Methods, Tips and Data for Inexpensive Audio Activities: Earbud-based Sound Experiments Beat Frequency Experiment This beat frequency experiment requires only Audacity and earbuds. In Audacity, set up one track that plays only a 3440-Hz tone. Set up another track that plays a patterns of tones: 3440-Hz for 1-s, 3441-Hz for 3-s and then 3442-Hz, 3443-Hz, 3445-Hz and 3450-Hz for 2-s each. When these two tracks are played simultaneously through the computer speakers and also recorded at the same time (a perk of Audacity), then complete beat frequency patterns are shown in the recorded track (Fig. 1a below). To obtain incomplete cancellation, play your frequency set-up though the earbuds and separate them by about 5 cm between both the earbuds and the computer microphone (Fig. 1b). The equation for these beat frequencies is:

𝑇!"#$ =1

𝑓! − 𝑓!

Where 𝑓! is the single 3440-Hz tone and 𝑓! is the multiple tone track. For this scenario, the beat periods are 0, 1.0000, 0.5000, 0.3333, 0.2000 and 0.0100±0.0003s. If you want a more conceptual experiment rather than an example of beat frequencies, give the students tones to put in Audacity and have them determine how they are related to each other (the inverse of the difference between the two frequencies). Then you can reveal the equation. You can also use more than two tracks in Audacity, generating different kinds of interference patterns like those in Fig. 1c. For a simulation (Fig. 1d) requiring input for up to three sound wave amplitudes ("Aj"), frequencies ("fj") and phases ("øj", used to shift simulated data to match real data), use the following time-dependent mathematical model:

𝐵𝑒𝑎𝑡𝑠 = 𝐴! ∗ cos 2𝜋𝑓!𝑡 + 𝜙!!

!!!

This simulation is freely downloadable on this website.

Figure 1: a) The beat frequency interference pattern created from the computer’s speaker. Note perfect cancellation since it was output from one source. b) Same frequencies as (a) but with the Track 1 in the left earbud and Track 2 in the right earbud at different separations from the microphone which generates an incomplete wave cancellation pattern. c) A 1-s beat pattern created by generating three Audacity tracks through an inexpensive earbud (left half of figure) and better quality earbud (right half figure). The three tones include a 0.8 amplitude 3440-Hz tone, a 0.4 amplitude 3445-Hz tone and a 0.4 amplitude 3450-Hz tone. The low quality earbud was 3-cm from the computer microphone and the high quality earbud was an "Appled iPod Original" located 12-cm from the computer microphone. d) Logger Pro beat frequency simulated with same frequencies and ratio of amplitudes. Note the simulation provides good agreement with experimental beat patterns but without overmodulation noise. Vertical axis on all graphs are arbitrary amplitude units. Horizontal time axis for Figures "a" and "b" are 10-s, "c" and "d" are 1-s. Interference Patterns Tape each earbud on a meter stick 40 cm apart from each other. This is what you will use to drag the earbuds at a constant speed past the computer’s internal microphone. Then create a 3440-Hz track on Audacity or a signal generator app (if using a smartphone as well) and plug the earbuds into the device you choose to use. We chose the frequency of 3440-Hz because the speed of sound in air for the temperatures of our lab is 344 m/s. Turn on the tone, and starting with one earbud directly over the microphone, pull the meter stick across to the other earbud. With this lab, it doesn’t particularly matter what speed you pull the earbuds at; what matters more is that the speed is constant. This can take some practice, so make sure students are given time to try this experiment multiple times. Once a clear pattern is produced (Fig. 2), the constant speed (𝑣) can be determined by the distance pulled (𝑥 = 40 cm) divided by the time it took you to pull it (𝑡).

𝑣   =𝑥𝑡

To add additional complexity to this experiment, in-phase and out-of-phase interference can be discussed. The patterns produced in the methods above obey constructive in-phase, or destructive out-of-phase, interference relationships when the difference between the position referenced to each earbud (xleft or xright) is a whole integer of the wavelength (𝑛):

𝒙!"#$ − 𝒙!"#!! = ∆𝒙 = 2𝑥 = 𝑛𝜆

The simplification, x = nλ/2, is used when the origin is taken to be half way between the two earbuds. We attempted a plausible simulation for the in-phase and out-of-phase speakers by superposing two sets of traveling waves moving towards each other:

𝐼𝑛  𝑃ℎ𝑎𝑠𝑒 = 𝑎𝑏𝑠 𝑗 𝐴! cos2𝜋 𝑗 𝐿2− 𝑥

𝜆 + 𝜙 exp−𝐶 ∗ 𝑎𝑏𝑠 𝑗 𝐿2− 𝑥

𝜆 + 𝜙  !

!!!!

Where the amplitude "Aj" is adjusted to fit the data, "L" is the length between the two earbuds, the origin of the position "x" is midway between the two earbuds, and "C" is a decay constant adjusted to fit the data. For out-of-phase interference, change the first "abs(j)" to "j". The wavelength "l" is calculated based on the speed of sound and frequency. The exponential decay is a simple attempt to model the position dependent amplitudes, but otherwise has no physical basis. The simulation's phase slider can be adjusted to make the two traveling waves appear to approach each other. The simulation also includes an arbitrary phase shift to make the simulation better match real data (Fig 2). We found a phase shift of about 90° gave the best result for our data.

