Lee Olsen Wrote:
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> >David Lubman Wrote:
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> > My informal finding? Chirps may be heard as few as 4-8 stairs. I'll bet this
> > experiment has been done before and probably better, and resides somewhere
> > in the psychoacoustics literature.

>"This wall at the Lawrence Berkeley National Lab produces unusual chirping echoes
> when I clap nearby. Berkeley professor Frank Crawford pointed out the phenomenon
> in a short note > to the American Journal of Physics in 1969. The wall acts as an
> acoustic diffraction grating, separating out the different frequencies of sound that
> go into a handclap."

Thanks Lee! I believe that Professor Crawford’s explanation is correct. My explanation specific to the Kukulkan pyramid staircase was developed later and independently (1999).

Others also have solved the staircase diffraction problem. My favorite is by a 3rd year undergraduate physics major at Bowdoin College in Maine. She made quite a hit at the college with her term paper! Her title (from memory) "how to make a staircase chirp like a bird"

I’m also credited for a chunk of the solution in Jearl Walker’s “Flying Circus of Physics” – an underground favorite of mathematical physics junkies. I'd guess that most physics professors have this book. I can’t find my copy, but the book index on Amazon’s website contents lists “ 3.65 Musical echoes from stairs and fences, p 172.”

My earlier work employed “picket fence effect”, which is better known in optical physics than acoustics, and is very simple. It predicts chirped echo frequency vs. time correctly, as well as the presence and frequency glide of harmonics. It is very useful for demonstrating the psychoacoustic effects. Such as:

"How can a sequence of impulses be perceived as a smoothly gliding tone?"

Students need only create and auralize a *.wav file of the impulse sequence to hear it
for themselves.

On Amazon's “Flying Circus table of contents ” I just noticed that the next entry is:

“3.66 SHORT STORY - Echoes from ancient structures, p 173”.

I didn’t remember that. I wonder if it is related. Amazon shows a copyright 2007 edition. My misplaced edition could be older.

I told the Bowdoin College student that the chirped echo problem can be solved by diffraction. But can be solved more accurately and completely using impulse response (IR) methods. IR methods employ the convolution theorem of linear system theory. IR methods have important practical and theoretical benefits over simple diffraction.
They are more complete and accurate. Unlike the simple diffraction solution, IR methods precisely account for echo spectrum. Most importantly given the IR, echoes of any other sound can easily be simulated (e.g., voice or ethnic sound instruments).

The IR of the Kukulkan pyramid is written as a summation of 91 terms, one for each step. This solution was first given publicly in 2002 at an acoustical conference in Cancun, Mexico. I provided analytical, graphical and sound simulation results, and explained its many benefits over the diffraction solution. Details are shown in my Acoustics 08 paper [intellagence.eu.com]

Archaeoacoustic use of IR methodology has since gained new practitioners. The IR of standing ruins (or intact structures) can be quickly captured on-site with sophisticated and increasingly affordable audio techniques. As reported in the August 2010 New Scientist, British workers did this at Stonehenge, and at an alleged full Stonehenge replica in Washington State.

My use at Kukulkan was different in that the impulse response of the original staircase construction was calculated (rather than captured on site) from reports of its original dimensions and materials.

Given the fame in the popular press, and the interest Kukulkan pyramid's chirping staircase has generated in the physics world, does it seem odd that mesoamericanists haven't looked into its archaeology as well?