Physics 1251 The Science and Technology of Musical Sound

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Physics 1251The Science and Technology of
Musical Sound

• Unit 3
• Session 26 MWF
• Sound in Pipes

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Physics 1251 Unit 3 Session 26 Sound in Pipes

• A standing wave on a string (tied at the ends) of
  length 3.0 m has two other nodes. What is the
  wavelength of the string wave?

There are 6 node-antinode distances. Therefore,
L 6 ? ?/4 3.0 m ? 2.0 m.
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Physics 1251 Unit 3 Session 26 Pipes

• 1' Lecture
• Sound in pipes can produce standing waves in the
  air column.
• Standing waves in air columns produce pressure
  nodes and displacement nodes (and antinodes) at
  different places.
• A change in the acoustic impedance of the air
  column produces a reflection.
• Organ pipes and the flute are examples of open or
  unstopped pipes.

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Physics 1251 Unit 3 Session 26 Sound in Pipes

• Standing Waves
• Strings reflected waves combine to produce
  cancellation through destructive interference at
  nodes and constructive interference at antinodes.

String Wave Demonstration ? redux
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Physics 1251 Unit 3 Session 26 Sound in Pipes

• Standing Waves in a Cylindrical Pipe
• A Closed or Stopped Pipe the pressure wave
  reflects without inversion, but the displacement
  wave inverts upon reflection.
• Thus, a pressure anti-node will occur at the
  wall but, on the other hand, a displacement node
  will occur at the same place.

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Physics 1251 Unit 3 Session 26 Sound in Pipes

• Reflection of a Sound Wave in a Stopped Pipe
• A pressure anti-node appears at a wall.

Pressure Wave
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Physics 1251 Unit 3 Session 26 Sound in Pipes

• Displacement Standing Wave
• Visualization

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Physics 1251 Unit 3 Session 26 Sound in Pipes

• Reflection of a Sound Wave in a Stopped Pipe
• A displacement node appears at a wall.

Displacement Anti-node
Displacement Wave
Displacement Node
?/4
?/4
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Physics 1251 Unit 3 Session 26 Sound in Pipes

• Comparison of Pressure and Displacement Standing
  Wave in a Double Stopped Pipe

?/4
?/4
Pressure Wave
Displacement Wave
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Physics 1251 Unit 3 Session 26 Sound in Pipes

• Standing waves in a pipe are an example of the
  property of Interference of sound
  waves.

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Physics 1251 Unit 3 Session 26 Sound in Pipes

• Recall
• f ? v
• Thus, f v/ ? L Nna ?/4
• So f Nna v/4L
• Nna 2 n fn 2 n v/4L
• fn n v/2L
• In double stopped pipe.

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Physics 1251 Unit 3 Session 26 Sound in Pipes

• Why does the sound wave reflect?

Because of an abrupt change in a property of the
medium.
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Physics 1251 Unit 3 Session 26 Sound in Pipes

• 80/20Acoustic Impedance
• Z p/U
• Acoustic Impedance is the ratio of the pressure p
  of a sound wave to the flow U ( u S) that
  results.
• For a plane wave in a tube of cross section S
  (m2) in air the acoustic impedance is
• Z ?v/S 415/ S rayl

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Physics 1251 Unit 3 Session 26 Sound in Pipes

• Acoustic Impedance and Reflection
• The pressure that reflects is
• pout R ? pin
• R (Z2 Z1 )/ (Z2 Z1)
• At an immoveable wall U 0
• (no displacement) irrespective of the pressure
  and, thus,
• Z ? 8
• R 1

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Physics 1251 Unit 3 Session 26 Sound in Pipes

• But what about an open pipe?
• Does the sound reflect?

Yes! Z p/U p drops suddenly near the end of
the pipe, since S ? 8. Thus, Z 0, R -1
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Physics 1251 Unit 3 Session 26 Sound in Pipes

• Comparison of Pressure Standing Wave in a Single
  Stopped and an Open Pipe

L
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Physics 1251 Unit 3 Session 26 Sound in Pipes

• 80/20For Stopped Pipe
• Nna odd number 2n-1, n1,2,3,4
• ?n 4L/ Nna 4L / (2n-1)
• fstopped f2n-1 v/ ?n (2n-1) v/ 4L
• 80/20Only odd harmonics of fstopped 1 v/4L.

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Physics 1251 Unit 3 Session 26 Sound in Pipes

• Comparison of Pressure Standing Wave in a Single
  Stopped and an Open Pipe

L
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Physics 1251 Unit 3 Session 26 Sound in Pipes

• Open versus Stopped Pipe
• Demonstration

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Physics 1251 Unit 3 Session 26 Sound in Pipes

• 80/20For Open Pipe
• Nna even number 2n, n1,2,3,4
• ?n 4L / Nna 4L/(2n) 2L/ n
• fopen fn v/ ?n n ? v/2L
• 80/20All harmonics of fopen 1 v/2L 2
  fstopped 1

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Physics 1251 Unit 3 Session 26 Sound in Pipes

• End Correction for Open Pipe
• d 0.6 a for a ? d 0 a for a gt ? / 4

a Radius
L d
d
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Physics 1251 Unit 3 Session 26 Sound in Pipes

• Organ Pipes

Open Pipes
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Physics 1251 Unit 3 Session 26 Sound in Pipes

• Organ Pipes
• Pipe Organs use both
• open and stopped pipes
• in different ranks because
• the timbre is different
• for each.

Open Pipes
Stopped Pipes
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Physics 1251 Unit 3 Session 26 Sound in Pipes

• Transverse Flute

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Physics 1251 Unit 3 Session 26 Sound in Pipes

• Transverse Flute
• 80/20The transverse flute is a cylindrical open
  pipe.

Mouthpiece is open
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Physics 1251 Unit 3 Session 26 Sound in Pipes

• Other Flute-like Instruments

Kaen
RenaissanceFlute
Penny whistle
Fife
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Physics 1251 Unit 3 Session 26 Sound in Pipes

• Stopped Pipe Flute-like Instruments

The Pan Pipe Only odd harmonics different
timbre Lower fundamental shorter pipe
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Physics 1251 Unit 3 Session 26 Sound in Pipes

• Summary
• fopen fn n ? v/2L
• fstopped f2n-1 (2n-1) v/ 4L
• Stopped and open cylindrical pipes have different
  timbres.
• Impedance Z p/U
• An abrupt change in Z is responsible for the
  reflections that lead to standing waves in pipes.