1. A “physical phenomenon that stimulates the sense of hearing.”
Sound 13-3
A “physical phenomenon that stimulates the
sense of hearing.”

2. What do you think?
A violin, a trumpet, and a clarinet all play the same note, a concert A. However, they all sound different.
What is the same about the sound?
Are the frequencies produced the same?
Are the wave patterns the same?
Why do the instruments sound different?
When asking students to express their ideas, you might try one of the following methods.
(1) You could ask them to write their answers in their notebook and then discuss them.
(2) You could ask them to first write their ideas and then share them with a small group of 3 or 4 students. At that time you can have each group present their consensus idea. This can be facilitated with the use of whiteboards for the groups.
The most important aspect of eliciting student’s ideas is the acceptance of all ideas as valid. Do not correct or judge them. You might want to ask questions to help clarify their answers. You do not want to discourage students from thinking about these questions and just waiting for the correct answer from the teacher. Thank them for sharing their ideas. Misconceptions are common and can be dealt with if they are first expressed in writing and orally.
This is a difficult concept. Comments may center on the string instrument vs. the wind instrument. If so, ask them to clarify why the instruments would sound different even if they play the same note. You could also ask why a violin and a viola (both string instruments) sound differently when playing the same note.
This elicitation may raise more questions than provide answers. That is a very worthwhile process. You may want to make a note of the questions raised and be sure they are answered before the end of this section.

3. Standing Waves
Standing waves are produced when two identical waves travel in opposite directions and interfere.
Interference alternates between constructive and destructive.
Nodes are points where interference is always destructive.
Antinodes are points between the nodes with maximum displacement.
If possible, use the web site below prior to this slide.
Choose “Wave” and then choose “Superposition Principle of Wave.”
Change the frequency of each wave to 5 (this will be easier to see).
Observe the two waves as they overlap. Point out to students that the standing wave pattern is the only thing an observer would see because it is the resultant of the two component waves. The red wave appears to be standing still (not moving right or left).
Pause the simulation and show them the nodes (points that never move because the two components always cancel) and the antinodes (points that have the maximum displacement in each direction).

4. Standing Waves on a String
There is a node at each end because the string is fixed at the ends.
The diagram shows three possible standing wave patterns.
Standing waves are produced by interference as waves travel in opposite directions after plucking or bowing the string.
The lowest frequency (one loop) is called the fundamental frequency (f1).
To show how standing waves are produced, go to the following web site:
Choose “Standing Waves (Explanation by superposition).”
When showing this, pause and point out that the nodes are produced by waves that always cancel.
This demonstration shows a six loop standing wave.

5. Standing Waves on a String
To the left is a snapshot of a single loop standing wave on a string of length, L.
What is the wavelength for this wave?
Answer:  = 2L
What is the frequency?
Answer:

6. Now that students understand why f1 = v/(2L), help them see why the wavelengths for the next three harmonics are as shown. It is helpful to look at just one segment of the wave instead of the four that are shown for each mode.
Point out the “harmonic” terminology for each mode. These are sometimes called overtones instead of harmonics. The second harmonic is called the 1st overtone, and so on.
Ask students to come up with a general equation for f in terms of v and L, using n where n represent the number of loops. Show them f1 = v/(2L), f2 = v/L, f3 = (3v)/(2L), and so on, and see if they see the pattern to write fn = ????
The answer is on next slide.

7. Harmonics n is the number of loops or harmonic number.
v is the speed of the wave on the string.
Depends on tension and density of the string
L is the length of the vibrating portion of the string.
How could you change the frequency (pitch) of a string?
Shorter strings (decreased L) have higher pitches.
Higher-tension strings (increased v) have higher pitchers.
More dense strings (decreased v) have lower pitches.
Point out to students that v in the equation is the speed of the waves as they move back and forth on the string. It is not the speed of the sound. The speed of waves on the string ranges from 100’s to 1000’s of m/s depending on the string tension and density. The vibrating string then produces sound waves that travel at 346 m/s (at 25°C) through the air.

