple scattering theory and numerical simulations based on a Boundary Element formulation have been made. In an anechoic chamber, arrays of (a) cylindrical acousticallyrigid scatterers (PVC pipes) and (b) thin elastic shells have been installed with their axis parallel to ground planes consisting either of Medium Density Fibreboard (MDF) plate or a sheet of partially reticulated polyurethane foam. Measurements of Insertion Loss (IL) spectra due to the arrays have been made without and with ground planes for several receiver heights. The data have been compared with predictions and numerical simulations. The minima in the excess attenuation spectrum due to the ground alone resulting from destructive interference between direct and ground-reflected sound waves, tend to have an adverse influence on the band gaps related to a periodic array in the free field when these two effects coincide. On the other hand, the presence of rigid ground may result in an IL for an array near the ground similar to or, in the case of the first Bragg band gap, greater than that resulting from a double array, equivalent to the original array plus its ground plane mirror image, in the free field.

The Journal of the Acoustical Society of America, 2011

The acoustical performances of regular arrays of cylindrical elements with their axes aligned and... more The acoustical performances of regular arrays of cylindrical elements with their axes aligned and parallel to a ground plane have been investigated through predictions and laboratory experiments. Semi-analytical predictions based on multiple scattering theory and numerical simulations based on Boundary Elements formulation have been made. In an anechoic chamber, 7 × 3 arrays of solid scatterers (hollow PVC cylinders of 55 mm diameter) have been located near to ground planes consisting of Medium Density Fibreboard (MDF) plate or sheets of porous materials. Measurements of Insertion Loss (IL) spectra due to the arrays have been made without and with ground planes for several source and receiver heights. The resulting data have been compared with predictions and numerical simulations. The introduction of a ground plane is shown to have noticeable influences on free field IL spectra. The effects are most adverse when the frequencies of the band gaps coincide with the minima in the excess attenuation spectrum due to the ground alone. On the other hand if the ground effect excess attenuation minimum coincides with an array pass band then it is retained in the insertion loss. Cases are identified where it is possible to obtain an insertion loss with the array in the presence of ground that is greater than that of the same array in the free field.

The use of sonic crystals (SCs) as environmental noise barriers has certain advantages from both ... more The use of sonic crystals (SCs) as environmental noise barriers has certain advantages from both the acoustical and the constructive points of view with regard to conventional ones. However, the interaction between the SCs and the ground has not been studied yet. In this work we are reporting a semi-analytical model, based on the multiple scattering theory and on the method of images, to study this interaction considering the ground as a finite impedance surface. The results obtained here show that this model could be used to design more effective noise barriers based on SCs because the excess attenuation of the ground could be modelled in order to improve the attenuation properties of the array of scatterers. The results are compared with experimental data and numerical predictions thus finding good agreement between them. (Some figures in this article are in colour only in the electronic version)

The Journal of the Acoustical Society of America, 2010

ABSTRACT Sonic crystals (SCs) are periodic arrangement of scatterers embedded in a fluid medium w... more ABSTRACT Sonic crystals (SCs) are periodic arrangement of scatterers embedded in a fluid medium with different elastic properties. One of the main acoustical properties of these materials is the existence of ranges of frequencies where the transmission is forbidden according to Bragg's law of diffraction. This property enables the use of these devices in the design of acoustic screens for environmental noise. One factor that can affect the attenuation of these systems is the presence of the ground. Experimental, numerical, and analytical results are shown in this work, related to the influence of the hard ground on the attenuation properties of 2-D SC consisting of rigid cylinder arrays with the cylinder axes parallel to the ground.

The acoustical performances of regular arrays of cylindrical elements with their axes aligned and... more The acoustical performances of regular arrays of cylindrical elements with their axes aligned and parallel to a ground plane have been investigated through predictions and laboratory experiments. Semianalytical predictions based on multiple scattering theory and numerical simulations based on a Boundary Element formulation have been made. In an anechoic chamber, arrays of (a) cylindrical acousticallyrigid scatterers (PVC pipes) and (b) thin elastic shells have been installed with their axis parallel to ground planes consisting either of Medium Density Fibreboard (MDF) plate or a sheet of partially reticulated polyurethane foam. Measurements of Insertion Loss (IL) spectra due to the arrays have been made without and with ground planes for several receiver heights. The data have been compared with predictions and numerical simulations. The minima in the excess attenuation spectrum due to the ground alone resulting from destructive interference between direct and ground-reflected sound waves, tend to have an adverse influence on the band gaps related to a periodic array in the free field when these two effects coincide. On the other hand, the presence of rigid ground may result in an IL for an array near the ground similar to or, in the case of the first Bragg band gap, greater than that resulting from a double array, equivalent to the original array plus its ground plane mirror image, in the free field.

