The physics of guitar string vibrations

Article (PDF Available)inAmerican Journal of Physics 84(1):38-43 · January 2016with 3,308 Reads
DOI: 10.1119/1.4935088
Cite this publication
Abstract
We describe laboratory experiments to study the harmonic content of standing waves in guitar strings. The experimental data were taken by using the magnetic pickup from a guitar and a digital oscilloscope with a Fast Fourier transform capability. The amplitudes of the harmonics in the measured signal depend on the location where the string is plucked, resulting in a different timbre of the sound. The relative amplitudes of transverse standing waves in a string were determined from the experimental data and also predicted from the wave equation with the boundary and initial conditions corresponding to the initial shape of the string.
The physics of guitar string vibrations
Polievkt Perov,
a)
Walter Johnson,
b)
and Nataliia Perova-Mello
c)
Department of Physics, Suffolk University, Boston, Massachusetts 02114
(Received 24 June 2014; accepted 20 October 2015)
We describe laboratory experiments to study the harmonic content of standing waves in guitar
strings. The experimental data were taken by using the magnetic pickup from a guitar and a digital
oscilloscope with a Fast Fourier transform capability. The amplitudes of the harmonics in the
measured signal depend on the location where the string is plucked, resulting in a different timbre of
the sound. The relative amplitudes of transverse standing waves in a string were determined from
the experimental data and also predicted from the wave equation with the boundary and initial
conditions corresponding to the initial shape of the string.V
C2016 American Association of Physics Teachers.
[http://dx.doi.org/10.1119/1.4935088]
I. INTRODUCTION
The research into student understanding and curriculum
development in the area of sound waves shows that many
students have misconceptions.
1,2
We believe innovative,
research-based materials are needed to supplement the cur-
riculum and create an environment where students can
actively participate and engage in the learning process
through meaningful activities. In this paper, we address the
harmonic content of resonant standing waves through the use
of the guitar, to facilitate student learning.
Sounds produced by different musical instruments are lon-
gitudinal waves propagating in air. In stringed instruments,
such as a guitar, violin, or piano, the sound waves in air are
excited by transverse resonant standing waves in the strings,
while in wind instruments the resonant standing waves are
longitudinal waves in air-filled pipes. One can easily distin-
guish different musical instruments when the same note is
played because various instruments have their own specific
timbre (or quality of sound). The sound of an instrument
playing, for example, the note E
4
, consists not only of the
fundamental frequency (or first harmonic; f
1
¼329.63 Hz for
the note E
4
), but also of some higher harmonics (f
n
¼nf
1
,
n¼1, 2, 3,…). The amount and relative intensities of the har-
monics define the particular tone of a musical instrument,
which distinguishes it from others. However, even for the
same instrument the tone depends on the way the notes are
played by a musician.
To determine the amplitudes of the fundamental and
higher harmonics in the oscillating component (for example,
a string) of a musical instrument, a Fourier analysis of the
string waveform is usually used. A very detailed study of the
vibrating string profile has been performed
3
using optical
detection of the string vibration. In these experiments, the
vibrating string created a vibrating shadow on the photode-
tector. The Fourier analysis of the signal measured at differ-
ent initial conditions (how the string was plucked) has shown
a good agreement with the calculations of the harmonics (up
to the ninth) in the vibrating string from the initial conditions
(the initial shape) of the string. An analysis of vibrating
string by another optical technique, high speed photography,
was recently reported,
4
which also demonstrated good agree-
ment between experiments and calculations.
The role of boundary conditions on the transverse standing
waves in a string with one or both ends fixed has been dem-
onstrated,
5
which clearly showed how the positions of nodes
and crests of the particular mode of oscillations changed
depending on whether the end of the string was free or fixed.
Laboratory exercises on oscillation modes in open, closed,
and conical pipes (air columns) have been suggested.
6
Placing a movable microphone inside the pipe allowed for
determination of the positions of nodes and crests of the
standing waves in the pipes of different types and of different
sizes.
In this paper, we describe experiments on the harmonic
content of resonant standing waves in a guitar string, plucked
at different locations along the string. Our method is based
on the Fourier analysis of the signal measured by the mag-
