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*To*: AUDITORY@xxxxxxxxxxxxxxx*Subject*: More cochlear mechanics.*From*: "reinifrosch@xxxxxxxxxx" <reinifrosch@xxxxxxxxxx>*Date*: Sun, 27 Aug 2006 16:28:25 +0000*Delivery-date*: Sun Aug 27 12:43:34 2006*List-help*: <mailto:LISTSERV@LISTS.MCGILL.CA?body=INFO AUDITORY>*List-owner*: <mailto:AUDITORY-request@LISTS.MCGILL.CA>*List-subscribe*: <mailto:AUDITORY-subscribe-request@LISTS.MCGILL.CA>*List-unsubscribe*: <mailto:AUDITORY-unsubscribe-request@LISTS.MCGILL.CA>*Reply-to*: reinifrosch@xxxxxxxxxx*Sender*: AUDITORY Research in Auditory Perception <AUDITORY@xxxxxxxxxxxxxxx>

Dear colleagues, 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 .

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