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More cochlear mechanics.
The formula for the basilar-membrane stiffness S presented
S = S_0 * [1 - x/(4d)]^4 (1)
(where x = longitudinal coordinate and d = a few mm), has
met with so much interest that I permit myself to add a
It is fairly easy to see why Equation (1) is compatible with
the LG (Liouville-Green) approximation [also called WKB
(Wentzel, Kramers, Brillouin) approximation], and thus with
weak reflected waves.
If the BM (basilar membrane) impedance is stiffness-
dominated (i.e., if the BM mass and friction are negligible)
and the cochlear half channels have equal, x-independent
rectangular cross-sections, then the long-wave complex
liquid-pressure wave equation has the following solution:
p(x,t) = a_p(x) * e^[i*phi(x)] * e^[i*omega*t] . (2)
In Equation (2), a_p(x) is a real amplitude.
In the LG approximation, a_p(x) is found to be as follows:
a_p(x) = a_p(0) * [k(0) / k(x)]^(1/2) . (3)
The local wave number k(x) is:
k(x) = omega * [2*rho/(H*S)]^(1/2) . (4)
Here, rho = 1000 kg / (m^3) is the liquid density, and H is the
"effective" half-channel height (i.e., cross section divided by
Equations (1), (3), and (4) yield:
a_p(x) = a_p(0) * [1 - x/(4d)] . (5)
Equation (5) guarantees that the condition for the accuracy
of the LG approximation,
(a_p)'' << (k^2) * a_p , (6)
is perfectly fulfilled at all angular frequencies omega
small enough to be compatible with the long-wave
approximation, k*H << 1 .
[ I plan to write another not-too-long text giving more details:
Reflections for S(x) = S_0 * e^(-x/d) ; possible validity of
Equation (1) above for the whole cochlea; I shall send that
text to those who have asked for the first one, and also to
"newcomers" of course.]
With best wishes,
Dr. phil. nat.,
r. PSI and ETH Zurich,
Phone: 0041 56 441 77 72.
Mobile: 0041 79 754 30 32.
E-mail: reinifrosch@xxxxxxxxxx .