Waves in the cochlea and in acoustic rainbow sensors

Abstract

A WKB solution to the cochlear wave equation is derived, which results from the interaction between the passive dynamics of the basilar membrane and the 1D fluid coupling in the scalae, including both fluid viscosity and compressibility. The effect of various nondimensional parameters on the form of this solution is discussed. A nondimensional damping parameter and a nondimensional phase-shift parameter are shown to have the greatest influence on the response under normal conditions in the cochlea, with the fluid viscosity and compressibility only playing a minor role. It is then shown that in the case of an acoustic rainbow sensor, comprised of a discrete series of Helmholtz resonators in a duct, the governing wave equation in the continuous limit has the same form as the cochlear wave equation. The nondimensional compressibility parameter in this case is governed by the ratio of the Helmholtz resonator volume to that of the connecting duct and this parameter can be much larger than in the cochlea, and so plays a more dominant role in determining the response.

On the problem of comparing graded metamaterials

“…This means that the effective model (2.1) can also be used to describe systems with local resonances, as is the case for several of the graded metamaterials explored in the literature, e.g. [13,17–19]. The information about the local resonances is contained within the expressions for T and α.…”

“…For this reason, many studies simply opt to use the obvious example of a linear gradient [10,13,17,22]. A notable exception to this is the field of cochlea-inspired graded metamaterials, where exponential gradients are the obvious choice to replicate the spatial frequency separation that occurs in the cochlea [19,20,23].…”

“…a piezoelectric patch) can extract a relatively large amount of energy [16,17]. Conversely, applications to machine hearing exploit the fact that a graded metamaterial exhibits a continuous relationship between the incoming frequency and the position of maximum amplitude [18][19][20]. This replicates the action of the cochlea in mammalian hearing, and means that the frequency of a single-frequency wave can be identified by measuring the location at which the maximum amplitude occurs.…”

Bioinspired acoustic metamaterials: From natural designs to optimized structures

“…Numerical computations were used to predict the variation of the acoustic pressure along the waveguide for three different frequencies (0.5, 1, and 2 kHz), with the pressure increasing along the waveguide until a peak is reached for a specific resonator, rapidly decaying after this maximum. The design of rainbow sensors consisting of an array of Helmholtz resonators with distinct resonant frequencies was later shown to be, in the limit case of small elements, equivalent in form to the cochlear wave equation, for which analytical solutions have been derived (Marrocchio et al, 2021). Rupin et al (2019) have proposed an active one-dimensional acoustic MM to mimic the behavior of a cochlea.…”

Design of FIR filters with magnitude specifications

“…Potential applications for GAFs include: multiplexers [13], designing rainbow sensors, [14], analyzing seismic signals [15], underwater sound classification [16], cochlear implants [17], and hearing aids [18]. Due to the nature of these applications, any discussion of GAFs must include a discussion of filterbank representations.…”