Some methodical „revision“ remarks on
measurements of air column oscillations
Václav Syrový, Pavel Dlask
Musical Acoustics Research Centre
Faculty of Music, Academy of Performing Arts, Prague
www.hamu.cz/sound
Some methodical „revision“ remarks on measurements of air column oscillations
1
Vienna Talk – 25 Years of IWK, 27th - 28th August, 2005
35
Length of tube = 985 [mm]
Length correction [mm]
30
25
20
15
10
Usually used correction:
0.3*D
0.41*D (Rayleigh)
5
0
10
20
30
40
Measured correction:
Mode 1
Mode 3
Mode 5
Mode 7
Mode 9
Mode 19
50
Diameter [mm]
Some methodical „revision“ remarks on measurements of air column oscillations
2
60
etc.
1. remark
∆ …open-end correction
Some known formulas for open-end correction
∆ ? 0,3 D (by Levine & Schwinger)
∆ ? 0,41 D (by Rayleigh)
1. The influence on pipe tone is observed very often
in distance larger than 0,3 D from the pipe end.
∆ ? 0,29 D (by Blaikley)
2. The known formulas don’t respect the tube length and the frequency too.
3. The influence of tube dimensions on resonance mode inharmonicities
∆ ? 0,33 D (by Boehm & Bate)
etc.
is unclear.
4. It’s impossible directly to calculate the acoustical length and open-end correction too.
5. The acoustical length is non existent, only there is the true resonance frequency value.
Some methodical „revision“ remarks on measurements of air column oscillations
3
On this picture we can see one of the tube
measurement
results
of
a
length
corresponding to one quarter of a wave
length. Then the results of measurement
inspired us to revise a calculation of the
length correction used, above all, by organ
makers. I asked Prof. Merhaut, a mentor and
friend of mine, to revise the calculation and
already the first results showed a very good
matching
with
measured
values.
Regrettably, Prof. Merhaut died last year, so
Pavel Dlask, our new colleague, whom I
would like to introduce to you, undertook
his work.
Vienna Talk – 25 Years of IWK, 27th - 28th August, 2005
The „acoustical“ length of wind musical instruments, what is it?
The best known image bears on the organ pipe:
The length of oscillating air column is longer then length of tube and
determines the 1. harmonic frequency of sounding tone.
Why?
Because the equalization of acoustic preassure inside and outside of the
tube is gradual. Is it true???
∆
The research of musical instruments has not
been only recognition of new findings but
also looking back from time to time. This
contribution
that
follows
up
our
measurement of cylindrical air column
resonance mode frequencies in 2002
presents also the looking back. The purpose
of the measurement was a specification of
well-known simple empiric relations for a
size of length correction as a difference
between a mechanical length of the
cylindrical tube and so-called air column
acoustic length in this tube. The
measurements were performed by the BIAS
system received from our Viennese friends.
Vienna Talk – 25 Years of IWK, 27th - 28th August, 2005
The results of measurement led us to the
first methodical remark that definitely does
not bring any new facts and may be
summarized in the next five points.
Length correction (m)
Now we are going to follow up the 5th point
and look again at the result of measurement.
Length of tube 985 mm
As you can see, the length correction
measured values, more precisely the
difference between the calculated and
measured resonance frequency do not
depend on a tube diameter linearly. It is
well known that this fact is related to an
influence of radiation impedance on the
open tube end and losses inside the tube.
However, in which parts of the curve these
losses do apply more than the radiation
Diameter (m)
impedance and vice versa? The answer must
be sought firstly in a type of considered
radiation impedance and secondly in a
sound velocity determined by a temperature, humidity, pressure and air chemical composition. The
sound velocity is changed with losses inside the tube and so it depends not only on climatic conditions
but also on the frequency and tube diameter.
