Cochlear mechanics with fluid viscosity and compressibility

We extend the one-dimensional cochlear model to include the effect of viscosity and compressibility of the cochlear fluid. The resulting boundary-value problem is solved exactly using numerical techniques and semianalytically using the WKB approximation. Our results indicate the general trend of basilar membrane response increasing with an increase in compressibility or a decrease in viscosity. However, in the physiologically relevant range of these parameters, the change in the response is insignificant, justifying the assumption made in one-dimensional cochlear models that these effects are unimportant. Using the semianalytical WKB algorithm, we also demonstrate the simultaneous existence of forward- and backward-traveling waves on the basilar membrane.

Figure 1

Schematic diagram showing a longitudinal section of the uncoiled cochlea. BM: basilar membrane; OWM: oval window membrane; and RWM: round window membrane.

Figure 2

Instantaneous waves (solid line) and wave envelope (dashed line) on the BM for (a) and (b) $f=1$ kHz and (c) and (d) $f=20$ Hz: (a) and (c) exact solution and (b) and (d) WKB approximation. Instantaneous waves are shown at uniformly spaced time instants over a period of oscillation. The time instants normalized by the period of oscillation $T$ are indicated in the legends. The parameter values are $L=35$ mm and $l=1$ mm from [5], $\overline{\rho }=1000\mathrm{kg}/{\mathrm{m}}^3$ from [6], $\mu =0$, $\kappa =0$, and a SPL of 40 dB. Note the shift in the location of the peak amplitude toward the apex of the BM at the lower frequency. The location of the resonant place is $x\approx 1.7$ cm for $f=1$ kHz and $x\approx 3.2$ cm for $f=20$ Hz.

Figure 3

Amplitude of the two WKB modes in Eq. (15) for $f=1$ kHz, a SPL of 40 dB, $\mu =0$, and $\kappa =0$. Note the smallness of the plus mode.

Figure 4

Effect of fluid viscosity on the peak amplitude of BM displacement for (a) $f=1000$ Hz and a SPL of 80 dB, (b) $f=1000$ Hz and a SPL of 40 dB, (c) $f=100$ Hz and a SPL of 80 dB, and (d) $f=100$ Hz and a SPL of 40 dB. Open symbols denote exact solutions and closed symbols WKB solutions. Here $\kappa =0$. The black arrow indicates roughly the value of fluid viscosity that is relevant to mammalian cochleae.

Figure 5

Effect of fluid compressibility on the peak amplitude of BM displacement for (a) $f=1000$ Hz and a SPL of 80 dB, (b) $f=1000$ Hz and a SPL of 40 dB, (c) $f=100$ Hz and a SPL of 80 dB, and (d) $f=100$ Hz and a SPL of 40 dB. Open symbols denote exact solutions and closed symbols WKB solutions. Here $\mu =0$. The black arrow indicates roughly the value of fluid compressibility that is relevant to mammalian cochleae.