PHYS 1441 – Section 501
Lecture #18
Wednesday, Aug. 4, 2004
Dr. Jaehoon Yu
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•
•
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Equation of Motion of a SHO
Waves
Speed of Waves
Types of Waves
Energy transported by waves
Reflection and Transmission
Superposition principle
Today’s Homework is HW#9, due 6pm, Thursday, Aug. 12!
Wednesday, Aug. 4, 2004
PHYS 1441-501, Summer 2004
Dr. Jaehoon Yu
1
Announcements
• Quiz #3 Results:
– Class Average: 56.4
• Quiz 1: 49.1
• Quiz 2: 54.4
– Top score: 90
• Exam 6-8pm, Wednesday, Aug. 11, in SH125
– Covers: Ch. 8.6 – Ch. 11
– Please do not miss the exam
• Monday’s review:
– Send me the list of 5 hardest problems in your practice test
by midnight Sunday
– I will put together a prioritized list of problems
– You will go through them in the class with extra credit
Wednesday, Aug. 4, 2004
PHYS 1441-501, Summer 2004
Dr. Jaehoon Yu
2
The SHM Equation of Motion
The object is moving on a circle with a constant angular speed w
How is x, its position at any given time expressed with the known quantities?
v0
x  A cos since    t and   2 f
x  A cos t  A cos 2 ft
k
How about its velocity v at any given time?
v0 
m
A
v  v0 sin   v0 sin  t   v0 sin  2 ft 
How about its acceleration a at any given time?
From Newton’s
Wednesday, Aug. 4, 2004
2nd
law a 
F  kx
 kA 




 cos  2 ft   a0 cos  2 ft 
m
m
 m
kA
a0 
m
PHYS 1441-501, Summer 2004
Dr. Jaehoon Yu
3
Sinusoidal Behavior of SHM
x  A cos  2 ft 
v  v0 sin  2 ft 
a  a0 cos  2 ft 
Wednesday, Aug. 4, 2004
PHYS 1441-501, Summer 2004
Dr. Jaehoon Yu
4
The Simple Pendulum
A simple pendulum also performs periodic motion.
The net force exerted on the bob is
 Fr  T  mg cos A  0
F
t
 mg sin  A  mg
Since the arc length, x, is
x  L
mg
F   Ft  
x
L
Satisfies conditions for simple harmonic motion!
It’s almost like Hooke’s law with. k  mg
L
The period for this motion is
T  2
m
 2
k
mL
 2
mg
L
g
The period only depends on the length of the string and the gravitational acceleration
Wednesday, Aug. 4, 2004
PHYS 1441-501, Summer 2004
Dr. Jaehoon Yu
5
Example 11-8
Grandfather clock. (a) Estimate the length of the pendulum in a grandfather clock that
ticks once per second.
Since the period of a simple
pendulum motion is
T  2
The length of the pendulum
in terms of T is
T 2g
L
4 2
Thus the length of the
pendulum when T=1s is
L
g
T 2 g 1 9.8
L

 0.25m
2
2
4
4
(b) What would be the period of the clock with a 1m long pendulum?
T  2
Wednesday, Aug. 4, 2004
L
1.0
 2
 2.0s
g
9.8
PHYS 1441-501, Summer 2004
Dr. Jaehoon Yu
6
Damped Oscillation
More realistic oscillation where an oscillating object loses its mechanical
energy in time by a retarding force such as friction or air resistance.
How do you think the
motion would look?
Amplitude gets smaller as time goes on since its energy is spent.
Types of damping
A: Overdamped
B: Critically damped
C: Underdamped
Wednesday, Aug. 4, 2004
PHYS 1441-501, Summer 2004
Dr. Jaehoon Yu
7
Forced Oscillation; Resonance
When a vibrating system is set into motion, it oscillates with its natural
frequency f0.
1
k
f0 
2
m
However a system may have an external force applied to it that has
its own particular frequency (f), causing forced vibration.
For a forced vibration, the amplitude of vibration is found to be dependent
on the different between f and f0. and is maximum when f=f0.
A: light damping
B: Heavy damping
The amplitude can be large when
f=f0, as long as damping is small.
This is called resonance. The natural
frequency f0 is also called resonant frequency.
Wednesday, Aug. 4, 2004
PHYS 1441-501, Summer 2004
Dr. Jaehoon Yu
8
Wave Motions
Waves do not move medium instead carry energy
from one place to another
Two forms of waves
– Pulse
– Continuous or
periodic wave
Mechanical Waves
Wednesday, Aug. 4, 2004
PHYS 1441-501, Summer 2004
Dr. Jaehoon Yu
9
Characterization of Waves
• Waves can be characterized by
– Amplitude: Maximum height of a crest or the depth of a trough
– Wave length: Distance between two successive crests or any
two identical points on the wave
– Period: The time elapsed by two successive crests passing by
the same point in space.
– Frequency: Number of crests that pass the same point in space
in a unit time
• Wave velocity: The velocity at which any part of the
wave moves
Wednesday, Aug. 4, 2004
PHYS 1441-501, Summer 2004
Dr. Jaehoon Yu
v