Some methodical „revision“ remarks on measurements of air column oscillations
∆f = fcalculated - fmeasured [Hz]
40
Vienna Talk – 25 Years of IWK, 27th - 28th August, 2005
L eng th o f tube = 3 27 mm
Diame ter of tub e = 13 .9 mm
50
45
4
red c urv e : c 0 = 344.95 m/s
blue c urv e : c 0 = 345.87 m/s
(ideal gas : p 0 = 101 325 Pa ; T = 296.15 K)
(air : p 0 = 101 325 Pa ; T = 296.15 K ; relativ e humidity = 50% ;
CO 2 c onc entration = 0.031 %)
35
30
25
20
15
10
5
0
-5
-10
-15
1
3
5
7
Mode [-]
9
Calc ulated with reduc ed radiation impedanc e of puls ativ e s phere
Calc ulated with reduc ed radiation impedanc e of puls ativ e s phere
On this picture we can see an instances of
the narrowest and the shortest measured
tube. The difference between the calculated
and measured resonance frequency for the
first five modes, naturally odd modes, rely to
a radiation impedance in form of a pulsative
sphere, then in form of a vibrating piston in
an infinite wall and, finally, when a radiation
impedance is not considered. It all applies
once for an ideal gas and for the second time
for air with values along the measuring.
Calc ulated with radiation impedanc e of plane v ibrating s ourc e loc ated in infinite wall
Calc ulated with radiation impedanc e of plane v ibrating s ourc e loc ated in infinite wall
Calc ulated without c ons idering radiation impedanc e
80
75
70
65
60
55
50
45
40
35
30
25
20
15
10
5
0
-5
-10
-15
-20
-25
1
Length of tube = 327 mm
Diameter of tube = 28.7 mm
red curve : c0 = 344.95 m/s (ideal gas : p0 = 101 325 Pa ; T = 296.15 K)
blue curve : c0 = 345.87 m/s (air : p0 = 101 325 Pa ; T = 296.15 K ; relative humidity = 50% ;
CO2 concentration = 0.031 %)
3
5
Mode [-]
∆f = fcalculated - fmeasured [Hz]
∆f = fcalculated - fmeasured [Hz]
Calc ulated without c ons idering radiation impedanc e
7
9
145
140
135
130
125
120
115
110
105
100
95
90
85
80
75
70
65
60
55
50
45
40
35
30
25
20
15
10
5
0
-5
-10
-15
1
Length of tube = 327 mm
Diameter of tube = 55.8 mm
red curve : c0 = 344.95 m/s (ideal gas : p0 = 101 325 Pa ; T = 296.15 K)
blue curve : c0 = 345.87 m/s (air : p0 = 101 325 Pa ; T = 296.15 K ; relative humidity = 50% ;
CO2 concentration = 0.031 %)
3
5
Mode [-]
7
Calculated with reduced radiation impedance of pulsative sphere
Calculated with reduced radiation impedance of pulsative sphere
Calculated with reduced radiation impedance of pulsative sphere
Calculated with reduced radiation impedance of pulsative sphere
Calculated with radiation impedance of plane vibrating source located in infinite wall
Calculated with radiation impedance of plane vibrating source located in infinite wall
Calculated with radiation impedance of plane vibrating source located in infinite wall
Calculated with radiation impedance of plane vibrating source located in infinite wall
Calculated without considering radiation impedance
Calculated without considering radiation impedance
Calculated without considering radiation impedance
Calculated without considering radiation impedance
9
If we merely increase the tube diameter only the differences between the mentioned type of calculation
on single resonance modes increase.
∆f = fcalculated - fmeasured [Hz]
Speed of sound = 345.87 m/s
To the calculation wasn't encountered the influence of surounding environment to the tube output
120
115
110
105
100
95
90
85
80
75
70
65
60
55
50
45
40
35
30
25
20
15
10
5
0
1
3
5
7
9
Now we will perform a similar calculation
for air without a speculation about the
influence of the radiation impedance. We
will get relations that are very close to the
already mentioned experiential relations but
at the same time very different from the
results of measurement.
