chapter goal: to learn the basic properties of traveling...
TRANSCRIPT
Chapter 20 Traveling Waves
Chapter Goal: To learn the basic properties of traveling waves.
Slide 20-2
• result from periodic disturbance
• same period (frequency) as source
• Longitudinal or Transverse Waves
• Characterized by
– amplitude (how far do the “bits” move from their
equilibrium positions? Amplitude of MEDIUM)
– period or frequency (how long does it take for each “bit” to
go through one cycle?)
– wavelength (over what distance does the cycle repeat in a
freeze frame?)
– wave speed (how fast is the energy transferred?) v f
1f
3Hz
5Hz
Wavelength and Frequency are Inversely related: v
f
The shorter the wavelength, the higher the frequency.
The longer the wavelength, the lower the frequency.
Wave speed: Depends on Properties of the Medium:
Temperature, Density, Elasticity, Tension, Relative Motion
v f
Transverse Wave • A traveling wave or pulse that causes the elements of
the disturbed medium to move perpendicular to the
direction of propagation is called a transverse wave
Longitudinal Wave
Pulse
Tuning Fork
Guitar String
A traveling wave or pulse that causes the elements of the disturbed medium to move parallel to the direction of propagation is called a longitudinal wave:
Wave PULSE:
• traveling disturbance
• transfers energy and momentum
• no bulk motion of the medium
• comes in two flavors
• LONGitudinal
• TRANSverse
Traveling Pulse
• For a pulse traveling to the right
– y (x, t) = f (x – vt)
• For a pulse traveling to the left
– y (x, t) = f (x + vt)
• The function y is also called the wave function: y (x, t)
• The wave function represents the y coordinate of any element located at position x at any time t
– The y coordinate is the transverse position
• If t is fixed then the wave function is called the waveform
– It defines a curve representing the actual geometric shape of the pulse at that time
Traveling Pulse
Wave Form
Space Snap Shots
2
2( , )
( 3 ) 1y x t
x t
2
2@ 0 , ( ,0)
( ) 1t s y x
x
2
2@ 1 , ( ,1)
( 3) 1t s y x
x
2
2@ 2 , ( ,2)
( 6) 1t s y x
x
Traveling Waves
The media moves in SHM. The wave travels at constant speed.
The wave has the same frequency as the ‘shaking’ source!
Traveling Waves
• The wave represented by the
curve shown is a sinusoidal
wave
• It is the same curve as sin q
plotted against q
• This is the simplest example of a
periodic continuous wave
– It can be used to build more
complex waves
• Each element moves up and
down in simple harmonic motion
• Distinguish between the motion
of the wave and the motion of the
particles of the medium
2
( , ) siny x t A x vt
( , ) sin( )y x t A kx t
Wave Functions are Solutions to the
Wave Equation
2k
22 f
T
( , ) sin 2x t
y x t AT
2
( , ) siny x t A x vt
( , ) sin( )y x t A kx t
v fT k
( , ) sin 2 ( )x
y x t A f tv
2 2
2 2 2
1y y
x v t
Derive these:
Speed of wave depends on
properties of the MEDIUM
Speed of particle in the
Medium depends on
SOURCE: SHM
v f
2
( , ) sin( )
( , ) cos( )
( , ) sin( )
y x t A kx t
v x t A kx t
a x t A kx t
Wave Speed
v f
This gives the relationship between the wavelength
and frequency for constant wave speed.
The frequency depends on the source and the speed
depends on the properties of the medium.
The speed of sound is independent of the frequency.
When traveling from one medium to another, if the
speed changes, the wavelength changes but the
frequency (energy) remains the same.
Time Plot
Snap shot in Space.
This is an image of
one piece of a string
and how it moves as
the waves goes by in
time. The one piece
oscillates in SHM.
( , ) sin( )y x t A kx t
Space Plot
Snap shot in TIME.
Time is fixed. This is an
image of the entire string
or the medium’s
displacement from
equilibrium at one
instant. Can represent
either transverse or
longitudinal waves!!
( , ) sin( )y x t A kx t
Wave 1
Ocean waves with a crest-to-crest distance of
10.0 m can be described by the wave
function
y(x, t) = (0.800 m) sin[0.628(x – vt)]
where v = 1.20 m/s.
(a) Sketch y(x, t) at t = 0.
(b) Sketch y(x, t) at t = 2.00 s.
