Transcript 投影片 1 - National Cheng Kung University

14. Wave Motion
1.
2.
3.
4.
5.
6.
7.
8.
Waves & their Properties
Wave Math
Waves on a String
Sound Waves
Interference
Reflection & Refraction
Standing Waves
The Doppler Effect & Shock Waves
Other kinds of waves:
• Sound
• Light
• Radio
• Ultrasound
• Microwave
• Earthquake / Tsunami
Wave:
Traveling disturbance that
Ocean waves travel thousands of kilometers
across the open sea before breaking on shore.
How much water moves with the waves?
None
transport energy but not matter.
14.1. Waves & their Properties
Mechanical waves: mechanical disturbances in material medium.
E.g., air, water, violin string, Earth’s interior, ….
Electromagnetic waves: EM disturbances anywhere (including vacuum)
E.g., Visible, infrared, & ultraviolet light, radio waves, X ray, …
Longitudinal & Transverse Waves
Longitudinal waves
Transverse waves
Transverse
1-D Vibration
Longitudinal
Water Waves
Water waves
mixed
Wave Amplitude
Wave amplitude = maximum value of the disturbance.
( w.r.t. undisturbed state )
Water wave: max height above undisturbed level.
Sound wave: max excess pressure.
Wave in coupled springs: max displacement from equilibrium position.
Wave Shape
Waveform = shape of waves.
Pulse = isolated disturbance.
Continuous wave
= ongoing periodic disturbance.
Wave train
= periodic disturbance of finite duration.
Wavelength, Period, & Frequency
A continuous wave is periodic in both time & space.
Wavelength  : distance over which the wave pattern repeats. ( length of 1 cycle )
Period T : duration over which the wave pattern repeats. ( time for 1 cycle )
Frequency f : number of wave cycles per unit time. ( f = 1 / T )
Wave Speed
Speed of wave depends only on the medium.
Sound in air  340 m/s  1220 km/h.
in water  1450 m/s
in granite  5000 m/s
Small ripples on water  20 cm/s.
Earthquake  5 km/s.
Wave speed
v

T
 f
GOT IT? 14.1.
A boat bobs up & down on a water wave, moving a vertical distance of 2 m in 1 s.
A wave crest moves a horizontal distance of 10 m in 2 s.
Is the wave speed
(a)
2 m/s, or
(b)
5 m/s ?
Explain.
( Speed of disturbance )
pk @ x = 0
pk @ x = v t
14.2. Wave Math
y  x,0  f  x 
At t = 0,
At t , y(0) is displaced to the right by v t.

y  x, t   f  x  v t 
For a wave moving to the left :
For a SHW (sinusoidal):
y  x,0  A cos k x
k
y  x, t   f  x  v t 
2

= wave number
SHW moving to the right :
y  x, t   A cos  k x   t 
  k x  t
v
 

T k
= phase

2
T
 k x v t
= wave speed
Waves
Example 14.1. Surfing
A surfer paddles to where the waves are sinusoidal with crests 14 m apart.
He bobs a vertical distance 3.6 m from trough to crest, which takes 1.5 s.
Find the wave speed, & describe the wave.
y  x, t   A cos  k x   t 
A
1
 3.6 m   1.8 m
2
T  2  1.5 s   3.0 s
  14 m
v
k
2


T
 4.7 m / s
 0.449 m
1

2
 2.09 s 1
T
y  x, t   1.8 cos  0.449 x  2.09 t 
GOT IT? 14.2.
Figure shows two waves propagating with the same speed.
Which has the greater
(a) amplitude,
U
(b) wavelength,
L
(c) period,
L
(d) wave number,
U
(e) frequency ?
U
v=/T
The Wave Equation
1-D waves in many media can be described by the partial differential equation
2 y
2 y
 2 2
2
x
v t
whose solutions are of the form
Wave Equation
y  x, t   f  x  v t 
v = velocity of wave.
y  x, t   Acos  k x   t 

v
E.g.,
• water wave ( y = wave height )
• sound wave ( y = pressure )
• …

k
( towards  x )
14.3. Waves on a String
A pulse travels to the right.
In the frame moving with the pulse, the entire string
moves to the left.
Top of pulse is in circular motion with speed v & radius R.
Centripedal accel:
m v2
ma
yˆ
R
Tension force F is cancelled out in the x direction:
Fy  2F sin   2F

( small segment )
m v2
2 R  v 2
2 F 

R
R
 = mass per unit length [ kg/m ]
F   v2
v
F

Example 14.2. Rock Climbing
A 43-m-long rope of mass 5.0 kg joins two climbers.
One climber strikes the rope, and 1.4 s later, the 2nd one feels the effect.
What’s the rope’s tension?
F v
2

m
L
v
L
t
m L   5.0 kg   43 m 
 110 N
F 2
2
t
1.4 s 
Wave Power
E
SHO :
1
m  2 A2
2
Segment of length x at fixed x :
P
E
1
 x  2 A2
2
1 x 2 2
1

