The brick paver maze labyrinth consists mainly of 44 concentric rings encircling a center stone. Between each concentric ring is a bevel.  If one claps sound bounces off of each bevel between concentric rings and returns to the labyrinth’s center. Because the labyrinth is overwhelmingly circularly symmetric the reflected hand clap is focused back to the center of the circle. A listener not in close proximity to the center only hears a hand clap. 

Each reflection returns to the center at slightly different times due to differences in sound path length.  These differences in path length cause the returning reflections to interfere with each other producing an entirely new sound.  The phenomenon of producing a higher pitched tone by adding a sound and the delayed version of itself together is called Repetition Pitch. Repetition Pitch Theory was pioneered by Frans Bilsen.  One can predict the frequency of the resulting tone using geometry.

Each bevel of the labyrinth was assigned a number which shall be known as n. The bevel between the center stone and 1^st ring was neglected because it was covered by the clapper’s feet. 

 The numbering scheme is as follows:

$ n=0 \ldots 43$

using the numbering system an average brick width was computed

$\langle w \rangle = \frac{r_{43}-r_0}{43} $

where \( r_{43}\) is the labyrinth's radius.

The distance to each paver gap from the center of the labyrinth was then written as

$r_n= r_0 + n \langle w \rangle \hspace{1cm} 0 \leq n \leq 43 $

Where \(r_0\), the distance from the center stone to the \(n=0 \) bevel, was held constant at 0.246m

It follows from the sound path diagram above that travel time from ones hands to the nth bevel and back to ones ears or microphone is given by:

$\tau (n)= \frac{\sqrt{r_n^2+Z_o^2}+ \sqrt{r_n^2+(Z'+Z_o)^2} }{v_s}$

where \(V_s\) is the temperature corrected speed of sound.

Inverting the delay time between successive reflections yields the frequency equation

$f_m(n)= \frac{m}{\tau(n+1)-\tau(n)}\hspace{1cm} m= 1,2,3,\ldots \mathrm{and} \ n=0,1,2,\ldots 41$

where $m$ is the harmonic number and $n$ is the boundary number.

Data was taken from the labyrinth using a microphone and wave recording program. The frequency of the produced high pitch tone predicted by the labyrinth's geometry matched the frequency calculated from the waveform data for three different microphone heights .

The fit of the experimental average fundamental frequency to the theoretical average fundamental frequency for each height. (a) $Z_{tot}$= 1.1 ~m, (b) $Z_{tot}$=1.65~m and (c) $Z_{tot}$=2.23~m

* Extracting the squeak’s frequency from the labyrinth’s waveform data is very involved so for the sake of space I am leaving that part out and jumping to the final results.

 content by Dante Dimidio Ursinus College Class of 2010