Wave propagation in non-uniform waveguides

Abstract

Non-uniform waveguides in which the waves slow down as they propagate have a variety of uses. First, the attenuation within the waveguide is enhanced as the waves slow down, which can be used for absorption of flexural or acoustic waves without excessive use of damping material in so-called 'acoustic black holes'. Second, the position of the peak response along the waveguide can vary with frequency, allowing for spatial frequency analysis, as occurs in the cochlea and also in recent 'rainbow' sensors.

In this thesis, four different examples of such non-uniform waveguides are studied using both analytical and numerical methods. The reflection from elastic wedges of different thickness profiles is first analysed using the WKB method, and the results are compared with those calculated with the Finite Element method. It is shown that higher orders of WKB approximation are required to capture the details of the frequency variation of the reflection coefficient, and that an exponential wedge presents the least reflection among the studied profiles. For an acoustic waveguide with fitted rings of tapered inner radius, the reflection coefficient is then calculated with a Transfer Function method and compared with experimental results in the literature.

A one-dimensional 'box model' of the cochlea with passive micromechanics is also analysed, using both the WKB method and the Finite Difference method. The dependence of the coupled cochlear response to a number of non-dimensional parameters is studied, and it is found that one such parameter, which has previously been used to define the overall phase shift, also determines the symmetry of the frequency response of the system. The design of an acoustic 'rainbow' sensor is discussed and shown to involve a trade-off between the selectivity of the response and the number of elements. The new design is then described, consisting of a main duct with Helmholtz-Resonator side branches of varying dimensions, in which the damping is optimised so that its main characteristics are similar to those of the cochlea, to give a tonotopic mapping and a smooth frequency response.

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