SOUND

BEATS

The teacher will demonstrate two tuning forks that can be tuned or de-tuned to produce beats. He will also demonstrate resonance and sympathetic vibrations.

The equation of a wave of frequency f and amplitude A is

We can tell this because sin repeats itself every 2p radians. So the equation above corresponds to a wave that repeats itself every time ft = 1 ie every time t = 1/f ie every time t = period of this wave [ since T = 1/f]

Also, sine varies from 1 to 1 so the function above varies from A to +A

With the x axis = time, this becomes

For a wave of amplitude 10 and frequency 200 Hz, graph this on your calculator or use Graphmatica on the computer. Adjust the x-scale so you can see about three complete waves

Now graph a second function of the same amplitude but of frequency 203 Hz.

Now graph the sum of the two functions/waves. Sketch all three graphs.

You should find that the graph of the sum varies periodically with a period much greater than each wave. This periodic variation is what we call beats.

Using your graphs, estimate the time on the x-axis from a minimum amplitude to the next minimum amplitude.

From this value find the frequency of the beats for these two waves.

Mathematical Calculation of Beat Frequency

Use the trig formula for sinA + sinB to find the mathematical equation for the combined wave in the form sinU x cosV. Do this for the general case of two waves of frequencies f1 and f2, with the same amplitude A.

From this equation can you see why the beat frequency is the difference of the individual frequencies? It's hard to see. Better ask the teacher if you can't see it for yourself.

It might surprise you but you probably have seen beats on TV too. When you are watching a TV monitor on TV or taking a picture of a TV with your video camera you may notice the picture in the picture seem to move up or see some lines moving up the screen. All this means is that the two TVs or the camera and the TV are producing their pictures at slightly different frequencies. The difference between the frequencies determines how quickly the pictures move in and out of synchronization.- beats!

DOPPLER EFFECT

The teacher will whirl a speaker (hooked to an audio generator) around his head in front of class. Listen to the sound as the speaker moves toward you, moves away from you, doesn't move.

Toward you

Away from you

At rest

You will run a simulation showing a source moving across the screen while emitting sound waves. You can vary the speed of the source, the speed of sound, the frequency of the sound.

Simulating the Doppler Effect

Run the program in the Virtual Labs Folder caller Doppler Effect for Sound.

You should see a source moving back and forth across the screen. The circles represent the leading edge of the emitted sound wave as it spreads out from the point where it was emitted.

What is the frequency of the emitted sound? What is the speed of the source in m/s? In Mach (Fraction of the speed of sound) ?

What is the frequency of the sound received by the observer as the source approaches? As the source moves away?

What do you think increasing the speed of the source will do to the frequency of the emitted sound? to the frequency of the observed sound when approaching/ receding?

Test your predictions by doubling the speed of the source. You can check the mathematics for the predicted frequencies by using the Doppler effect formulas.[Make sure you understand the derivation of the formula] Do one calculation and compare with the computer screen.

Shock Waves

Now try increasing the speed of the source to 300 m/s; almost the speed of sound. Run and observe that the circles nearly fall on top of each other in front of the source.

Then decrease the speed of sound to 300 m/s and run again. This time the circles do fall on top of each other in front of the source. When the source speed attains or exceeds the speed of the waves, the waves cannot travel forward away from the source. If the observer were located just above the path(on the positive y axis) then how long would he/she hear the sound (continuous or just one pulse). Would it be louder or the same loudness it was when the source moved slower than the speed of sound?

You should have noticed that he/she hears the sound once, only for a short time and very loud since all of the generated pulses hit his ear at about the same time; all the waves hit him basically at once. This piling up of the waves is known as a shock wave or "sonic boom".

Questions to ponder: Has anyone ever created a sonic boom in a car? Can passengers hear the Concorde's engines when its going faster than sound? Why doesn't the Concorde fly fast over highly populated areas?

The EAR

Observe Model.

See how we hear

MUSIC

MUSICAL SOUNDS

To complete this activity you must bring or make a simple musical instrument on which you can either play a simple tune or the musical scale. Making your own instrument from bottles, rubber bands, wine glasses, a saw or whatever can illustrate you creativity.

You will use the computer interface to display the waves from your musical instrument and compare it with others.

Musical sounds

• Hook up the microphone to Channel A and load the FFT software. FFT stands for Fast Fourier Transform. This means the computer dissects your sound wave to display the frequencies and relative amplitudes of each frequency. It is sort of a "sound print". The computer automatically starts recording when the sound is loud enough to trigger it.
• Use a tuning fork or whistle to generate a single frequency. Sketch the sound wave and its FFT in the cart below.
• Now each of you should try to play the same note on your musical instruments. Sketch each of your waves and FFT's.
• How does the fundamental frequency compare for these different instruments playing the same note?
• What is it about the sounds that make the notes sound different?

You should be able to see that the FFTs of string instruments are similar. So they have the same sort of sound.

Similarly, all brass instruments have fairly similar FFTs.

• Now, on one instrument play the same note an octave apart.

What do you notice about the fundamental?

Why do you think that instruments that play lowpitched notes have to be large?

• Which instrument has the richest set of harmonics?

Which instrument produces the most pure note?

Notice that some instruments play all harmonics and some only play some (like open and closed pipes)

• Observe a discordant sound such as a bell scaling.

You should see that discordant sounds contain many frequencies mathematically unrelated whereas harmonious sounds only contain frequencies mathematically related.

• Observe noise eg static from a poor radio

Pure noise contains all frequencies.

SAMPLERS and SYNTHESIZERS (optional)

Digital Recording of Sound

Quantization Errors

Storage and Manipulation

Midi files

Synthesizers

Demo + play around if you like