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*To*: AUDITORY@xxxxxxxxxxxxxxx*Subject*: Re: propagation speed of the traveling wave*From*: Reinhart Frosch <reinifrosch@xxxxxxxxxx>*Date*: Sat, 9 Apr 2005 13:18:31 +0000*Comments*: To: "Rossi Mark (PA-ATMO1/EES21) *" <Mark.Rossi@DE.BOSCH.COM>*Delivery-date*: Sat Apr 9 09:30:44 2005*In-reply-to*: <5C26DA67C8DFC343AEBE7BFB9F19F0BD010956B4@si-mail20.de.bosch.com>*Reply-to*: Reinhart Frosch <reinifrosch@xxxxxxxxxx>*Sender*: AUDITORY Research in Auditory Perception <AUDITORY@xxxxxxxxxxxxxxx>

On April 8, 2005, Mark Rossi posted a message about frequency-dependent propagation speeds of cochlear travelling waves. A simple model is treated, e.g., in Fig. 4.7C of the article by Robert Patuzzi on page 211 of the book "The Cochlea" (Springer, 1996). There, a mass-loaded liquid-surface wave with spring restoring force is considered. At low frequencies, i.e., at wavelengths much greater than the channel depth h times two pi, the phase velocity [e.g., the velocity of a wave crest] and the group velocity [e.g., the velocity of a wave group caused by a click] are equal to each other, and do not depend on the frequency. At higher frequencies both velocities become smaller, and the group velocity becomes smaller than the phase velocity. Both velocities drop to zero at a "critical" angular frequency of omega = square-root of (E / M) [where E is the spring constant per square meter, and M is the surface layer mass per square meter]. In the most realistic cochlear models, sine waves do not get as far as the point on the BM where the critical frequency is equal to the frequency of the sine wave. The phase velocity is omega / k (where k is the wave number, k = 2 pi / lambda; lambda = wavelength). The group velocity is d omega / d k. The above- mentioned figure of Patuzzi contains the formula giving omega as a function of k. Addendum 1, for readers who want to study Patuzzi's graphs: In the figure caption, it says: "All values are in arbitrary units". The following values are consistent with the graphs: Water depths: h = 0.01, 0.1, 1, and 10 mm. "Stiffness" (spring constant per m^2): E = 10^4 kg / (s^2 m^2). Liquid density: d = 10^3 kg / m^3. Surface-layer mass per m^2: M = 10 kg / m^2. Velocities given in m / s. Wavelengths given in m. Angular frequencies omega given in radians / second. Addendum 2, concerning the question whether cochlear waves at the base are at the long-wavelength limit: At the base, h(effective) = half-channel cross-section / BM width = about 1 mm^2 / 0.1 mm = 10 mm. The long-wavelength case implies a wave number k << k_0 = 1 / h = 100 m^-1. E = 10^10 kg / (s^2 m^2) [see Egbert de Boer's article in the above-mentioned book "The Cochlea"]. M = about 0.1 kg / m^2. For k = k_0, the formula in Patuzzi's figure yields omega = about 27500 s^-1, i.e., a frequency of about 4400 Hz; thus the long-wavelength condition is fulfilled only for frequencies much lower than 4.4 kHz. Reinhart Frosch, CH-5200 Brugg. reinifrosch@xxxxxxxxxx >-- Original-Nachricht -- >Date: Fri, 8 Apr 2005 12:21:19 +0200 >Reply-To: "Rossi Mark (PA-ATMO1/EES21) *" <Mark.Rossi@xxxxxxxxxxxx> >From: "Rossi Mark (PA-ATMO1/EES21) *" <Mark.Rossi@xxxxxxxxxxxx> >Subject: propagation speed of the traveling wave >To: AUDITORY@xxxxxxxxxxxxxxx > >Dear list, > >in order to estimate the propagation time of the >traveling wave on the BM for certain frequencies I >need to know the propagation speed of the wave. > >[...] > >I found that the propagation speed is frequency >dependent. [...] > >Mark Rossi

**References**:**propagation speed of the traveling wave***From:*Rossi Mark (PA-ATMO1/EES21) *

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