2
( , ) siny x t A x vt
When x = 5n, we get a node!!! x = 5, 10, 15, 20
Space Snap Shots in Time
Note how the entire wave form has shifted
2.40 m in the positive x direction in this time
interval: x= vt =(1.2m/s)(2s)=2.4m!!!
Wave Speed is Constant! Medium Accelerates!!
String
2
( , ) sin( )
( , ) cos( )
( , ) sin( )
y x t A kx t
v x t A kx t
a x t A kx t
y max = A
vy, max = A
ay, max = 2 A
COMPARE: Motion Equations for
Simple Harmonic Motion Chapter 15
x is fixed!! Cos or Sin changes phase!
22
2
( ) cos ( )
sin( t )
cos( t )
x t A t
dxv A
dt
d xa A
dt
2a x
Notice:
y = (15.0 cm) cos(0.157x – 50.3t).
At a certain instant, let point A be at the origin
and point B be the first point along the x axis
where the wave is 60.0 out of phase with point
A. What is the coordinate of point B?
22 2 ( )
vt f t t v t k x
/ 36.67
0.157x cm
k
2k x x
2
x
Wave Function
4
6
10 /
1.67
A cm
cm
v cm s
f Hz
( , ) sin( )y x t A kx t
2k
22 f
T
( , ) 4sin( 10.5 )3
y x t x t
3k
2 10.5f
Problem:
The displacement of a vibrating string vs position along the
string is shown. The wave speed is 10cm/s.
D) If the linear density of the string is .01kg/m, what is the
tension of the string?
v f
Problem:
The displacement of a vibrating string vs position along the
string is shown. The wave speed is 10cm/s.
D) If the linear density of the string is .01kg/m, what is the
tension of the string?
2( / )F v m L
2 5(.1 ) (.01 / ) 10F m kg m N
/
Fv
m L
Problem:
The displacement of a vibrating string vs position along the
string is shown. The wave speed is 10cm/s.
e) If the the tension doubles, how does the wave speed change?
Frequency? Wavelength?
/
Fv
m L
22
/
Fv
m L
22
/
Fv
m L
Wave speed increases by a factor of 2
Problem:
The displacement of a vibrating string vs position along the
string is shown. The wave speed is 10cm/s.
e) If the the tension doubles, how does the wave speed change?
Frequency? Wavelength?
/
Fv
m L
v f
Problem:
The displacement of a vibrating string vs position along the
string is shown. The wave speed is 10cm/s.
e) If the the tension doubles, how does the wave speed change?
Frequency? Wavelength?
/
Fv
m L
v f Frequency depends on
the SOURCE of vibration
Wavelength depends on BOTH!
Wave speed depends
on the MEDIUM
Problem:
The displacement of a vibrating string vs position along the
string is shown. The wave speed is 10cm/s.
e) If the the tension doubles, how does the wave speed change?
Frequency? Wavelength?
/
Fv
m L
2 2 22 2 v v f f
v f
Sound Generation Energy is transmitted as a pressure wave.
There is no net motion of the medium.
The medium oscillates in simple harmonic motion.
The frequency of the wave is the same as the vibrating source.
Vibrating
String
Spherically Symmetric
Sound Source (bell).
Representations of Waves
• Wave fronts are the concentric arcs
– The distance between successive
wave fronts is the wavelength
• Rays are the radial lines pointing
out from the source and
perpendicular to the wave fronts
• Far away from the source, the
wave fronts are nearly parallel
planes
• The rays are nearly parallel lines
Echo vs Reverb
A reverberation is perceived when the reflected sound wave reaches your ear in less than 0.1 second after the original sound wave. Since the original sound wave is still held in memory, there is no time delay between the perception of the reflected sound wave and the original sound wave. The two sound waves tend to combine as one very prolonged sound wave.
Diffract We can hear around corners.
Why can’t we see around corners?
If the size of the wave (wavelength) is close in size to the
object (door way) then the wave will diffract (bend).
Refract Sound waves refract (bend) when moving between
mediums in which it travels at different speeds.
343 m/s in Air @ 20 C
5960 m/s in Steel @ 20 C
1522 m/s in Ocean Water @ 20 C
Speed of Sound in a Vacuum?
Speed of Sound: Temperature
C(331 m/s) 1273 C
Tv
WARNING!
