An Experimental Investigation of the
End Effects for Blue Man Group" Pipes
M. E. Bacon and Steven Torok,
Thiel College, Greenville, PA
O
ne of us (ST) constructed a Blue Man Group"
instrument! from 2-in PVC piping/ (internal radius
r = 2.54 em) as an undergraduate project. The
instrument itself is shown in Fig. 1.
In designing such an instrument, it is important to determine the lengths of the pipes that are necessary to give the required notes. For pipes open at both ends, it is well known that
the wavelength of the fundamental is twice the effective length
of the pipe. In addition, it is also well known that the effective
length extends beyond the measured length of the pipes and a
so-called end correction is necessary. In designing the instrument, the total end correction for the pipes was assumed to be
twice the generally accepted value ' of 0.6133r for a single open
end. In the case of the PVC piping used by the present authors,
the measured length of the pipes should be shorter than the effective lengths by 3.12 cm.
The instrument is played by tapping on one of the open
ends of the pipes with a paddle made from a soft synthetic
material and a drumstick. As is customary with this type of
instrument, the pipes for the lowest frequencies (pipes with
lengths greater than one meter) were bent using elbows (see
Fig. 1). In addition, couplings were used at the top ends to
suspend the pipes through holes in the horizontal mounting
pieces. The use of the elbows and couplings made the cutting
of the pipe to the correct lengths somewhat uncertain, and
in general the lengths of the straight pieces of the bent pipes
were adjusted by trial and error. However, even for the straight
pipes, where there were no elbows, the frequencies of the notes
deviated Significantly from those expected-so much so that
we decided to investigate further.
Basic theory
For pipes open at both ends, the wavelength of the fundamental mode is nominally twice the length of the pipe. The
frequency (j = c! A) of the note for a pipe of effective length Lis
therefore given by
I= c12L,
(1)
where c is the speed of sound.
Assuming that the air is dry and behaves as an ideal gas, the
speed of sound c as a function of temperature can be written in
the well-known form
Fig. 1. A 15-note PVC musical instrument.
f= 331.3(1 + Tc/273)1I2
(3)
2L
Let the effective length L = Lo + DL, where Lo is the measured length of the pipe and DL is the total end correction.
Substituting for L in Eq. (3) yields
f= 331.3(1+ Tc/273t2
(4)
2(Lo +DL)
As already mentioned, it is well established theoretically (see
Fig. 2 of Ref. 2) that, provided ttr / L «1, the end correction
for an open-ended pipe is 0.6133r, where r is the radius of
the pipe. Ballpark results for the end correction of a quarterwavelength pipe (i.e., a single open end) have been given
previously in the undergraduate literature.v'' Most recently,
the end correction for a single open end has been measured
experimentally by Krumm and Matteson," who found agreement with the theoretical predictions of Ref. 2. In designing
the instrument, the physical lengths needed for the various notes were therefore determined assuming a total end
correction of twice that due to a single open end (i.e., DL=
1.227r = 3.12 cm). However, as mentioned above, discrepancies between the expected and measured frequencies were
observed.
Experiment and results
(2)
Tc is the temperature in degrees Celcius.
Substituting Eq. (2) into Eq. (1) gives
152
THE PHYSICS
TEACHER.
Vol. 49, MARCH 2011
In order to investigate the troubling deviation of the observed frequencies of the pipes from their expected frequencies, an additional set of straight pipes was cut from the same
2-in (internal diameter 5.08 cm) PVC pipe. No elbows or
couplings were used in order to eliminate any possible com001: 10.1119/1.3555499
Table I. Measured lengths of the pipes.
Temperature 22.loC
speed of sound 344.4m/s
LO=I.OI2m
1.012
164.3
3.61
0.89
185.8
0.796
207.3
3.68
3.47
0.747
220.5
3.4
:?
':
=
Peak Frequency
164.3 Hz
=
-e
~
FFT
1:
;:
~
2
0-
0.661
246.8
3.67
"0
0.587
276.6
3.56
'"
0.521
310.3
3.39
==
o
o
500
1000
Frequency (Hz)
o
2.00E·1
O.OOEO
4.00E·1
6.00E·1
I.OOEO
8.00E·1
Time(s)
Fig. 3. A typical sound pressure-vs-time
imposed.
350
graph with FFT super-
i
I
2: 0.2°C
Tc= 22.1
300
f=344.4!{2*(Lo+DL»)
'N
:J:
::::::
DL= 3.50 ±0.04 em
~
5
250
~
u,
200
ISO
Fig. 2. One of seven PVC pipes used in
measuring the end effect of open Blue
Man Group4!>pipes. Shown to the right
is the paddle used to tap the top of the
pipe. At the left is the LabQuest data
logger used to record the temperature
and sound pressure.
plication due to their use. The measured lengths (Lo) of these
pipes are given in Table I. The least count uncertainty (LCU)
in these physical lengths was 0.5 mm. The pipes were held
vertically in a retort stand with a clearance of at least 30 cm
from the floor (see Fig. 2).
As in the instrument itself, the top of a given pipe was
tapped with a paddle in order to excite the fundamental
mode. The tap was accomplished using a down-up motion
lasting a fraction of a second (as with drumming). The experiments were performed in an interior windowless room, where
the ambient temperature was between 22°C and 23°C during
the day. The temperature was monitored using Vernier's'' Direct Temperature Probe, and the sound pressure was recorded
using the internal microphone of a LabQuest data logger.8
The temperature was also monitored by a liquid-in-glass thermometer, which agreed with the Vernier temperature probe
to within O.S°c. The sound pressure data were downloaded
to Logger Pro,8 where a fast Fourier transform (FFT) was performed and the peak frequency of the sound pressure signal
I
0.5
I!
