SPH3U1
Lesson 06
Waves and Sound
STANDING WAVES AND RESONANCE
LEARNING GOALS
Students will learn:
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

That vibrating objects have a natural frequency
That resonance selectively amplifies certain frequencies
That standing waves are an example of resonance
PREPARATION AT HOME
Reading

Nelson Physics 11 – Sections 9.2 & 9.4 - Pages 422-425, 430-431
Videos
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
Standing Wave Demo
Earl Haig Physics Review
o Standing Waves
o Natural Frequency and Resonance
Reading Quiz
NATURAL FREQUENCY & RESONANCE
All solid objects will vibrate when they are struck or shaken. Each object vibrates at some
characteristic frequency called its natural frequency. A guitar string, for example will
vibrate at a specific frequency when it is plucked. Resonance is the amplification of a
vibration due to a periodic applied force that matches the natural frequency.
EXAMPLE 1: PENDULUMS
Observe the pendulums set up at the front. Record your observations
EXAMPLE 2: MASSES ON SPRINGS
Observe the springs set up at the front. Record your observations
a) Pushing a child on a swing is a good example of natural frequency and resonance.
Explain how.
SPH3U1
Lesson 06
Waves and Sound
STANDING WAVES
Strings and springs also have natural frequencies. If you make a spring vibrate at its
natural frequency, you can create standing waves. Even though the medium vibrates
back and forth, the wave appears not to move.
first harmonic
Standing waves are in reality the result of two waves
travelling in opposite directions interfering with each
other. If they are timed correctly, then the nodes and
antinodes appear always in the same place. In the
case of standing waves in a string or spring, a wave
travels towards one end of a medium and reflects
back. The reflected wave then interferes with the
incident (incoming) wave. If the wavelength is such
that the length of the string or spring (L) is an integer
multiple of half the wavelength (), then standing
waves will occur. Nodes (N) and antinodes (A) will be
created as in the diagram to the right.
second harmonic
third harmonic
CONDITION FOR STANDING WAVES:
In your textbook you will see the equations for standing waves, but you really don’t need
them. All you need is to know what a wavelength is and how to draw a picture . Let’s try
this:
1. On the diagram to the side, draw a
horizontal line to show ONE wavelength
().
2. Now compare your answer to #1 with the standing wave diagrams above. If you
look at only one string, which harmonic shows ONE wavelength?
3. With this in mind, how many wavelengths are shown in the first harmonic? the third
harmonic? (Hint: halves are ok)
4. Draw a standing wave that has 2½ wavelengths. What harmonic would this be?
5. Which harmonic in the diagram above has the longest wavelength?
SPH3U1
Lesson 06
Waves and Sound
EXAMPLE
Consider a guitar string that is 75 cm long. It is vibrating in the first harmonic.
a. Sketch a diagram of the string showing the standing wave. Label the length
of the string.
b. Since the first harmonic only shows half a wavelength, how long is one entire
wavelength?
c. If the waves travels at 660 m/s, what is the frequency of the wave?
d. Now say it is vibrating in the second harmonic. Draw a new sketch.
Assuming it’s on the same guitar string, how long is the wavelength?
Calculate the frequency of this note.
2. 2nd harmonic 3. half; 1½
4. fifth 5. first harmonic
6. b) 150 cm = 1.50 m c) 440 Hz (the note is an A)
d) 0.75 m; 880 Hz
STANDING WAVES IN AIR COLUMNS
If you blow over the top of a pop bottle, it will make a noise. This is because a standing
wave is being created in the air of the bottle. If you blow with more force, the note will be
higher. A simplified diagram of the standing waves in the air is shown below:
a) Which harmonic shows the highest frequency?
Which shows the longest wavelength?
b) Under the diagram, label how many
wavelengths are shown in each harmonic.
There will be fractions.
c) If the tube is 2.0 m long, how long is each
wavelength?
1st harmonic
2nd harmonic
a) 3rd; 1st
b)
3rd harmonic
SPH3U1
Lesson 06
Waves and Sound
Standing waves can also exist in air columns that are open at both ends. (For example, a
flute)
1st harmonic
d) how are these waves different than the waves
in air columns closed at one end?
2nd harmonic
e) Beside the diagram, label how many
wavelengths are shown in each harmonic.
There will be fractions.
f)
If the tube is 3.0 m long, how long is each
wavelength?
3rd harmonic
PRACTICE PROBLEMS
1. A train’s whistle acts like an air column that is closed at one end. In its second
harmonic, it sounds a note that is 240 Hz. It is 1.15 m long.
a. Sketch the standing wave in the air column
b. How many wavelengths are present in the whistle?
c. Based on (b) and the length of the whistle, what is the wavelength of the
wave?
d. What is the speed of this sound wave?
2. A guitar string vibrating in the first harmonic is playing a C note at 523 Hz. If the
string is 0.85 m long, how fast is the wave travelling on the string? (Hint: follow the
steps from #1)
3. A grandfather is making a flute for his granddaughter. He wants it to play an A note
(440 Hz) in the first harmonic.
a. If the speed of sound will be 344 m/s, how long should he make the flute?
(Hint: start with a diagram of the standing wave and work backwards from
the speed)
b. Give two other frequencies that this flute could play. (Hint: draw new wave
diagrams)
4. page 460 #4 (fundamental means first harmonic), 8, 9, 10 Do P425 Q1-3, P426 Q27 (hint for question 2 think of an echo) and P457 Q1-2
1b. ¾ c. 1.53 m d. 368 m/s 2. 889 m/s 3a. 0.39 m b. 880 Hz, 1320 Hz
HOMEWORK
1. Go onto the internet. Research the following examples of resonance and describe
them:
a)
A child on a swing pumping her legs
b)
Buildings in an earthquake
c)
Tacoma Narrows Bridge (for this one watch a video on youtube)
d)
An unbalanced wheel on a car