Waves Wrap Up and Review
Doppler Effect
Optics
Nature of Light
Lana Sheridan
De Anza College
June 5, 2017
Last time
• standing waves and sound
• musical instruments
• standing waves in rods and membranes
• beats
• nonsinusoidal waves
• intensity of a wave
Overview
• sound level
• the Doppler effect
• bow and shock waves
• talk about waves
• light as a wave
Intensity of a Wave
Intensity
the average power of a wave per unit area
I=
Pavg
A
Intensity is used for waves that move on 3 dimensional media, such
as sound or light.
The waves travel in one direction, and the area A is arranged
perpendicular to the direction of the wave travel.
Power and Intensity of Sound Waves
Power of a sound wave:
1
2
Poweravg = ρv ω2 A smax
2
Dividing this by the area gives the intensity of a sound arriving on
that area:
1
I = ρv (ωsmax )2
2
This can be written in terms of the pressure variation amplitude,
∆Pmax = ρv ωsmax :
(∆Pmax )2
I=
2ρv
Decibels: Scale for Sound Level
The ear can detect very quiet sounds, but also can hear very loud
sounds without damage.
(Very, very loud sounds do damage ears.)
As sound that has twice the intensity does not “sound like” it is
twice as loud.
Many human senses register to us on a logarithmic scale.
Roughly, noise A sounds twice as loud noise B if noise A is 10
times as loud as noise B.
Decibels: Scale for Sound Level
Decibels is the scale unit we use to measure loudness, because it
better represents our perception than intensity.
The sound level β is defined as
β = 10 log10
I
I0
• I is the intensity
• I0 = 1.00 × 10−12 W/m2 is a reference intensity at the
threshold of human hearing
The units of β are decibels, written dB.
Question
Quick Quiz 17.31 Increasing the intensity of a sound by a factor
of 100 causes the sound level to increase by what amount?
(A) 100 dB
(B) 20 dB
(C) 10 dB
(D) 2 dB
1
Serway & Jewett, page 515.
Question
Quick Quiz 17.31 Increasing the intensity of a sound by a factor
of 100 causes the sound level to increase by what amount?
(A) 100 dB
(B) 20 dB
←
(C) 10 dB
(D) 2 dB
1
Serway & Jewett, page 515.
Perception of Loudness and Frequency
Human hearing also depends on frequency.
1
Figure from R. L. Reese, University Physics, via Serway & Jewett.
Perception of Loudness and Frequency
Human hearing also depends on frequency.
Humans can only hear sound in the range 20-20,000 Hz.
17.4 The Doppler Ef
Sound level
b (dB)
Infrasonic
Sonic
Ultrasonic
frequencies
frequencies
frequencies
220
Large rocket engine
Underwater communication
200
(Sonar)
180
Jet engine (10 m away) Rifle
160
Threshold of
pain
140
Rock concert
120
Car horn
School cafeteria
100 Thunder
Motorcycle
overhead
80
Urban traffic
Shout
Birds
60
Conversation
Bats
40
Whispered speech
Threshold of
20
hearing
0
Frequency f (Hz)
1
10
100
1 000
10 000
100 000
Low frequency sounds need to be louder to be heard.
1old
of pain. Here the boundary of the white area appears straight because the psyFigure from
R. L.isReese,
University
Physics,
via Serway
Jewett.
chological
response
relatively
independent
of frequency
at this &
high
sound level.
Figure 1
ranges of
level of va
normal hu
the white
University
Brooks/C
observe the op
stern) of the
3.0 s has
The frequency of a sound counts how many wavefronts than
(pressure
peaks) arrive per second.
you observe a
If you are moving towards a source of sound, you encounterThese
more effec
depends on th
wavefronts per second → the frequency you detect is higher!
When you are
than that of t
quency. When
the observed f
O
S
Let’s now e
waves become
becomes an o
and a sound s
S
vO
ary and the o
moves with a
Figure 17.9 An observer O
means at rest
The Doppler Effect
you observe a lower fr
These effects occu
The Doppler Effect
depends on the direct
When you are moving
than that of the wave
quency. When you tur
the observed frequenc
O
S
Let’s now examine
waves become sound
becomes an observer
and a sound source S
S
vO
ary and the observer
moves with a speed v
Figure 17.9 An observer O
means at rest with res
(thewaves
cyclist)traveling
moves withrelative
a speed to you is
The speed you see the
If a point source em
v
toward
a
stationary
point
O
v 0 = v + v0 , while relative
to the source the speed is v .
at the same speed in
source S, the horn of a parked
hears
truck. The0observer
spherical wave as men
v + v0 a frev
0
quency
f = f 9 that
=is greater thanf the fronts equals the wav
λ
v
source frequency.
these three-dimension
We take the freque
(v and v0 are positive numbers.)
and the speed of soun
The Doppler Effect
The speed you see the waves traveling relative to you is
v 0 = v + v0 , while relative to the source the speed is v .
v0
v + v0
0
f =
=
f
λ
v
(v and v0 are positive numbers.)
