SOUND

Read Sections 12-1 and 12-4

Sound: The speed of sound in air at 25o C is 343 m/s (often rounded to 340 m/s). The speed of sound changes with temperature since the density and elasticity of air change as temperatures fluctuate. At higher temperature, particles are less dense, but they are moving faster, allowing the sound to transfer faster.

v = 331 + 0.6T, so at 20 oC, the speed of sound is 343 m/s.

Speed depends on the mediums elasticity. When a wave travels from one medium to a different medium the speed & wavelength change. However the frequency remains the same.

Pitch: Frequency High Frequency = High Pitch Low Frequency = Low Pitch

Loudness: Amplitude

Wave Type: Longitudinal

Sound waves can originate from vibrating strings or in tubes. This is the basis for musical instruments.

There are two types of tubes: those open at both ends & those closed at one end.

Open Tubes

Same as strings, multiples of ½ waves. But the waves look a little different, since the ends aren’t fixed. Either of the following images could be used to represent a standing wave in an open tube.

1.

This is the image of the fundamental frequency in an open tube. Consider the length of the tube to be L. Then since this is ½ a wave, λ = 2L.

2.

This is the image of the 2nd harmonic for an open tube, ½ wavelength is added to the wave. For this tube, again let the length be L, λ = L.

3.

For the 3rd harmonic, then length of the tube would contain another half wave. So, using this as an example, the wavelength of for this tube on the 3rd harmonic would be 0.8 m.

Problems.

1. You buy a tube with length 2.4 m.

a. Determine the wavelength of the 4th harmonic. Determine the frequency of for this harmonic.

b. Determine the wavelength of the fundamental frequency. Determine the fundamental frequency.

2. Determine the length of pipe that would have a 2nd harmonic for a note of 512 Hz.

Closed Tubes

Closed Tubes: Closed tubes hold multiples of ¼ waves. Closed tubes begin with a fundamental frequency at ¼ wave. If L is the length of the closed air column, then at the fundamental, the wavelength, λ = 4L.

I like the image to the right. It shows the use of a standing tube of water to determine the speed of sound in air. What is different about a closed tube is this, you start at ¼ of a wave rather than ½ like strings and open tubes. You still have to add ½ of a wave for the next harmonic, however, THERE ARE NO EVEN

HARMONICS for a closed tube. 3𝜆

4 is the 3rd harmonic and

5𝜆

4 is

the 5th harmonic.

Review. Comparison of Open vs. Closed Tubes

Watching this video will make you a better person: http://www.youtube.com/watch?v=Ude8pPjawKI

3. A vibrating tuning fork is held above a column of air, as shown in the diagrams above. The reservoir is raised and lowered to change the water level, and thus the length of the column of air. The shortest length of air column that produces a resonance is L1 = 0.072 m, and the next resonance is heard when the air column is L2 = 0.24 m long. The speed sound in water is 1490 m/s.

a. Calculate the wavelength of the standing sound wave produced by this tuning fork.

b. Calculate the speed of sound for this experiment.

c. Calculate the wavelength of the sound waves produced by this tuning fork in the water.

d. The water level is lowered again until a third resonance is heard. Calculate the length L3 of the air column that produces this third resonance.

4. A hollow tube of length 3.6 m, open at both ends as shown, is held in midair. A tuning fork with a frequency 48 Hz vibrates at one end of the tube and causes the air in the tube to vibrate at its fundamental frequency.

a. Determine the wavelength of the sound.

b. Determine the speed of sound in the air inside the tube.

c. Determine the next higher frequency at which this air column would resonate.

d. If we cut some of the tube off, what would happen to the frequency of the fundamental note?

___ Higher ___ Lower ___ Remain the same. Justify.

5. A tuning fork is used to establish a standing wave in an open ended pipe filled with air at a temperature of 20º C where the speed of sound is 343 m/s, as shown above. The sound wave resonates at the 2nd harmonic frequency of the pipe. The length of the pipe is 33 cm.

a. Sketch the standing wave in the figure above. (For simplicity sketch a transverse wave to represent the standing wave.)

b. Determine the wavelength of the resonating sound wave.

c. Determine the frequency of the tuning fork.

d. Determine the next higher frequency that will resonate in a pipe of this length.

e. If the open pipe is replaced with a pipe which is closed at one end, what would have to be the shortest length of the closed pipe for the original tuning fork to resonate at its fundamental frequency?

Read Sections 12-5 and 12-6 on interference, superposition, and beats.

Beats

When the waves of 2 sound sources at different frequencies interact, you get a pattern of constructive and destructive interference, called beats.

If the two frequencies are f1 and f2, then the beat frequency is |f2 – f1|, and the wave interference pattern looks like the image to the right.