netic guitar pickup placed at a selected location along the
string. With such, the students can clearly see that the ampli-
tudes of the fundamental and higher harmonics in the string
with both ends fixed strongly depend on the location where
the string is plucked. A magnetic pickup connected to an am-
plifier, while being an indirect method, allows students to
perform measurements directly with a guitar, selecting any
string or even several strings played in chords. This is unlike
the optical arrangement with a single string, combined with
an optical system for a direct determination of the mechani-
cal displacement as described in Refs. 3and 4.
The expected harmonic content of the standing waves in a
string plucked at a particular location can be predicted from
a Fourier analysis of the initial shape (triangle) of the string
pulled at this location. From the Fast Fourier Transform
(FFT) of the magnetic pickup signal, the amplitudes of the
harmonics in the transverse oscillations of the sting can be
determined, and a quantitative comparison of the experimen-
tal and calculated results can be performed. This work has
been running at our school as an advanced physics lab for
undergraduate physics majors, giving them an opportunity to
experience wave phenomena, sound waves, standing waves
in strings, and the role of harmonics in the timbre of musical
instruments. The computational part of the lab requires that
the students perform a Fourier analysis of a periodic signal,
allowing for quantitative analysis of experimental data in
determining the harmonic composition of standing waves in
strings.
In Sec. II, the amplitudes of the first and higher harmonics
of resonant standing waves in a string are calculated as the
functions of the location where the string is plucked. In Sec.
III, the experimental setup is described in sufficient detail for
setting up and running the lab. Section IV represents the ex-
perimental results on relative amplitudes of harmonics in the
signals picked up by a sensor (magnetic pickup) while play-
ing a string of the guitar. The results are compared with
38 Am. J. Phys. 84 (1), January 2016 http://aapt.org/ajp V
C2016 American Association of Physics Teachers 38
calculated amplitudes of the harmonic, in the standing waves
of the guitar string. Finally, we conclude by discussing
results and the methods from the point of view of the impact
this laboratory work can have on a student’s interest in
physics, better understanding of concepts of wave phenom-
ena, and mastering their skills in experimenting and analyti-
cal work.
II. FOURIER SERIES OF AN INITIAL SHAPE OF A
STRING AND HARMONICS OF A STANDING WAVE
When the same note is played on a guitar by plucking a
string at different locations, one can hear that the timbre of
the instrument is notably different. The sound produced by
the string contains the fundamental frequency and higher
harmonics. Relative amplitudes of the harmonics depend on
the location where the string has been plucked.
The frequencies and amplitudes of the harmonics can be
experimentally measured using an FFT analysis of the oscil-
lations measured with a sensor. The expected initial ampli-
tudes of the harmonics when the string is plucked in a
particular manner can be calculated as the Fourier coeffi-
cients of the infinite harmonic series representing the initial
shape of the string. While the initial shape of a guitar string
is not exactly triangular due to the bending stiffness of a
metal string, we assume that the bending stiffness can be
neglected when analyzing the role of the position where the
string is plucked.
For a string of length Lfixed at both ends (x¼0 and
x¼L), the transverse displacement y(x,t) depends on both
the position xand the time tand satisfies the one-
dimensional wave equation
7
@2y
@x2¼1
v2
@2y
@t2;(1)
where v¼ffiffiffiffiffiffiffiffiffiffi
FT=l
pis the wave speed, F
T
is the tension force
in the string, and l¼m=Lis the mass of the string per unit
length.
For the case of the string plucked at some position and
released at t¼0, the initial conditions can be described as
y
0
(x,0) ¼f(x), where f(x) is the initial shape of the string, and
ð@y=@tÞt¼0¼0 for any x. The solution of Eq. (1) with such
initial conditions is
7
yx;t
ðÞ
¼X
1
n¼1
bnsin npx
Lcos npvt
L¼X
1
n¼1
yncos npvt
L;(2)
where yn¼bnsinðnpx=LÞand bnis the amplitude of the nth
harmonic. The resonant frequencies of the harmonics are
given by the equation for a string fixed at both ends
fn¼nv
2L:(3)
Such frequencies of the oscillations (harmonics) produced
by a string can be varied by changing the force of tension in
the string (as when tuning the instrument) or the length Lof
the string’s vibrating section (by pressing a finger against the
string at different locations). For each harmonic, the dis-
placement y
n
(x,t) due to this harmonic is zero at x¼0, x ¼L,
and at x¼L/n. The coefficients b
n
represent the amplitudes
of the harmonics and can be calculated from the initial shape
of the string y
0
(x,0) ¼f(x) as
bn¼2
LðL
0
fx
ðÞ
sin npx
L

