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Re: A new paradigm?(On pitch and periodicity (was "correction to post"))
I want to express my complete agreement with your response and have little to add.
Just the remark that the statement made by dr Heerens that transmission line models would be specifically designed to support the traveling wave is incorrect. Such models are based on physics and derived from the Navier-Stokes equations. No assumptions about possible or impossible wave propagation are made in the derivation of the equations of motion, not in the numerical implementation. Only the compressibility of the fluid is generally - not always - neglected, which leads to an infinite velocity of compression waves.
2011/10/31 Richard F. Lyon <DickLyon@xxxxxxx>
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.