An amphibian hair bundle is floppier than the human bundle used as the worked example…

PHYSICS · Multi-part · Fermi University Physics · Mechanics & Dynamical Systems · Biophysics & Medical Physics · Bioelectricity, the senses and the physics of the body

Forces, Machines and Signals, Ch Hearing: from air pressure to a gated channel

Problem

An amphibian hair bundle is floppier than the human bundle used as the worked example for the hearing noise limit, with stiffness

\kappa = 0.40\,\mathrm{mN\,m^{-1}}

, and is tuned to

f_0 = 500\,\mathrm{Hz}

with quality factor

Q=15

. Take body temperature,

k_BT = 4.28\,\mathrm{pN\,nm}

, and treat the bundle's thermal motion as an overdamped spring whose noise is flat up to a corner frequency comparable to

f_0

, exactly as for that human bundle. ((a)) State the assumptions needed and estimate the bundle's total rms thermal displacement. ((b)) Estimate the rms displacement equivalent to the noise passed by a filter of bandwidth

f_0/Q

centred on

f_0

, following the same reasoning used for the human bundle. ((c)) Compare this with a detection threshold of order

1\,\mathrm{nm}

and comment on whether the frog's ear should be able to detect signals near this size.

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