Music

1
Music

• Physics 202
• Professor Vogel
• (Professor Carkners notes, ed)
• Lecture 8

2
Intensity of Sound

• The loudness of sound depends on its intensity,
  which is the power the wave delivers per unit
  area
• I P/A
• The units of intensity are W/m2
• The intensity can be expressed as
• I ½rvw2sm2
• Compare to expression for power in a transverse
  wave
• Depends directly on r and v (medium properties)
• Depends on the square of the amplitude and the
  frequency (wave properties)

3
Intensity and Distance

• Consider a source that produces a sound of
  initial power Ps
• As you get further away from the source the
  intensity decreases because the area over which
  the power is distributed increases
• The total area over which the power is
  distributed depends on the distance from the
  source, r
• I P/A Ps/(4pr2)
• Sounds get fainter as you get further away
  because the energy is spread out over a larger
  area
• I falls off as 1/r2 (inverse square law)

4
Inverse Square Law
Source
r
A14pr2 I1 Ps/A1
2r
A24p(2r)2 16pr2 4A1 I2 Ps/A2 ¼ I1
5
The Decibel Scale

• The human ear is sensitive to sounds over a wide
  range of intensities
• To conveniently handle such a large range, a
  logarithmic scale is used known as the decibel
  scale
• b (10 dB) log (I/I0)
• Where b is the sound level (in decibels, dB)
• I0 10-12 W/m2 (at the threshold of human
  hearing)
• log is base 10 log (not natural log, ln)
• There is an increase of 10 dB for every factor of
  10 increase in intensity

6
Sound Levels

• Hearing Threshold
• 0 dB
• Whisper
• 10 dB
• Talking
• 60 dB

• Rock Concert
• 110 dB
• Pain
• 120 dB

7
Human Sound Reception

• Humans are sensitive to sound over a huge range
• A pain level sound is a trillion times as intense
  as a sound you can barely hear
• Your hearing response is logarithmic
• A sound 10 times as intense sounds twice as loud
• Thus the decibel scale
• Why logarithmic?
• Being sensitive to a wide intensity range is more
  useful than fine intensity discrimination
• Similar to eyesight
• Your ears are also sensitive to a wide range of
  frequencies
• About 20 20000 Hz
• You lose sensitivity to high frequencies as you
  age

8
Generating Musical Frequencies

• Many devices are designed to produce standing
  waves
• e.g., Musical instruments
• Frequency corresponds to note
• e.g., Middle A 440 Hz
• Can produce different f by
• changing v
• Tightening a string
• Changing L
• Using a fret

9
Music

• A musical instrument is a device for setting up
  standing waves of known frequency
• A standing wave oscillates with large amplitude
  and so is loud
• We shall consider an generalized instrument
  consisting of a pipe which may be open at one or
  both ends
• Like a pipe organ or a saxophone
• There will always be a node at the closed end and
  an anti-node at the open end
• Can have other nodes or antinodes in between, but
  this rule must be followed
• Closed end is like a tied end of string, open end
  is like a string end fixed to a freely moving ring

10
Sound Waves in a Tube
11
Harmonics

• Pipe open at both ends
• For resonance need a integer number of ½
  wavelengths to fit in the pipe
• Antinode at both ends
• L ½ l n v lf
• f nv/2L
• n 1,2,3,4
• Pipe open at one end
• For resonance need an integer number of ¼
  wavelengths to fit in the pipe
• Node at one end, antinode at other
• L ¼l n v lf
• f nv/4L
• n 1,3,5,7 (only have odd harmonics)

12
Harmonics in Closed and Open Tubes
13
Beat Frequency

• You generally cannot tell the difference between
  2 sounds of similar frequency
• If you listen to them simultaneously you hear
  variations in the sound at a frequency equal to
  the difference in frequency of the original two
  sounds called beats
• fbeat f1 f2

14
Beats
15
Beats and Tuning

• The beat phenomenon can be used to tune
  instruments
• Compare the instrument to a standard frequency
  and adjust so that the frequency of the beats
  decrease and then disappear
• Orchestras generally tune from A (440 Hz)
  acquired from the lead oboe or a tuning fork