physics 1251 the science and technology of musical sound
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Physics 1251 The Science and Technology of Musical Sound. Unit 3 Session 26 MWF Sound in Pipes. Physics 1251Unit 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?. - PowerPoint PPT PresentationTRANSCRIPT
Physics 1251Physics 1251The Science and The Science and
Technology of Musical Technology of Musical SoundSound
Physics 1251Physics 1251The Science and The Science and
Technology of Musical Technology of Musical SoundSound
Unit 3Unit 3
Session 26 MWFSession 26 MWF
Sound in PipesSound in Pipes
Unit 3Unit 3
Session 26 MWFSession 26 MWF
Sound in PipesSound in Pipes
Physics 1251Physics 1251 Unit 3 Session 26Unit 3 Session 26Sound in PipesSound in Pipes
A standing wave on a string (tied at A standing wave on a string (tied at the ends) of length 3.0 m has two the ends) of length 3.0 m has two other nodes. What is the other nodes. What is the wavelength of the string wave?wavelength of the string wave?
There are 6 node-antinode There are 6 node-antinode distances. Therefore, L = 6 distances. Therefore, L = 6 ‧ ‧ λ/4 = λ/4 = 3.0 m; λ =2.0 m3.0 m; λ =2.0 m..
Physics 1251Physics 1251 Unit 3 Session 26Unit 3 Session 26PipesPipes
11′ Lecture:′ Lecture:• Sound in pipes can produce standing Sound in pipes can produce standing
waves in the air column.waves in the air column.• Standing waves in air columns produce Standing waves in air columns produce
pressure nodes and displacement nodes pressure nodes and displacement nodes (and antinodes) at different places.(and antinodes) at different places.
• A change in the acoustic impedance of A change in the acoustic impedance of the air column produces a reflection.the air column produces a reflection.
• Organ pipes and the flute are examples Organ pipes and the flute are examples of open or unstopped pipes.of open or unstopped pipes.
Physics 1251Physics 1251 Unit 3 Session 26Unit 3 Session 26Sound in Pipes Sound in Pipes
Standing Waves:Standing Waves:• Strings – reflected waves combine Strings – reflected waves combine
to produce cancellation through to produce cancellation through destructive interference at nodes destructive interference at nodes and constructive interference at and constructive interference at antinodes.antinodes.
String Wave Demonstration String Wave Demonstration ‒‒ reduxredux
Physics 1251Physics 1251 Unit 3 Session 26Unit 3 Session 26Sound in Pipes Sound in Pipes
Standing Waves in a Cylindrical Pipe:Standing Waves in a Cylindrical Pipe:
• A Closed or Stopped Pipe – the pressure A Closed or Stopped Pipe – the pressure wave reflects without inversion, but the wave reflects without inversion, but the displacement wave inverts upon displacement wave inverts upon reflection.reflection.
• Thus, a Thus, a pressurepressure anti-nodeanti-node will occur at will occur at the wall; but, on the other hand, a the wall; but, on the other hand, a displacementdisplacement nodenode will occur at the same will occur at the same place.place.
Reflection of a Sound Wave in a Stopped Reflection of a Sound Wave in a Stopped Pipe:Pipe:
A pressure anti-node appears at a wall.A pressure anti-node appears at a wall.
Physics 1251Physics 1251 Unit 3 Session 26Unit 3 Session 26Sound in Pipes Sound in Pipes
λ/λ/44
λ/λ/44
Pressure WavePressure Wave Pressure Anti-nodePressure Anti-nodePressure NodePressure Node
+ - + -
+ -+ -
Physics 1251Physics 1251 Unit 3 Session 26Unit 3 Session 26Sound in PipesSound in Pipes
Displacement Standing WaveDisplacement Standing Wave
VisualizationVisualization
Physics 1251Physics 1251 Unit 3 Session 26Unit 3 Session 26Sound in Pipes Sound in Pipes
Reflection of a Sound Wave in a Stopped Reflection of a Sound Wave in a Stopped Pipe:Pipe:
A displacement node appears at a wall.A displacement node appears at a wall.