 A   v  2 A2
2 t
2
v = phase velocity of wave
Wave Intensity
Intensity = power per unit area  direction of propagation [ W / m2 ]
Wave front = surface of constant phase.
Plane wave : planar wave front.
Spherical wave : spherical wave front.
Plane wave :
I  const
Spherical wave :
I
P
4  r2
Example 14.3. Reading Light
A book 1.9 m from a 75-watt light bulb is barely readable.
How far from a 40-W bulb the book should be to provide the same intensity at the page.
I
P75 P40
 2
2
r75 r40
P
4  r2
r40  r75
P40
P75  1.9 m 
40 W
75 W
 1.4 m
GOT IT? 14.3.
The intensity of light from the more distant one of two identical stars is only 1%
that of the closer one.
(a)
twice
(b) 100 times
(c) 10 times
(d) 10 times
as far away.
Is the more distant star
14.4. Sound Waves
Sound waves = longitudinal mechanical waves
through matter.
P,  = max , x = 0
Speed of sound in air :
v
P

P = background pressure.
 = mass density.
 = 7/5 for air & diatomic gases.
 = 5/3 for monatomic gases, e.g., He.
P,  = eqm , |x| = max
P,  = min , x = 0
Sound & the Human Ear
Audible freq:
20 Hz ~ 20 kHz
Bats: 100 kHz
Ultrasound: 10 MHz
db = 0 :
Hearing Threshold
@ 1k Hz
Decibels
Sound intensity level :
 I 
  10 log10  
 I0 
I0  1012 W / m2
[  ] = decibel (dB)
I  I0 10 /10
 Threshold of hearing at 1 kHz.
 I2 
 2  1  10 log10  
 I1 
2  1  10 dB

I 2  10 I1
2  1  3 dB

I 2  103/10 I1  2 I1
I2
 10 2  1  / 10
I1
Nonlinear behavior: Above 40dB, the ear percieves  = 10 dB as a doubling of loudness.
Example 14.4. TV
A TV blasts at 75 dB.
If it’s then turned down to 60 dB, by what factor has the power dropped ?
 I2 
 P2 
 2  1  10 log10    10 log10  
 I1 
 P1 
I
P
4  r2
P2
     / 10
1
1
 10 2 1
 106075 / 10  103 / 2 
 0.032

P1
30
10 10
10 db drop  ½ in loudness
15 db drop  between ½ & ¼ in loudness
14.5. Interference
Principle of superposition: tot = 1 + 2 .
constructive interference
destructive interference
Interference
Fourier Analysis
Fourier analysis:
Periodic wave = sum of SHWs.
square wave  A

1
sin  n t 

n  0 2n  1
Fourier Series
E note from
electric guitar
Dispersion
Non-dispersive medium
Dispersion:
wave speed is wavelength (or freq) dependent
Dispersion
Surface wave on deep water:
v
Dispersive medium
g
2
 long wavelength waves reaches shore 1st.
Dispersion of square wave pulses determines max
length of wires or optical fibres in computer networks.
Conceptual Example 14.1. Storm Brewing
It’s a lovely, sunny day at the coast,
but large waves, their crests far apart, are crashing on the beach.
How do these waves tell of a storm at sea that may affect you later?
crests far apart
 long wavelength
 v =  (  g / 2 ) large
 storm that generates the waves are not far behind
Note: tsunamis generate shallow-water waves that do not obey
v
g
2
Making the Connection
A storm develops 600 km offshore & starts moving towards you at 40 km/h.
Large waves with crests 250 m apart are your 1st hint of the storm.
How long after you observe these waves will the storm hit?
Time for storm to reach you =
Speed of wave =
g

2
Time for wave to reach you =
600 km
 15 h
40 km / h
 250 m    9.8 m / s 2 
2
600 km
 8.45 h
71.0 km / h
The storm is 15  8.45 = 6.55 h  6.6 h away.
 19.7 m / s  71.0 km / h
Beats
Beats: interference between 2
waves of nearly equal freq.
Destructive
Constructive
y t   A cos 1 t  A cos 2 t
1

1

 2 A cos  1  2  t  cos  1  2  t 
2

2

Freq of envelope = 1  2 .
smaller freq diff  longer period between beats
Applications:
Synchronize airplane engines (beat freq  0).
Tune musical instruments.
High precision measurements (EM waves).
Interference in 2-D
Destructive
Constructive
Nodal lines:
amplitude  0
path difference = ½ n 
Water waves from two sources with separation  
Interference
Example 14.5. Calm Water
Ocean waves pass through two small openings, 20 m apart, in a breakwater.
75 m from the breakwater & midway between the openings, water is rough.
33 m parallel to the breakwater away, the water is calm.
What’s the wavelength of the waves?
AP  BP 