I
!
I
0.6
0.7
0.8
Measured
length
!
I
I!
0.9
I
!!
!
1.0
r
1.1
LO(m)
Fig. 4. Frequency f vs the measured length Lo of the pipes. Dots
are data points. Solid line is the nonlinear fit of the data to Eq. (7)
with DL as an adjustable parameter.
determined with an uncertainty of +1- 0.2 Hz. A typical waveform of the recorded sound pressure, together with its FFT, is
shown in Fig. 3. The strength of the overtones varied with the
tapping and was not investigated.
Table I (column 2) lists the measured peak frequencies for
the various measured lengths (Lo) of the pipes. The measurements shown in Table I were taken at a temperature of22.1 0c.
The corresponding speed of sound was therefore 344.4 mls
[Eq. (2)]. Column 3 of Table I shows the total end correction
(DL) calculated from Eq. (4) using the appropriate speed of
sound and measured lengths (Lo). The mean value and standard deviation of column 3 yields DL = 3.54 ± 0.12 ern. We
note that the values of DL in Table I show no systematic variation with frequency. DL was also determined by plotting a
graph of frequency f versus Lo (Fig. 4) and performing a nonlinear fit of the data to Eq. (4) using a scientific spreadsheet.
This gave DL = 3.50 ± 0.04 ern, in good agreement with the
mean value from column 3, and this is the value used in the
discussion below.
THE PHYSICS
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Vol. 49, MARCH 2011
153
Discussion and conclusion
References
With two open ends, one might have expected the total end
effect to be 3.12 cm (l.56 em per end). Instead we determined
experimentally a total end effect of 3.50 :::t::: 0.04 cm. If we assume that the expected value of the end correction is correct,
the speed of sound would have to be 343.2 mls (corresponding to a temperature of 20°C for dry air) given the measured
frequencies. This temperature is about 2° below our measured
temperature, which is significantly more than the uncertainty
in our temperature measurements. It should also be noted that
the effect of any humidity would worsen the situation.
Since one end of the pipe is definitely "open" (the end farthest from the tapped end), we assume that its end effect is in
agreement with the theoretical prediction of Ref. 2 (i.e.,
l.56 ern), Therefore, the end effect of the tapped end is about
l.94 ern, or about 0.4 em greater than the expected value for
an open end. We suggest that the transient position of the
paddle during tapping serves to effectively increase the end
effect of the tapped end. This is perhaps not surprising to
those who have studied the acoustics of flutes and clarinets,"
but was a revelation to the present authors.
In conclusion it would be interesting to repeat the above
measurements at significantly different temperatures. It would
also be interesting to investigate the effect on the resonant
frequencies of an obstruction at the "open" end. This could be
accomplished by placing a paddle at various distances from
the "open" end. Such an investigation could lead to a practical
way of tuning a Blue Man Group" -type instrument.
Finally, by assuming that there is an asymmetric end effect of the magnitude suggested by the present experiment,
the basic apparatus, together with a LabQuest data logger,S
could serve as a simple and convenient way of determining
the speed of sound in air at various temperatures with a high
degree of accuracy using very inexpensive equipment.
1.
2.
Blue Man Group": www.blueman.com/experience/instruments.
Nathan True, "Building a pve Instrument"; devices.natetrue.
com/pvc!pvcphase1.htm.
It should be noted that the equation
used to guess the lengths of pipes on this website is incorrect.
3.
Harold Levine and Julian Schwinger, "On the radiation of
sound from an untlanged circular pipe;' Phys. Rev. 73 (4),
383-406 (1948). See also N. H. Fletcher and T. D. Rossing, The
Physics of Musical Instruments, 2nd ed. (Springer, New York,
1999).
Hugh D. Young and Roger A. Freedman, University Physics,
12th ed. (Addison- Wesley, 2006), Vol. 1,p. 533.
154
THE PHYSICS
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Vol. 49, MARCH 2011
4.
5.
6.
7.
8.
9.
Michael C. LoPresto, "Measuring end correction for a quarterwave tube;' Phys. Teach. 43, 380 (Sept. 2005).
Don Easton, "Speed of sound in air;' letter to the editor, Phys.
Teach. 43,567 (Dee. 2005).
Matthew Krumm and Sam Matteson, "Frequency dependence
of end corrections for a pipe of circular cross section;' 2009
Spring Meeting of the Texas Section of the APS, AAPT, and
SPS meeting, April 2-4 (2009). Also private communication
Sam Matteson, September 2009.
Vernier Software, 13979 SW Millikan Way, Beaverton, OR
97005; www.vernier.com.
www.phys.unsw.edu.au/jw/tlutes.v.clarinets.html#endeffect.
Mike Bacon obtained his BSc (honors) degree from the University of Natal
(now the University of Kwa-Zulu Natal) in South Africa and his PhD from
Colorado State University. His initial work was in plasma spectroscopy
and plasma physics. For the past 35 years he has enjoyed working with
undergraduates on a variety of undergraduate research projects. He has
taught in South Africa, Lesotho, and the U.S. He enjoys traveling, particularly in the Southern Hemisphere, and snorkeling in the Caribbean.
Thiel College, 75 College
edu
Ave.,
Greenville, PA 16125; mbacon@thiel.
Steve Torok is currently an IT support analyst with Education
Management Corp., a leading provider of private post-secondary education in North America. Prior to joining EDMC, Stephen studied both at
the University of Pittsburgh and Thiel College, where he graduated in
December 2009 with a BS in applied physics and mathematics minor.
Stephen's longtime interest in music helped lead him to combine his
music background and scientific studies for this research project.