λ/λ/44
λ/λ/44
Displacement Displacement WaveWave
Displacement Anti-Displacement Anti-nodenode
Displacement NodeDisplacement Node
Physics 1251Physics 1251 Unit 3 Session 26Unit 3 Session 26Sound in Pipes Sound in Pipes
Comparison of Pressure and Displacement Comparison of Pressure and Displacement Standing Wave in a Double Stopped PipeStanding Wave in a Double Stopped Pipe
Displacement Displacement WaveWave
Pressure WavePressure Wave λ/λ/44
λ/λ/44
Physics 1251Physics 1251 Unit 3 Session 26Unit 3 Session 26Sound in PipesSound in Pipes
Standing waves in a pipe are an Standing waves in a pipe are an example of the property of example of the property of
Interference of sound Interference of sound waves.waves.
Physics 1251Physics 1251 Unit 3 Session 26Unit 3 Session 26Sound in PipesSound in Pipes
Recall:Recall:• f λ = v f λ = v • Thus, f = v/ λ; L = NThus, f = v/ λ; L = Nnana λ/4 λ/4
• So f = NSo f = Nnana v/4L v/4L
• NNnana = 2 n ; = 2 n ; ffnn = 2 n v/4L = 2 n v/4L
ffnn = n v/2L = n v/2L
In double stopped pipe.In double stopped pipe.
Physics 1251Physics 1251 Unit 3 Session 26Unit 3 Session 26Sound in PipesSound in Pipes
Why does the sound wave reflect?Why does the sound wave reflect?
Because of an abrupt change Because of an abrupt change in a property of the medium.in a property of the medium.
Physics 1251Physics 1251 Unit 3 Session 26Unit 3 Session 26Sound in Pipes Sound in Pipes
80/2080/20Acoustic Impedance:Acoustic Impedance:
Z = p/UZ = p/U
Acoustic Impedance is the ratio of the Acoustic Impedance is the ratio of the pressure p of a sound wave to the flow pressure p of a sound wave to the flow U (= u S) that results.U (= u S) that results.
For a plane wave in a tube of cross section S (mFor a plane wave in a tube of cross section S (m22) ) in air the acoustic impedance is:in air the acoustic impedance is:
Z = ρv/S = 415/ S raylZ = ρv/S = 415/ S rayl
Physics 1251Physics 1251 Unit 3 Session 26Unit 3 Session 26Sound in Pipes Sound in Pipes
Acoustic Impedance and Reflection:Acoustic Impedance and Reflection:
The pressure that reflects is The pressure that reflects is
ppoutout = R = R ‧ ‧ ppinin
R = (ZR = (Z22 – Z – Z1 1 )/ (Z)/ (Z22 + Z + Z11))
At an immoveable wall U = 0 At an immoveable wall U = 0 (no displacement) irrespective of the pressure (no displacement) irrespective of the pressure and, thus, and, thus,
Z Z → ∞→ ∞
R R ≈ 1≈ 1
Physics 1251Physics 1251 Unit 3 Session 26Unit 3 Session 26Sound in PipesSound in Pipes
But what about an open pipe? But what about an open pipe?
Does the sound reflect?Does the sound reflect?
Yes! Z = p/UYes! Z = p/U
p drops suddenly near the p drops suddenly near the end of the pipe, since S end of the pipe, since S → ∞.→ ∞.
Thus, Z ≈ 0, R ≈ -1Thus, Z ≈ 0, R ≈ -1
LL
Physics 1251Physics 1251 Unit 3 Session 26Unit 3 Session 26Sound in Pipes Sound in Pipes
Comparison of Pressure Standing Wave in a Single Comparison of Pressure Standing Wave in a Single Stopped and an Open PipeStopped and an Open Pipe
λ/λ/44
λ/λ/44
λ/λ/44
4λ/44λ/4
3λ/43λ/4
Physics 1251Physics 1251 Unit 3 Session 26Unit 3 Session 26Sound in PipesSound in Pipes
80/2080/20For For Stopped PipeStopped Pipe::NNna na = odd number = 2n-1, n=1,2,3,4 …= odd number = 2n-1, n=1,2,3,4 …
λλnn = 4L/ N = 4L/ Nna na = 4L / (2n-1) = 4L / (2n-1)
ffstopped stopped = f= f2n-12n-1 = v/ λ = v/ λnn = (2n-1) v/ 4L = (2n-1) v/ 4L
80/2080/20Only Only odd odd harmonics of fharmonics of fstopped 1stopped 1 = v/4L. = v/4L.
