Mathematically Tuned into Sound Analyzing the Sound of a Tuning Fork Name: Group Members: When you throw a rock in a pond you probably notice the rippling effect of the water outward. Oftentimes these ripples come across an object floating in the water (like a leaf), causing it to vibrate up and down on the water. The vibrations of an object produce pressure oscillations in the surrounding air which travel outward. When the pressure waves reach the eardrum they cause it to vibrate. These vibrations are translated into nerve impulses and interpreted by people as sounds. The pressure waves are called sound waves, and the voices and sounds you hear every day are a combination of many different sound waves. Today we will analyze the sound of a tuning fork, which is a single tone that can be modeled mathematically using a sine or cosine function. You will need: 1 CBL per group 1 TI-83 Plus per group with link cable 1 laptop computer with TI-Connect program and USB cable (to save resulting data graphs) 1 Vernier Microphone 1 256 Hz Tuning Fork TI Program: TUNED Instructions for activity SETUP: 1. Make sure your calculator is in radian mode. 2. Connect the microphone to channel 1 of the CBL (located at the top-right of the CBL). 3. Connect the TI-83 Plus calculator to the CBL with a link-to-link cable. Insert the cable in the bottom of the calculator and the bottom of the CBL. 4. Run the TUNED program on your calculator. a. Upon pressing ENTER to start the program, select 1:COLLECT DATA from the OPTIONS menu and press ENTER. b. Select 2:NO under the SEE DIRECTIONS menu and press ENTER. c. You are now ready to collect the data. Make sure to strike the tuning fork against a rubber object (such as the sole of a shoe). Hold the fork close to the microphone and press ENTER on the calculator to record the sound. d. Your data should be sinusoidal and centered about the x-axis. If you are not satisfied with your data, press CLEAR, then ENTER and perform another trial. e. Once satisfied with your data, connect your calculator to the TI-Connect USB cable to save your picture in your folder. You will need to print your picture and attach it to this activity.
Analyzing the Data: To model the curve produced by the tuning fork, one of the following equations can be used: y = a sin(b(x - c)) + d y = a cos(b(x - c)) + d where a = amplitude (vertical stretch of graph) b = number of cycles data completes over the course of the natural period of 2 p 2p the function. = Period, therefore, = b. b Period c = phase shift (horizontal shift of data) vertical shift of graph (center line) d = 1. To begin, trace to the first maximum point on the graph and write the coordinates below. Do not round your answers. Maximum = (, ) or Now trace to the minimum that follows the maximum you just found. Write the coordinates for the minimum below (do not round). Minimum = (, ) Using the two points you just found, you should have enough information to write a cosine equation which models the data. 2. Please explain how to find the amplitude given the space below. a = 3. Please explain how to find the b value given the space below. b = 4. Please explain how to find the c-value given the space below. c =
5. Please explain how to find the d-value given the space below. d = 6. Write the cosine equation that models the data using your values for a, b, c, and d. y = 7. Enter your equation into Y1 of your graphing calculator and press GRAPH. Download your graph (which should run over the top of your data graph) to your folder using TI-Connect. Place a hard copy of your graphs below. Put the data graph on the left and your model graph on the right. 8. The frequency of a sound wave is the number of cycles per second. The period of a sound wave is the number of seconds per cycle. Explain the relationship between frequency and period. 9. Use the period to calculate the frequency of the sound wave and record it below. Frequency =
10. Standard tuning forks are imprinted with their frequency. Check the tuning fork that you used in this activity and record its frequency below. Tuning fork frequency = How does this compare with the frequency you found in question 9? Explain possible reasons for any discrepancies. 11. The amplitude of a sound wave increases with the loudness of the sound. Explain how you could alter the value of a if you repeated this investigation. 12. Pitch is associated with the frequency of the tuning fork. A higher pitched tone would have a higher frequency. Explain how your graph would change if you used a tuning fork of higher frequency (hint: think about how the period is affected). 13. Other values of c would work in your equation. Explain how you know this, and calculate two other c values.
DOWNLOADING AND SAVING TI-83 PLUS CALCULATOR IMAGES USING TI-CONNECT TO DOWNLOAD AN IMAGE TO TI-CONNECT: 1. Connect your calculator to the TI USB Cable that is connected to the computer. 2. On the computer desktop, run the TI-Connect program by clicking on the TI-Connect icon. 3. Make sure that the screen on your calculator displays the image or graphic you want to download to the computer. From the home page of TI-Connect, click on the Screen Capture Icon (Camera icon). 4. You should now see the image from your TI screen on your computer. To add the border, pull down the IMAGE menu and select Add Border. TO COPY AN IMAGE IN TI-CONNECT: You can pick one of three different ways to copy from TI-Connect: 1. From the EDIT menu, select COPY, or 2. Press CTRL+C, or 3. Select the icon displaying a clipboard with an arrow. TO PASTE AN IMAGE FROM TI-CONNECT TO MICROSOFT WORD OR OTHER: 1. Once you have copied the image in TI-Connect, open a word document. Press CTRL+V to paste the copied image to your document. TO SAVE AN IMAGE TO YOUR FOLDER: 1. Capture the picture of your screen using TI-Connect (see DOWNLOAD AN IMAGE TO TI-CONNECT above). Click on the blue disk icon. 2. Fill in the location where you would like to save the picture (or use the drop-down menu). 3. Create a new folder by clicking on the New Folder icon. Call the new folder TI Pictures. 4. Save your image to this folder by giving the image a filename. Make sure the image is saved as type BMP (*.bmp), which should be the default.