Some textbooks and teachers mistakenly state that the speed of sound increases with increasing density. This is usually illustrated by presenting data for three materials, such as air, water and steel. With only these three examples it indeed appears that speed is correlated to density, yet including only a few more examples would show this assumption to be incorrect. All other things being equal, sound will travel more slowly in denser materials, and faster in stiffer ones.
• The speed of sound waves in a medium depends on
the elasticity, density and temperature: decreases with
increasing density and increases with Temperature!
• The compressibility can be expressed in terms of the
elastic modulus of the material:
Bv
elastic property
inertial propertyv
Yv
Liquid or Gas:
1-D String: Fv
Solid Rod:
Speed of Sound: Medium
All other things being
equal, sound will travel
more slowly in denser
materials, and faster in
stiffer ones.
i
Fvolume stress AB
Vvolume strainV
Bv
elastic property
inertial propertyv
Yv
Fv
Speed of Sound: Medium
All other things being equal, sound will travel more slowly in
denser materials, and faster in stiffer ones.
For instance, sound will travel faster in iron than uranium,
and faster in hydrogen than nitrogen, due to the lower density
of the first material of each set. At the same time, sound will
travel faster in iron than hydrogen, because the internal
bonds in a solid are much stronger than the gaseous bonds
between hydrogen molecules. In general, solids will have a
higher speed of sound than liquids, and liquids will have a
higher speed of sound than gases – not because of greater
density, but stronger bonds!
Toy Model
The transmission of sound can be illustrated by using a toy model consisting of an array of balls interconnected by springs. For real material the balls represent molecules and the springs represent the bonds between them. Sound passes through the model by compressing and expanding the springs, transmitting energy to neighboring balls, which transmit energy to their springs, and so on. The speed of sound through the model depends on the stiffness of the springs (stiffer springs transmit energy more quickly). In a real material, the ‘stiffness of the springs’ is called the elastic modulus, and the mass corresponds to the density. All other things being equal, sound will travel more slowly in denser materials, and faster in stiffer ones.
elastic property
inertial propertyv
Bv
Yv
The power transmitted through a closed surface by a wave is
proportional to the amplitude of the wave.
Sound Power
2
W
m
owerI
Area
The intensity of a wave, the power per unit area, is the rate at
which energy is being transported by the wave through a unit
area A perpendicular to the direction of travel of the wave:
Intensity
2 2
W
4 m
PI
r
2
~Intensity Amplitude
Spherical Waves
• A spherical wave propagates radially outward from the oscillating sphere
• The energy propagates equally in all directions
• To compare intensities at two locations, the inverse square relationship can be used
2
1 2
2
2 1
I r
I r
2 2
W
4 m
av avIA r
Intensity
A point source emits sound with a power output of
100 watts. What is the intensity (in W/m2) at a
distance of 10.0 m from the source?
2 2
W
4 m
PI
r
0
10 logI
dBI
12 2
0Threshold of hearing : 10 /I W m
Decibel Index:
Whisper: 20db
Conversation: 60db
Loud Music: 120 db
Jet: 140 dB
Rocket: 250dB
At 90db, wear ear plugs!!!
12 2
0Threshold of hearing : 10 /I W m
4 2Bursting of eardrums : 10 /I W m
6 2Normal Conversation: 10 /I W m
10 2Whisper: 10 /I W m
2
0
10
WhisperI
I
0
log 2WI
I 2 bels
10 1decibels bel 20 decibels
0 dB
20 dB
60 dB
160 dB
0
10 logI
dBI
OSHA Safety Standards
OSHA - Occupational Safety and Health Act - The OSHA criteria document reevaluates and reaffirms the Recommended Exposure Limit (REL) for occupational noise exposure established by the National Institute for Occupational Safety and Health (NIOSH) in 1972. The REL is 85 decibels, A-weighted, as an 8-hr time-weighted average (85 dBA as an 8-hr TWA). Exposures at or above this level are hazardous.
Whisper: 20db
Conversation: 60db
Loud Music: 120 db
Jet: 140 dB
Rocket: 250dB
At 90db, wear ear plugs!
You Try
A point source emits sound with a power output of
100 watts. What is the intensity (in W/m2) at a
distance of 10.0 m from the source in dB?
0
10 logI
dBI
2 2
W
4 m
PI
r
11
0
10 logI
dBI
If a sound is twice as intense, how much greater is the sound
level, in db?