Moving away from the source, the relative velocity of the detector
to the source decreases v 0 = v − v0 .
v − v0
0
f
f =
v
The Doppler Effect
A point source is moving
to the right with speed v
A similar thing happens if the source of the waves is moving.
B
S
Observer B
a
S
vS
l!
A
Observer A
C
b
In the diagram, the source is moving toward the wavefronts it has
created on the right and away from the wavefronts it has created
toward the source, and a negative value is substituted when the
on the left.
away from the source.
This changes
thesuppose
wavelength
of the is
waves
aroundand
the the
source.
Theyis at re
Now
the source
in motion
observer
moves
directly
observer
inleft.
Figure 17.10a, each new wave is
are shorter
on the
right,toward
and longer
on A
the
position to the right of the origin of the previous wave. As a result,
heard by the observer are closer together than they would be if the
The Doppler Effect
B
S
Observer B
a
Observer A
For A:
S
vS
l!
A
Observer A
C
Courtesy of the Educational
Development Center, Newton, MA
A point source is moving
to the right with speed vS .
b
detects the wavelength as λ 0 = λ − vs T = λ − vfs .
toward the source, and a negative value is substituted when the observer m
away from the source.
the source is in
motion
and theobserver is at rest. If the sou
Now suppose
vdirectly toward vobserver A in Figure v17.10a, each new wave is emitted fro
0
=
f
=
f moves
=
position
origin of thevprevious
λ 0 to the right
v /f of−the
vs /f
− vs wave. As a result, the wave fr
heard by the observer are closer together than they would be if the source were
moving. (Fig. 17.10b shows this effect for waves moving on the surface of wa
As a result, the wavelength l9 measured by observer A is shorter than the w
length l of the source. During each vibration, which lasts for a time interval T
period), the source moves a distance vST 5 vS /f and the wavelength is shortene
this amount. Therefore, the observed wavelength l9 is
vS
lr 5 l 2 Dl 5 l 2
f
The Doppler Effect
B
S
Observer B
a
S
vS
l!
A
Observer A
C
Courtesy of the Educational
Development Center, Newton, MA
A point source is moving
to the right with speed vS .
b
Observer A detects the wavelength as λ 0 = λ − vs T = λ − vfs .
toward the source, and a negative value is substituted when the observer m
away from the source.
For A:
the source is in
motion
and theobserver is at rest. If the sou
Now suppose
vdirectly toward vobserver A in Figure v17.10a, each new wave is emitted fro
0
=
f
=
f moves
=
position
origin of thevprevious
λ 0 to the right
v /f of−the
vs /f
− vs wave. As a result, the wave fr
heard by the observer are closer together than they would be if the source were
moving. (Fig. 17.10b shows this effect for waves moving on the surface of wa
As a result, the wavelength
l9 measured
by observer A is shorter than the w
For Observer B:
length l of the source. During
v each vibration, which lasts for a time interval T
0
period), the source
f =moves a distancef vST 5 vS /f and the wavelength is shortene
+observed
vs
this amount. Therefore, vthe
wavelength l9 is
vS
lr 5 l 2 Dl 5 l 2
f
The Doppler Effect
In general:
0
f =
v ± v0
v ∓ vs
f
The top sign in the numerator and denominator corresponds to the
detector and/or source moving towards on another.
The bottom signs correspond to the detector and/or source
moving away from on another.
The Doppler Effect
A point source is moving
to the right with speed vS .
B
S
S
vS
l!
A
Observer A
C
Courtesy of the Educational
Development Center, Newton, MA
erver B
Quick Quiz 17.42 Consider detectors of water waves
at three
17.4 The Doppler Effect
locations A, B, and C in the picture. Which of the following
statements is true?
Figure 17.10
(a) A sourc
ing with a speed vS toward
tionary observer A and awa
a stationary observer B. Ob
A hears an increased frequ
and observer B hears a dec
frequency. (b) The Dopple
in water, observed in a ripp
Letters shown in the photo
to Quick Quiz 17.4.
b
(A) The wave speed is highest at location C.
detected
largest
at location
C.
source,(B)
and aThe
negative
value wavelength
is substituted is
when
the observer
moves
he source.
