A fully coupled subwavelength resonance approach to filtering auditory signals

Abstract

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

Mimicking the active cochlea with a fluid-coupled array of subwavelength Hopf resonators

“…It is known that the frequency-location (tonotopic) map in the cochlea is exponential [5] and it was shown in [13] that if an array of subwavelength resonators has a similar sizegrading, we reproduce this phenomenon. Thus, we will consider a domain D in R 2 which is the disjoint union of N ∈ N circular subdomains {D 1 , .…”

“…Since we want our structure to be of similar dimensions to the cochlea and exhibit a resonant response to audible frequencies, we need subwavelength resonant modes to exist. This is known to occur in the case that δ ≪ 1 [13,20,54]. For the simulations performed here we use material parameters corresponding to air-filled resonators surrounded by water, giving that δ ≈ 10 −3 .…”

“…The value of this approach is that solving (2.4) is reduced to finding φ, ψ such that the two transmission conditions on ∂D hold [37,54]. An asymptotic analysis of the resonant frequencies and eigenmodes, based on their layerpotential representations (2.6), was performed in [13]. It was shown that a system of N coupled resonators has N subwavelength resonant modes u 1 (x), .…”

High‐order exceptional points and enhanced sensing in subwavelength resonator arrays

“…Moreover, we say that a resonant frequency is a subwavelength resonant frequency if as . It is known that a system of subwavelength resonators has subwavelength resonant frequencies, up to multiplicity 22 …”

“…As is typically the case with structures composed of many subwavelength resonators (see, e.g., Refs. 22, 25), the maxima of the resonant modes occur on the resonators themselves. Thus, for optimal enhanced sensing of a small particle, the particle should be placed close to one of the resonators.…”