Wave Motion
Wave Types
• Longitudinal
– Motion parallel to
energy transport
• Transverse
– Motion perpendicular
to energy transport
Properties of Waves
• Wavelength
– l, distance between crests
• Frequency
– f, # oscillations per second
• Speed
1
f 
T
– How quickly the disturbance moves
A
v l f
l
Equilibrium
CDR Radio
• What is the wavelength of CDR Radio?
For FM stations the call numbers
is the frequency of the station in megahertz
f = 90.3×106 Hz
Velocity for radio waves is the same
as the speed of light.
v = 3.0×108 m/s
v l f
Time Dependence
• A moving function has both time and
position dependence.
y = f(x-v·t) (travel to the right)
y = f(x+v·t) (travel to the left)
 1, 0  x  0.5
• Ex. Haar Wavelet f ( x)  
T = 2s
-2 m
-1 m
 1,
T = 0s
0m
1m
0.5  x  1
2m
Wave Number
• Time dependent wave
• Wave number k 
y  A cos 2l x  v  t 
2
l
2
• k is to l as w is to T w 
T
• Another form y  A cosk  x  w  t   
Waves on a String
• Wave velocity
v
T

T = Tension
 = mass/length
• Ex. What is the velocity of a
wave pulse on a 1 mm diameter
Copper wire w/ tension of 230 N?
(rcu = 8.92 × 103 kg/m3)
Sound Waves
• Compression (High Pressure)
• Rarefaction (Low Pressure)
Comp.
Equil.
Rare.
• Frequency Ranges
– Infrasonic < 20 Hz
– Audible 20 – 20k Hz
– Ultrasonic > 20k Hz
Speed of Sound
• Air
v
B
r
v  (331m / s)
T
273K
– Bulk modulus (B) - How easy it is to compress
a volume of air.
– At 20°C and sea level v = 343 m/s (air)
• Solids v 
Y
r
– Young’s modulus (Y) - How easy it is to
compress a solid.
Sound Level
• Intensity - Power transmitted per unit area.
power
P
I

2
area
4r
(Spherical Wave)
• Intensity Level – Perceived intensity of sound.
– Measured in decibels (dB)
– 10 times the intensity is perceived as plus 10 dB
– 2 times the intensity is perceived as plus 3 dB
I 
  10 log  
 I0 
• Threshold of hearing, I0 = 1012 W/m2
Fireworks
• 100 m away from an explosion the intensity
level is 120 dB. What is the intensity level
at 500 m?
The Ear
• Mechanical energy is converted
into a neural signal in the cochea.
– Nerve cells are triggered by the
displacement of the basilar
membrane.
Superposition
Principle
• When 2 or more waves are
present, the resulting wave is an
algebraic sum of all the waves.
yT  y1  y2
• Interference
Constructive - Waves
in phase
Destructive - Waves
180° out of phase
A
2A
-A
A
+
-A
-2A
A
2A
-A
A
+
-A
-2A
Adding Waves
At t = 0s, the crests of two waves are located at the
origin. If the first wave has an amplitude of 5 cm
and wavelength of 2 m and the second wave has
an amplitude of 2 cm and a wavelength of 0.5 m,
what does the resulting wave look like?
Amplitude (cm)
Superposition
10
5
0
-5
-10
0
1
2
Position (m)
3
4
Beats
• Periodic variation in intensity with two
waves close in frequency.
y2  A cos2f 2t 
y1  A cos2f1t 
From Trig: cos a  cos b  2 cos a 2 b cos a 2 b 

Therefore, y  2 A cos 2
f1  f 2
2
 
t cos 2
f1  f 2
2
t

Modulation
Oscillate
Envelope Average freq.
Beat Frequency
• One speaker is transmitting a 10 Hz signal
and a second is transmitting a 11 Hz signal.
What beat frequency is experienced?
f b  f1  f 2
f b  1Hz
Amplitude (cm)
f b  10 Hz  11Hz
Beats
15
10
5
0
-5
-10
-15
0
0.2 1
0.4
2
0.6
Position
Position (m)
(m)
30.8
14
Interference
• Remember the Superposition Principle
Constructive
r2  r1  nl (n  0,1,2,)
Destructive
r2  r1  n  12 l (n  0,1,2,)
r1
Source
#1
#2
r2
Effect
Earthquakes
• S waves – Transverse
– Shear
• P waves – Longitudinal
– Compression
Northridge, CA 1994
Standing Waves
• Two identical waves traveling in opposite
directions.
y1  A sin kx  wt 
y2  A sin kx  wt 
yT  y1  y2  2 A sin kxcoswt 
• Antinodes - Max Amplitude
nl
x
n  1, 3, 5
4
• Nodes - Zero Amplitude
nl
x
n  0, 2, 4
4
Harmonic Series
• Have a node at each
end of the string.
• Possible wavelengths
2L
ln 
n
• Frequency
f 
v
l
L
1st harmonic
n=1
fundamental
2nd harmonic
n=2
1st overtone
3rd harmonic
n=3
2nd overtone
Waves on a String
• General Form for normal modes
– Nth harmonic
2L
ln 
n
n
fn 
2L
F

(n=1, 2, 3, …)
Ex. A 4.0 g wire is stretched to its full length
of 1.75 meters under a tension of 400 N.
What frequency is heard as it vibrates in the
wind?
Double Open
Ended Pipe
• Air oscillates in and
out of the pipe’s ends.
2L
ln 
n
nv
fn 
2L
Fundamental
2nd Harmonic
(n=1, 2, 3, …)
3rd Harmonic
Air Displacement
Single Open
Ended Pipe
• Air oscillates in and
out of the pipe’s ends.
4L
ln 
n
nv
fn 
4L
Fundamental
1st Overtone
(n=1, 3, 5, …)
2nd Overtone
Air Displacement
Organ Pipe
Ex. A pipe organ needs to
play a low Bb (116.54 Hz).
If a single open ended pipe is
used, how long should the
pipe be?
What is the frequency of the
2nd overtone for this pipe?
Doppler Effect
• If a source is stationary wave
are emitted radially outward.
• Wavelength in direction of
motion is compressed.
• Therefore, frequency heard is
f ' f
v  vO 
v  vS 
(vS and vO considered positive
when approaching each other)
Doppler Effect
• While standing at
a corner, the siren
of the police car
goes from 530 Hz
to 470 Hz as it
passes by. How
fast is the car
traveling?
Stationary Source
Moving Source
Shock Waves
• Supersonic speeds
– Mach Number
vs
Mach # 
v
– Angle of shock wave
v
sin  
vs
v – velocity of sound
vs – velocity of the supersonic object
v

vs