T
f
10
Waves vs Particle Velocity
Is the velocity of a wave moving along a cord the
same as the velocity of a particle of the cord?
No. The two velocities are
different both in magnitude
and direction. The wave
on the rope moves to the
right but each piece of the
rope only vibrates up and
down.
Wednesday, Aug. 4, 2004
PHYS 1441-501, Summer 2004
Dr. Jaehoon Yu
11
Speed of Transverse Waves on Strings
How do we determine the speed of a transverse pulse traveling on a string?
If a string under tension is pulled sideways and released, the tension is responsible for
accelerating a particular segment of the string back to the equilibrium position.
The acceleration of the
So what happens when the tension increases?
particular segment increases
Which means?
The speed of the wave increases.
Now what happens when the mass per unit length of the string increases?
For the given tension, acceleration decreases, so the wave speed decreases.
Which law is this hypothesis based on?
Newton’s second law of motion
Based on the hypothesis we have laid out
v
above, we can construct a hypothetical
formula for the speed of wave
Is the above expression dimensionally sound?
Wednesday, Aug. 4, 2004
T


T
m L
T: Tension on the string
: Unit mass per length
T=kg m/s2. =kg/m
(T/)1/2=[m2/s2]1/2=m/s
PHYS 1441-501, Summer 2004
Dr. Jaehoon Yu
12
Example 11 – 10
Wave on a wire. A wave whose wavelength is 0.30m is traveling down a d 300-m long
wire whose total mass is 15 kg. If the wire is under a tension of 1000N, what is the
velocity and frequency of the wave?
The speed of the wave is
v
1000

 140m / s
15 / 300

T
The frequency of the wave is
v
140m / s
f  
 470 Hz

0.30m
Wednesday, Aug. 4, 2004
PHYS 1441-501, Summer 2004
Dr. Jaehoon Yu
13
Example for Traveling Wave
A uniform cord has a mass of 0.300kg and a length of 6.00m. The cord passes over a
pulley and supports a 2.00kg object. Find the speed of a pulse traveling along this cord.
5.00m
1.00m
M=2.00kg
Since the speed of wave on a string with line v  T

density  and under the tension T is
0.300kg
The line density  is  
 5.00 10  2 kg / m
6.00m
The tension on the string is
provided by the weight of the
object. Therefore
T  Mg  2.00  9.80  19.6kg  m / s 2
Thus the speed of the wave is
v
Wednesday, Aug. 4, 2004
T


19.6
 19.8m / s
2
5.00 10
PHYS 1441-501, Summer 2004
Dr. Jaehoon Yu
14
Type of Waves
• Two types of waves
– Transverse Wave : A wave whose media particles move
perpendicular to the direction of the wave
– Longitudinal wave : A wave whose media particles move
along the direction of the wave
• Speed of a longitudinal wave
For solid v 
E

Wednesday, Aug. 4, 2004
E:Young’s modulus
: density of solid
v
Elastic Force Factor
inertia factor
liquid/gas v 
PHYS 1441-501, Summer 2004
Dr. Jaehoon Yu
B

E: Bulk Modulus
: density
15
Example 11 – 11
Sound velocity in a steel rail. You can often hear a distant train approaching by putting
your ear to the track. How long does it take for the wave to travel down the steel track if
the train is 1.0km away?
The speed of the wave is
v
E


2.0 10 N / m
3
 5.110 m / s
3
3
7.8 10 kg / m
11
2
The time it takes for the wave to travel is
1.0 10 m
l
t  
 0.20s
3
v
5.110 m / s
3
Wednesday, Aug. 4, 2004
PHYS 1441-501, Summer 2004
Dr. Jaehoon Yu
16
Earthquake Waves
•
Both transverse and longitudinal waves are produced when an
earthquake occurs
–
–
•
Using the fact that only longitudinal waves goes through the core of
the Earth, we can conclude that the core of the Earth is liquid
–
•
S (shear) waves: Transverse waves that travel through the body of the Earth
P (pressure) waves: Longitudinal waves
While in solid the atoms can vibrate in any direction, they can only vibrate
along the longitudinal direction in liquid due to lack of restoring force in
transverse direction.
Surface waves: Waves that travel through the boundary of two
materials (Water wave is an example) This inflicts most damage.
Wednesday, Aug. 4, 2004
PHYS 1441-501, Summer 2004
Dr. Jaehoon Yu
17
The Richter Earthquake Scale
• The magnitude of an earthquake is a measure of the amount of
energy released based on the amplitude of seismic waves.
• The Richter scale is logarithmic, that is an increase of 1
magnitude unit represents a factor of ten times in amplitude.
However, in terms of energy release, a magnitude 6 earthquake
is about 31 times greater than a magnitude 5.
–
–
–
–
M=1 to 3: Recorded on local seismographs, but generally not felt
M=3 to 4: Often felt, no damage
M=5: Felt widely, slight damage near epicenter
M=6: Damage to poorly constructed buildings and other structures within
10's km
– M=7: "Major" earthquake, causes serious damage up to ~100 km (recent
Taiwan, Turkey, Kobe, Japan, California and Chile earthquakes).
– M=8: "Great" earthquake, great destruction, loss of life over several 100
km (1906 San Francisco, 1949 Queen Charlotte Islands) .
– M=9: Rare great earthquake, major damage over a large region over 1000
km (Chile 1960, Alaska 1964, and west coast of British Columbia,
Washington, Oregon, 1700).
Wednesday, Aug. 4, 2004
PHYS 1441-501, Summer 2004
Dr. Jaehoon Yu
18
Energy Transported by Waves
Waves transport energy from one place to another.
As waves travel through a medium, the energy is transferred as vibrational energy
from particle to particle of the medium.
1 2
For a sinusoidal wave of frequency f, the particles move in
E  kA
SHM as a wave passes. Thus each particle has an energy
2
Energy transported by a wave is proportional to the square of the amplitude.
energy / time power
Intensity of wave is defined as the power transported across

