AW: Re: Cochlear mechanics.

["reinifrosch@xxxxxxxxxx" <reinifrosch@xxxxxxxxxx>]

Hello Dick and List !

In the meantime, you have probably seen my 
1.5-page text on the cochlear reflections;
nevertheless, I would like to react to your comments.

----Ursprüngliche Nachricht----
Von: DickLyon@xxxxxxx
Datum: 25.08.2006 17:26
An: <AUDITORY@xxxxxxxxxxxxxxx>
Betreff: Re: Cochlear mechanics.

[...]
>>S = S_0 * e^(-x/d) ,

>Reinhart, I think it would be more fair to call that a 
convenient 
>approximation more than an assumption.  It's well known that for 
>large enough d to get into a low-CF region, it decreases too 
slowly.

My present calculations are only about the basal
(i.e., high-charactaristic-frequency) region of the cochlea.

>>[...]  Yesterday I found that absence
>>of significant reflections is predicted even for low
>>frequencies if a different function is used instead:
>>
>>S = S_0 * [1 - x/(4d)]^4 .

>The use of [1 - x/N]^N as an approximation for exp(-x) is often 
very 
>useful, for example in approximating Gaussians or high powers of 
>cosines, in my experience, but I had not looked at it in this 
>particular application.  It looks like a useful way to get the 
>stiffness and CF to go to 0 for finite d while preserving the 
desired 
>behaviour at low d.

The exciting property of "my" formula (i.e., N=4) is
that it reduces the left-hand side of the inequality
which if fulfilled guarantees accuracy of the WKB
(or LG) approximation and weakness of reflections
to zero !

>What do you mean by "even for low frequencies"?  That was true of 
the 
>original function as well, right?  So you just mean it doesn't 
mess 
>up the no-reflections property?

In the case of the function S(x) = S_0 * e^(-x/d)
the mentioned inequality yields

d^-2 << 16 k^2 ;

Low frequency leads to low k and thus to a
violation of the inequality.

With best wishes,

Reinhart Frosch.