22
0
10 logI
dBI
2 12 1
0 0
10 log 10 logI I
dB dBI I
2 12 1
0 0
10 log /I I
dBI I
2
1
10 logI
dBI
2 1 10 log2dB
3.01dB
53 dB is twice as intense as 50dB. Log Scale!!
Quiz: Increasing the intensity of sound
by a factor of 100 causes the sound
level to increase by what amount?
1. 100dB
2. 10dB
3. 20dB
4. 200dB
5. 2 dB
22
1
10 logI
dBI
2 10 log 100 10 2 20dB x
11
0
10 logI
dBI
The decibel level of a jackhammer is 130 dB relative to
the threshold of hearing. Determine the sound intensity
produced by the jackhammer.
1
0
130 10 logI
dB dBI
1
0
13 logI
I
1
0
log1310 10
I
I
13 1
0
10I
I
13
1 010I I 13 1210 10 210 /W m
is a What you Hear
The Pressure Wave sets the Ear Drum into Vibration.
The ear converts sound energy to mechanical energy to a nerve impulse which is transmitted to the brain.
Drum to Stirrup: Simple Machine
Amplification Since the pressure wave striking the large area of the eardrum is concentrated into the smaller area of the stirrup, the force of the vibrating stirrup is nearly 15 times larger than that of the eardrum. This feature enhances our ability of hear the faintest of sounds.
Resonance of the Cilia Nerves The inner surface of the cochlea is lined with over
20 000 hair-like cilia connected to nerve cells, each differing in length by minuscule amounts. Each hair cell has a natural sensitivity to a particular frequency of vibration. When the frequency of the sound wave matches the natural frequency of the nerve cell, that nerve cell will resonate with a larger amplitude of vibration which induces the cell to release an electrical impulse along the auditory nerve towards the brain.
Cochlear Cilia Nerve Damage
Normal Ear Damaged Ear
Excessive exposure to loud sound can damage your cilia.
Sonic: 20 Hz 20 kHz
INFRAsonic: 20Hz
ULTRAsonic: 20kHz
f
f
A middle C vibrates 252 times per second.
Sound Frequencies
Scientists first detected infrasound in
1883, when the eruption of the
Krakatoa volcano in Indonesia sent
inaudible sound waves careening
around the world, affecting barometric
readings. 310dB estimated
The eruption of the Fuego volcano in
Guatemala last year generated high-
amplitude infrasound, mostly below 10
hertz. The pressure readings show that
the strength of these sound waves can
reach the equivalent of 120 decibels.
Infrasonic: < 20Hz
Ultrasound
Intensity of reflected sound wave (echo) is
related to change in density in target.
Ultrasound beam:
-2
7 1 detail
~ 10
MHz mm
I W
"A Womb With a View" and
"Fetal Fotos” “Peek in the Pod”
Hi Cost Hi-Definition Ultrasound
Are there RISKS?
"We do know in animal
studies, certain levels of
ultrasound can cause
damages in growing bones,
in developing bones," said
Dr. Dan Schultz of the Food
and Drug Administration.
Animal Perception of Sound
•domestic cats •100-32,000 Hz
•domestic dogs •40-46,000 Hz
•African
elephants •16-12,000 Hz
•bats •1000-150,000
Hz
•rodents •70-150,000 Hz
Human: 20-20,00Hz
Infrasonic Contact Calls
Female African elephants use "contact calls" to communicate
with other elephants in their bands (usually a family group).
These infrasonic calls, with a frequency of about 21 Hz and a
normal duration of 4-5 seconds, carry for long distances (several
kilometers), and help elephants to determine the location of
other Elephants. Calls vary among individual elephants, so that
others respond differently to familiar calls than to unfamiliar
calls. Perhaps elephants can recognize the identity of the caller.
Echolocation: Sonic Vision
Dolphins produce high frequency (100kHz) clicks that pass through
the melon. These sound waves bounce off objects in the water and
return to the dolphin in the form of an echo. The brain receives the
sound waves in the form of nerve impulses. By this complex system
of echolocation, dolphins can determine size, shape, speed, distance,
direction, and even some of the internal structure of objects in the
water.
Dolphin
Vocalization
The LFAS system consists of a 35-
ton block of 18 huge underwater
speakers and dozens of
microphones. The speakers emit a
consistent low-frequency tone,
between 100 and 500 Hertz, at
240dB, which travels out into the
water at a depth of several hundred
meters. The low frequency permits
the sound to travel tremendous
distances, detecting objects many
hundreds of miles away by
echolocation.
