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14. Wave Motion
1. Waves & their Properties
2. Wave Math
3. Waves on a String
4. Sound Waves
5. Interference
6. Reflection & Refraction
7. Standing Waves
8. The Doppler Effect & Shock Waves
Ocean waves travel thousands of kilometers
across the open sea before breaking on shore.
How much water moves with the waves?
Other kinds of waves:
• Sound
• Light
• Radio
• Ultrasound
• Microwave
• Earthquake / Tsunami
Wave:
Traveling disturbance that
transport energy but not matter.
None
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 wavesTransverse waves
Water waves
LongitudinalTransverse
mixed
1-D Vibration
Water Waves
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.
vT
fWave speed
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 )
14.2. Wave Math
At t = 0, ,0y x f x
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 : ,y x t f x v t
For a SHW (sinusoidal):
,0 cosy x A k x2
k
= wave number
SHW moving to the right :
, cosy x t A k x t 2
T
k x t = phase
vT k
= wave speed
k x v t
pk @ x = 0 pk @ x = v t
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.
, cosy x t A k x t
13.6
2A m 1.8m
14 m 2 1.5T s 3.0 s
120.449k m
12
2.09 sT
4.7 /v m sT
, 1.8 cos 0.449 2.09y x t x t
GOT IT? 14.2.
Figure shows two waves propagating with the same speed.
Which has the greater
(a) amplitude, (b) wavelength, (c) period, (d) wave number, (e) frequency ?
U LL U U
v = / T
The Wave Equation
1-D waves in many media can be described by the partial differential equation
,y x t f x v t
2 2
2 2 2
y y
x v t
Wave Equation
whose solutions are of the form
v = velocity of wave.
E.g., •water wave ( y = wave height )•sound wave ( y = pressure )•…
, cosy x t A k x t vk
( towards x )
14.3. Waves on a String
= mass per unit length [ kg/m ]
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:
2
ˆm v
mR
a y
Tension force F is cancelled out in the x direction:
2 sinyF F 2F ( small segment )
2
2m v
FR
22 R v
R
Fv
2F v
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?
m
L L
vt
110 N2
m LF
t
2
5.0 43
1.4
kg m
s
2F v
Wave Power
SHO :
Segment of length x at fixed x : 2 21
2E x A
2 21
2
xP A
t
2 21
2v A
v = phase velocity of wave
2 21
2E m A
Wave Intensity
Wave front = surface of constant phase.
Plane wave : planar wave front.
Spherical wave : spherical wave front.
Intensity = power per unit area direction of propagation [ W / m2 ]
Plane wave : I const
Spherical wave :24
PI
r
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.
24
PI
r 75 40
2 275 40
P P
r r
4040 75
75
Pr r
P 40
1.975
Wm
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. Is the more distant star
(a) twice
(b) 100 times
(c) 10 times
(d) 10 times
as far away.
14.4. Sound Waves
Sound waves = longitudinal mechanical waves through matter.
Speed of sound in air :P
v
P = background pressure.
= mass density.
= 7/5 for air & diatomic gases.
= 5/3 for monatomic gases, e.g.,
He.
P, = max , x = 0
P, = min , x = 0
P, = eqm , |x| = max
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 :
100
10 logI
I
12 20 10 /I W m Threshold of hearing at 1
kHz.
[ ] = decibel (dB)/10
0 10I I
22 1 10
1
10 logI
I
2 1 / 102
1
10I
I
2 110I I2 1 10 dB
3/102 110I I2 1 3 dB 12 I
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 ?
60 75 / 1010
22 1 10
1
10 logI
I
2
101
10 logP
P
24
PI
r
2 1 / 102
1
10P
P 3 / 210 0.0321
30
1
10 10
10 db drop ½ in loudness
15 db drop between ½ & ¼ in loudness
14.5. Interference
constructive interference
destructive interference
Principle of superposition: tot = 1 + 2 .
Interference
Fourier Analysis
Fourier analysis:
Periodic wave = sum of SHWs.
E note from electric guitar
0
1sin
2 1n
square wave A n tn
Fourier Series
Dispersion
Non-dispersive medium
Dispersive medium
Dispersion:wave speed is wavelength (or freq) dependent
Surface wave on deep water:
2
gv
long wavelength waves reaches shore 1st.
Dispersion of square wave pulses determines max
length of wires or optical fibres in computer networks.
Dispersion
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 obey2
gv
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 = 600
1540 /
kmh
km h
Speed of wave =2
g
2250 9.8 /
2
m m s
19.7 /m s 71.0 /km h
Time for wave to reach you = 600
8.4571.0 /
kmh
km h
The storm is 15 8.45 = 6.55 h 6.6 h away.
Beats
Beats: interference between 2 waves of nearly equal freq.
1 2cos cosy t A t A t
1 2 1 2
1 12 cos cos
2 2A t t
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).
ConstructiveDestructive
Interference in 2-D
Water waves from two sources with separation
Nodal lines:amplitude 0
path difference = ½ n
Destructive Constructive
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?
2 275 33 10AP m m m
2AP BP
86.5m
2 275 33 10BP m m m 78.4 m
2 86.5 78.4m 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
Fixed end
Free end
Partial Reflection
A = 0;reflected wave inverted
A = max;reflected wave not inverted
light + heavy ropes
Rope
Partial reflection + oblique incidence
refraction
Partial reflection + normal incidence
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
String with both ends fixed:
2L n
, cos cosy x t A k x t B k x t
Superposition of right- travelling & reflected waves:
, 2 sin siny x t A k x t
1 1cos cos 2 sin sin
2 2A
standing wave
sin 0kL 1,2,3,n
Allowed waves = modes or harmonics
n = mode numbern = 1 fundamental moden > 1 overtones
y = 0 node y = max antinode
2L n
0, 0y t B = A
Standing Waves
1 end fixed node,
1 end free antinode.
2 14
L n
cos 0kL
1,2,3,n
22 1
2L n
, cos cosy x t A k x t B k x t
0x L
dy
dx
B A
sin sin 0kA kL t kA kL t
cos sin 0kL t
Standing Waves
Standing Wave Resonance
vf
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 on a violin, imaged using holographic interference of laser light waves.
Standing waves in wind instruments:
(a)open at one end L = (2n1) / 4
(b) open at both ends L = n / 2
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.
vf
343 /
2 5.5
m s
m
31Hz ~ B0
/ 2
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 ?
2 14
L n
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.
vf
f Doppler effectApproaching source:
.
t = T
u T
t = 2T 2 uT = uT
t = 0
approach u T
u = speed of source
uv
1u
v
recede u T 1u
v 1 /recede
ff
u v
T = period of wave
Moving Source
1 /approachapproach
v ff
u v
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?
392343 / 1
440
Hzm s
Hz
37.4 /m s
1app
ff
uv
1app
fu v
f
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.
1toward
uf f
v
1away
uf f
v
Prob. 76
For u/v << 1:
1app
ff
uv
1u
fv
towardf
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.
1app
u
c
correct to 1st order in u/c
1app
uf
c
Shock Waves
1app
u
v
0app if u v Shock wave: u > v
Mach number = u / v
Mach angle = sin1(v/u)
E.g.,
Bow wave of boat.
Sonic booms.
Solar wind at ionosphere
Shock wave front
Source, 1 period ago
Moving Source