Physics 4BL Experiment 4

Laboratory 4

Speed of Sound and Light

Max Woo
11/12/14
Lab 11, Michael Ip
Partners: Zoe Zhu

Introduction

The purpose of this lab is to measure the speed of sound and light. We will be using a speaker and a microphone to measure the speed of sound, utilizing two methods: measuring phase shift and measuring standing waves. For the second part, we will be using a cordless telephone and a “Yagi” antenna, measuring the standing waves to determine the speed of light.

Experimental Data

Sound

To measure sound, we first set up our microphone on a movable track facing towards the speaker, as shown in Fig 1:

Fig 1: Speaker and microphone setup for the sound part

We hooked up our speaker to both a 12V DC voltage supply and to a function generator. We then configured the function generator to output a sine wave with an amplitude of ~3V and a frequency between 2kHz-20kHz. Finally, we hooked up our oscilloscope to measure the output of the function generator in CH1, and the output of the microphone in CH2.

Next, we moved the microphone until we shifted the phase of the microphone (CH2) by half a wavelength and one whole wavelength, recording the movement distance for both. We repeated this process for 4 additional function generator frequencies, which in total gave us 2 wavelength values for each of 5 frequencies. We then calculated the angular velocity $\omega$ and wave vector $k$ using the equations:

$\omega = 2\pi f$ (Eq. 1)
$k = 2\pi / \lambda$ (Eq. 2)

And plotted them against each other, called the dispersion relation, on a graph shown below:

Graph 1 dispersion relation for sound waves

Next, to calculate the speed of sound using standing waves, we first added a mobile reflector behind the speaker and facing towards the microphone, as shown in Fig 2:

Fig 2: Setup for the reflector

The mobile reflector had a potentiometer attached to the wheels to help convert distance measurements to voltage measurements. To utilize this, we connected a 2V DC power supply to the input of the potentiometer, and then connected the output to the speaker. We then connected CH0 of the myDAQ to the position voltage, given by the output of the potentiometer, and connected CH1 to the microphone. Thus, we had a way to measure the sound received by the microphone as a function of the reflector distance from the microphone.

In order to measure position from the output voltage of the potentiometer, we first moved the reflector back and forth and measured the relationship between the position of the reflector and the output voltage, using linear regression to determine the conversion factor between the two. We then measured the microphone output voltage as we varied the position of the reflector, giving us a sinusoidal graph like the one shown below:

Graph 2: Microphone output voltage vs position of the reflector

Because we knew the frequency we had set the function generator to (3kHz), the only other information we needed to know was the wavelength. Measuring the distance between adjacent maxima or minima in the graph gave us half the wavelength of the sound wave, so doubling the values gave us the full wavelengths, shown in the table below:

Table 1: Wavelength values determined from the distance between adjacent extrema

Light

The setup for this part of the experiment was almost identical to the standing wave part of the sound section, except instead of using a speaker and microphone, we used a cordless phone to transmit the light waves and a “Yagi” antenna to recieve them.

We first noticed that the “Yagi” antenna recieved signal the best when pointed towards the source (with the cross-wires getting shorter as they got closer to the source). Thus, when we replaced the speaker with the cordless phone and replaced the microphone with the “Yagi” antenna, we made sure to point the antenna towards the cordless phone. We also made sure the keep the reflector in place, and then disconnected CH1 of the myDAQ from the microphone and connected it to the “Yagi” antenna.

To ensure the most accurate results, we re-calibrated the potentiometer, measuring another set of voltage-position values and using linear regression to find the conversion factor. Then we repeated the process used in the standing wave portion of the sound section, measuring the output voltage of the “Yagi” antenna as a function of reflector position. Likewise, measuring the distance between extrema gave us the wavelengths of the light waves, shown in the table below:

Table 2: The measured wavelengths of light waves

Data Analysis

Sound

For the phase-shift method, we measured the angular frequency and wave vectors of different sound waves. The dispersion relation gives us the relationship between the speed $v$, angular frequency $\omega$, and wave vector $k$ of a sound wave:

$v = \omega / k$ (Eq. 3)

Thus, we can calculate speed using a linear regression of angular frequency against wave vector, which gives us $v = 355.42 \pm 19.99 \ m/s$. The speed of sound in air is generally 340.29 m/s, which is within the error bounds of our result, confirming our method.

For the standing waves section, we measured several values for the wavelength of 3 kHz sound waves. We can get a single value by taking the average of these results, with error calculated using the equation for standard deviation of mean:

$\sigma_{\overline{V}_N} = \frac{1}{\sqrt{N}} \sigma_N$ (Eq. 4)

This gives us $\lambda = 0.109 \pm 0.008 \ m$. Then, using the relationship between speed $v$, frequency $f$, and wavelength $\lambda$:

$v = f\lambda$ (Eq. 5)

and the associated propagation of error formula:

$\sigma_{v} = v \sqrt{(\frac{\sigma_{f}}{f})^2 + (\frac{\sigma_{\lambda}}{\lambda})^2}$ (Eq. 6)

We get $v = 327.14 \pm 2.80 \ m/s$. While close, the accepted value for the speed of sound (340.29 m/s) does not fall within the error bounds of our result. One possible reason for this is that the reflector was not restricted to a straight line of movement, so unsteady forward/backward movement during calibration and measurement could have shifted results both upwards and downwards. This shows that our first method, the phase-shifting method, is more accurate than the moving reflector method. However, it is important to note that the moving reflector method gives more precise results.

Light

The procedure for this section is almost identical to the moving-reflector procedure of the sound wave section, so the data analysis is also very similar. Thus, combining the measured values for wavelength with Eq. 4, we get an average value $\lambda = 0.12785 \pm 0.00245$. Then, using Eq. 5 and 6, we get $v = 313239 \pm 769 \ km/s$. The accepted value for the speed of light (299,792 km/s) falls within the error bounds of our result, confirming the accuracy of our method.

Conclusion

To conclude, we were able to achieve the purpose of our experiment, which was to calculate the speed of sound and light. In the phase-shift part of the sound wave section, we were able to obtain a value with error bounds that the accepted speed of sound fell within. Likewise, for the light section, the accepted value for the speed of light also fell within the error bounds of our measurements. The only part of our lab that was unsuccessful was the moving-reflector method of measuring the speed of sound, which could have been caused by imprecise movement of the reflector that introduced experimental error into our procedure.