2
AP 
 75 m   33 m  10 m
BP 
 75 m  33 m 10 m
2
2
2
 86.5 m
2
 78.4 m
  2 86.5 m  78.4 m  16 m
GOT IT? 14.4.
Light shines through two small holes onto a screen in a dark room.
The holes spacing is comparable to the wavelength of the light.
Looking at the screen, will you see
(a)
two bright spots
(b) a pattern of light & dark patches?
Explain.
14.6. Reflection & Refraction
light + heavy ropes
A = 0;
reflected
wave
inverted
A = max;
reflected
wave not
inverted
Partial Reflection
Fixed end
Rope
Free end
Partial reflection + normal incidence
Partial reflection + oblique incidence
 refraction
Application: Probing the Earth
P wave = longitudinal
S wave = transverse
S wave shadow
 liquid outer core
P wave partial reflection
 solid inner core
Explosive thumps
 oil / gas deposits
14.7. Standing Waves
Superposition of right- travelling & reflected waves:
y  x, t   A cos  k x   t   B cos  k x   t 
y  0, t   0  B =  A
1

1

cos   cos   2 A sin       sin     
2

2


y  x, t   2 A sin k x sin  t
String with both ends fixed:
sin k L  0 
2

L  n
standing wave
Ln

2
n  1, 2,3,
Allowed waves = modes or harmonics
n = mode number
n = 1  fundamental mode
n > 1  overtones
Standing Waves
y = 0  node
y = max  antinode
y  x, t   A cos  k x   t   B cos  k x   t 
B  A
1 end fixed  node,
dy
dx
1 end free  antinode.
0
xL
kAsin  kL   t   kAsin  kL   t   0
cos k L sin  t  0
cos k L  0

L   2n  1

4
2

L   2n  1

2
n  1, 2,3,
Standing Waves
Standing Wave Resonance
f 
v

v = const 
fundamental mode ~ lowest freq
overtones ~ multiples of fund. freq
Skyscraper ~ string with 1 free end & 1 fixed end.
Tacoma bridge: resonance of torsional standing waves.
Other Standing Waves:
• Water waves in confined spaces (waves in lake).
• EM waves in cavity (microwave oven).
• Sound wave in the sun.
• Electrons in atom.
Musical Instruments
Standing waves in wind instruments:
(a) open at one end
L = (2n1)  / 4
(b) open at both ends L = n  / 2
Standing waves on a violin,
imaged using holographic
interference of laser light waves.
Example 14.6. Double Bassoon
Double bassoon is the lowest pitched instrument in most orchestra.
It’s “folded” to achieve an effective open-ended column of 5.5 m long.
What is the fundamental freq, assuming sound speed is 343 m/s.
/2
f 
v


343 m / s
2   5.5 m 
 31 Hz
~ B0
GOT IT? 14.5.
A string 1 m long is clamped tight at one end & free to slide up & down at the other.
Which of the following are possible wavelengths for standing waves on it:
4/5 m, 1 m, 4/3 m, 3/2 m, 2 m, 3 m, 4 m, 5 m, 6 m, 7 m, 8 m ?
L   2n  1

4
14.8. The Doppler Effect & Shock Waves
Point source at rest in medium radiates uniformly in all directions.
When source moves, wave crests bunch up in the direction of motion (   ).
Wave speed v is a property of the medium & hence independent of source motion.
f 
v


Approaching source:

f 
Doppler effect
t=0
uT
T = period of wave
u = speed of source
t=T
2 uT = uT
t = 2T
.
recede

 u
  1  
v
 v
u
   u T   1  
 v
approach    u T    u
f approach 
f recede 
v
approach

f
1 u / v
f
1 u / v
Moving Source
Application of the Doppler effect:
• Ultrasound: measures blood flow & fetal heartbeat.
• High freq radio wave: speeding detector.
• Starlight: stellar motion.
• Light from galaxies: expanding universe.
Example 14.7. Wrong Note
A car speeds down the highway with its stereo blasting.
An observer with perfect pitch stands by the roadside, & as the car approach,
notices that a musical note that should be G ( f = 392 Hz ) sounds like A ( 440 Hz ).
How fast is the car moving?
f app 
f
1
u
v

 392 Hz 
f 
u  v 1 
   343 m / s  1 


440
Hz
f


app 

 37.4 m / s
 134 km / h
Moving Observers
An observer moving towards a point source at rest in medium sees a faster moving wave.
Since  is unchanged, observed f increases.
 u
ftoward  f 1  
 v
f away
For u/v << 1:
Prob. 76
 u
 f 1  
 v
f app 
f
1
u
v
 u
 f 1    ftoward
 v
Waves from a stationary source that reflect from a moving object undergo 2 Doppler effects.
1. A f toward shift at the object.
2. A f approach shift when received at source.
Doppler Effect for Light
Doppler shift for EM waves is the same whether the source or the observer moves.


u
app   1  
c

correct to 1st order in u/c
 u
f app   1  
 c
Shock Waves


u
app   1  
v


app  0
if
u v
Shock wave: u > v
Mach number = u / v
Mach angle = sin1(v/u)
Source,
1 period ago
Shock wave front
Moving Source
E.g.,
Bow wave of boat.
Sonic booms.
Solar wind at ionosphere