Midterm II Results
Physics 111
Lecture 29 (Walker: 14.1-4)
Waves
Sound
Nov. 18, 2009
Lecture 29
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Wave Motion
• A wave is a traveling disturbance.
• The wave transports energy, but does not
transport matter.
• Some common waves:
•Sideways motion in stretched rope
•Water waves
•Sound waves
•Electromagnetic waves (radio, TV, cell phone)
• Source of waves -- a vibrating object
•Hand shaking rope
•Vocal cords; loudspeaker cone
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• Approximate grade ranges
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Types of Waves
Categorize waves by
Type of disturbance (mechanical, air
pressure, electric field)
Direction of disturbance relative to
direction of wave motion
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Types of Waves:
Transverse and Longitudinal
Types of Waves
Direction of disturbance can be either
perpendicular to wave motion (transverse wave)
or parallel to wave motion (longitudinal wave).
In a longitudinal wave, the displacement is
along the direction of wave motion.
v
v
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Characteristics of Wave
Types of Waves
Wavelength λ: distance over which wave repeats
Period T: time for one wavelength to pass a
given point
Water waves are a
combination of transverse
and longitudinal waves.
Frequency f: # waves that pass by per second
Speed v of a wave:
Amplitude A: Size of disturbance
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Wave Motion
Example
Wave characteristics:
• Amplitude, A
• Wavelength, λ
• Frequency f and period T
• Wave velocity
• Wave velocity in stretched rope is 20 m/s.
What is wavelength of wave with 5 Hz
frequency?
• v = λf, so λ = v/f
• λ = (20 m/s) /(5 s-1) = 4 m
x
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Mechanical Waves on a String
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Waves on a String
The total mass of the string depends on how
long it is; what matters in the speed is the mass
per unit length. We expect that a larger mass
per unit length results in a slower wave speed.
The speed of a wave is determined by the
properties of the material through which it
propagates.
For a string, the wave speed is determined by:
1. the tension in the string (restoring force), and
2. the mass of the string.
As the tension in the string increases, the speed
of waves on the string increases as well.
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SI Unit: kg/m
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Example
Waves on a String
• A string of length 6 m has mass 0.6 kg and is
stretched to a tension of 36 N.
What is the speed of transverse waves on
the string?
• µ = m/l = 0.6kg/6m = 0.1 kg/m
For a stretched rope with tension F and
mass per unit length µ:
Speed increases when tension increases, and
decreases when mass per unit length increases.
v=
F
µ
=
36 N
0.1kg / m
=19 m/s
Speed does not depend on wave amplitude or
on frequency (at least for an ideal “medium”)
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Waves on a String: Reflections
Waves on a String: Reflections
When a wave reaches the end of a string, it will
be reflected. If end is fixed, reflected wave will be
inverted (direction of disturbance reversed):
If end of string is free to move transversely, the
wave will be reflected without inversion.
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Harmonic Waves
Disturbance vs.Position @ Fixed Time
• If the vibrating source of a wave vibrates in
simple harmonic motion (SHM), the wave shape,
or waveform will be like a sine or cosine.
• We get the same waveform whether we look at:
•“Snapshot” of the disturbance at different
places at one instant of time
•Disturbance at one position along wave as
function of time
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Wavelength?
Disturbance vs.Time @ Fixed Position
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Harmonic Wave Functions
Since the wave has the same disturbance at
(x + λ) as it does at x, and at (t + T) as at t, the
wave function must be of the form
Disturbance
or
2π
⎛ 2π
= A sin ⎜
x−
T
⎝ λ
⎞
t⎟
⎠
Waveform is Sine Wave. Frequency?
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Nature of Sound
Sound Waves
• Pressure/density wave in air (or other medium)
In a sound wave, density and pressure of the
air (or other medium carrying the sound) are
the quantities that oscillate.
Sound waves are longitudinal.
Compression
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Producing a Sound Wave
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Sound Wave Characteristics
• Source: Vibrating object that can push air
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Rarefaction
Physical
Frequency f (Hz)
Wave Amplitude A (Pa)
Waveform (or shape)
----------------------------(Phase)
Wavelength λ (m)
Wave speed vS (m/s)
Loudspeaker cone
Top plate of violin or guitar
Vocal cords
Air column in organ pipe
• Vibrating object causes air pressure and
density disturbances that travel as a wave
Perceptual
Pitch (“Middle C”)
Loudness (Sones)
Timbre (Tone Quality)
--------------------------(Not perceived)
(Not perceived)
(Not perceived)
Sound wave speed in air vS = 343 m/s (@20°C)
Note wavelength –frequency relation: f λ = vS
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Frequency of Sound Waves
Speed of Sound Waves
Sound waves can have any frequency. The
human ear can hear sounds between about 20 Hz
and 20,000 Hz.
In general, the
denser the material,
the faster sound
travels through it.
Sounds with frequencies greater than 20,000 Hz
are called ultrasonic; sounds with frequencies
less than 20 Hz are called infrasonic.
Ultrasonic waves are familiar from medical
applications (ultrasound imaging, etc.)
Infrasonic waves are used by elephants and
whales to communicate, and are found in
earthquakes.
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Frequency Ranges of Sound
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Ultrasonic Application: Medical Imaging
20 kHz
0 20 Hz
Audible (Audio) Frequencies
Infrasonic
Ultrasonic
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Ultrasound is also
used for medical
imaging. Repeated
traces are made
as the transducer
is moved, and a
complete picture
is built.
Intensity
Ultrasound Medical Imaging
Ordinary ultrasound gives a good picture; highresolution ultrasound is excellent.
• Intensity (I) =
Energy crossing
unit area per sec.
E
Power
I ≡
=
A∆t
A
Intensity proportional to
square of wave amplitude
Unit: W/m2
Example: Microphone
element (A=0.05 m2)
receives 0.01 J of energy
in 10 seconds. I = ? Lecture 29
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Intensity
I ≡
E
Power
=
A∆t
A
P = 0.01J/10s = .001W
I = P/A = .001W/0.05m2
= 0.02 W/m2
Threshold of
Hearing:
∆P=2x10-5 Pa
(Patm/1012)
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Intensity Falloff with Distance
from Source
Intensity Level β
• Also called decibel level; usually equal to
Sound Pressure Level (SPL)
⎛ I ⎞
β = 10 log 10 ⎜⎜ ⎟⎟
⎝ I0 ⎠
where I0 = 10-12 W/m2 (threshold of hearing)
• Unit of β is deciBels (dB)
In open space, sound intensity from small source:
I (r ) =
Psource
4π r 2
This is the “Inverse Square Law” –Twice as far from
source, sound intensity is ¼ as much. (Loudness
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reduced less than half.)
•Note that an increase in intensity level of 10 dB
requires a 10 times increase in sound power (!);
change of 20 dB requires 100 x power, etc.
• Ear requires 10 dB intensity level change for
sound to seem twice asLecture
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End of Lecture 29
•
For Friday, Nov. 20, read Walker 14.5-6.
• Homework Assignment 14a is due at 11:00 PM on
Sunday, Nov. 22.
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