Activity 1 – Creating a musical octave with water and bottles
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* use mathematical formulas to calculate area, volume, and frequency
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* determine the frequency of a sound
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* create different sounds using the mathematical formulas
Prerequisites: Students need to know what frequency means and be given the formula for finding the frequency in a Helmholtz resonator, f = (v/2()(sq. root of a/Vl) where v equals the speed of sound, a is the area of the neck, l is the length of the neck, V is the volume of the resonator. They will also need the formulas for the volume of a sphere or cylinder and the area of a circle. Students also need to know what a Helmholtz resonator is and how it actually works.
Materials Needed: Per group, you will need water, a measuring cup, a tape measure or ruler, a calculator and approximately 7 to 12 identical size bottles. You will also need a few bottles of varying shapes and sizes. They will also need the included chart that relates frequency to a piano scale. (see appendix A)
Activity: Students will begin by taking the different size bottles and blow across the opening to hear the sound that is created by the Helmholtz resonator. Once the students have heard the sound they have created, they need to calculate the frequency, f, for each of the empty bottles using the appropriate formulas. Since the equation given in the sound objective #5 uses the AIR volume of the bottle, students should calculate this for their bottles. To change the air volume in the bottle they will need to add different amounts of water. The total volume (Vt) of the bottle below the neck, minus the liquid volume (Vl), will give the students the volume of air (Va) in the bottle. This is demonstrated by the equation Vt – Vl = Va. Before the students create a musical octave it might be nice practice to have them fill the bottles with varying amounts of liquid and calculate the frequency of the note created. When they are ready to create their octave, students will need to work the equation for frequency of a Helmholtz resonator backward to determine the air volume they need for each note. I would have my students actually fill their seven bottles appropriately and then test their creation.
Homework: Students could be given a worksheet that has them practice using the formulas. They could be given a note and told how to calculate the liquid needed to create the appropriate air volume. It would be a nice conclusion to ask the students to discuss what would make a note flat or sharp given the formula and a frequency.
Activity 2 – Relating frequency to period of a graph
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*discuss the relationship of frequency to period
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* calculate the period of a sound wave generated from a musical note
Prerequisites: Students will need a good understanding of what frequency means. They will also to have knowledge of what a period represents in a graph…it does NOT have to be a trigonometric graph although if it is, their connection to the trigonometric concepts will be greatly increased. Finally they will need to be able to determine the frequency of a played musical note. This could come directly from the previous experiment or from the chart in Appendix A.
Materials Needed: The students will need something that will create a musical note. A keyboard, flute, or one of the bottles from the previous experiment will be necessary. They will also need Appendix A and a calculator.
Activity: This is a much shorter activity and shouldn’t take the entire period. It very easily could be combined with Lesson 3 on amplitude. During this lesson, one student in the group should play a note. The other students will then determine the frequency of the note from the appendix. After having determined the frequency they then would calculate the period of the function. In a pre-calculus class I would finally have the students convert the period to radians so that we were ready to graph a function after lesson 3.
Homework: For homework I would have students practice working in both directions…that is to say I would give them two types of problems. One type would give them the period of the graph and ask them to determine both the frequency and the note. The other type of problem would ask the students to determine the period of the graph generated from a given musical note.
Activity 3 – Finding the amplitude of a sound wave generated from a note
Objectives:
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* use logarithms to solve equations
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* use and interpret data gathered from a sound level meter
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* calculate the amplitude of the sine wave generated from a note
Prerequisites: Students will need to have some understanding of and practice with logarithms. They will also need to know how a sound level meter works and what its data output means.
Materials Needed: A sound level meter, graph paper, and a calculator are all needed for this experiment. In addition there needs to be something present to create a musical note such as a keyboard etc. Less expensive sound level meters only go as low as 50 decibels. If your musical instrument won’t register in this range or higher, you either need a louder instrument or a more sensitive sound level meter.
Activity: This too is a relatively shorter activity and involves more pen and paper work than some of the other experiments. Students should turn on their sound level meters using the C weighting. A second student should play a note from the given musical source. A third person should hit DH on the sound level meter when they feel the note is being played consistently. The sound level meter will then “freeze” on a reading. This number is then placed in to the following equation for SPL, where SPL stands for the sound pressure level. SPL = 10 log A and A stands for the amplitude of the generated sound wave. Using their knowledge of logarithmic functions students should then calculate A. Because they already know how to calculate the period from the frequency, I would recommend having the students graph the sine wave that models the note. For example, say I heard a musical note C (256Hz) and the sound level meter read 54dB. I could then find the period of the graph from activity 2 finding that p = 1/256 = .00390625. I could also solve the log function 54 = 10 log A finding that A = 251188.6432. Now I have enough information to graph the sine wave. Students should follow these steps and graph the sine wave for the note they played. A beautiful extension to this experiment would be to have the students use the same note only at a louder volume. They will result in a different amplitude, which in turn changes the graph. This should generate a good deal of discussion.
