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Cochlear liquid-particle trajectories.

Dear List, 

The calculations on the "new" basilar-membrane stiffness
formula take longer than expected. If they give interesting
results, I shall communicate them in a week or two.

Yesterday I received a cochlear-mechanics report on a
transmission-line model, containing a graph of liquid
streamlines which disagree with what I have learned.

In the case of small-displacement surface waves on
the ocean it is helpful to change to a coordinate system
which moves along with the wave crests:

x' = x - c * t 

(where x = longitudinal coordinate, t = time, c = phase
velocity). In the primed coordinate system, the liquid
particles make a stationary flow in the -x'-direction.
At the crests, the streamlines are far apart from each other,
so that the velocity v_x' of the particles is comparatively low.

Transformation back to the lab system yields closed,
elliptical liquid-particle trajectories (circular for short 
In long waves (e.g., tsunamis far from the coast)  the particle 
trajectories are oblong ellipses; at the ocean floor, the length 
of the short ellipse axis is zero.

In cochlear waves, the phase velocity c decreases if x
increases; the liquid-particle trajectories, however, are
similar (I believe) to those in ocean waves, i.e., closed 
and approximately elliptical.

The elliptical particle trajectories do not cross the basilar
membrane (BM). At the BM, the liquid particles move
along the BM. That is why the mass of the organ-of-Corti
cells (which in the idealistic models do not move along
the BM) plays a role even if their density is equal to
that of the liquid.

A good introduction into cochlear waves was written
by the late G.K. Yates: Chapter 2 of the book "Hearing",
B.C.J. Moore, ed., Academic Press, San Diego,
1995, Section II.B, pages 49-53.

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 .