A number of physical arrangements for acoustic rainbow sensors have been suggested, where the aim is to separate different frequency components into different physical locations along the sensor. Although such spatial discrimination has been achieved with several designs of sensor, the resulting frequency responses at a given position along the sensor are generally not smoothly varying. In contrast, the cochlea provides an interesting natural example of a rainbow sensor, which has an exponential frequency distribution and whose response does vary smoothly with frequency. The design of a rainbow sensor is presented that has a number of discrete resonators and an exponential frequency distribution. We discuss the conditions for a smoothly varying frequency response in such a sensor, as part of a broader design strategy. It is shown that the damping within the resonators determines the trade-off between the frequency resolution and the number of elements required to achieve a smooth response. The connection is explained between this design and that of an effective acoustic absorber. The finite number of hair cells means that the cochlea itself can be thought of as being composed of discrete units and the conditions derived above are compared with those that are observed in the cochlea.
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.
Acoustic black holes (ABHs) can provide effective damping of the reflected wave component when used to terminate a beam. The behaviour of an ABH is characterised by its local modes, which produce narrow frequency bands of high absorption. To enhance the performance of ABH terminations, a multi-taper ABH has previously been proposed and analytical results demonstrate that the use of two or more tapers produces a compound effect on the reflection coefficient, resulting in more bands of low reflection. This paper extends this work and presents an experimental realisation of a multi-taper ABH confirming the previous analytical results.
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