Spatial Laplace transform for complex wavenumber recovery and its application to the analysis of attenuation in acoustic systems

Plane-wave expansions (PWEs) based on Fourier transform and their physical interpretation are discussed for the case of homogeneous and isotropic lossy media. Albeit being mathematically correct, standard Fourier-based definition leads to nonphysical properties, such as the absence of homogeneous plane waves, lack of dissipation along transversal directions and inaccurate identification of single plane waves. Generalizing the PWE definition using Laplace transform, which amounts to switching to complex spectral variables, is shown to solve these issues, reinstating physical consistency. This approach no longer leads to a unique PWE for a field distribution, as it allows an infinite number of equivalent definitions, implying that the interpretation of the individual components of a PWE as physical plane waves does not appear as justified. The multiplicity of the generalized definitions is illustrated by applying it to the near-field radiation of an elementary electric dipole, for different choices of Laplace cuts, showing the main differences in the generalized PWEs.

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“…This property underlies a recent application of Laplace transform for parameter identification of plane waves in acoustic fields [17].…”

Section: Generalized Plane-wave Expansions and Propagatorsmentioning

confidence: 86%

Implications in the interpretation of plane-wave expansions in lossy media and the need for a generalized definition

Cozza

Aberbour²,

Derat³

2018

Wave Motion

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“…It is known for instance that in highly damped systems the flat branches of the dispersion curves tend to disappear (i.e. they are non-propagative solutions) [71] while band gaps show a lower attenuation capacity. This could affect also the efficiency of the conversion presented in this article.…”

Section: Discussionmentioning

confidence: 99%

The multi-physics metawedge: graded arrays on fluid-loaded elastic plates and the mechanical analogues of rainbow trapping and mode conversion

et al. 2018

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We consider the propagation and mode conversion of flexural-acoustic waves along a fluid-loaded graded array of elastic resonators, forming a metasurface. The multi-physics nature of the problem, coupling two disparate physical systems, brings both challenges and novel features not previously seen in so-called bifunctional metamaterials. In particular, by using an appropriately designed graded array of resonators, we show that it is possible to employ our metasurface to mode-convert sub-sonic surface flexural waves into bulk acoustic waves and vice-versa; transferring energy between two very different physical systems. Whilst the sub-sonic mechanical surface wave is dispersive, the bulk acoustic wave is dispersionless and radiates energy at infinity. We also show that this bifunctional metasurface is capable of exhibiting the classical effect of rainbow trapping for sub-sonic surface waves.An active area of research in metasurfaces [25] focuses upon graded resonator metasurfaces, where the general concept is that for a graded surface, or waveguide, different wavelengths are trapped at different spatial positions. Sub-wavelength microstructures are commonly employed in two ways: the first approach is to create effective macroscopic wavespeeds that vary spatially, thus achieving the required control of wave propagation. The second approach involves using deep sub-wavelength resonances to obtain the desired effects. Whilst the latter approach is significantly more difficult to create, it is far more powerful; hence our aim in the present paper is to extend this latter concept to fluid-loaded compliant structures. These ideas are being widely adopted in photonics and phononics due to their excellent abilities to control, manipulate and filter waves in compact devices. Graded and chirped designs include: trapping in rainbow devices [26][27][28][29], flat focussing mirrors and lenses in optics, plasmonics and acoustics [30][31][32][33][34][35], gradient index lens for acoustic and flexural waves focussing [36,37], acoustic absorbers [38][39][40] and sound enhancement [41,42].In the absence of the fluid-loading, it is only very recently that graded sub-wavelength structures for thin elastic plates has been considered as a chirped graded array [43], thin beams [44] or in the context of gradient index (GRIN) lenses created by graded structuration to control elastic symmetric (S 0 ) and antisymmetric (A 0 ) waves [45] and to obtain deeply sub-wavelength focussing [46], cloaking [47] and GRIN lenses (e.g. Luneburg, Maxwell-fisheye and Eaton lenses [44, 48]) using resonator arrays [37]. There is also an active community in so-called platonics [49] studying the generalisations of phononic crystals to elastic plates in-vacuo with considerable progress in pulling out features associated with Dirac cones [50], dynamic anisotropy and lensing /shielding effects [51]. Much of this is concerned with mass-loaded or constrained plates, but recent work on resonator arrays on plates [52] has moved this into contemporary areas of physics...

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“…The wave propagation within an elastic solid can be controlled by the interaction of the elastic wave with nonlinear isolated resonators, namely elastic spheres in contact. These features have been used to design an acoustic rectifier [3], tunable functional switches [4], tunable phononic crystals [5] or to study the attenuation of surface waves in a colloidal based metamaterial [6,7]. Nevertheless, the fine design of a metamaterial based on the nonlinear behavior of granular materials requires a quantitative agreement between the model of contact and the actual experimental behavior.…”

Section: Introductionmentioning

confidence: 99%

Testing a bead-rod contact with a nonlinear resonance method

Merkel

Theocharis

Allein

et al. 2019

Journal of Sound and Vibration

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We study the dynamics of an elastic structure composed of a cylindrical rod in contact with a bead at one extremity. Wave propagation within the cylindrical rod is considered linear and dispersionless while the bead-rod contact shows a highly nonlinear behavior as expected from the Hertz's model of contact. The resonance curves of the nonlinear contact depend on the excitation amplitude, where a downshift of the resonance frequency with increasing excitation amplitude is observed. The prediction of the resonance frequency shift by the Hertz's model is compared to the experimental results and shows a disagreement. A better agreement is found by considering the losses with a viscoelastic model, namely the Kuwabara and Kono or Brilliantov model. The observation of the nonlinear effects linked to the resonance of the massspring system can lead to the design of nonlinear elastic metamaterials, where the wave propagation is controlled by nonlinear isolated resonators.

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Spatial Laplace transform for complex wavenumber recovery and its application to the analysis of attenuation in acoustic systems

Cited by 26 publications

References 48 publications

Implications in the interpretation of plane-wave expansions in lossy media and the need for a generalized definition

Implications in the interpretation of plane-wave expansions in lossy media and the need for a generalized definition

The multi-physics metawedge: graded arrays on fluid-loaded elastic plates and the mechanical analogues of rainbow trapping and mode conversion

Testing a bead-rod contact with a nonlinear resonance method

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