Mode [-]
∆f = fcalculated - fmeasured [Hz]
Diameter of tube = 55.8 mm ; Length of tube = 985 mm
Diameter of tube = 28.7 mm ; Length of tube = 985 mm
Diameter of tube = 13.9 mm ; Length of tube = 985 mm
Diameter of tube = 55.8 mm ; Length of tube = 656 mm
Diameter of tube = 28.7 mm ; Length of tube = 656 mm
Diameter of tube = 13.9 mm ; Length of tube = 656 mm
Diameter of tube = 55.8 mm ; Length of tube = 327 mm
Diameter of tube = 28.7 mm ; Length of tube = 327 mm
Diameter of tube = 13.9 mm ; Length of tube = 327 mm
Speed of sound = 345.87 m/s
To the calculation was encountered the influence of surounding environment to the tube output using
reduced radiation impedance of pulsative sphere
40
38
36
34
32
30
28
26
24
22
20
18
16
14
12
10
8
6
4
2
0
1
3
5
7
9
Now we will perform a calculation for the
radiation impedance in form of a pulsative
sphere. Here, the instances of the shortest and
simultaneously the broadest tube will
distinctively vary. Perhaps it is not a
waveguide but rather a Helmholtz´s resonator.
Mode [-]
Diameter of
Diameter of
Diameter of
Diameter of
Diameter of
Diameter of
Diameter of
Diameter of
Diameter of
tube = 55.8 mm ; Length of
tube = 28.7 mm ; Length of
tube = 13.9 mm ; Length of
tube = 55.8 mm ; Length of
tube = 28.7 mm ; Length of
tube = 13.9 mm ; Length of
tube = 55.8 mm ; Length of
tube = 28.7 mm ; Length of
tube = 13.9 mm ; Length of
tube = 985 mm
tube = 985 mm
tube = 985 mm
tube = 656 mm
tube = 656 mm
tube = 656 mm
tube = 327 mm
tube = 327 mm
tube = 327 mm
∆f = fcalculated - fmeasured [Hz]
Speed of sound = 345.87 m/s
To the calculation was encountered the influence of surounding environment to the tube output using
radiation impedance of plane vibrating source located in infinite wall
6
4
2
0
-2
-4
-6
-8
-10
-12
-14
-16
1
3
5
7
Mode [-]
Diameter of
Diameter of
Diameter of
Diameter of
Diameter of
Diameter of
Diameter of
Diameter of
Diameter of
tube = 55.8 mm ; Length of
tube = 28.7 mm ; Length of
tube = 13.9 mm ; Length of
tube = 55.8 mm ; Length of
tube = 28.7 mm ; Length of
tube = 13.9 mm ; Length of
tube = 55.8 mm ; Length of
tube = 28.7 mm ; Length of
tube = 13.9 mm ; Length of
tube = 985 mm
tube = 985 mm
tube = 985 mm
tube = 656 mm
tube = 656 mm
tube = 656 mm
tube = 327 mm
tube = 327 mm
tube = 327 mm
9
Finally, we will perform a calculation for the
radiation impedance in form of a plane piston
vibrating in an infinite wall. Here even a
difference polarity between the short tube
calculated and measured resonance frequency
will vary.
The next pictures bring the same although otherwise sequenced results.
D ia m e te r o f tub e = 1 3 .9 m m
S p e e d o f s o und = 3 4 5 .8 7 m /s
blac k c ur v e
red c ur v e
blue c ur v e
blac k c ur v e
red c urv e
blue c ur v e
60
55
28
26
∆f = fcalculated - fmeasured [Hz]
∆f = fcalculated - fmeasured [Hz]
32
30
D ia m e te r o f tub e = 2 8 .7 m m
S p e e d o f s o und = 3 4 5 .8 7 m /s
65
: lenght of t ube = 985 m m
: lenght of tube = 656 mm
: lenght of t ube = 327 m m
24
22
20
18
16
14
12
10
8
6
4
2
0
-2
: lenght of t ube = 985 m m
: lenght of t ube = 656 m m
: lenght of tube = 327 m m
50
∆f = fcalculated - fmeasured [Hz]
36
34
45
40
35
30
25
20
15
10
5
0
-5
-10
-4
-6
-15
-8
1
∆f = fcalculated - fmeasured [Hz]
3
5
7
Mode [-]
-20
9
1
3
5
7
Mode [-]
9
115
110
105
100
95
90
85
80
75
70
65
60
55
50
45
40
35
30
25
20
15
10
5
0
-5
-10
-15
D ia m e te r o f tub e = 5 5 .8 m m
S p e e d o f s o und = 3 4 5 .8 7 m /s
blac k c ur v e
red c urv e
blue c ur v e
1
: lenght of t ube = 985 m m
: lenght of t ube = 656 m m
: lenght of tube = 327 m m
3
5
Mode [-]
7
Calc ulated with r educ ed r adiat ion im pedanc e of puls ativ e s pher e
Calc ulated with r educ ed r adiation impedanc e of puls ativ e s phere
Calc ulated with r educ ed r adiation impedanc e of puls ativ e s phere
Calc ulated with r adiat ion im pedanc e of plane v ibr ating s ourc e loc at ed in infinite wall
Calc ulated with r adiation impedanc e of plane v ibr ating s our c e loc ated in infinite wall
Calc ulated with r adiation impedanc e of plane v ibr ating s our c e loc ated in infinite wall
Calc ulated without c ons ider ing r adiat ion im pedanc e
Calc ulated without c ons ider ing r adiation im pedanc e
Calc ulated without c ons ider ing r adiation im pedanc e
Length of tube = 985 mm
Diameter of tube = 13.9 mm
To the calculation was encountered the influence of surounding environment to the tube output using
radiation impedance of plane vibrating source located in infinite wall
18
17
16
15
14
13
12
11
10
9
8
7
6
5
4
3
2
1
0
1
3
5
7
9
11
13
15
17
19
9
The influence of temperature and air
humidity on the mentioned results is shown
on this picture.
