Quasi-static non-linear characteristics
of double-reed instruments
André Almeida, Christophe Vergez and René Caussé
http://arxiv.org/physics/0607011,
submitted to the Journal of the Acoustical Society of America.
Dated July 4th 2006
Almeida et al. (0607011)
Double-reed characteristics
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Outline
1
Introduction
2
Measurement
3
Results
4
Analysis
5
Conclusions
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Double-reed characteristics
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Introduction
About the IRCAM
Institut de recherche et coordination acoustique/musique founded in
1969 by Georges Pompidou and directed by Pierre Boulez (69-92),
Laurent Bayle (92-02) and Bernard Stiegler (02-)
Contemporary musical research and production (together with the
Centre Pompidou)
Pompidou’s goal:“bring science and art together in order to widen the
instrumentarium and rejuvenate musical language”
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Double-reed characteristics
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Introduction
Modelling a reed instrument
(
exciter (non-linear) → reed
Instrument =
resonator (linear)
→ air column
Oscillation occurs because of coupling between reed and air column.
Exciter must be non-linear to create oscillation from continous source of
energy.
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Double-reed characteristics
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Introduction
The elementary reed model
The reed’s role is to controle and modulate the air volume flow (q)
entering the instrument by means of a pressure drop (∆p).
Assuming quasi-static conditions, one can use a simple model (pm is the
pressure inside the mouth and pr the one inside the reed):
(∆p)r = pm − pr =: ks (S0 − S) .
Here, S0 is the reed opening area when at rest and ks a stiffness constant.
Bernoulli:
ρ
ρ 2
pr + vr2 = pm + vm
≈ pm
2
2
with vm ≈ 0 because of volume flow conservation.
Almeida et al. (0607011)
Double-reed characteristics
,
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Introduction
The elementary reed model
Therefore:
r
q = S · vr = S ·
pm − pr
pM − (∆p)r
2
=
ρ
ks
s
2(∆p)r
ρ
,
(1)
using the definition of the so-called static reed beating pressure
pM := ks S0 required to close the reed completely.
Changing to dimensionless quantities by means of
(∆p)r
∆p̃ :=
pM
q̃ :=
ks
3/2
pM
r
ρ
·q
2
,
one ends up with:
p
q̃ = (1 − ∆p̃) ∆p̃
Almeida et al. (0607011)
Double-reed characteristics
.
(2)
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Introduction
The elementary reed model
Flow (q) in l/s
Plotting eq. 2 gives
0.25
0.2
0.15
0.1
0.05
0
0
5
10
15
20
Pressure (∆p)r in kPa
Almeida et al. (0607011)
Double-reed characteristics
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Introduction
The elementary reed model
Flow (q) in l/s
Plotting eq. 2 gives
0.25
0.2
0.15
0.1
0.05
0
0
5
10
15
20
Pressure (∆p)r in kPa
ERM works very well for single-reed instruments → can it be applied to
double-reed instruments as well?
Main difference: Local minima in cross section ⇒ separation of flow from
wall ⇒ multi-valued relation possible.
Almeida et al. (0607011)
Double-reed characteristics
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Measurement
Principle of measurement
Problem: Volume flow velocity can be measured only in one point at the
time, not on a whole area ⇒ use indirect measurement with a diaphragm
(flow resistance; cross-section Sd ).
(∆p)s
(∆p)r
(∆p)d
p1
p2
patm
1
2
The pressure flow can now be approximated by the Bernoulli law:
ρ q 2
(∆p)d = pr − patm =
.
2 Sd
(3)
⇒ q can be calculated from measurement of pr .
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Double-reed characteristics
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Measurement
Practical issues
Diaphragm reduces range over which (∆p)r can be measured:
Q (l/s)
B
B’
0.3
0.2
0.1
C
A
10
20
30
40
50
60
p (kPa)
−0.1
Solution: Empirically find the best diaphragm cross-section.
Almeida et al. (0607011)
Double-reed characteristics
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Measurement
Practical issues
Diaphragm reduces range over which (∆p)r can be measured:
Q (l/s)
B
B’
0.3
0.2
0.1
C
A
10
20
30
40
50
60
p (kPa)
−0.1
Solution: Empirically find the best diaphragm cross-section.
Reed auto-oscillations have to be prevented in order to maintain
quasi-static flow.
Solution: Increase the reed’s acoustic resistance by increasing its mass
(with duct tape).