(C) The detected frequency is highest at location C.
pose the source is in motion and the observer is at rest. If the source
(D)observer
The detected
highest
location
ctly toward
A in Figurefrequency
17.10a, eachisnew
wave isatemitted
fromA.
a
the right of2the origin of the previous wave. As a result, the wave fronts
Serway
Jewett,than
pagethey
520.would be if the source were not
e observer are
closer&together
Pitfall Prevention 17.1
Doppler Effect Does Not D
on Distance Some people
that the Doppler effect de
on the distance between th
source and the observer. A
The Doppler Effect
A point source is moving
to the right with speed vS .
B
S
S
vS
l!
A
Observer A
C
Courtesy of the Educational
Development Center, Newton, MA
erver B
Quick Quiz 17.42 Consider detectors of water waves
at three
17.4 The Doppler Effect
locations A, B, and C in the picture. Which of the following
statements is true?
Figure 17.10
(a) A sourc
ing with a speed vS toward
tionary observer A and awa
a stationary observer B. Ob
A hears an increased frequ
and observer B hears a dec
frequency. (b) The Dopple
in water, observed in a ripp
Letters shown in the photo
to Quick Quiz 17.4.
b
(A) The wave speed is highest at location C.
detected
largest
at location
C.
source,(B)
and aThe
negative
value wavelength
is substituted is
when
the observer
moves
he source.
(C) The detected frequency is highest at location C.
pose the source is in motion and the observer is at rest. If the source
(D)observer
The detected
highest
location
ctly toward
A in Figurefrequency
17.10a, eachisnew
wave isatemitted
fromA.
a
the right of2the origin of the previous wave. As a result, the wave fronts
Serway
Jewett,than
pagethey
520.would be if the source were not
e observer are
closer&together
Pitfall Prevention 17.1
←
Doppler Effect Does Not D
on Distance Some people
that the Doppler effect de
on the distance between th
source and the observer. A
The Doppler Effect Question
A police car has a siren tone with a frequency at 2.0 kHz.
It is approaching you at 28 m/s. What frequency do you hear the
siren tone as?
Now it has passed by and is moving away from you. What
frequency do you hear the siren tone as now?
The Doppler Effect Question
A police car has a siren tone with a frequency at 2.0 kHz.
It is approaching you at 28 m/s. What frequency do you hear the
siren tone as?
Now it has passed by and is moving away from you. What
frequency do you hear the siren tone as now?
approaching: 2.2 kHz
receding (moving away): 1.8 kHz
The Doppler Effect for Light
Electromagnetic radiation also exhibits the Doppler effect,
however, the equation needed to describe the effect is different.
For em radiation:
p
1 + v /c
f =p
f
1 − v /c
0
Relativity is needed to derive this expression (hold on for Phys4D).
This can be used to determine how fast distant objects in space
are moving toward or away from us.
The Doppler Effect and Astronomy
1
Image from Wikipedia by Georg Wiora.
Bow Waves and Shock Waves
Bow waves and shock waves can be detected by nearby observers
when the speed of the wave source exceeds the speed of the waves.
This effect happens when an aircraft transitions from subsonic
flight to supersonic flight.
1
Figure from Hewitt, 11ed.
Bow waves
Supersonic transition
Shock Waves
The envelope of the wave
fronts forms a cone whose
apex half-angle is given by
sin u ! v/vS .
Notice the shock wave in
the vicinity of the bullet.
Omikron/Photo Researchers/Getty Images
entaed
0 to
t is
d v in
opic
ing at
he hot
S
0
vS
1
vt
2
u
S0 S1 S2
vS t
a
b
The angle which the shockwave makes is called the Mach angle.
Shock Waves
v
= vS of a source exceeds the wave speed v.
Now consider what happens when sin
the θspeed
v
s 17.11a. The circles represent spheriThis situation is depicted graphically in Figure
cal wave fronts emitted by the source at various times during its motion. At t 5 0,
The
ratio viss at
/vS 0isand
called
thetoward
Mach
the source
moving
thenumber.
right. At later times, the source is at S1,
and then S 2, and so on. At the time t, the wave front centered at S 0 reaches a radius
Shock Waves Question
Quick Quiz 17.63 An airplane flying with a constant velocity
moves from a cold air mass into a warm air mass. What happens
to the Mach number?
