116
Joseph A. C. Humphrey eta!.
We can now fix all known physical properties, as well as the length, diameter and
resonance frequency of a particular hair, and use Eq. (19a') to calculate S as a
function of R. The result is a series of (R, S) pairs all of which satisfy the values
fixed in the equation. Figure 5 shows the result of doing this for the trichobothria
of Section 2.3.1. Each curve connecting a series offilled symbols corresponds to a
particular hair in air. A striking feature of these curves is that they overlap in the
neighborhood of R = 1 10" 15 N m s rad- 1 and S = 4 10 -lz N m rad- 1 (the values used
for the physically approximate calculations shown in Figs. 1-4). This is not a
coincidence. While R and S are expected to vary with hair length, they are
mechanical properties of the hair-supporting apparatus which consists of cuticular
material (chitin fibers embedded in a protein matrix). Thus, we expect R and S to
vary, but within limited ranges (of 1 - 2 orders of magnitude) with limits fixed by
the molecular structure of the cuticular material.
Support for this hypothesis is provided by the data for R and S in Barth et al.
(1993). Figure 5 shows three points (filled diamonds) connected by a dotted curve.
The two end points correspond to the 250- and 750-~m hairs investigated by these
authors and the one in the center is the result of an interpolation for a 500-~m hair.
The correspondence is good between the theoretically predicted overlap region
and the experimental results for R and S.
We apply the above principle to hairs in water. From the limited information
available we expect to find shorter hairs responding to lower frequency stimuli
and, for illustration purposes, we assume two hairs of the following
characteristics: d = 5 ~m, L = 250 ~m andfres(IIJ =50 Hz; and d = 6 ~m. L = 500
~m, andfres(I?J = 10Hz. Using these values to calculateS as a function of R yields
the two curves connecting the unfilled symbols in Fig. 5. As for the hairs in air,
these two curves also cross, very near to the point given by R = I 10· 14 N m s rad" 1
and S = 2 10 -II N m rad- 1 . Even if approximate, these results are sufficiently
different from those for hairs in air to conclude that for hairs to perform as motion
sensors in water, with sensitivities comparable to those found in air, they must
possess larger values of R and S than required in air.
3 Discussion
3.1 Parameter Dependence of Hair Motion
It is possible to solve Eq. (l) numerically for very general conditions and
analytically for the special case of the velocity profile given by Eq. (8), the result
being Eq. ( 1 0) (Humphrey et al. 1993). However, even this analytical solution
requires numerical evaluation to determine hair deflection angles, velocities, and
frequencies. Hence the motivation for the physically approximate solution derived
here, given by Eqs. (19a, 19b and 20a, 20b), which explains several of the
observed dependencies of hair behavior on the physical parameters that affect that
behavior and allows us to uncover new ones. For example, inspection of Eqs.
(19a, 19b) shows that, in principle, the damping constant, R, and the torsional
Joseph A. C. Humphrey eta!.
We can now fix all known physical properties, as well as the length, diameter and
resonance frequency of a particular hair, and use Eq. (19a') to calculate S as a
function of R. The result is a series of (R, S) pairs all of which satisfy the values
fixed in the equation. Figure 5 shows the result of doing this for the trichobothria
of Section 2.3.1. Each curve connecting a series offilled symbols corresponds to a
particular hair in air. A striking feature of these curves is that they overlap in the
neighborhood of R = 1 10" 15 N m s rad- 1 and S = 4 10 -lz N m rad- 1 (the values used
for the physically approximate calculations shown in Figs. 1-4). This is not a
coincidence. While R and S are expected to vary with hair length, they are
mechanical properties of the hair-supporting apparatus which consists of cuticular
material (chitin fibers embedded in a protein matrix). Thus, we expect R and S to
vary, but within limited ranges (of 1 - 2 orders of magnitude) with limits fixed by
the molecular structure of the cuticular material.
Support for this hypothesis is provided by the data for R and S in Barth et al.
(1993). Figure 5 shows three points (filled diamonds) connected by a dotted curve.
The two end points correspond to the 250- and 750-~m hairs investigated by these
authors and the one in the center is the result of an interpolation for a 500-~m hair.
The correspondence is good between the theoretically predicted overlap region
and the experimental results for R and S.
We apply the above principle to hairs in water. From the limited information
available we expect to find shorter hairs responding to lower frequency stimuli
and, for illustration purposes, we assume two hairs of the following
characteristics: d = 5 ~m, L = 250 ~m andfres(IIJ =50 Hz; and d = 6 ~m. L = 500
~m, andfres(I?J = 10Hz. Using these values to calculateS as a function of R yields
the two curves connecting the unfilled symbols in Fig. 5. As for the hairs in air,
these two curves also cross, very near to the point given by R = I 10· 14 N m s rad" 1
and S = 2 10 -II N m rad- 1 . Even if approximate, these results are sufficiently
different from those for hairs in air to conclude that for hairs to perform as motion
sensors in water, with sensitivities comparable to those found in air, they must
possess larger values of R and S than required in air.
3 Discussion
3.1 Parameter Dependence of Hair Motion
It is possible to solve Eq. (l) numerically for very general conditions and
analytically for the special case of the velocity profile given by Eq. (8), the result
being Eq. ( 1 0) (Humphrey et al. 1993). However, even this analytical solution
requires numerical evaluation to determine hair deflection angles, velocities, and
frequencies. Hence the motivation for the physically approximate solution derived
here, given by Eqs. (19a, 19b and 20a, 20b), which explains several of the
observed dependencies of hair behavior on the physical parameters that affect that
behavior and allows us to uncover new ones. For example, inspection of Eqs.
(19a, 19b) shows that, in principle, the damping constant, R, and the torsional