Oscillations (continues)
Vibrating systems
What we studied before?
• Simple harmonic motion (one mode)
• Mass on spring
• Simple pendulum (SHM for small angles)
More complicated systems
• More than one mode of oscillations
• Oscillations that are not harmonic
1
1. Two-mass vibrator
Longitudinal vibrations
m
m
1
fa 
2
m
m
k
m
1
fa 
2
3k
m
Modes (a) and (b) are independent
Transverse vibrations
m
m
(Violin string normally vibrates transversely)
m
m
Two-mass vibrator has 2 longitudinal and 4 transverse modes of vibration
2
2. Systems with many modes
N-mass vibrator has N longitudinal and 2N transverse modes of vibration
3
3. Standing waves on a string
Vibrating string can be thought of as the limit of the mass-spring system
when the number of masses becomes very large
4
4. Vibrations in musical instruments
1.
2.
3.
4.
5.
Strings
Membranes
Bars
Plates (Chladni patterns)
Tuning forks
5. Vibration Spectra (Fourier Analysis)
xn (t )  An cosn t  n   an cosn t   bn sinn t 
xtot (t )   xn (t )
n
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Waves
1. There are many examples of waves:
- Seismic waves
- Ripples on a pond
- Sound
- Electromagnetic wave including light
2. Types of Waves
•
A wave can have a short duration – just a
quick pulse; these are called “wave pulses”
•
A wave can also repeat; these are called
continuous waves
•
y( x, t )  A cos(t  kx)
A transverse wave is a wave in which the oscillations are transverse
to the direction the wave travels
A longitudinal wave is a wave in which the oscillations are along the
direction the wave travels
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3. Wavelength and Speed of Waves
λ - wave length
T - period
f - frequency
v   / T  f
•
•
•
“Speed” of a wave is the rate of movement of the disturbance.
It is not the speed of the individual particles!
Speed is determined by the properties of the medium
Sinusoidal waves
y
y( x, t )  A cos(t  kx)
y
T
λ
t
Angular frequency:
Wave number:
  2 / T  2f
k  2 / 
x
v   /T   / k
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Question 1: A sinusoidal wave has a wavelength of 4.00 m and a period
of 2.00s. What is the speed of the wave?
v   / T  4.00m 2.00s   2.00m s
Question 2: A sinusoidal wave has a wavelength of 4.00 m and an
angular frequency of 3.14 rad/s. What is the speed of the wave?


v  f   2  4.00m 3.14s 1 2  2.00m s
Question 3: If you double the wavelength  of a periodic wave and the
wave speed does not change, what happens to the wave frequency f ?
1) f is unchanged
2) f is halved
3) f is doubled
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4. Speed of waves in strings
v
F
m

l

m  m ass
l  length
F  tension
v  speed
The speed of the wave increases if we increase the tension (F) in the string,
and decreases if we increase the linear mass density (μ) of the string.
Example: A metal string has a mass of 0.020 kg, a length of 40 cm, and is
under 80 N tension.
1. What is the speed of a wave in this metal string?

m 0.020 kg

 0.05 kg m
l
0.40 m
v
F


80N
 40 m s
0.05kg / m
2. What happed if we increase a tension up to 120 N?
v
F


120N
 49 m s
0.05kg / m
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Question 1: A wave pulse can be sent down a rope by jerking sharply on the
free end. If the tension of the rope is increased, how will that affect the speed
of the wave?
1) speed increases
2) speed does not change
3) speed decreases
v F 
Question 2: A wave pulse is sent down a rope of a certain thickness and a
certain tension. A second rope made of the same material is twice as thick,
but is held at the same tension. How will the wave speed in the second rope
compare to that of the first?
1) speed increases
2) speed does not change
3) speed decreases
Thicker rope means more massive rope
Question 3: A length of rope L and mass M hangs from a ceiling. If the bottom
of the rope is jerked sharply, a wave pulse will travel up the rope. As the wave
travels upward, what happens to its speed? Keep in mind that the rope is not
massless.
1) speed increases
2) speed does not change
3) speed decreases
The tension in the rope is not constant in the case of a massive rope! The
tension increases as you move up higher along the rope, because that part of
the rope has to support all of the mass below it! Since the tension increases
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as you go up, so does the wave speed.