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Wave Motion
Wave Types
• Longitudinal– Motion parallel to
energy transport
• Transverse– Motion perpendicular
to energy transport
Properties of Waves
• Wavelength– , distance between crests
• Frequency– f, # oscillations per second
• Speed– How quickly the disturbance moves
EquilibriumA
fv
Tf
1
CDR Radio• What is the wavelength of CDR Radio?
For FM stations the call numbersis the frequency of the station in megahertz
f = 90.3×106 Hz
Velocity for radio waves is the sameas the speed of light.
v = 3.0×108 m/s
fv
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)
• Ex. Haar Wavelet
15.0,1
5.00,1)(
x
xxf
0 m 1 m 2 m-1 m-2 m
T = 2s T = 0s
Wave Number• Time dependent wave
• Wave number
• k is to as is to T
• Another form
2
k
tvxAy 2cos
T
2
txkAy cos
• Wave velocity
• Ex. What is the velocity of a wave pulse on a 1 mm diameter Copper wire w/ tension of 230 N?
(cu = 8.92 × 103 kg/m3)
Waves on a String
T
v T = Tension = mass/length
Sound Waves• Compression (High Pressure)• Rarefaction (Low Pressure)
• Frequency Ranges– Infrasonic < 20 Hz
– Audible 20 – 20k Hz
– Ultrasonic > 20k Hz
Equil.
Comp.
Rare.
Speed of Sound
• Air– 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– Young’s modulus (Y) - How easy it is to
compress a solid.
Bv K
Tsmv 273)/331(
Yv
Sound Level• Intensity - Power transmitted per unit area.
• 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
• Threshold of hearing, I0 = 1012 W/m2
0
log10I
I
24 r
P
area
powerI
(Spherical Wave)
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.
• Interference
Constructive - Waves in phase
+A
-A
A
-A2A
-2A
+A
-A
A
-A2A
-2A
Destructive - Waves 180° out of phase
21 yyyT
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?
Superposition
-10
-5
0
5
10
0 1 2 3 4
Position (m)
Am
pli
tud
e (c
m)
Beats• Periodic variation in intensity with two
waves close in frequency.
tfAy 11 2cos tfAy 22 2cos
From Trig: 22 coscos2coscos bababa
Therefore, ttAy ffff22
2121 2cos2cos2
ModulationEnvelope
OscillateAverage 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?
21 fffb HzHzfb 1110
Hzfb 1
Beats
-15-10-5051015
0 1 2 3 4
Position (m)
Am
pli
tud
e (c
m)
Beats
-15-10-5051015
0 0.2 0.4 0.6 0.8 1
Position (m)
Am
pli
tud
e (c
m)
Interference• Remember the Superposition Principle
Constructive
Destructive
),2,1,0( 12 nnrr
),2,1,0( 21
12 nnrr
#1
#2
r1
r2Source
Effect
Earthquakes
• S waves – Transverse– Shear
• P waves – Longitudinal– Compression
Northridge, CA 1994
Standing Waves• Two identical waves traveling in opposite
directions. tkxAy sin1 tkxAy sin2
tkxAyyyT cossin221
• Antinodes - Max Amplitude5 3, 1,n
4
nx
• Nodes - Zero Amplitude4 2, 0,n
4
nx
Harmonic Series
• Have a node at each end of the string.
• Possible wavelengths
• Frequency
n
Ln
2
v
f
L
n=1
n=2
n=3
fundamental
1st overtone
2nd overtone
1st harmonic
2nd harmonic
3rd harmonic
Waves on a String• General Form for normal modes
– Nth harmonic
n
Ln
2
F
L
nfn 2
(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 OpenEnded Pipe
• Air oscillates in and out of the pipe’s ends.
n
Ln
2
L
nvfn 2
(n=1, 2, 3, …)
Air Displacement
2nd Harmonic
3rd Harmonic
Fundamental
Single Open Ended Pipe
• Air oscillates in and out of the pipe’s ends.
n
Ln
4
L
nvfn 4
(n=1, 3, 5, …)
Air Displacement
1st Overtone
2nd Overtone
Fundamental
Organ PipeEx. 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
S
O
vv
vvff
' (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
– Angle of shock wave
sv
vsin
v – velocity of sound
vs – velocity of the supersonic object
v
vMach s#
vs
v