Problems.

6. What is the beat frequency of two sound sources of 512 Hz and 504 Hz.

7. How many beats would you hear in 8 seconds if two interfering notes of 328 Hz and 325 Hz were played?

8. If a 480 Hz tuning fork is held next to a guitar string, you hear 15 beats in 3 seconds. What are the possible frequencies of the string?

Watch this video: http://www.youtube.com/watch?v=gHOC58L-53Q

2-Dimensional Interference

When two wave sources have the same frequency, they can still interact and produce an interference pattern.

Go to the Phet Colorado applet: http://phet.colorado.edu/en/simulation/sound

Either download or run on your browser. You will need updated Java to run it. Once it is running, select the tab in the middle at the top, “Two Source Interference”. Turn the amplitude completely up to max and enable audio. When you press the play button, you should get an interference pattern as seen below.

When you press pause, hopefully you can see the faint lines that represent total destructive interference. Between these ripples, you should be able to see the “spots” that represent the crests and troughs of the constructive interference regions between the lines.

If you click on the man and put him along the destructive interference lines, the sound will be canceled out. If you click on the man and put him in the constructive interference region, the sound will be at a maximum loudness. Play around with where you put him and see the results.

If you don’t have sound when pausing and starting the app, reselect the audio enabled check box and it will come back on.

I have tried to draw lines where there would be destructive interfence and placed the guy at a point where the sound would be minimized due to the interference.

On the image below, I have placed the guy in a region of constructive interference ripples. I tried to show this with the dotted line.

A second applet to see the wave interference properties is also at Phet Colorado.

Go to the Phet Colorado applet: http://phet.colorado.edu/en/simulation/wave-interference

Turn the frequency and amplitude all the way up. Select two drips and play it. You should get an image like

The constructive and destructive interference regions are more visible in this view.

Read Sections 12-7 and 12-8 on the Doppler Effect, Shock Waves, and Sonic Booms.

Doppler Effect

You’ve probably noticed that when an ambulance passes you with its siren going, or a plane flies overhead, you can hear the sound change.

● I'm not talking about louder or quieter, but rather the pitch of the sound.

● As it comes closer to you the pitch (frequency) of the siren will seem to increase.

● When it is moving away, it seems like the frequency decreases.

● When the true frequency of a source of sound is changed so that an observer hears a different apparent frequency as they move relative to each other, it is known as the Doppler Effect.

“Bob the Swimming Bug” (aka Why Frequency Changes)

To see why this change in frequency happens, I want you to imagine a very special bug named “Bob.”

● I have trained Bob to do a very special trick. He can tread water!

● When I first trained him to do this, he could only do it while staying in exactly one spot. If we were to look down on Bob as he treads water, we would see waves that look like Figure 1…

Notice how Bob is in the middle and all the waves are spreading out evenly from him in the water.

If I asked you to stand on the right side of Bob, and then later on the left of Bob, and measure the frequency of the waves (how many pass you per second) you would give the same number.

Let’s say that Bob is bobbing up and down at exactly 4 Hz, and he always treads water at this frequency. No matter where you are standing, you will measure the same frequency for Bob's waves. No Doppler Effect yet.

After years of training I am able to get Bob to tread water while slowly moving to the right.

Each time he starts to make a new wave, he’ll be in a spot slightly to the right of where he used to be.

That means the center of each new circle will be slightly to the right of where he made his previous wave circle.

The red circle is the first wave he made, then blue, pink, and finally black.

Each wave ripple continues to spread out as Bob moves to the right.

At one second, Bob is at the center of the red wave he has created.

At two seconds, he is at the center of the next wave he is creating, a blue one. His red wave continues to grow outwards with its center where he was at one second.

At three seconds, the pink circle is created with Bob at the center. The blue and red circles are still expanding with their centers at the same spots as when they were originally created.

This pattern would continue for as long as Bob is bobbing up and down and moving to the right.

Now let’s look at what you would measure about Bob’s frequency if you were standing on the left side compared to the right side. Remember that Bob is truly treading water at 4 Hz.

On the left side the waves have been spread further apart.

Spread out more means that the wavelength is bigger. Since wavelength and frequency are inversely related, you would say the waves have a lower frequency (maybe something like 2Hz). The waves are not passing you as frequently.

On the right hand side the waves are squished closer together.

If I was standing on the right side, I would say that the waves have a smaller wavelength, so they must also have a higher frequency (like maybe 6Hz). They are passing you more frequently.

In both cases you measured an apparent frequency of Bob’s treading that is different from his true frequency, but you have no way of knowing this unless you have measured his true frequency when he was motionless relative to you.