At 100 mile radius from the ship the noise only drops to 160
db which causes shearing of the tissues in the air sack behind
whales' and dolphins' brain. This air sack is highly sensitive
since it is used in echolocation.
“Sound bombing" of ocean floors
to test for oil and gas for National
Security?
2004: More than 100 whales and dolphins died in two separate
beachings in 24 hours on remote Australian islands after US and
Australian navies sound bombed the ocean nearby.
Sea Quakes produce
powerful pressure
waves that rupture
the sinuses and
middle ear of whales
and dolphins.
•Cosmological Redshift: Expanding Universe
•Stellar Motions: Rotations and Radial Motions
•Solar Physics: Surface Studies and Rotations
•Gravitational Redshift: Black Holes & Lensing
•Extra-solar Planets via Doppler Wobbler
S O ?v
Speed of a wave is determined
by the properties of the Medium!
Case 1: Moving Source Stationary Observer 0Ov
What is the speed of sound to the observer?
wavev v
S O
Speed of a wave is determined
by the properties of the Medium!
Case 1: Moving Source Stationary Observer 0Ov
What is the speed of sound to the observer?
v v
wavev v
What is wavelength and frequency to the observer?
, f f
Sv
is shortened by
= Sv
Case 1: Moving Source Stationary Observer 0Ov
= (1 )Sv
v
= (1 )Sv
v
= (1 )Sv
= ( )sv v
v
?f
Case 1: Moving Source Stationary Observer 0Ov
Use v v
= sv v
v
( )s
f f fv v
v
'f f
1
1 S
f fv
v
Sv
What if ?Sv v
When the duck speed is equal or greater than
the speed of waves in water, the waves form a bow wave.
Case 2: Observer Moving & Stationary Source
S Ov
Observer Moving TOWARD (+) and
AWAY (-) from Source
ov vf f
v
ov v v
0(1 )v
f fv
In Sum, if both Source and Observer
are moving…..
sound
observer
source
o
s
O
S
v vf f
v v
v
v
v
0(1 )v
f fv
1
(1 )S
f fv
v
Only Source Moving: Only Observer Moving:
Both: The signs depend on
the relative motion.
For the velocity:
Moving away: minus
Moving toward: plus
Doppler Shift
A car approaches a stationary police car at 36 m/s. The frequency of the siren (relative to the police car) is 500 Hz. What is the frequency (in Hz) heard by an observer in the moving car as he approaches the police car? (Assume the velocity of sound in air is 343 m/s.)
a. 220
b. 448
c. 5264
d. 552
e. 383
sound
observer
source
o
s
O
S
v vf f
v v
v
v
v
?
?
?
v
f
Doppler Shift
A truck moving at 36 m/s passes a police car moving at 45 m/s in the opposite direction. If the frequency of the siren relative to the police car is 500 Hz, what is the frequency heard by an observer in the truck as the police car approaches the truck? (The speed of sound in air is 343 m/s.)
a. 396
b. 636
c. 361
d. 393
e. 617
sound
observer
source
o
s
O
S
v vf f
v v
v
v
v
?
?
?
v
f
Doppler Shift
A truck moving at 36 m/s passes a police car moving at 45 m/s in the opposite direction. If the frequency of the siren is 500 Hz relative to the police car, what is the frequency heard by an observer in the truck after the police car passes the truck? (The speed of sound in air is 343 m/s.)
a. 361
b. 636
c. 393
d. 396
e. 383 sound
observer
source
o
s
O
S
v vf f
v v
v
v
v
?
?
?
v
f
Windy Wave Speed Question
How does the wind affect the sound of a fog horn you hear
on a windy day? What changes?
a) Frequency b) wavelength c) speed d) nothing
Ultrasound Question 8. How far apart are two layers of tissue that produce echoes
having round-trip times that differ by 0.750s? What minimum
frequency must the ultrasound have to see detail this small?
The speed of sound in human tissue is 1540m/s.
6
4w1540 m s 0.750 10 s
5.78 10 m2 2
v td
v f fv
ww 6m s
m2.67 10 Hz
1540
578 10 4.
2 1 s 2 s 1 s / 2d d d v t v t v t
You Try
Calculate the intensity level in dB of a sound
wave that has an intensity of 15 10–4 W/m2.
a. 20
b. 200
c. 92
d. 9
e. 10
0
10 logI
dBI
2 2
W
4 m
PI
r