Homework: I think it would be wonderful to have the students write about what makes the same note produce two different graphs. In addition, I might ask my students to calculate a few examples. Some of these I would ask them to find the SPL and others I would ask them to find the amplitude.
Activity 4 – Given a graph of a sound wave, finding the frequency and amplitude.
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* use a microphone, CBL unit, and a graphing calculator to create data
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* interpret the graph of a note recorded from the materials
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* determine the period and amplitude of a graph
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* calculate the frequency and then determine the note analyzed
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* compare and discuss the accuracy of experiment
Prerequisites: The student needs to understand the relationship between frequency and period. They also should be able to calculate the frequency if given the period. They will have to be introduced to the CBL unit and how it works. This can be done during the lab if they are unfamiliar with the apparatus. Students will need to have seen a sine graph before and have some knowledge of how to find the amplitude and period. Finally students will need to be able to use the graphing calculators, understanding how to use the trace key.
Materials Needed: You will need for every group a microphone, a CBL unit, tuning forks of various pitches or something that creates musical notes, TI-82 graphing calculators with the program SOUND already programmed in.
Activity: Students will link the CBL unit to the microphone and TI-82. They will go to PRGM (program) key on the calculator and run the program entitled SOUND. They should follow the directions given from the program. The students should be ready with the musical instrument or tuning fork, as they will need to be ready to play the note at the appropriate time. The musical note should be generated BEFORE the CBL unit is started. Once the CBL unit is receiving data, a graph will appear on the calculator screen. It will not be a perfect sine wave, but it will be close. The teacher should then be prepared to ask the students to calculate the period of the graph using the trace key. They should then use this data to determine the frequency (activity #2). Similarly they should determine the amplitude of the function. After they have found the frequency they need to determine the note played. Finally they should compare the note they have found from the data and the actual note of the keyboard or tuning fork. These will not always match exactly and students hopefully will have questions as to why this occurs. This inaccuracy provides a great opportunity to discuss precision in experiments. If time permits it would be nice to have them re-try the experiment with the same note a second time to compare data. You could ask them to determine which experiment they think was better/more accurate and why. As a final extension the students could use a different musical note or tuning fork and try the experiment again.
Homework: Perhaps the students could write up a one-page essay discussing the accuracy or inaccuracy of the experiment and ways to improve for the next time.
Activity 5 – Phase Shifts of Sine Functions
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* to determine how a phase shift occurs when dealing with sound
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* to find the equation that represents a sound wave that has been delayed (that is shifted horizontally)
Prerequisites: The student will need to be already familiar with how to find the period and amplitude of a function from its graph. They will also need to know the general equation of a trigonometric sine wave so they will be able to insert all the appropriate constants and give an exact equation for the sound wave. The formula is explained above, but is also given again here: y = a sin b(( - c).
Materials Needed: Each group will need either a tuning fork or a musical instrument, a CBL unit, a microphone, and a TI-82 calculator.
Activity: This activity is very similar to Activity 4. The key difference between these two activities is that in this activity the students will start the CBL unit BEFORE they generate the note. Doing so will create a brief time when there is no noise. It is important that the students do not talk during this time, as the microphone will pick up anything noises nearby. They shouldn’t wait too long to start the musical note or else the CBL unit will have completed its data collection before it has time to register the note. If the activity is done correctly there should be a straight line and the generated sine curve should start a ways into the screen. Students should measure this distance compare to the period of the function. For example, if the period is .45 and the lag distance is 1.3, then we would not say the phase shift is 1.3. They will need to calculate the remainder that occurs when .45 is divided into 1.3. The remainder is the phase shift. The mathematical reasoning behind this is that a pure true sine wave continues on forever in both directions. When it is not shifted, the sine wave always hits the point (0,0). If the lag distance is greater than the period, then we have to imagine that the function did continue to the left as well. And if it did, every distance of .45 would contain another complete period of the sine wave. Thus we need to find out the remainder to know how much we have actually been shifted from (0,0). Students will also have to find the amplitude and period from the function as we did in Activity 4. The activity could be expanded to include multiple graphs with different lag times.
Homework: Have students practice finding the equations from given graphs. This could include both sine waves and cosine waves. Similar problems could include giving them the equations and having them practice creating the graphs.
Activity 6 – Addition of two or more Sine Functions
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* find the sum of two trigonometric functions
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* relate how two pure sounds can create a new sound that no longer looks like a pure sine wave
Prerequisites: Students need to know what sine waves look like as well as how to calculate their value for a given independent value. They also need to have a basic understanding of how a pure sound can create a sine function. If the experiments are followed in order, then students should already have a nice grasp of these concepts.
Materials Needed: At this point the students will need an oscilloscope. Perhaps either the physics department or a nearby college might allow you to borrow an oscilloscope. The class will have to work as a group or they will need to come up one group at a time. You will also need calculators and graph paper.
