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.
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
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 = sin1(v/u)
E.g.,
Bow wave of boat.
Sonic booms.
Solar wind at ionosphere
Shock wave front
Source,
1 period ago
Moving Source
