Subject: AW: Re: Cochlear mechanics. From: "reinifrosch@xxxxxxxx" <reinifrosch@xxxxxxxx> Date: Fri, 25 Aug 2006 17:56:16 +0000Hello 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@xxxxxxxx Datum: 25.08.2006 17:26 An: <AUDITORY@xxxxxxxx> 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.