Math and Sound Investigation

1) Open up Audacity. Go to Generate : Tone to create a tone also known as a sound wave. You can delete a sound wave by highlighting it and pressing delete. It is not recommended to have more than two sounds waves in your workspace at once. Try generating sound waves with different amplitudes, waveform and frequency.

2) At a fixed point in space, a basic sound wave (pure tone) can be represented by a sinusoidal function. One of the functional representation is Asin(kt).

Fill in the blanks with the following words: amplitude, frequency, volume, pitch

The _________________ of the function represents the loudness or _________________ of the sound.

The _________________ of the wave function represents the _________________ or how high or low the note is.

Copy and paste the above two statements with their answers onto separate file. You will print your answers at the end of class to hand in.

2) Investigate:

Generate a sine wave tone. Choose a frequency (call this n)
Generate another sine wave with frequency n+1.
Play the result.
What does it sound like? Record your answer.

You may hear a pitch change and a beating frequency.
Delete the wave with frequency n+1
Repeat the above and investigate with frequency n+2, n+3, n+10, n+20, n+100, etc
How does the pitch change as the second frequency increases?
How does the rate / frequency of the beat change?
Record your answers.

3) On graph paper, hypothesis what the wave would look like.

4) When you are listening to the waves together, it is like adding two sin waves.
sin2πf1+sin(2πf2) =
What is the trigonometric identity associated with this sum?

5) Hypothesis which part of the wave is associated with the amplitude (the beating) of the wave and which part is associated with the pitch.

6) Clear your workspace. Make two sound waves with different frequencies. Record this frequency. Using your hypothesis in question 5 predict the frequency of the beat and the frequency of the pitch. Use appropriate frequencies in which you can hear the beats.

7) Test your prediction by listening to the waves. Record how many beats you hear in 10 seconds. What are the experimental values of the beats? Is your hypothesis in 5 valid?

8) With the trigonometric identity found in 4, label which part is responsible for the beats and which part is responsible for the pitch of the waveform.

9) Plot out the sum in a graphing program or graphing calculator. You may need to plot twice: choose an appropriate scale for the beat and another for the pitch. Include your plots with your answers to hand in at the end of class.

10) Find two frequencies f1, f2, in which a period of the beat is 10 times larger than the period of the oscillations (part responsible for the pitch). Show your work. Graph the sum of sin2πf1+sin(2πf2) .

Extension Questions:

Go to http://www.mindspring.com/~j.blackstone/dist101.htm. Scroll down to the applet. Play around with the applet and record how the sound changes. Does the perceived pitch change? In other words, if you tried to hum the note does it change as you change the wave?