Short Version :
14. Wave Motion
Wave Properties
Wave amplitude
Waveform
•Pulse
•Continuous wave
•Wave train
Periodicity in space :
Wavelength
Wave number

Periodicity in time :
Period
Frequency
T
k = 2/
 = 2/T
Longitudinal & Transverse Waves
Longitudinal waves
Transverse waves
Transverse
1-D Vibration
Longitudinal
Water Waves
Water waves
mixed
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
pk @ x = 0
pk @ x = v t
14.2. Wave Math
At t = 0,
y  x,0  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 :
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 

2
T
  k x   t = phase  k  x  v t 
v
 

T k
= wave speed
Waves
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 
( towards  x )
v = velocity of wave.
y  x, t   A cos  k x   t 

E.g.,
•water wave ( y = wave height )
•sound wave ( y = pressure )
•…
v

k
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  2 F sin   2F

( small segment )
m v2
2 R  v 2
2 F 

R
R
 = mass per unit length [ kg/m ]
F   v2
v
F

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
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 
I 0  1012 W / m 2
[  ] = decibel (dB)
I  I 0 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.
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.
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
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
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
kA sin  k L   t   kA sin  k L   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
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
Moving Source

f
1 u / v
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
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
f toward  f 1  
 v
For u/v << 1:
Prob. 76
f away
 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