Lesson 4
Announcements
 HW # 4 (standing waves) due next Tuesday in class
AP Physics B Standards
IV.A.3. Standing waves
Students should understand the physics of standing waves, so they can:
a) Sketch possible standing wave modes for a stretched string that is fixed at both ends, and determine the amplitude, wavelength, and frequency of such standing waves. b) Describe possible standing sound waves in a pipe that has either open or closed ends, and determine the wavelength and frequency of such standing waves.
Lesson Objectives
Students will be able to
1.
2.
3.
predict the wavelength of standing waves produced by strings and pipes (both open and closed ended). draw standing waves and identify the nodes & antinodes.
calculate wavelengths and frequencies of multiple harmonics of standing waves.
REVIEW: Reflection of a Wave
String Fixed on Both Ends
½
L=½
=2L
String Fixed on Both Ends
Incident Wave
Reflected Wave
String Fixed on Both Ends
Standing Wave
 A standing wave is a wave which is reflected back and forth between fixed ends (of a string or pipe, for example).
 Reflection may be closed or open‐ended.
 Superposition of the wave upon itself results in constructive interference and an enhanced wave.
 Standing waves are only possible for a set of specific frequencies.
Fixed‐end standing waves
(violin string)
st
1 harmonic
2nd harmonic
3rd harmonic
Animation available at:
http://id.mind.net/~zona/mstm/physics/waves/standingWaves/standingWaves1/StandingWaves1.html
Fixed‐end standing waves
L
(violin string)
Fundamental or
First harmonic
 = 2L
First Overtone or
Second harmonic
=L
Second Overtone
or Third harmonic
 = 2L/3
Standing Waves in Air
 Standing waves are a result of changes in air pressure.
 Atmospheric pressure at an open end is a constant, resulting in a node.
 Air can be compressed or rarefacted against closed
ends, resulting in a antinode.
Standing Waves in Air
At one moment
Half-cycle later
Standing Waves in Air
Open‐end standing waves
(organ pipes)
Fundamental or
First harmonic
 = 2L
First Overtone or
Second harmonic
=L
Second Overtone or
Third harmonic
 = 2L/3
Mixed standing waves
(some organ pipes)
First harmonic
 = 4L
Second harmonic
 = (4/3)L
Third harmonic
 = (4/5)L
Sample Problem 4.1
How long do you need to make string that produces a high C (512 Hz)? The speed of the waves on the string 1040 m/s.
A) Draw the situation.
B) Calculate the string length.
C) What is the wavelength and frequency of the 2nd harmonic?
Sample Problem 4.2
How long do you need to make an organ pipe whose fundamental frequency is a middle C (256 Hz)? The pipe is closed on one end, and the speed of sound in air is 340 m/s.
A) Draw the situation.
B) Calculate the pipe length.
C) What is the wavelength and frequency of the 2nd harmonic?
Sample Problem 4.3
How long do you need to make an organ pipe whose fundamental frequency is a middle C (256 Hz)? The pipe is open on both ends, and the speed of sound in air is 340 m/s.
A) Draw the situation.
B) Calculate the pipe length.
C) What is the wavelength and frequency of the 2nd harmonic?
Resonance
 Resonance occurs when a vibration from one oscillator occurs at a natural frequency for another oscillator.
 The first oscillator will cause the second to vibrate.
 See the Tacoma Narrows bridge clip
Mechanical Universe (2:52 – 8:15 Tacoma Narrows intro and explanation of resonance)
(20:55 – 24:05 bridge collapse and explanation of why it happened)
Source: http://www.learner.org/vod/login.html?pid=566
Beats
 Beats is the word physicists use to describe the characteristic loud‐soft pattern that characterizes two nearly (but not exactly) matched frequencies.
 Musicians call this “being out of tune” or “bad intonation.”
Modeling Beats