120
Joseph A. C. Humphrey et al.
The plots for hairs in air show clearly that when R = Jd} hair sensitivity
maximizes at a value L/ Ores(V)- 0( 1) (meaning of order 1 since the approximations
made do not allow a more accurate claim than this). This finding agrees nicely
with the experiments of Barth et al. (1993 ), who find that L/ Ores(O) - 0(1 ), where
Ores(O) - (Jli'PWres((})) 112 , for the trichobothria of Cupiennius salei. The results for R
<< pL 3 and R >> pL 3 in air show that increases in the damping factor significantly
decrease hair sensitivity, but that this effect is much more pronounced for LIO,.es((})
< 0(1) (hairs embedded in the boundary layer) than for L!Ores((}) > 0(1) (hairs
poking through the boundary layer). Bearing in mind that we have used constant
values for RandS for hairs in water, the sensitivity results for LIOres(V) - 0(10)
show the same dependence on the damping constant as for hairs in air. However,
in water the dependence vanishes as Llores(V) decreases. Over the entire parameter
range investigated in water, sensitivity always increases with decreasing LIOres(O)·
3.2 What Can Physics Say About Adaptive Evolution?
It is of special interest to explore the sensitivity of hair maximum deflection angle
resonance frequency, Wres(O), with respect to the physical parameters that affect it.
(The same can be done for Wres(V)> Oren and Vres·) Mathematically speaking, we
seek the values of the partial derivatives in the expression
(21)
where the x; are x 1 = d, x2 = L, x3 = R, x4 = S, X5 = p, and x6 = p. Call S; =
OWres(O)It3x; the absolute sensitivity of Wres((}) with respect to the parameter X;.
Values of the different S; can be determined numerically via fmite difference
approximation (S; = L1Wres((})IL1x;) using the physically approximate solution. More
relevant, however, is to compare relative sensitivities. For this, we rewrite Eq. (21)
to read
(22)
where the R; = X; ( OWres(O) I Ox;) = X; ( LIWres((}) I L1x; ) are the relative sensitivities.
Values for the R; are plotted in Fig. 7 for two hairs of equal dimensions (d = 7 f.!m,
L = 500 f.!m) using R = I 10" 15 N m s rad- 1 and S = 4 10 - 12 N m rad- 1 for the hair in
air and R = I 10" 14 N m s rad- 1 and S = 2 10 -II N m rad- 1 for the hair in water.
(Note that in the plot the sign of the relative sensitivity associated with the
increase of a particular physical parameter is indicated in parentheses after that
parameter.) The plot shows for the conditions calculated that the relative
sensitivities in air are larger by an order of magnitude or more than the
corresponding values in water, and that they are largest for the parameters L, R, S,
and p.
Joseph A. C. Humphrey et al.
The plots for hairs in air show clearly that when R = Jd} hair sensitivity
maximizes at a value L/ Ores(V)- 0( 1) (meaning of order 1 since the approximations
made do not allow a more accurate claim than this). This finding agrees nicely
with the experiments of Barth et al. (1993 ), who find that L/ Ores(O) - 0(1 ), where
Ores(O) - (Jli'PWres((})) 112 , for the trichobothria of Cupiennius salei. The results for R
<< pL 3 and R >> pL 3 in air show that increases in the damping factor significantly
decrease hair sensitivity, but that this effect is much more pronounced for LIO,.es((})
< 0(1) (hairs embedded in the boundary layer) than for L!Ores((}) > 0(1) (hairs
poking through the boundary layer). Bearing in mind that we have used constant
values for RandS for hairs in water, the sensitivity results for LIOres(V) - 0(10)
show the same dependence on the damping constant as for hairs in air. However,
in water the dependence vanishes as Llores(V) decreases. Over the entire parameter
range investigated in water, sensitivity always increases with decreasing LIOres(O)·
3.2 What Can Physics Say About Adaptive Evolution?
It is of special interest to explore the sensitivity of hair maximum deflection angle
resonance frequency, Wres(O), with respect to the physical parameters that affect it.
(The same can be done for Wres(V)> Oren and Vres·) Mathematically speaking, we
seek the values of the partial derivatives in the expression
(21)
where the x; are x 1 = d, x2 = L, x3 = R, x4 = S, X5 = p, and x6 = p. Call S; =
OWres(O)It3x; the absolute sensitivity of Wres((}) with respect to the parameter X;.
Values of the different S; can be determined numerically via fmite difference
approximation (S; = L1Wres((})IL1x;) using the physically approximate solution. More
relevant, however, is to compare relative sensitivities. For this, we rewrite Eq. (21)
to read
(22)
where the R; = X; ( OWres(O) I Ox;) = X; ( LIWres((}) I L1x; ) are the relative sensitivities.
Values for the R; are plotted in Fig. 7 for two hairs of equal dimensions (d = 7 f.!m,
L = 500 f.!m) using R = I 10" 15 N m s rad- 1 and S = 4 10 - 12 N m rad- 1 for the hair in
air and R = I 10" 14 N m s rad- 1 and S = 2 10 -II N m rad- 1 for the hair in water.
(Note that in the plot the sign of the relative sensitivity associated with the
increase of a particular physical parameter is indicated in parentheses after that
parameter.) The plot shows for the conditions calculated that the relative
sensitivities in air are larger by an order of magnitude or more than the
corresponding values in water, and that they are largest for the parameters L, R, S,
and p.