(A) it increases
(B) it decreases
(C) it stays the same
3
Serway & Jewett, page 522.
Shock Waves Question
Quick Quiz 17.63 An airplane flying with a constant velocity
moves from a cold air mass into a warm air mass. What happens
to the Mach number?
(A) it increases
(B) it decreases
(C) it stays the same
Hint:
r
v = 331
3
Serway & Jewett, page 522.
1+
TCel
[m/s]
273
Shock Waves Question
Quick Quiz 17.63 An airplane flying with a constant velocity
moves from a cold air mass into a warm air mass. What happens
to the Mach number?
(A) it increases
(B) it decreases
←
(C) it stays the same
Hint:
r
v = 331
3
Serway & Jewett, page 522.
1+
TCel
[m/s]
273
Waves
Any questions about waves?
Light
We are now moving on to chapters 35-38.
Light is also a wave.
Speed of Light
Light travels very fast.
We can figure out how fast it goes from Maxwell’s laws, by
deriving a wave equation from them.
Maxwell’s Equations
I
E · dA =
I
qenc
0
B · dA = 0
I
E · ds = −
I
B · ds = µ0 0
dΦB
dt
dΦE
+µ0 Ienc
dt
In free space (a vacuum) with no charges qenc = 0 and Ienc = 0.
1
Strictly, these are Maxwell’s equations in a vacuum.
Maxwell’s Equations Differential Form
∇·E =
ρ
0
∇·B = 0
∂B
∂t
∂E
+ µ0 J
∇ × B = µ0 0
∂t
∇×E = −
In free space with no charges ρ = 0 and J = 0.
Notation
The differential operators in 3 dimensions.
Gradient of a scalar field at a point, f :
∇f =
∂f
∂f
∂f
i+
j+
k
∂x
∂y
∂z
Divergence of a vector field at a point v = [vx , vy , vz ]:
∇·v =
∂vy
∂vz
∂vx
+
+
∂x
∂y
∂z
Curl of a vector field at a point v:
∂vy
∂vy
∂vz
∂vx
∂vz
∂vx
∇×v =
−
i+
−
j+
−
k
∂y
∂z
∂z
∂x
∂x
∂y
Maxwell’s Equations and the Wave Equation
∇·E = 0
∇·B = 0
∂B
∂t
∂E
∇ × B = µ0 0
∂t
∇×E = −
Take the curl of both sides of equation 3:
∇ × (∇ × E) = −
∂
(∇ × B)
∂t
Using the vector triple product rule, a × (b × c) = b(a · c) − c(a · b)
∇(∇ · E) − (∇ · ∇)E = −
∂
(∇ × B)
∂t
Maxwell’s Equations and the Wave Equation
∇(∇ · E) − ∇2 E = −
∂
(∇ × B)
∂t
Using the 1st equation:
∇2 E =
∂
(∇ × B)
∂t
Using the 4th equation,
∇2 E = µ0 0
∂2 E
∂t 2
Another Implication of Maxwell’s Equations
Using all 4 equations it is possible to reach a pair of wave
equations for the electric and magnetic fields:
∇2 E =
∇2 B =
1 ∂2 E
c 2 ∂t2
1 ∂2 B
c 2 ∂t2
with wave solutions:
E = E0 sin(k · r − ωt)
B = B0 sin(k · r − ωt)
where c =
ω
k .
Another Implication of Maxwell’s Equations
For E-field varying in x direction:
∂2 Ex
1 ∂2 Ex
=
∂x 2
c 2 ∂t2
The constant c appears as the wave speed and
1
c=√
µ0 0
c = 3.00 × 108 m/s, is the speed of light.
The values of 0 and µ0 together predict the speed of light!
0 = 8.85 × 10−12 C2 N−1 m−2 and µ0 = 4π × 10−7 kg m C−2
Another Implication of Maxwell’s Equations
Wave solutions:
E = E0 sin(k · r − ωt)
B = B0 sin(k · r − ωt)
where c =
ω
k .
These two solutions are in phase. There is no offset in the angles
inside the sine functions.
The two fields peak at the same point in space and time.
At all times:
E
=c
B
Summary
• sound level
• the Doppler effect
• bow and shock waves
• light as a wave
Homework Serway & Jewett:
• Ch 17, onward from page 523. Probs: 19, 25, 37, 41, 47, 54,
71
• Ch 18, onward from page 555. OQs: 3, 5; CQs: 3; Probs: 59,
77