physics 1251 the science and technology of musical sound

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


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.


λ/λ/44

λ/λ/44

Pressure WavePressure Wave Pressure Anti-nodePressure Anti-nodePressure NodePressure Node

+ - + -

+ -+ -


Displacement Standing WaveDisplacement Standing Wave

VisualizationVisualization


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


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


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.


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.


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.


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


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


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


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


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


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


Open versus Stopped PipeOpen versus Stopped Pipe

DemonstrationDemonstration


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


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


Organ PipesOrgan Pipes

Open PipesOpen 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


Transverse FluteTransverse Flute


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


Other Flute-like InstrumentsOther Flute-like Instruments

Penny Penny whistlewhistle FifeFife

RenaissanRenaissanceFluteceFlute

KaenKaen


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


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.

physics 1251 the science and technology of musical sound

Documents

sound waves

pipes reflection

pipes comparison of

pipes acoustic impedance

pressure nodes

displacement wave inverts

organ pipes

unstopped pipes