Mode [-]
c0 = 344.951 m/s - speed of sound (ideal gas : p 0 = 101 325 Pa ; T = 296.15 K)
c0 = 345.115 m/s - speed of sound (dry air : p 0 = 101 325 Pa ; T = 296.15 K ; relative humidity = 0% ; CO2 concentration = 0.0314%
c0 = 345.874 m/s - speed of sound (air : p0 = 101 325 Pa ; T = 296.15 K ; relative humidity = 50% ; CO2 concentration = 0.0314%)
c0 = 343.994 m/s - speed of sound (air : p0 = 101 325 Pa ; T = 293.15 K ; relative humidity = 50% ; CO2 concentration = 0.0314%)
Length of tube = 985 mm
Speed of sound = 345.87 m/s
36
Now there is the most important thing. Let
us compare the measured and calculated
values of the length correction. These lines
correspond merely to the influence of the
radiation impedance but these curves
include also correction of sound velocity
respectively according to Thomas Moore´s
publication in Acta Acoustica this year.
Considering the radiation impedance in
form of a plane piston source vibrating in
an infinite wall, the accordance with the
results of measurement is very good. The
Diameter / Length [-]
differences that might be thought to be
measurement errors also rely, among
others, to uneven distribution of air temperature inside the tube. (The tubes have been made from
duralloy.)
b la c k
red
34
: w it ho ut c or r e c t ion of s pe ed o f s o un d
: inc lud in g c or r e c t ion of s pe ed of s o un d b y T h om a s R . M o or e an d a l.
32
30
28
Length correction [mm]
26
24
22
20
18
16
14
12
10
8
6
4
2
0
0.01
0.02
0.03
0.04
0.05
0.06
M e a s u r e d v a lu e s - w ith m e a s u r in g s y s te m B IA S 5 .1
C a lc u la te d v a lu e s - c a lc u la t e d w ith r e d u c e d r a d ia tio n im p e d a n c e o f p u ls a tiv e s p h e r e
C a lc u la te d v a lu e s - c a lc u la t e d w ith r a d ia tio n im p e d a n c e o f p la n e p is to n v ib r a tin g s o u r c e lo c a t e d in in fin it e w a ll
The second remark may be summarized in the following points:
1. The radiation impedance in form of a vibrating piston in an infinite wall is preferential for a
calculation of the cylindrical air column resonance frequencies.
2. The calculation must cover the influence of all the losses that occur inside the tube.
18
17
16
15
14
13
12
11
10
9
8
7
6
5
4
3
2
1
0
-1
10
Length of tube = 985 mm
S peed of s ound = 345.87 m/s
M ode 9
Inharmonicity [Hz]
Let us make two additional remarks. The
first one also refers to a notorious fact
that the higher resonance modes
frequency positions are not harmonically
distributed. On this picture we can see a
frequency deviation of the ninth
resonance mode for various tube
diameters. Leaving out the radiation
impedance influence, the deviation will
be zero. In case of radiation impedance
20
30
40
50
60
Diameter [mm]
as a pulsative sphere or a plane piston
source in an infinite wall the
inharmonicities increase with increasing
of the tube diameter. However, correcting the sound velocity will essentially change the dependence of
inharmonicity sizes. This calculated relation does not differentiate the considered type of radiation
impedance very much and matches the values measured in 2002.
b la c k
re d
: w ith o u t c o rre c tio n o f s p e e d o f s o u n d
: in c lu d in g c o rre c tio n o f s p e e d o f s o u n d b y T h o m a s R . M o o re a n d a l.
