Dear Dick,I came across these auditory patterns due to pure tones in a text book on Physics where these figures have been attributed "courtesy of Dr. Harvey Fletcher". Ref: Mechanics, Heat and Sound by Francis Weston Sears, Library of Congress card No. 51-899, Addison-Wesley Publishing Company, Inc., Second Edition, Seventh printing-June 1958 (pretty long ago!). I don't have the history on how Fletcher derived these figures. I have over the years have thought about these figures with the results of ISI statistics and came to the conclusion that the only way a result such as shown in these figures could be explained is by showing that it was possible to describe a frequency analysis method that analyzes a pure tone as sum of a harmonic series in the energy domain. But that is neither here or there, it was simply my approach. I don't quite understand the term "volume compliance", and am quite happy to accept that not much of a change of BM geometry is required to provide the required change of volume range for a TW, but was more concerned with the restoring force available from the BM, and hence "stiffness", I guess I could have called it "springiness?". This does not vary as much as required, at least as reported. As far as "complexity of the problem" is concerned, each to their own. However, I do feel that a narrow focus using some very elaborate mathematical models to explain TWs, misses the higher level need to explain the more fundamental psycho-acoustical results that are generally accepted.
Thanks very much, happy hunting, Randy Randhawa On 11/9/2011 10:14 PM, Richard F. Lyon wrote:
Andrew Bell wrote:... between proponents of traveling wave models and those who are uncomfortable with its complexity.Is "uncomfortable with its complexity" really the problem? It's not that complicated, really. You have some fluid whose only relevant property is mass, and a compliant membrane whose only property is compliance, and a bit of geometry, leading to a simple boundary-value partial differential equation, the solution of which is traveling waves. Yes, you need a basic education in mathematical physics or engineering to understand it, but not any complicated math.You will of course see apparent traveling waves in any collection of resonators or other filters with graded time constants. But there are reasons why those are not good models of how the cochlea works. You can make such a parallel filterbank work moderately well as a functional model, but only if the filter order is at least 6 or 8 (like an N=3 or N=4 gammatone filterbank); basic (second-order) resonators have been often tried and rejected as giving bad predictions for masking and such. So now if you want to connect to some underlying physics, you need 6 or 8 state variables at each location, not just displacements and velocities as in the traveling-wave systems. It just seems too unlikely that there are that many undiscovered energy storage elements at each place in the cochlea. Perhaps the "squirting wave" can do it if there are a bunch of very compressible elements in there somewhere -- but I am uncomfortable with pushing all that complexity into unknown micromechanics, when we have a simpler paradigm that fits the data pretty well.Ranjit Randhawa wrote:... auditory patterns published by Dr. Harvey Fletcher, which showed that for pure tones the maximum peak of activity occurs at the CF location and decreasing peaks of activity at harmonic locations.I'd be interested in knowing what you're referring to. I've never seen anything like that in Fletcher's papers.... the range of stiffness of the BM only varies by a factor of 6 or soI'm not sure what that report is, or exactly what the "stiffness" means there, but the "volume compliance", the ratio of volume displacement to pressure, is proportional to some (5th?) power of the membrane width, and inversely as some power (3rd?) of membrane thickness; the powers mean that the thickness and width don't have to vary by a huge ratio to span several orders of magnitude of tuning. I don't have the numbers handy, but I'm pretty sure that they're out there, and consistent with the entire range of hearing (this paper by Cole, 1977: http://www.springerlink.com/content/p32n73j266g632r7/ shows a factor of 1000 tuning range from a factor-of-6 width variation). Physical models have even been built, and they show traveling waves consistent with their membrane properties (e.g. http://www.tufts.edu/~rwhite07/PRESENTATIONS_REPORTS/Liu-IMECE2008.pdf ).Dick