The aim of this paper is to understand the behaviour of a large number of coupled subwavelength resonators. We use layer potential techniques in combination with numerical computations to study the acoustic pressure field due to scattering by a graded array of subwavelength resonators. Using this method, we study a graded-resonance model for the cochlea. We compute the resonant modes of the system and explore the model's ability to decompose incoming signals. We are able to offer mathematical explanations for the cochlea's so-called "travelling wave" behaviour and tonotopic frequency map.
Mathematics subject classification: 35R30, 35C20
When sources of energy gain and loss are introduced to a wave scattering system, the underlying mathematical formulation will be non-Hermitian. This paves the way for the existence of exceptional points, whereby eigenmodes are linearly dependent. The main goal of this work is to study the existence and consequences of exceptional points in the setting of high-contrast subwavelength metamaterials. We begin by studying a system of two paritytime-symmetric subwavelength resonators and prove that this system supports exceptional points. Using homogenization theory, we study a large ensemble of resonators and show that this behaviour can be replicated at the macroscale. Finally, we study a metascreen of subwavelength resonators and prove that there are frequencies at which this system exhibits unidirectional reflectionless transmission.
We present a design for an acoustic metamaterial that mimics the behaviour of the active cochlea. This material is composed of a size-graded array of cylindrical subwavelength resonators, has similar dimensions to the cochlea and is able to per- form frequency separation of audible frequencies. Nonlinear amplification is introduced to the model in order to replicate the behaviour of the cochlear amplifier. This formulation takes the form of a fluid-coupled array of Hopf resonators. We seek solutions in the form of a modal decomposition, so as to retain the physically derived coupling between resonators.
Photodetachmentfrom negative ions in a static electric field has been studied using Cl and S These experiments were done with greater precision than previous studies and allow a quantitative comparison of data with theory. A 10-keV ion beam was sent through a region with fields of up to 1.5 kV/cm applied parallel to the beam velocity. A pulsed dye-laser beam, perpendicular to the ion beam, photoneutralized the ions. The relative cross sections were measured by detecting the resulting fast neutral atoms. Detachment below threshold and oscillations on the cross section above threshold were observed near the S threshold at 16269.5 cm ' and near the Cl threshold at 29138.3 cm '. The phase of the oscillations in the data is in good agreement with predictions for s-wave photodetachment in a static electric field, but the amplitude of the oscillations is found to be slightly reduced.PACS number(s): 32.80.Fb, 32.60.+i
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