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Re: propagation speed of the traveling wave

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

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

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.

>-- 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