Acoustic Characterization of Echo Amphitheater, Abiquiu, New Mexico

Standing in front of the amphitheater

Abstract

I recently visited Echo Amphitheater, a naturally formed amphitheater near Abuiquiu, New Mexico. Despite being central to wild west folklore, the site has never (to my knowledge) been studied for its acoustical properties. To gain a better sense of the exotic acoustics of the amphitheater, I recorded myself clapping my hands four times with a TASCAM DR-05 field recorder. When I returned to Dallas, I analyzed the echo, decay, and bandpass of the amphitheater to study its size, shape, and composition.

Table of Contents


Data

Below are the four handclaps. Feel free to download, share, analyze, etc.

Clap 1

Clap 2

Clap 3

Clap 4


Echo

The annotated waveforms below identify the first four echoes and indicate their order.

Fig. 1. Annotated waveform for the first clap. The top waveform corresponds to the right channel, and the bottom to the left.

Fig. 2. Annotated waveform for the second clap. Note the significant anharmonic noise due to wind (which can be heard in the data above).

Fig. 3. Annotated waveform for the third clap. This was the "cleanest" of claps.

Fig. 4. Annotated waveform for the fourth clap.

In the table below, the time between each echo from the previous echo is recorded. The initial clap is set at t = 0.

Clap # 1st order echo time (s) 2nd order echo time (s) 3rd order echo time (s) 4th order echo time (s)
1 0.379 0.196 0.165 0.149
2 0.395 0.261 0.186 0.158
3 0.401 0.213 0.187 0.174
4 0.406 0.269 0.225 0.132
Average 0.39525 0.23475 0.19075 0.15325

Fig. 5. The positive concavity of this plot of average echo time vs. echo order reflects the curvature of the Echo Amphitheater. Generally, a plot with positive concavity arises due to a concave echoic chamber, a plot with no concavity arises due to a rectilinear echoic chamber, and a plot with negative concavity arises due to a convex echoic chamber.

We can learn more about the size and geometry of the Echo Amphitheater by considering the speed of sound (v ≃ 343 m/s).

Fig. 6. This is a schematic of the simplest model that explains the first-fourth order echoes. The lengths of the paths traveled by the sound are denoted by la, b, c, d, e, f, g.

Now considering that the distance traveled l = vt,



The path defined by the lengths lb, ld, and lf can help us find the curvature of the canyon. We have two ways to find the lengths (depending on the experimental setup):

Near-field (applicable to my setup):

If we set up the microphone and source near the center of the amphitheater wall, lblc, ldle, and lflg. The above system of undetermined equations then becomes determined:


Far-field (maybe for my next visit to NM, but requires a stronger source and more sensitive microphone):

If we set up the microphone and source a good distance away from the amphitheater wall (lalclelg), the above system of undetermined equations becomes determined:



Proceeding with the near-field limit, we find la = 67.78 m, lb = 40.25 m, ld = 32.71 m, and lf = 26.28 m. la tells us that the first-order depth of the amphitheater is 67.78 m. We can use the other path lengths to construct a 2D model of the amphitheater's curvature. Considering 45-degree-incidence, for example, three points along the amphitheater wall would be (lb , 0), ((lb-ld) cos 45, la/3), and ((lb-ld+lf) cos 45, 2la/3):
We can then compute the radius of curvature of the amphitheater to be
This is a useful parameter that can be used to model the echo and geometry of the amphitheater.

Decay

We can find the decay factor (measured in dB per echo order) of the canyon by analyzing the intensity of the initial claps and the first-third-order echoes (the fourth order echo was too faint to accurately measure the intensity). The decay factor is average of the slope of the graphs below.


Fig. 6. Echo order vs dB for clap 1

Fig. 7. Echo order vs dB for clap 2

Fig. 8. Echo order vs dB for clap 3.

Fig. 9. Echo order vs dB for clap 4.

The decay factor was calculated to be -12.225 dB/echo order. The decay factor is a reflection of the canyon's composition. You can read about the geological history of the Echo Amphitheater here.


Bandpass

In addition to modulating the delay and amplitude of the input signal, and the amphitheater modulates the spectrum, acting as a low-pass filter. This can be seen below:


Fig. 10. In this plot of clap 1, the waveform and spectrum are plotted side-by-side. The intensity of frequencies is indicated by the red color. Note that the first-order echo is devoid of the 5000-8000 Hz frequencies abundant in the initial clap.

Since the peak of the initial clap was around 1000 Hz, the bandwidth (BW) is about three octaves:

Fig. 11. Zoomed-in spectrum of the initial clap and first-order echo for clap 1.

We see a similar ~three octave bandwidth (BW) for claps 2-4:

Fig. 12. Zoomed-in spectrum of the initial clap and first-order echo for clap 2.

Fig. 13. Zoomed-in spectrum of the initial clap and first-order echo for clap 3.

Fig. 14. Zoomed-in spectrum of the initial clap and first-order echo for clap 4.

We can quantify this calculate low-pass effect with the so-called Q-factor:


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