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Double-reed characteristics
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Measurement
Experimental set-up
pm
Camera
Lens
pr
Reed
Controllable
leak
Diaphragm
Manometer
Compressed
air source
Artificial
mouth
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Double-reed characteristics
Humidifier
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Measurement
Typical run
50
pr
pressure (kPa)
40
pm
30
20
10
0
−10
0
Almeida et al. (0607011)
20
40
60
80
time (s)
Double-reed characteristics
100
120
140
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Results
Results
0.35
0.3
Decreasing pressure: q
follows different path due to
memory effects in the reed.
flow (q) in l/s
0.25
0.2
0.15
0.1
0.05
0
−10
0
10
20
30
pressure difference (∆ p)r in kPa
Almeida et al. (0607011)
40
50
Double-reed characteristics
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Results
Results
0.35
0.3
Decreasing pressure: q
follows different path due to
memory effects in the reed.
0.2
0.15
0.1
0.05
0
−10
0
10
20
30
pressure difference (∆ p)r in kPa
40
−1.4
10
50
No total closing of reed!
If capillary (viscous) flow
through open channels:
q ∝ ∆p
√
Analysis shows q ∝ ∆p
⇒ reed not yet stable?
−1.5
10
−1.6
flow (l/s)
flow (q) in l/s
0.25
10
0
10
−1.7
10
−2
10
−1.8
10
experimental
fitted Bernoulli
0
10
−1.9
10
1.3
10
1.5
1.7
10
10
pressure difference (kPa)
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10
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Results
Different instruments
Pressure flow characteristic
Normalized pressure flow characteristic
1.2
0.5
Clarinet
Oboe
Bassoon
0.4
Clarinet
Oboe
Basoon
1
Flow
Flow (l/s)
0.8
0.3
0.6
0.2
0.4
0.1
0.2
0
−5
0
5
10
15
20
Pressure difference (kPa)
25
30
0
0
1
2
3
4
Pressure difference
5
6
Important differences:
S0 (ob) ≈ 2mm2 , S0 (bsn) ≈ 7mm2 ⇒ use different diaphragms.
Exciting mechanism of clarinets differs geometrically and
mechanically.
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Double-reed characteristics
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Analysis
Comparing theory and experiment
In principle, fit of two parameters (ks and S0 ) to experimental data.
Determination of qmax from ERM differs from experimental value:
0.35
experimental
increasing
decreasing
0.3
flow (l/s)
0.25
0.2
0.15
0.1
0.05
0
−10
0
10
20
30
pressure difference (kPa)
40
50
ERM gives qmax = 13 pM , experiment however yields qmax ≈ 15 pM
⇒ improvement of model needed!
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Double-reed characteristics
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Analysis
Accounting for turbulence
Displacement of maximum value can be analysed in terms of pressure
recovery due to flow deceleration inside the reed duct.
dimension (mm)
8
height
width
6
4
2
0
0
10
20
0
10
20
30
40
axial distance (mm)
50
60
70
50
60
70
area (mm2)
15
10
5
0
30
Almeida et al. (0607011)
40
Using energy and flow conservation doesn’t really work because of
turbulent flow! E.g. at reed out4q
≈ 5000. Because of
put Re = πdν
Re ∝ 1/d, turbulence occurs already at low flow volumes.
⇒ Use phenomenological model of
“conical diffuser” to account for
pressure recovery after expansion
of cross section.
Double-reed characteristics
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Analysis
The “conical diffuser” model
Pressure recovery (neglecting distributed losses due to laminar viscosity) is
accounted for by
pout − pin
,
Cp := 1 2
2 ρvin
which ranges from 0 (no recovery) to 1 (full recovery).
Splitting the reed in two, one doesn’t consider recovery between m and c,
leading to
r
pm − pc
q=S· 2
(∆p)c =: ks (S0 − S) .
ρ
The total pressure difference then follows (with Sc the c.s. at c) from
ρ q 2
0
(∆p)r = (∆p)c − Cp
.
(4)
2 Sc
Almeida et al. (0607011)
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Analysis
The “conical diffuser” model
Fitting the improved model to the experimental data:
0.35
experimental
increasing
decreasing
0.3
flow (l/s)
0.25
0.2
0.15
0.1
0.05
0
−10
0
10
20
30
pressure difference (kPa)
40
50
where the same values for ks and S0 have been used as before, using Cp as
fitting parameter (Cp = 0.8).
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Double-reed characteristics
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Conclusions
Conclusions
The quasi-static non-linear characteristics were measured for double-reed
instruments using a similar device as for single-reed instruments.
The obtained curves are similar to single-reed ones and no
multi-valued relations have been found.
There exist substantial qualitative differences between experiment and
the ERM for high volume flows, which can be explained by pressure
recovery.
Modelling the complete instrument as shown is not entirely valid! It’s
only sensible if the length of the mouthpiece is negligible compared to
the typical wavelength. But since ` ≈ 7cm and λtypical ≈ 50cm ⇒
should the staple be considered a part of the resonator? ⇒ No
pressure recovery in exciter but resonator
⇒ modelling the complete instrument.
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