Let λ be a countably additive vector measure with values in a separable real Hilbert space H. We ... more Let λ be a countably additive vector measure with values in a separable real Hilbert space H. We define and study a pseudo metric on a Banach lattice of integrable functions related to λ that we call a λ-weighted distance. We compute the best approximation with respect to this distance to elements of the function space by the use of sequences with special geometric properties. The requirements on the sequence of functions are given in terms of a commutation relation between these functions that involves integration with respect to λ. We also compare the approximation that is obtained in this way with the corresponding projection on a particular Hilbert space.

SUMMARY In this article we present a program of interdisciplinary practices for a first engineeri... more SUMMARY In this article we present a program of interdisciplinary practices for a first engineering course. We followed this program during the course 1996-1997, and we give several examples of the kind of exercises that we used. We also present some remarks and conclusions that we have achieved in our experience. Our purpose is to introduce a teaching strategy for Mathematics and Physics at the University level.

Sonic crystals are periodic structures that have acoustic band gaps centred at frequencies depend... more Sonic crystals are periodic structures that have acoustic band gaps centred at frequencies depending on the lattice constant of the array and on the direction of the incident acoustic wave. To eliminate this dependence, this work presents designed mixed structures constructed with rigid scatterers and resonators embedded in air. Specifically, balloons filled with a blend of air and helium were used as resonators, showing experimental evidence about the resonant behavior of an array formed with these balloons. As a result, the authors obtain full band gaps in a predetermined range of frequencies desired.

Sonic crystals are defined as materials consisting of periodic distributions of acoustic scattere... more Sonic crystals are defined as materials consisting of periodic distributions of acoustic scatterers in another medium with different elastic properties. These materials present two main acoustical features: they present ranges of frequencies where the acoustic wave propagation is forbidden and, moreover, they can be used to construct acoustical refractive devices. We present here an interferometric method of characterizing directly the refraction index of these materials. We focused our research on sonic crystals constructed with a periodic distribution of rigid cylinders in air. The interferometric method used is based on the acoustical analog of an optical device called Fresnel's biprism. Our numerical predictions are in good agreement with the experimental results obtained and allow us to estimate the refraction index for very low frequencies. Moreover, in this range we show the nonlinear behavior in the dispersion relation refraction index versus frequency. Finally, we analyze the relationship between the refraction index and some typical sonic crystal parameters. In view of the results we can conclude that this method offers certain advantages with respect to the classical method used to obtain the refraction index in sonic crystals with airborne propagation: the phase delay method.

An improvement in the attenuation capabilities of acoustic metamaterials by means of the creation... more An improvement in the attenuation capabilities of acoustic metamaterials by means of the creation of defects is considered here as a multiobjective optimization problem. From this point of view, it is possible to define the optimum strategy in the creation of defects to achieve an important increase in acoustic attenuation in a predetermined range of frequencies. A powerful multiobjective optimization algorithm called evMOGA has been used to solve this problem. The study has been restricted to the case of a two-dimensional sonic crystal formed by rigid cylinders in air, the defects being vacancies in the initial structure.

This paper shows a promising method for acoustic barrier design using a new acoustic material cal... more This paper shows a promising method for acoustic barrier design using a new acoustic material called Sonic Crystals (SCs). The configuration of these SCs is set as a multiobjective optimization problem which is very difficult to solve with conventional optimization techniques. The paper presents a new parallel implementation of a Multiobjective Evolutionary Algorithm called ev-MOGA (also known as-MOGA) and its application in a complex design problem. ev-MOGA algorithm has been designed to converge towards a reduced, but well distributed, representation of the Pareto Front (solution of the multiobjective optimization problem). The algorithm is presented in detail and its most important properties are discussed. To reduce the ev-MOGA computational cost when objective functions are substantial, a basic paralleliza-tion has been implemented on a distributed platform.

An exhaustive study has been made into the potential improvement in attenuation and focusing of p... more An exhaustive study has been made into the potential improvement in attenuation and focusing of phononic crystal arrays resulting from the deliberate creation of vacancies. Use is made of a stochastic search algorithm based on evolutionary algorithms called the epsilon variable multi-objective genetic algorithm which, in conjunction with the application of multiple scattering theory, enables the design of devices for effectively controlling sound waves. Several parameters are analyzed, including the symmetries used in the distribution of holes and the optimum number of holes. The validity and utility of the general rules obtained have been confirmed experimentally.

