WKB solutions to the wave equation for the cochlea and for acoustic rainbow sensors

The WKB method is used to derive an approximate solution to the cochlear wave equation, 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 resulting WKB solution can be expressed in terms of a few nondimensional parameters, and their physical meaning is discussed. Notably, a nondimensional phase parameter, N, changes the nature of the resonance, which is symmetric for low values of N, indicating weak fluid coupling, and asymmetrical, with a characteristic peak, for high values of N, indicating strong fluid coupling. On the other hand, the contribution from nondimensional parameters derived from the fluid viscosity and fluid compressibility is negligible. It is then shown that recent designs of acoustic rainbow sensor, comprised of an array of Helmholtz resonators of increasing size coupled to a duct, are described by a wave equation which has the same form of that of the cochlea. However, in this case, the nondimensional compressibility parameter is much larger than in the cochlea, and so plays a more dominant role in determining the response.