The same thing happens with sound. If a sound source is moving towards you, you will detect a higher than real frequency. As it moves away, you will detect a lower than normal frequency.

In addition to that, you might move relative to the sound source. If you run at the source, you will shorten

the wavelength, creating a higher pitched observed sound. If you run away from the source, then you will increase the wavelength, creating a lower than actual frequency.

Thus, the Doppler effect is determined by the relative motion of an observer and a source. If the source and the observer are moving towards one another, the frequency increases. If the source and observer and getting further apart, then the frequency will appear to be lower.

You must watch this video: https://www.youtube.com/watch?v=h4OnBYrbCjY

And this one: http://www.youtube.com/watch?v=a3RfULw7aAY

Doppler Effect Math

Where v – velocity of sound 𝑓′ – observed frequency

vs – velocity of source f – emitted frequency

vo – velocity of observer

How to use this formula?

Draw the observer of the sound on the LEFT side of your paper ALWAYS. Define positive motion as two the right, always. Draw arrows for the direction of motion of the observer and of the source. Substitute the correct values and signs into the equation above.

Problems.

9. A car approaches a stationary observer traveling at 30 m/s and blowing its horn at 690 Hz. What frequency will the observer experience if the speed of sound is 330 m/s?

10. A car travelling 45 m/s passes a runner who is running 5 m/s in the same direction as the car is traveling. If the car was blowing its 690 Hz horn as it passes the runner, what frequency did the runner experience while the car was

a. Approaching?

b. After it had passed?

Sonic Boom

Assume an object is emitting a wave, like the ambulance, or the arrow in the image below. The sound fronts are building up in front of the arrow creating an area of high density waves. If the object breaks across the wave front, or begins to move faster than the waves they are creating, they create a powerful trail of dense wave behind them. This is what creates a shock wave, a.k.a., sonic boom.

To better see how these wave fronts are formed, go to

http://galileoandeinstein.physics.virginia.edu/more_stuff/flashlets/doppler.htm

This is a Doppler effect demo, but I like if for generating wave diagrams.

Look at the slider and adjust it to some numbers less than 1, in this case, subsonic. Press play and look at the wave diagrams generated. You can press stop, change the value on the slider, and then restart at any time.

Hopefully you will see, that as you move to numbers closer to 1, the wave fronts on the right get closer and closer, and at 1, the wave fronts are all overlapping. If you were riding in a boat, this is where the nose of the boat would be pointing up slightly out of the water.

When you go over 1, you begin to make shock waves (V waves in a boat) and the wave is dragged behind you.

This video is RIDICULOUS! : http://www.youtube.com/watch?v=gWGLAAYdbbc

11. When an automobile moves towards a listener, the sound of its horn seems relatively

a. low pitched b. high pitched c. normal

12. When the automobile moves away from the listener, its horn seems

a. low pitched b. high pitched c. normal

13. Circle the letter of each statement about the Doppler Effect that is true.

a. It occurs when a wave source moves towards an observer.

b. It occurs when an observer moves towards a wave source.

c. It occurs when a wave source moves away from an observer.

d. It occurs when an observer moves away from a wave source.

14. True / False: A moving wave source does not affect the frequency of the wave encountered by the observer.

15. True / False: A higher frequency results when a wave source moves towards an observer.

16. Two fire trucks with sirens on speed towards and away from an observer as shown below.

a) Which truck produces a higher than normal siren frequency?

b) Which truck produces a lower than normal siren frequency?

17. What is the frequency heard by a person driving at 15 m/s toward a blowing factory whistle (800. Hz) if the speed of sound is 343 m/s?

18. A car approaching a stationary observer emits 450. Hz from its horn. If the observer detects a frequency of 470. Hz, how fast is the car moving? The speed of sound is 343 m/s.

19. Minnie Sota, a piano tuner strikes a key and a tuning fork at the same time. The tuning fork is 440 Hz. If a beat frequency of 5 Hz is produced, what are the possible frequencies for the piano? Which of these is the correct frequency if tightening the string reduces the beats to zero?

20. The figure above shows two wave pulses that are approaching each other. Draw the shape of the

resultant pulse when the centers of the pulses, points A and B coincide?

21. A train whistle has a frequency of 200 hertz as heard by the engineer on the train. Assume that the

velocity of sound in air is 340 meters per second. If the train is approaching a stationary listener on a

windless day at a velocity of 40 meters per second, what is the whistle frequency that the listener

hears?

22. A standing wave of frequency 8 hertz is set up on a string 1.6 meters long with nodes at both ends

and in the center, as shown above.

a. What is the speed of the wave?

b. What is the frequency of the 3rd harmonic?

c. What is the wavelength of the fundamental frequency?

d. If the tension in the string is 12 N, what is the linear density of the string?