11.3.3.2: Explorations

Exploration 1: Creating Sounds by Adding Harmonicas

Begin by choosing the first harmonic (represented by the H#, with H1 being the fundamental or first harmonic) and drag the slider to add that harmonic to the total wave. As you do this, note that the frequency remains the same, but the amplitude slowly decreases. Continue to decrease the value of H1 so that it is negative. Notice that the negative sign simply inverts the shape of the sound wave. Therefore, the slider controls the amplitude and phase (0 or π only) of the harmonic of the sound wave. In addition to the overall wave form, the relative size of the components of the wave is shown in the graph on the right. Restart.

1. Measure the fundamental's period.
2. What is the fundamental frequency?
   Consider the following values for the harmonics:

   \(H\)	Case A	Case B	Case C	Case D
   \(1\)	\(1.000\)	\(1.000\)		\(1.000\)
   \(2\)			\(0.500\)	\(0.500\)
   \(3\)	\(-0.111\)	\(0.333\)		\(0.333\)
   \(4\)			\(0.250\)	\(0.250\)
   \(5\)	\(0.040\)	\(0.20\)		\(0.20\)
   \(6\)			\(0.166\)	\(0.166\)
   \(7\)	\(-0.020\)	\(0.142\)		\(0.142\)
   \(8\)			\(0.125\)	\(0.125\)
   \(9\)	\(0.0123\)	\(0.111\)		\(0.111\)
   \(10\)			\(0.100\)	\(0.100\)

   Table \(\PageIndex{1}\)

3. What wave patterns develop from these values?
4. Can you write down a mathematical formula describing each case? (Hint: it is a sum.)

Exploration authored by Morten Brydensholt, Wolfgang Christian, and Mario Belloni.
Script authored by Morten Brydensholt, Wolfgang Christian, and Mario Belloni.

Exploration 2: Creating Sounds by Adding Harmonics

Begin by setting the slider for \(H5\) (Harmonic 5) to the value of \(1\). You should hear a pure tone. Restart.

1. What is the frequency of the fifth harmonic?

Now slowly decrease the value of \(H5\) to zero. As you do this, note that the frequency remains the same, but the amplitude slowly decreases. Continue to decrease the value of \(H5\) so that it is negative. Notice that the negative sign simply inverts the shape of the sound wave. Therefore, each slider controls the amplitude and phase (\(0\) or \(\pi\) only) of the harmonic of the sound wave.

2. What happens to the sound you hear as the slider value goes from \(+1\) to zero?
3. What happens to the sound as the slider goes from \(0\) to \(-1\)?
4. Can you hear a difference in the sound at \(+1\) and \(-1\)?

At this point you should have determined that loudness is related to amplitude. Now let's figure out what determines the pitch (musical note). Set \(H5\) to zero. (You can do this using the slider or by typing zero into the text box beside the slider.) Use the slider to turn on some other harmonics. The sound you will hear is the sum of all harmonics that are on, so if you want to hear an individual harmonic, you will need to set all other values to zero.

5. Based on your experimenting, what determines the pitch of a tone? Specifically, is this quantity larger or smaller for high vs. low notes?

Now comes the really cool part. How does an electronic keyboard mimic the sounds of individual instruments? To understand this, we first need to understand why (mathematically) a trumpet sounds different from a clarinet. When you had only one harmonic turned on, you heard a pure tone. When you play a note on a trumpet, you do not create a pure tone. You set the trumpet vibrating, and this sets up resonant standing waves (pure tones). More than one resonant standing wave can exist at any one time. All of the resonant standing waves (harmonics) will add together, and what you hear is the sum of all these individual pure tones. The relative magnitudes of the individual harmonics are different for the clarinet and the trumpet, which is why the two instruments sound different even when they play the same note.

Try the following values for the harmonics.

\(H\)	Clarinet	Trumpet
\(1\)	\(0.91\)	\(0.53\)
\(2\)	\(0.51\)	\(1\)
\(3\)	\(0.71\)	\(0.94\)
\(4\)	\(0.86\)	\(0.95\)
\(5\)	\(1\)	\(0.66\)
\(6\)	\(0.71\)	\(0.58\)
\(7\)	\(0.54\)
\(8\)	\(0.2\)
\(9\)	\(0.18\)
\(10\)

Table \(\PageIndex{2}\)

Do the resulting tones sound like the clarinet and trumpet? Well, sort of. You should be able to hear some similarities to the instruments they are supposed to represent, and you should be able to tell that the sounds are different, but it does not exactly sound like the real thing.

6. Can you think of some reasons why the sound you produced is not exactly like the real thing?

Exploration authored by Melissa Dancy.
Script authored by Morten Brydensholt, Wolfgang Christian, and Mario Belloni.

Exploration 3: A Microphone between Two Loudspeakers

A microphone is placed between two loudspeakers (position is given in centimeters and time is given in seconds). The speakers are connected to two sources of sound that have variable frequencies, \(f_{1}\) and \(f_{2}\). The graph shows the sound waves arriving at the microphone, as a function of time, from each speaker and also shows the sum of the two waves. Change the frequency of either sound source \((25\text{ Hz} < f_{1}, f_{2} < 30\text{ Hz})\), and watch the changing interference between the two sound waves.   Study the phenomenon of beats and verify that the beat frequency is correct. Restart.

1. What happens as the two frequencies get closer together?
2. What happens as the two frequencies get farther apart?
3. Does it matter which speaker has the higher frequency?
4. What happens if the two frequencies are identical?

Remember that it is the difference between the two sound frequencies that determines the beat frequency.

Exploration authored by Steve Mellema and Chuck Niederriter.
Script authored by Steve Mellema and Chuck Niederriter.

Exploration 4: Doppler Effect and the Velocity of the Source

This example shows the Doppler effect. The black dot represents the source of the sound wave and travels with a speed set by the slider. That speed is given in terms of the speed of sound: hence \(v = 1\) corresponds to the speed of sound. Restart.

Vary the speed of the source from zero to the speed of sound and then to the maximum value allowed by the slider. Watch the animation and answer the questions below.

1. How does the pattern of the wave fronts change according to \(v_{\text{source}}\)?
2. For \(v_{\text{source}} > v_{\text{sound}}\) (slider values \(> 1\)) how does the V-shaped shock wave vary according to \(v_{\text{source}}\)?

Exploration 5: An Ambulance Drives by with Its Siren On

When the ambulance is moving (position is given in meters and time is given in seconds), use the animation to guide your answers to the following questions. Restart.

1. How does the wavelength of the sound wave change in relation to the woman at the right?
2. How does the frequency of the sound wave change in relation to the woman at the right?
3. How does the wavelength of the sound wave change in relation to the man at the left?
4. How does the frequency of the sound wave change in relation to the man at the left?
5. How does the wavelength of the sound wave change in relation to the patient in the ambulance?
6. How does the frequency of the sound wave change in relation to the patient in the ambulance?

Exploration authored by Morten Brydensholt, Wolfgang Christian, and Mario Belloni.
Script authored by Morten Brydensholt, Wolfgang Christian, and Mario Belloni

Physlets were developed at Davidson College and converted from Java to JavaScript using the SwingJS system developed at St. Olaf College.