Figure 2: The Audacity horizontal time axis was converted to a spatial interval using the earbuds' speed. a) in-phase signals from each earbud pair with b) computer simulation below. c) out-of-phase signals from each earbud pair with d) simulation below. The spatial axis for real and simulated data is identical for both graphs. Resonance For the resonance experiments, students use different lengths of tubes and estimate fundamental frequencies of each of the different lengths. The half wavelength experiment, cut the Pixie Stix tubes at different lengths (accuracy to ±1 mm) and give the following equation to your students:

𝑓 =𝑛𝑣2𝐿

Where “L” is the length, “n” is the fundamental frequency (and in this case n=1 because it will be the easiest for students to hear), “v” is the speed of sound in air and “f” is the frequency. Once students have calculated a frequency, they can test it out by placing an earbud at one end of the tube and generate their calculated frequency into the tube. They can then adjust their frequency until they hear the loudest note produced at the other end of the Pixie Sitx. The quarter wavelength calculation to solve for frequency is very similar but with 4 in the denominator of the equation instead of 2. And instead of using tubes of different lengths, keep one tube long, tape a ruler to it, and immerse it in a large jug of water (accuracy to ±0.5 mm).

The ruler helps you to see the length of the tube and you can create an open-closed system of different lengths by moving the tube up and down in the water. Once students have calculated fundamental frequencies at different lengths they can test out those frequencies on the lab setup and then enter their actual data into a graphing program like LoggerPro (used for the graph below).

Figure 3: Student ears were used to determine the maximum sound amplitude that was recorded for the resonance length. If students have a hard time finding the loudest point, they can record it through Audacity and measure the highest amplitude that way. The slopes plotted above are close to half the speed of sound (green line) (a) for the half-wavelength resonance and quarter the speed of sound (blue line) (b) for the quarter-wavelength resonance predicted. The speed of sound was about 344m/s at the temperature of the room on the day the data was collected. The slopes are less than predicted by around 20-m/s, however when larger (3cm) smooth bore tubes (data not shown) are used the slopes are within experimental error of about ±4.0-m/s. The negative intercepts are less than the uncertainty and within the end corrections expected for the finite diameter (1.0cm) of the Pixie Stix tube.

The y-intercepts are related to the tube diameter, and a simple end correction on the equations above can account for that, where “d” is the tube diameter. 𝐿 = !!!"#$%

!!− 0.8𝑑   Half wavelength (λ/2)

𝐿 = !!!"#$%

!!− 0.4𝑑   Quarter wavelength (λ/4)

Doppler Shift For this experiment you will need Audacity and a free sound generating app if you choose to use that instead for tones instead of Audacity. Tape the two earbuds together with the speakers facing

outward and connect them with a smartphone generating a 3000-Hz tone. Mark a place on the cord to hold the earbuds so you know what the radius of your circle is. Swing them in a horizontal circular motion past the computer microphone and record the sound with Audacity. The data below (Fig. 4) was collected with a constant radius of 0.534 ± 0.025-m. If you are good at constant rotation, the period is accurate to better than 1-ms and thus contributes negligibly to the error of the speed calculation. The radius is actually the leading source of error in this manual experiment since the earbuds are rotated by hand. The frequency shift increases with tangential speed and can be extracted to a resolution of about 13-Hz on the log plot Blackman-Welch spectra within Audacity. The raw signal versus time spectrum (Fig. 4) provides the period "T" used to calculate the tangential speed of the earbud rotation. The frequency shift "∆f" can be extracted using the Fast Fourier Transform analysis by taking the difference between the two shoulders of the FFT spectra (Fig. 4). The frequency shift versus tangential velocity was plotted to determine the speed of sound for the rotating earbud:

𝑣 =2𝜋𝑟𝑇  𝑎𝑛𝑑  ∆𝑓 =

𝑓

1− 𝑣𝑣!"#$%

−𝑓

1+ 𝑣𝑣!"#$%

=2𝑓

𝑣!"#$%𝑣

Earbuds have advantage of providing a loud, single frequency point source well above the noise level found in typical Audacity FFT spectra. Also, higher frequencies ensure a larger frequency shift that has a smaller error than found at lower frequencies.

a) Raw spectrum 4.429 ± 0.001s wide of 20 peaks (T = 0.221±0.001s) of the Doppler shifted signal from the earbuds controlled by the smartphone app. The spacing between sound "blips" comes about when the earbuds pass near the internal pc microphone while rotating. b) Zoom in on Blackman-Harris window log-plot spectra showing left shoulder (moving away from microphone) and right shoulder (moving toward microphone) frequency shift of central

frequency from earbuds of about 240-Hz. The red arrow is a 3000-Hz audio signal from a stationary computer speaker for comparison. c) Frequency shift versus tangential speed, with a slope of 16.9±0.6-Hz, close to the expected value of 2f/vsound = 17.4-Hz.