Journal of Fluid Mechanics



Acoustic streaming in the ear itself


James  Lighthill a1
a1 Department of Mathematics, University College, Gower Street, London WC1E 6BT, UK

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Abstract

Just as mean motions, usually described as acoustic streaming, can be generated by sound waves, so also those cochlear travelling waves into which incident sound waves are converted in the liquid-filled mammalian inner ear are capable of generating mean motions. These predominate, for acoustic components of each frequency ω, near the characteristic place where the wave energy E per unit length rises rather steeply to a maximum Emax before dropping precipitously to zero.

Even though the nature of cochlear travelling waves, as determined (above all) by the sharply and continuously falling distribution of stiffness for the basilar membrane vibrating within the cochlear fluids, is very different from that of ordinary sound waves (see §§2, 3 and 4 respectively for energy distribution along the length of the cochlea, over a cochlear cross-section and within boundary layers), nevertheless a comprehensive analysis of mean streaming motions in the cochlea shows them to be governed by remarkably similar laws. The expression \[ {\textstyle\frac{1}{4}}V^2c^{-1}-{\textstyle\frac{3}{4}}V({\rm d}V/{\rm d}x)\,\omega^{-1} \] (equation (1)) appropriate to a wave travelling in the x-direction with velocity amplitude V(x), as obtained by Rayleigh (1896) for the mean acoustic-streaming velocity just outside a boundary layer due to wave dissipation therein, remains a good approximation (see §§5 and 6 – with some modest corrections, at low or at high wavenumbers respectively, analysed in §§7 and 8) for travelling waves in the cochlea; where, however, the decrease of their phase velocity c to low values near the characteristic place conspires with the increase of V to enhance streaming there.

Farther from the boundary layer attached to the basilar membrane, the mean streaming is derived (§9) as a low-Reynolds-number motion compatible with the distribution (1) of ‘effective slip velocity’ at the boundary. This velocity's precipitous fall to zero at the characteristic place is shown (§§9 and 10) to produce there a mean volume outflow \[ q = \frac{0.15 E^{\max}}{\rho(\omega\nu)^{\frac{1}{2}}L} \] (equation (160)) per unit length of the basilar membrane into the scala media; here, ρ and ν are the endolymph's density and kinematic viscosity (essentially, those of water) and L is the e-folding distance for basilar-membrane stiffness.

Equation (160), derived here for a freely propagating wave (and so not allowing for enhancements from any travelling-wave amplification – discussed qualitatively in §3 – due to forcing by vibrations of outer hair cells) is the main conclusion of this paper. Physiological questions of whether this flow q may be channelled through the space between the tectorial membrane and inner hair cells, whose stereocilia may therefore be stimulated by a mean deflecting force, are noted here but postponed for detailed consideration in a later paper.

(Received June 17 1991)



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