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Still more cochlear mechanics.

Hello !

Mysteriously, I received a message stating that my posting 
"More cochlear machanics" (August 27, 2006) was rejected 
because an equal posting had been distributed before. 
So, second try:

The formula for the basilar-membrane stiffness S presented 
last Friday,

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 
new insight:

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 
BM width).

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,

Reinhart Frosch. 

Reinhart Frosch,
Dr. phil. nat.,
r. PSI and ETH Zurich,
Sommerhaldenstr. 5B,
CH-5200 Brugg.
Phone: 0041 56 441 77 72.
Mobile: 0041 79 754 30 32.
E-mail: reinifrosch@xxxxxxxxxx .