8. Fundamental Frequency
Click below to watch the Visual Concept.
Visual Concept

9. Standing Waves in an Air Column
Wind instruments also use standing waves.
Flutes, trumpets, pipe organs, trombones, etc.
Some instruments have pipes open at both ends while others have one end closed.
Air is free to move at open ends so antinodes occur.
Closed ends are nodes.
The velocity of the wave is now the velocity of sound in air (346 m/s at 25°C).
At this point use the web site:
and choose “Standing Longitudinal Waves”
This is an excellent demonstration showing the particles in the air column as well as the wave pattern.
Ask students what they expect the wave to look like when switching to higher harmonics.
Ask them how it will appear if you close one end.
This simulation also allows the calculation of the frequency for the various harmonics by changing the length of the tube.

10. Both Ends Open
The relationship is the same as that for strings. Instead of a node at each end, an antinode exists at each end. Therefore, f1 = 2L, f2 = L etc.

11. Closed at One End
Ask students why the harmonics only include the odd-numbered ones.
In order to have a frequency that corresponded to n = 2, the air column would need to have one-half a wave. This is not possible because one end is a node and the other an antinode. Therefore, you can only have 1/4 of a wave or the odd multiples of 1/4.

12. Wind Instruments Wind instruments are not as simple as organ pipes.
The shape is not always cylindrical.
The holes change the wave patterns as well.
The size of the “pipe” varies along the length.

13. Practice Problems One string on a toy guitar is 34.5 cm long.
What is the wavelength of the first harmonic or the fundamental wavelength?
Answer: 69.0 cm or m
The string is plucked and the speed of the waves on the string is 410 m/s. What are the frequencies of the first three harmonics?
590 Hz, 1200 Hz, 1800 Hz
Note: The use of significant figures causes the multiples of 590 to be 1200 and 1800 because only two significant figures are present in the answer.
For problems, it is a good idea to go through the steps on the overhead projector or board so students can see the process instead of just seeing the solution. Allow students some time to work on problems and then show them the proper solutions. Do not rush through the solutions. Discuss the importance of units at every step. Problem solving is a developed skill and good examples are very helpful.
Show students that they can get the other harmonics from the first one without using the equation with v and L, by using f2 = 2f1 and f3 = 3f1.

14. Practice Problems An organ pipe open at both ends is 34.5 cm long.
What is the wavelength of the first harmonic or the fundamental wavelength?
Answer: 69.0 cm or m
What are the frequencies of the first three harmonics if the air temperature is 25.0°C?
Answers: 501 Hz, 1000 Hz, 1500 Hz
Answer the same questions if the pipe is closed at one end.
Answers: 251 Hz, 753 Hz, 1250 Hz
For problems, it is a good idea to go through the steps on the overhead projector or board so students can see the process instead of just seeing the solution. Allow students some time to work on problems and then show them the proper solutions. Do not rush through the solutions. Discuss the importance of units at every step. Problem solving is a developed skill and good examples are very helpful.
f1 = v/(2L) = (346 m/s)/(2 x m) = 501 Hz; f2 = 2f1 and f3 = 3f1
f1 = v/(4L) = 251 Hz; f3= 3f1 and f5 = 5f1

15. Practice Problems
A violin string that is 50.0 cm long has a fundamental frequency of 440 Hz. What is the speed of the waves on this string?
For problems, it is a good idea to go through the steps on the overhead projector or board so students can see the process instead of just seeing the solution. Allow students some time to work on problems and then show them the proper solutions. Do not rush through the solutions. Discuss the importance of units at every step. Problem solving is a developed skill and good examples are very helpful.
f1 = v/(2L) = (346 m/s)/(2 x m) = 501 Hz; f2 = 2f1 and f3 = 3f1
f1 = v/(4L) = 251 Hz; f3= 3f1 and f5 = 5f1