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Re: A new paradigm?(On pitch and periodicity (was "correction to post"))
At 4:57 PM -0400 10/31/11, Willem Christiaan Heerens wrote:
... I really must remind you to the fact that a mechanical vibration
-- and the sound stimulus is such a vibration -- in a fluid, or in
this case water like perilymph, will always propagate with the speed
of sound, which has typically here the value of 1500 m/s. That is
just one of those constraints dictated by general physics.
No, not "always"; that 1500 m/s wave mode is for longitudinal
pressure waves only. Your conception of "general physics" needs a
slight extension to cover other types of waves. Then the problem
won't be so over-constrained.
In the ear, the stapes doesn't couple much energy into this fast
pressure-wave mode. A much slower propagating vibration mode is
involved in the cochlear traveling waves that use the compliance of
the basilar membrane, as opposed to compression of the fluid, as the
displacement-based restoring force that leads to the wave equations.
This mode has a very different form, doesn't depend on fluid
compressibility, requires a membrane with motion in a suitable
symmetry across it, etc. This is what the physics describes, and
what the models model.
Gravity waves on water are a related, but different, example of
mechanical vibrations that propagate much more slowly than 1500 m/s.
These modes use gravity as the restoring force, and can be put into
analogy with what the membrane does in the cochlea (though it's not
such a close analogy as to give the same wave equations).
Of course, until one acknowledges the basic physics of waves in
incompressible fluids, as described by Lamb and Rayleigh and others
over a hundred years ago, it will not be possible to converge on an
understanding of cochlear models and their traveling waves.
The physics and math are pretty simple, relying only on f=ma for
fluid elements, and conservation of volume for incompressibility, and
something to make a restoring force. To get waves, you need
something to hold potential energy and push back against displacment,
to trade that energy against the kinetic energy of moving fluid.
Fluid compression is one such mechanism, but there are others that
your approach is ignoring. This is what the membrane is about:
springiness, or compliance. The membrane compliance has been
measured, and the measurements fit the physical models and the
observed wave speeds.
Adding some compressibility to the model is also possible, and is
needed to get that fast pressure mode as well, which I agree is
involved in getting the round window to be pushed out when the oval
window is pushed in. But that can be approximated well enough with
incompressible and infinite-velocity pressure waves, since the
wavelengths are so long, as you point out. These pressure waves
don't create any differential pressures around the basilar membrane,
and have negligible associated displacements and velocities
everywhere (even at the windows), compared to the traveling-wave
modes, so they are typically ignored in the discussion of cochlear
hydrodynamics, where the motions are what we care about.
Sorry to be so long-winded.