Chapter 14
Waves and Sound (Cont.)
Dr. Jie Zou PHY 1151G
Department of Physics
1
Outline
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Standing waves
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Standing waves on a string
Vibrating columns of air
Sound intensity
Doppler effect
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Moving observer
Moving source
Dr. Jie Zou PHY 1151G
Department of Physics
2
A standing wave
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A standing wave is one that
oscillates with time, but remains
fixed in its location. It is in this
sense that the wave is said to be
“standing”.
Example:
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(a) A string tied down at both ends;
(b) When plucked in the middle, a
standing wave results-the fundamental
mode of oscillation of the string.
1 = 2L (wavelength of the fundamental)
Dr. Jie Zou PHY 1151G
Department of Physics
3
Node (N) Antinode (A)
Standing waves on a string
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First harmonic (fundamental) frequency and
wavelength: f1 = v/(2L); 1 = 2L
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v = (F/)1/2: wave speed on the string; F: tension in the string; :
mass per unit length of the string; L: length of the string
Frequency and wavelength of the nth harmonic,
v
n
with n = 1, 2, 3,…: fn = nf1 = 2 L ; n = 1/n = 2L/n
Dr. Jie Zou PHY 1151G
Department of Physics
4
An application – guitar
Dr. Jie Zou PHY 1151G
Department of Physics
5
Vibrating columns of air
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Standing waves in a
column of air closed at
one end:
f1 = v/(4L);
v
n
fn = nf1 = 4 L , n = 1, 3,
5,… odd harmonics only
n = 1/n = 4L/n
v: speed of sound in air
L: length of the air column
Dr. Jie Zou PHY 1151G
Department of Physics
6
Example
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An empty soda pop bottle is to be
used as a musical instrument in a
band. In order to be tuned properly,
the fundamental frequency of the
bottle must be 440.0 Hz.
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(a) If the bottle is 26.0 cm tall, how high
should it be filled with water to produce the
desired frequency? Treat the bottle as a pipe
that is closed at one end (the surface of the
water) and open at the other end. (6.5 cm)
(b) What is the frequency of the next higher
harmonic for this bottle? (1320 Hz)
Dr. Jie Zou PHY 1151G
Department of Physics
7
Sound intensity
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The loudness of a sound is determined
by its intensity.
Definition of intensity: If the energy
E passes through the area A in the time
t, the intensity, I, of the wave is
 I = E/(At). Since the power P = E/t,
2
 I = P/A. SI unit: W/m .
The intensity falls off with the
square of the distance:
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I = P/(4r2). SI unit: W/m2.
Dr. Jie Zou PHY 1151G
Department of Physics
8
Example: Intensity of sound
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Two people relaxing on a deck listen to a songbird sing.
One person, only 1.00 m from the bird, hears the sound
with an intensity of 2.80 10-6 W/m2 .
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What intensity is heard by the second person, who is 4.25 m from
the bird? Assume that no reflected sound is heard by either
person.
Dr. Jie Zou PHY 1151G
Department of Physics
9
Doppler effect
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Doppler effect: The change
in pitch, due to the relative
motion between a source of
sound and the receiver, is
called the Doppler effect.
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The pitch increases when the
observer and the source are
moving closer together.
The pitch decreases when the
observer and source are
separating.
Dr. Jie Zou PHY 1151G
Department of Physics
10
Doppler effect for a moving
observer
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For moving observer:
f´= (1  u/v)f
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f´= frequency to the
observer.
f = frequency of the source of
the sound.
u = speed of the observer.
v = speed of the sound.
“+”: when the observer
moves toward the source.
“-”: when the observer moves
away from the source.
Dr. Jie Zou PHY 1151G
Department of Physics
11
Doppler effect for a moving
source
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For moving source:
f’ = [1/(1  u/v)] f
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u = speed of the source.
v = speed of the sound.
“-”: when the source
moves toward the
observer.
“+”: when the source
moves away from the
observer.
Dr. Jie Zou PHY 1151G
Department of Physics
12
Example: Doppler effect
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A train sounds its whistle as it
approaches a tunnel in a cliff.
The whistle produces a tone of
650.0 Hz, and the train travels
with a speed of 21.2 m/s.
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(a) Find the frequency heard by an
observer standing near the tunnel
entrance.
(b) The sound from the whistle
reflects from the cliff back to the
engineer in the train. What
frequency does the engineer hear?
Dr. Jie Zou PHY 1151G
Department of Physics
13
Homework
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See online homework assignment on
www.masteringphysics.com
Dr. Jie Zou PHY 1151G
Department of Physics
14