The transmission of acoustic waves in a fractal distribution of rigid scatterers embedded in air ... more The transmission of acoustic waves in a fractal distribution of rigid scatterers embedded in air is reported in this work. The Sierpinsky fractal is used to produce a compact small device containing several periodicities, therefore the fractal distribution contains several finite sonic crystals. The attenuation band produced by the fractal distribution results from the sum of the Bragg peaks of each periodicity. On the other hand, bandgaps of sonic crystal depend on the well-known filling fraction, thus the radii of the scatterers in the fractal distribution has been optimized using genetic algorithm in order to overlap the bandgaps of each periodicity obtaining a wide and full attenuation band.

We show both experimentally and theoretically the evanescent behavior of modes in the band gap of... more We show both experimentally and theoretically the evanescent behavior of modes in the band gap of finite phononic crystal PC. Based on experimental and numerical data we obtain the imaginary part of the wave vector in good agreement with the complex band structures obtained by the extended plane wave expansion. The calculated and measured acoustic field of a localized mode out of the point defect inside the PC presents also evanescent behavior. The correct understanding of evanescent modes is fundamental for designing narrow filters and waveguides based on PCs with defects. During the past few years, there has been a great deal of interest in studying propagation of waves inside periodic structures. These systems are composites made of inhomoge-neous distribution of some material periodically embedded in other with different physical properties. Phononic crystals PC 1,2 are one of the examples of these systems. PCs are the extension of the so-called photonic crystals 3 when elastic and acoustic waves propagate in periodic structures made of materials with different elastic properties. When one of these elastic materials is a fluid medium, then PC are called sonic crystals SC. 4,5 For these artificial materials, both theoretical and experimental results have shown several interesting physical properties. 6 In the homogenization limit, 7 it is possible to design acoustic metamaterials that can be used to build re-fractive devices. 8 In the range of wavelengths similar to the periodicity of the PC a, multiple scattering process inside the PC leads to the phenomenon of so called band gaps BGs, which are required for filtering sound, 5 trapping sound in defects, 9,10 and for acoustic wave guiding. 11 Propagating waves inside a periodic media are a set of solutions of the wave equations satisfying the translational symmetry property. However, periodic media with point defects where the translational symmetry is broken, or finite periodic media, can support evanescent modes as well. Recently Laude et al. 13 have analyzed the evanescent Bloch waves and the complex band structure of PC. Complex band structures show bands that are simply not revealed by the traditional k method. By means of the complex band structures, BG can be defined as ranges of frequencies where all Bloch waves must be evanescent. The goal of the paper is to characterize the evanescent behavior of waves with frequencies in the BG inside of PC. Analytical, numerical, and experimental data show the evi-dences for the exponential-like decay of these modes. En-gelen et al. 12 shown that modes inside BG in photonic crystals decay multiexponentially. Supercell approximation in extended plane wave expansion EPWE 13–16 has been used in the present work to determine the imaginary part of the wave vector of evanescent modes. Specifically, we have deduced both analytically and experimentally the imaginary part of the wave vector observing that only the first harmonic contributes substantially to the decay of the acoustic field inside complete SC. In all cases we have obtained a very good agreement between theoretical and experimental results. We have performed experiments in an echo-free chamber of dimensions 8 6 3 m 3. The finite two-dimensional SC used in this paper forms a square array with lattice constant a = 22 cm. The size of the SC is 5a 5a and the radius of the cylinders is r = 10 cm. A prepolarized free-field 1 / 2 microphone Type 4189 B&K has been used throughout the experiments. The diameter of the microphone is 1.32 cm, which is approximately 0.06a. Our system 3DReAMS three-dimensional Robotized e-Acoustic Measurement System is capable of sweeping the microphone through a 3D grid of measuring points located at any trajectory inside the echo-free chamber. Motion of the robot is controlled by NI-PCI 7334. Figure 1 shows the complex and real band structures for the SC with a point defect. The complex band structures and the value of the k number for the modes inside the BG can be obtained by EPWE and it becomes in a purely real value for the localized mode. We can observe the localized mode at 920 Hz continuous line. That value exactly coincides with the value obtained by PWE with supercell approximation. We have compared these results with experimental data measuring the insertion loss IL behind the SC with and without the point defect. In Fig. 1, we can observe that the experimental IL for the localized mode at frequency 920 Hz continuous line is lower than the case of the complete SC dashed line, i.e., it can be concluded that there is a passing mode. This results because the localized mode is not killed completely by the SC around the point defect see also Ref. 10. In fact, although the localized mode has an evanescent behavior, as we will see later, in this case there is not enough number of rows around the point defect to kill it. For frequencies in the BG, the borders of the point defect act as perfect mirrors producing the localization in this a Electronic mail: virogar1@mat.upv.es.