PSC 1351

1
PSC 1351

• CHAPTER 5

2
Hookes Law
x
k
F(s)
F(s) - k x But F(s) m a a
-(k/m) x (Basic SHM equation) T (2p)
? (m/k) is the period of vibration f (1/ T)
is the frequency of vibration
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Wave

• A periodic vibration that travels through a
  medium and transports energy
• Type 1 Transverse Vibration direction is
  perpendicular to direction of motion (Waves on a
  string)
• Type 2 Longitudinal Vibration direction is
  parallel to direction of motion (Sound waves)
• Type 3 Complex Combination of 1 and 2 (Water
  waves)

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Wavelength
?
The distance between two nearest points of equal
phase in a wave is called the wavelength and
given the symbol ?. For example, it is the
distance between adjacent crests in a transverse
wave. For a transverse wave v ? (F(t) / µ) ,
where F(t) is the tension in the string and µ is
the mass/unit length of it. For all wave types
v ? f
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Standing Waves on a String
If a string is tied at one end to an oscillator
and at the other end to a load M that passes over
a pulley, and L is the distance between the
oscillator and pulley, then, a standing wave will
be set up in the string, provided that ? (2L
/ n) or f n (v / 2L), where n is an integer
having a value 1, 2, 3, ..,etc. These are the
resonance conditions. By measuring ? and knowing
the oscillator frequency, f, we may find the wave
velocity v ? f. Also, v ? (F(t) / µ , where
F(t) M g is the tension and µ is the mass/unit
length of the string.
Pulley
Antinode
Node
L
Oscillator
Load M
(?/2)
This is case n 4
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Sound Waves

• They are longitudinal waves vibration direction
  of air molecules is parallel to direction of
  travel of the wave
• Their velocity does not depend on frequency
• Velocity does vary with temperature
• v 331.6 (0.6) T (m/sec), where T is the
  temperature in ºC.
• v ? f (as is the case for all waves)

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Standing Waves in a Gas(1)
Case 1 Tube is closed at one end, open at the
other. If the air column has length L, then the
resonance conditions in wavelength and frequency,
respectively, are ? (4L / n) and f n(v
/(4L)), where n is an odd integer having values
1, 3, 5, 7,,etc. There must always be an
antinode at the open end.
Antinode at open end
?/2
Case shown is n 11
Node
L
Antinode
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Standing Waves in a Gas (2)
Case 2 Tube is open at both ends. If the tube
has length L, then the resonance conditions in
wavelength and frequency, respectively, are ?
(2L / n) and f n(v / (2L)), where n is an
integer with values n 1, 2, 3, 4,.., etc.
Note that these equations are the same as for a
string tied at both ends. There must be antinodes
at both ends.
L
Antinode
Antinode at open end
Antinode at open end
?/2
Case shown is n 4
Node
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Intensity
The energy transported by a wave per unit area,
per unit time is called the intensity of the
wave. I (?W)/(?A ?t) Since Power is P
(?W/?t) we may rewrite I as I (P/?A) in
units of (W/m²) (Only metric units are used)
For spherical wavefronts I (P/4pR²) , which
indicates that intensity obeys an inverse
square law.
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Example of Intensity
Let the observer be a distance R 4 meters away
from the speaker. Let the speaker have a
constant power of 4 W. The intensity at the
observation point will be I P/(4pR²)
(4)/(4p(4²)) I 0.0199 (W/m²) which is actually
a very loud sound.
Observer
R
Speaker
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Doppler Effect

• If there is any relative motion between a wave
  source and an observer, then the frequency of the
  waves as measured by the observer will be higher
  than the emitted waves if the source and observer
  are approaching each other, and lower if they are
  moving away from each other.
• Example the observed red shift of light arriving
  from very far away galaxies.

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Shock Waves
If a wave source travels through a medium at a
speed greater than the speed of the waves in
that medium, then a shock wave will be
produced. Example 1 A boat traveling on a
lake produces both a bow shock wave and a
stern shock wave
Boat velocity exceeds wave velocity
Bow shock wave
Example 2 An airplane flying at a speed
greater than sound produces a shock wave we
call a sonic boom.
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Beats

• If 2 sources of approximately equal frequencies
  are excited simultaneously, a periodic variation
  in sound intensity will be observed. This sound
  intensity variation is called beats.
• Example Two nearby window air conditioners
  operating at slightly different frequencies.
• The beat frequency equals the difference between
  the 2 frequencies.
• F(b) f(1) f(2)