Activity: Students should play one pure note so that the oscilloscope records the data. A sine wave will appear on the screen. Students will need to find the amplitude and period of this sine wave from the screen and record this data. Next students will need to play a different pure tone and follow the same procedures for it…recording amplitude and period. I would have them assume no phase shift. At this point I would send the students back into their groups and have them graph the two sine waves on separate graphs. I would ask them to guess what the graph would look like if the two sounds were played simultaneously. I might even hint to them that the two sounds were “added” together. Once the group has drawn a new graph that they think is represented from the two graphs I would bring them back up to the oscilloscope and play the two tones together. They would then be asked on a handout if their prediction was correct or not. If it was, I would have them explain how they know they are right. If their prediction was inaccurate I would then ask them to think about how the graph on the screen from the two notes was created from the two individual notes. They may need a little pushing at this point to figure things out. Once they think they know how the graph was created I would send them back to their work area and have them draw the graph and then come and check it against the oscilloscope. This idea could be expanded upon in two ways. One way would be for them to try more than two notes and see if they can predict the combined graph. A second method would be to show them a combined graph and see if they could determine the two individual notes that created the graph. A final question to extend the lesson would be to see if they could determine two functions that would totally cancel each other out leaving a straight line.
Homework: I would have them practice problems that ask them to graph several combined functions.
Appendix A
TEMPERED CHROMATIC SCALE
American Standard pitch. Adopted by the American Standards Association in 1936.
|
Note
|
Frequency
|
Note
|
Frequency
|
Note
|
Frequency
|
Note
|
Frequency
|
|
C0
|
16.352
|
|
C2
|
65.406
|
|
C4
|
261.63
|
|
C6
|
1046.5
|
|
C#0
|
17.324
|
|
C#2
|
69.196
|
|
C#4
|
277.18
|
|
C#6
|
1108.7
|
|
D0
|
16.354
|
|
D2
|
73.416
|
|
D4
|
293.66
|
|
D6
|
1174.7
|
|
D#0
|
19.445
|
|
D#2
|
77.782
|
|
D#4
|
311.13
|
|
D#6
|
1244.5
|
|
E0
|
20.602
|
|
E2
|
81.407
|
|
E4
|
329.63
|
|
E6
|
1318.5
|
|
F0
|
21.827
|
|
F2
|
87.307
|
|
F4
|
349.23
|
|
F6
|
1396.9
|
|
F#0
|
23.125
|
|
F#2
|
92.499
|
|
F#4
|
369.99
|
|
F#6
|
1480.0
|
|
G0
|
24.500
|
|
G2
|
97.999
|
|
G4
|
392.00
|
|
G6
|
1568.0
|
|
G#0
|
25.957
|
|
G#2
|
103.83
|
|
G#4
|
415.30
|
|
G#6
|
1661.2
|
|
A0
|
27.500
|
|
A2
|
110.00
|
|
A4
|
440.00
|
|
A6
|
1760.0
|
|
A#0
|
29.135
|
|
A#2
|
116.54
|
|
A#4
|
466.16
|
|
A#6
|
1864.7
|
|
B0
|
30.868
|
|
B2
|
123.47
|
|
B4
|
493.88
|
|
B6
|
1975.5
|
|
C1
|
32.703
|
|
C3
|
130.81
|
|
C5
|
523.25
|
|
C7
|
2093.0
|
|
C#1
|
34.648
|
|
C#3
|
138.59
|
|
C#5
|
554.37
|
|
C#7
|
2217.5
|
|
D1
|
36.708
|
|
D3
|
146.83
|
|
D5
|
587.33
|
|
D7
|
2349.3
|
|
D#1
|
38.891
|
|
D#3
|
155.56
|
|
D#5
|
622.25
|
|
D#7
|
2489.0
|
|
E1
|
41.203
|
|
E3
|
164.81
|
|
E5
|
659.26
|
|
E7
|
2637.0
|
|
F1
|
43.654
|
|
F3
|
174.61
|
|
F5
|
698.46
|
|
F7
|
2793.8
|
|
F#1
|
46.249
|
|
F#3
|
185.00
|
|
F#5
|
739.99
|
|
F#7
|
2960.0
|
|
G1
|
48.999
|
|
G3
|
196.00
|
|
G5
|
783.99
|
|
G7
|
3136.0
|
|
G#1
|
51.913
|
|
G#3
|
207.65
|
|
G#5
|
830.61
|
|
G#7
|
3322.4
|
|
A1
|
55.000
|
|
A3
|
220.00
|
|
A5
|
880.00
|
|
A7
|
3520.0
|
|
A#0
|
58.270
|
|
A#2
|
233.08
|
|
A#4
|
932.33
|
|
A#6
|
3729.3
|
|
B1
|
61.735
|
|
B3
|
246.94
|
|
B5
|
987.77
|
|
B7
|
3951.1
|
|
|
|
|
|
|
|
|
|
|
C8
|
4186.0
|
