We consider the problem of approximating a function using Herglotz wave functions, which are a superposition of plane waves. When the discrepancy is measured in a ball, we show that the problem can essentially be solved by considering the function we wish to approximate as a source distribution and time reversing the resulting field. Unfortunately this gives generally poor approximations. Intuitively, this is because Herglotz wave functions are determined by a two-dimensional field and the function to approximate is three-dimensional. If the discrepancy is measured on a plane, we show that the best approximation corresponds to a low-pass filter, where only the spatial frequencies with length less than the wavenumber are kept. The corresponding Herglotz wave density can be found explicitly. Our results have application to designing standing acoustic waves for self-assembly of micro-particles in a fluid.
Willis coupling is a recently recognized material property that couples the pressure-strain and momentum-velocity equations and can lead to novel applications of metamaterials. We present an approach for optimizing the asymmetry factor (a non-dimensional measure of Willis coupling's effect on the specific acoustic impedance) in a metamaterial with lumped-element hidden degrees of freedom. The representative material element in the system we examine is a length of tube, which could be considered as a unit cell in a periodic system. Lumped-element features including side-branch resonators and membranes may be attached to the tube at various locations. We aim to determine optimal locations for each type of lumped-element feature, as well as optimal parameters in the design of the features themselves (e.g., for a resonator: volume, neck length, and neck cross-sectional surface area). An ideal design for a metamaterial exhibiting significant Willis coupling would result in a broadband non-negligible asymmetry factor.
Willis coupling is a recently recognized material property that couples the pressure-strain and momentumvelocity equations and can lead to novel applications of metamaterials. We present an approach for optimizing the asymmetry factor (a non-dimensional measure of Willis coupling's effect on the specific acoustic impedance) in a metamaterial with lumped-element hidden degrees of freedom. The representative material element in the system we examine is a length of tube, which could be considered as a unit cell in a periodic system. Lumped-element features including side-branch resonators and membranes may be attached to the tube at various locations. We aim to determine optimal locations for each type of lumped-element feature, as well as optimal parameters in the design of the features themselves (e.g., for a resonator: volume, neck length, and neck cross-sectional surface area). An ideal design for a metamaterial exhibiting significant Willis coupling would result in a broadband non-negligible asymmetry factor.Published by the Acoustical Society of America
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