LL
Physics 1251Physics 1251 Unit 3 Session 26Unit 3 Session 26Sound in Pipes Sound in Pipes
Comparison of Pressure Standing Wave in a Single Comparison of Pressure Standing Wave in a Single Stopped and an Open PipeStopped and an Open Pipe
λ/λ/44
λ/λ/44
λ/λ/44
4λ/44λ/4
3λ/43λ/4
Physics 1251Physics 1251 Unit 3 Session 26Unit 3 Session 26Sound in PipesSound in Pipes
Open versus Stopped PipeOpen versus Stopped Pipe
DemonstrationDemonstration
Physics 1251Physics 1251 Unit 3 Session 26Unit 3 Session 26Sound in PipesSound in Pipes
80/2080/20For For Open PipeOpen Pipe::NNna na = even number = 2n, n=1,2,3,4…= even number = 2n, n=1,2,3,4…
λλnn = 4L / N = 4L / Nna na = 4L/(2n) = 2L/ n= 4L/(2n) = 2L/ n
ffopenopen = f = fnn = v/ λ = v/ λnn = n = n ‧ ‧ v/2Lv/2L
80/2080/20AllAll harmonics of harmonics of ffopenopen 11 = v/2L [= 2 f = v/2L [= 2 fstopped 1 stopped 1
]]
L + L + δδ
Physics 1251Physics 1251 Unit 3 Session 26Unit 3 Session 26Sound in Pipes Sound in Pipes
End Correction for Open PipeEnd Correction for Open Pipe
δ δ ≈ 0.6 a for a ≈ 0.6 a for a ≪≪ λ ; δ λ ; δ ≈ 0 a for a > ≈ 0 a for a > λ / 4 λ / 4
δδ
aa Radius Radius
Physics 1251Physics 1251 Unit 3 Session 26Unit 3 Session 26Sound in PipesSound in Pipes
Organ PipesOrgan Pipes
Open PipesOpen Pipes
Physics 1251Physics 1251 Unit 3 Session 26Unit 3 Session 26Sound in PipesSound in Pipes
Organ PipesOrgan Pipes
Pipe Organs use bothPipe Organs use both
open and stopped pipes open and stopped pipes
in different ranks becausein different ranks because
the timbre is different the timbre is different
for each.for each.StoppeStoppe
ddPipesPipes
OpenOpenPipesPipes
Physics 1251Physics 1251 Unit 3 Session 26Unit 3 Session 26Sound in PipesSound in Pipes
Transverse FluteTransverse Flute
Physics 1251Physics 1251 Unit 3 Session 26Unit 3 Session 26Sound in PipesSound in Pipes
Transverse FluteTransverse Flute
80/2080/20The transverse flute is a cylindrical The transverse flute is a cylindrical open pipe.open pipe.
Mouthpiece is Mouthpiece is openopen
Physics 1251Physics 1251 Unit 3 Session 26Unit 3 Session 26Sound in PipesSound in Pipes
Other Flute-like InstrumentsOther Flute-like Instruments
Penny Penny whistlewhistle FifeFife
RenaissanRenaissanceFluteceFlute
KaenKaen
Physics 1251Physics 1251 Unit 3 Session 26Unit 3 Session 26Sound in PipesSound in Pipes
Stopped Pipe “Flute-like” Stopped Pipe “Flute-like” InstrumentsInstruments
The Pan Pipe:The Pan Pipe:
Only odd harmonics = Only odd harmonics = different different timbretimbreLower fundamental = Lower fundamental = shorter shorter pipepipe
Physics 1251Physics 1251 Unit 3 Session 26Unit 3 Session 26Sound in PipesSound in Pipes
Summary:Summary:• ffopenopen = f = fnn = n = n ‧ ‧ v/2L v/2L
• ffstopped stopped = f= f2n-12n-1 = (2n-1) v/ 4L = (2n-1) v/ 4L
• Stopped and open cylindrical pipes Stopped and open cylindrical pipes have different timbres.have different timbres.
• Impedance: Z = p/UImpedance: Z = p/U• An abrupt change in Z is responsible An abrupt change in Z is responsible
for the reflections that lead to for the reflections that lead to standing waves in pipes.standing waves in pipes.