Waves in the cochlea and in acoustic rainbow sensors

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