This document discusses various properties and behaviors of waves. It begins by defining key wave properties like amplitude, wavelength, period, and frequency. It then distinguishes between longitudinal and transverse waves, and describes how wave speed depends only on the medium. Later sections cover topics like interference, reflection, refraction, standing waves, and the Doppler effect. The document also introduces concepts such as the wave equation, dispersion, beats, and shock waves formed when the source speed exceeds the wave speed.

1. Short Version : 14. Wave Motion

2. Wave Properties
Wave amplitude
Waveform
•Pulse
•Continuous wave
•Wave train
Periodicity in space :
Wavelength 
Wave number k = 2/
Periodicity in time :
Period T
Frequency  = 2/T

3. Longitudinal & Transverse Waves
Longitudinal waves
Transverse waves
Water waves
Longitudinal
Transverse
mixed
1-D Vibration
Water Waves

4. 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.
v
T

 f


Wave speed

5. 14.2. Wave Math
At t = 0,    
,0
y 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 cos
y x A kx

2
k


 = wave number
SHW moving to the right :
   
, cos
y x t A kx t

 
2
T

 
k x t
 
  = phase
v
T k
 
  = wave speed
 
k x v t
 
pk @ x = 0 pk @ x = v t
Waves

6. 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 )
• …
   
, cos
y x t A k x t

  v
k



( towards  x )

7. 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
ˆ
m v
m
R
 
a y
Tension force F is cancelled out in the x direction:
2 sin
y
F F 
  2F
  ( small segment )

2
2
m v
F
R
 
2
2 R v
R
 

F
v


2
F v



8. Wave Power
SHO :
Segment of length x at fixed x : 2 2
1
2
E x A
 
 
2 2
1
2
x
P A
t
 



2 2
1
2
v A
 

v = phase velocity of wave
2 2
1
2
E m A



9. 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 :
2
4
P
I
r



10. 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

11. Sound & the Human Ear
Audible freq:
20 Hz ~ 20 kHz
Bats: 100 kHz
Ultrasound: 10 MHz
db = 0 :
Hearing Threshold
@ 1k Hz

12. Decibels
Sound intensity level :
10
0
10 log
I
I

 
  
 
12 2
0 10 /
I W m

  Threshold of hearing at 1 kHz.
[  ] = decibel (dB)
/10
0 10
I I 

2
2 1 10
1
10 log
I
I
 
 
   
 
 
2 1 / 10
2
1
10
I
I
 


2 1
10
I I

2 1 10 dB
 
  
3/10
2 1
10
I I

2 1 3 dB
 
   1
2 I

Nonlinear behavior: Above 40dB, the ear percieves  = 10 dB as a doubling of loudness.

13. 14.5. Interference
constructive interference
destructive interference
Principle of superposition: tot = 1 + 2 .
Interference

14. Fourier Analysis
Fourier analysis:
Periodic wave = sum of SHWs.
E note from
electric guitar
 
0
1
sin
2 1
n
square wave A n t
n






Fourier Series

15. Dispersion
Non-dispersive medium
Dispersive medium
Dispersion:
wave speed is wavelength (or freq) dependent
Surface wave on deep water:
2
g
v



 long wavelength waves reaches shore 1st.
Dispersion of square wave pulses determines max
length of wires or optical fibres in computer networks.
Dispersion

16. Beats
Beats: interference between 2
waves of nearly equal freq.
  1 2
cos cos
y t A t A t
 
 
   
1 2 1 2
1 1
2 cos cos
2 2
A 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).
Constructive
Destructive
Beats

17. Interference in 2-D
Water waves from two sources with separation  
Nodal lines:
amplitude  0
path difference = ½ n 
Destructive Constructive
Interference

18. 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

19. Partial reflection + oblique incidence
 refraction
Partial reflection + normal incidence

20. 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

21. 14.7. Standing Waves
String with both ends fixed:
2
L n


     
, cos cos
y x t A k x t B k x t
 
   
Superposition of right- travelling & reflected waves:
 
, 2 sin sin
y x t A kx t


   
1 1
cos cos 2 sin sin
2 2
A
     
   
    
   
   
 standing wave
sin 0
kL  
1,2,3,
n 
Allowed waves = modes or harmonics
n = mode number
n = 1  fundamental mode
n > 1  overtones
y = 0  node y = max  antinode
2
L n




 
0, 0
y t   B =  A
Standing Waves

22. 1 end fixed  node,
1 end free  antinode.
 
2 1
4
L n

 
cos 0
kL  
1,2,3,
n 
 
2
2 1
2
L n
 

 
     
, cos cos
y x t A k x t B k x t
 
   
0
x L
dy
dx 

B A
 
   
sin sin 0
kA kL t kA kL t
 
    
cos sin 0
kL t
 
Standing Waves

23. 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.
v
f

    f  Doppler effect
Approaching source:

24. .
t = T
u T
t = 2T 2 uT = uT
t = 0
approach u T
 
 
u = speed of source
u
v


  1
u
v

 
 
 
 
recede u T
 
  1
u
v

 
 
 
 
1 /
approach
approach
v f
f
u v

 

1 /
recede
f
f
u v


T = period of wave
Moving Source

25. .
t = T
u T
t = 2T 2 uT = uT
t = 0
approach u T
 
 
u = speed of source
u
v


  1
u
v

 
 
 
 
recede u T
 
  1
u
v

 
 
 
  1 /
recede
f
f
u v


T = period of wave
Moving Source
1 /
approach
approach
v f
f
u v

 


26. Moving Observers
An observer moving towards a point source at rest in medium sees a faster moving wave.
Since  is unchanged, observed f increases.
1
toward
u
f f
v
 
 
 
 
1
away
u
f f
v
 
 
 
 
Prob. 76
For u/v << 1:
1
app
f
f
u
v


1
u
f
v




 
 
toward
f

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.

27. Doppler Effect for Light
Doppler shift for EM waves is the same whether the source or the observer moves.
1
app
u
c
 
 
 
 
 
correct to 1st order in u/c
1
app
u
f
c

 
 
 
 

28. Shock Waves
1
app
u
v
 
 
 
 
 
 0
app
  if u v

Shock wave: u > v
Mach number = u / v
Mach angle = sin1(v/u)
E.g.,
Bow wave of boat.
Sonic booms.
Solar wind at ionosphere
Shock wave front
Source,
1 period ago
Moving Source