C a lc u la te d v a lu e s - c a lc u la te d w ith re d u c e d ra d ia tio n im p e d a n c e o f p u ls a tiv e s p h e re
C a lc u la te d v a lu e s - c a lc u la te d w ith ra d ia tio n im p e d a n c e o f p la n e p is to n v ib ra tin g s o u rc e lo c a te d in in fin ite w a ll
C a lc u la te d v a lu e s - c a lc u la te d w ith o u t c o n s id e rin g ra d ia tio n im p e d a n c e
M e a s u re d v a lu e s - w ith m e a s u rin g s y s te m B IA S 5 .1
The last remark is devoted to cylindrical
tube air column transversal oscillations,
more precisely oscillations oriented outside
the tube axis. Practically, we met these
oscillations already over 15 years ago
during a development of pulse methods for
measurement of wind instrument input
impedance. On this picture, the results of
measurement of four clarinets with all side
holes closed are being compared. These
holes themselves already facilitate a creation
of a marked transversal resonance mode.
Frequency position of the mode coheres
with the inside tube diameter of the clarinet,
the shape of the mode coheres with the
variable depth, diameter and position of side holes of the clarinet.
Measurement of non-axis resonance modes
Hz
Length of tube 985 mm
Red line – BIAS measuring head in the axis of tube
Diameter of tube 71 mm
Black line – BIAS measuring head out the axis of tube (25 mm)
Some methodical „revision“ remarks on measurements of air column oscillations
19
Vienna Talk – 25 Years of IWK, 27th - 28th August, 2005
The next picture presents the results of
measurement of these transversal modes for
two positions of the BIAS measuring head.
At axial head position merely a slight
undulation occurs, at off - axial head
position these modes are very marked. The
existence of transversal modes may be
proved by a calculation of tube resonance
modes for three-dimensional solution in
cylindrical coordinates.
Length of tube = 985 mm
D iameter of tube = 71 mm
The result of calculation basically matches the
results of measurement.
16
14
10
8
Z
a11′
 [MΩ ]
12
6
4
2
0
0
500
1000
1500
2000
F re q ue nc y [H z ]
2500
3000
3500
4000
The mentioned remarks are interesting for us, above all, from a view of organ pipes sound timbre and
thereby the whole organ sound research. Although the cylindrical organ pipe presents the simplest
wind instrument, we do not get ahead very much with a simple solution of the Webster´s equation for
ideal gas, let alone in case of wood and brass instruments.
Nevertheless, We hope that our small
looking back would have been inspiring for
the research of these instruments, too.
Thank you for your attention!
Some methodical „revision“ remarks on measurements of air column oscillations
21
Vienna Talk – 25 Years of IWK, 27th - 28th August, 2005
Acknowledgements
This research was supported by the Ministry for Education, Youth and Sport of Czech
Republic (Project No. 1M6138498401).
References
V.Syrový, R.Jindra, J.Štěpánek (2002): Air Column Resonance Frequencies and Their Dependencies in
Cylindrical Tubes, 32nd International Acoustical Conference, European Acoustics Association (EAA)
Symposium "ACOUSTICS BANSKÁ ŠTIAVNICA 2002", 133-136
Syrový, V. (1989): Impulse measurements on woodwind musical instruments. Proceedings of ISMA ‘89,
Mittenwald, 64
Moore, T.R., et al. (2005): The Effects of Bell Vibrations on the Sound of the Modern Trumpet, Acta
Acustica, Vol.9, 578-589
Cremer O. (1993): The variation of the specific heat ratio and the speed of sound in air with temperature,
pressure, humidity, and CO2 concentration, J. Acoust. Soc. Am. 93 (5), 2510-2516