I 
unit area perpendicular to the direction of energy flow.
area
area
Since E is
2
2
proportional to A .
I1
I2
power
P
For isotropic medium, the I 

wave propagates radially
area
4 r 2
Ratio of intensities at I 2 P 4 r22  r  2
A2 r1
1


Amplitude

2 
two different radii is
A1 r2
I1 P 4 r1  r2 
IA
Wednesday, Aug. 4, 2004
PHYS 1441-501, Summer 2004
Dr. Jaehoon Yu
19
Example 11 – 12
Earthquake intensity. If the intensity of an earthquake P wave 100km from the source is
1.0x107W/m2, what is the intensity 400km from the source?
Since the intensity decreases as the square of the distance from the source,
I 2  r1 
 
I1  r2 
2
The intensity at 400km can be written in terms of the intensity at 100km
2
2
 r1 
I 2    I1   100km  1.0 107 W / m 2  6.2 105W / m 2


r
400
km
 2


Wednesday, Aug. 4, 2004
PHYS 1441-501, Summer 2004
Dr. Jaehoon Yu
20
Reflection and Transmission
A pulse or a wave undergoes various changes when the medium
it travels changes.
Depending on how rigid the support is, two radically different reflection
patterns can be observed.
1.
2.
The support is rigidly fixed (a): The reflected pulse will be inverted to the original
due to the force exerted on to the string by the support in reaction to the force on
the support due to the pulse on the string.
The support is freely moving (b): The reflected pulse will maintain the original
shape but moving in the reverse direction.
Wednesday, Aug. 4, 2004
PHYS 1441-501, Summer 2004
Dr. Jaehoon Yu
21
2 and 3 dimensional waves and the Law of
Reflection
•
•
•
Wave fronts: The whole width of
wave crests
Ray: A line drawn in the direction
of motion, perpendicular to the
wave fronts.
Plane wave: The waves whose
fronts are nearly straight
The Law of Reflection:
The angle of reflection
is the same as the angle
of incidence.
Wednesday, Aug. 4, 2004
PHYS 1441-501, Summer 2004
Dr. Jaehoon Yu
i=r
22
Transmission Through Different Media
If the boundary is intermediate between the previous two extremes,
part of the pulse reflects, and the other undergoes transmission,
passing through the boundary and propagating in the new medium.
When a wave pulse travels from medium A to B:
1. vA> vB (or A<B), the pulse is inverted upon reflection
2. vA< vB(or A>B), the pulse is not inverted upon reflection
Wednesday, Aug. 4, 2004
PHYS 1441-501, Summer 2004
Dr. Jaehoon Yu
23
Superposition Principle of Waves
If two or more traveling waves are moving through a
medium, the resultant wave function at any point is the
algebraic sum of the wave functions of the individual waves.
Superposition
Principle
The waves that follow this principle are called linear waves which in general have
small amplitudes. The ones that don’t are nonlinear waves with larger amplitudes.
Thus, one can write the
resultant wave function as
Wednesday, Aug. 4, 2004
n
y  y1  y2      yn   yi
i 1
PHYS 1441-501, Summer 2004
Dr. Jaehoon Yu
24
Wave Interferences
Two traveling linear waves can pass through each other without being destroyed or altered.
What do you think will happen to the water
waves when you throw two stones in the pond?
They will pass right through each other.
The shape of wave will
change Interference
Constructive interference: The amplitude increases when the waves meet
What happens to the waves at the point where they meet?
Destructive interference: The amplitude decreases when the waves meet
Wednesday, Aug. 4, 2004
In phase  constructive
PHYS 1441-501, Summer 2004
Out of phase
by /2 Yu
 destructive
Dr. Jaehoon
Out of phase not25by /2
 Partially destructive