dx:(4)
For the string of length Lpulled at the position x
1
from the
end of the string a distance y¼A(see Fig. 1), f(x) is given by
fx
ðÞ
¼A
x1
xfor 0 xx1(5)
and
fx
ðÞ
¼A
Lx1
Lx
ðÞ
for x1xL:(6)
Hence, the amplitudes of the Fourier components (harmonics)
of the initial shape of the string can be calculated as
bn¼2A
Lx1ðx1
0
xsin npx
L

dx
þ2A
LLx1
ðÞ
ðL
x1
Lx
ðÞ
sin npx
L

dx;(7)
with the final result
bn¼2Asin npx1=L
ðÞ
x1
L1x1
L

p2n2
:(8)
As a consistency check for Eq. (8), we observe that by
symmetry the Fourier amplitude for each harmonic should
have the same magnitude when the string is plucked at equal
distances on either side of the middle of the string. This
means that if we write x
1
in terms of distance from the mid-
dle as
x1¼L
2þaL
2¼L
21þa
ðÞ
;(9)
where 1<a<1 is the fraction of the distance from the
middle to the end of the string, then b
n
should have the same
magnitude for either sign of a. Inserting Eq. (9) into Eq. (8),
we obtain
bn¼
8Asin np
2þanp
2

1a2
ðÞ
p2n2:(10)
The denominator is clearly unchanged for either sign of a;as
is the numerator because the magnitude of the sine function
is symmetric about np=2 for any n.
We saw earlier that the spatial dependence of the nth har-
monic is given by ynðxÞ¼bnsinðnpx=LÞ. We can therefore
write the amplitude of the nth harmonic as
Fig. 1. Initial shape of the string of length Lpulled a distance Aat the posi-
tion x
1
from its end.
39 Am. J. Phys., Vol. 84, No. 1, January 2016 Perov, Johnson, and Perova-Mello 39
bn¼ynx
ðÞ
sin npx=L
ðÞ
:(11)
Using Eq. (8), we can write the ratio of the nth to the first
Fourier amplitudes as
bn
b1
¼sin npx1=L
ðÞ
n2sin px1=L
ðÞ
;(12)
where x
1
indicates the location at which the string was
plucked.
We can obtain an experimental value of this ratio if we
use a magnetic pickup at location x
m
and do an FFT to deter-
mine the amplitudes of vibration ynðxmÞand y1ðxmÞ. From
Eq. (11), the experimental ratio of Fourier amplitudes in
terms of these measurements is
Bn
B1
¼ynxm
ðÞ
sin pxm=L
ðÞ
y1xm
ðÞ
sin npxm=L
ðÞ
:(13)
Here, we have used the notation Bn=B1to indicate measured
Fourier amplitude ratios to distinguish from predicted ratios
bn=b1, based on initial shape of the plucked string.
The idea of this lab is to measure relative amplitudes of
the harmonics in the sensor’s signals while plucking the
string at different locations along the string. We then calcu-
late the amplitudes of harmonics in the standing waves in the
strings from the amplitudes of the harmonics in the measured
signals to determine the ratio B
n
/B
1
, which are compared to
the theoretically determined amplitudes of harmonics, calcu-
lated from the initial shape of the string as given by Eq. (12).
III. EXPERIMENTAL SETUP
The experimental setup is shown in Fig. 2. An electric gui-
tar was used in our experiments, but the active built-in pick-
ups of the electric guitar were not used, to exclude the effect
of the frequency response of the built-in amplifiers on the
recorded waveform from the pickup coil. Instead, a single
coil Neo-D passive pickup
8
was placed above the strings at a
selected distance (14.8 cm in this study) from the guitar’s
bridge using a stand with a clamp. The pickup’s signal was
amplified by a Studio Linear Amplifier SLA-1 (Applied
Research and Technology), and the amplified signal was
recorded using an Agilent DSO6012A digital oscilloscope,
which has a Fast Fourier Transform (FFT) feature for analy-
sis of measured signals. In addition to the recorded wave-
form, the oscilloscope displayed the FFT spectrum of the
signal, with the frequency (in Hz) along the x-axis and the
harmonics amplitude (in dBV) along the y-axis. The external
trigger of the oscilloscope was connected to a 6 -V battery
via a manual switch, so that a single run of the experiment
could be initiated by pressing the switch after the string was
plucked.
IV. RESULTS AND DISCUSSION
As a first step, the frequencies f
n
of the harmonics of the
strings of different lengths (pressed on different frets) were
measured, using the FFT on the oscilloscope, to confirm that
the measured frequencies were well described by Eq. (3).
Next, the open string #1 (the thinnest one) was plucked at
a particular location, and the voltage from the magnetic
pickup over one time sweep of the oscilloscope was used to
produce the FFT signal. An example of the FFT spectra is
shown in Fig. 3as an oscilloscope screen shot. The horizon-
tal axis is frequency and the vertical axis is signal intensity
bV, measured in decibel-volts (dBV). The intensity of the
signal is proportional to the square of the voltage, so we
have
bV¼20 log V
V0
:(14)
If the difference in two amplitudes in dBV is taken from the
FFT, then the ratio of the voltages at the oscilloscope input
can be calculated from
bVnbV1¼20 log Vn
V1
:(15)
This equation tells us that a 20 dBV difference in amplitudes
of two harmonics would correspond to the ratio V
2
/V
1
¼10
of the two amplitudes.
Fig. 2. Experimental setup. The pickup is held by a stand and connected to
the input of the linear amplifier SLA-1. The output of SLA-1 is connected to
the input of the oscilloscope, which has the FFT Math function selected. The
upper trace shows the FFT of the recorded signal (lower trace).
Fig. 3. The top trace on the screen displays the FFT of the guitar pickup sig-
nal on a logarithmic scale. The scales: 500 Hz/div (horizontal) and 20 dBV/
div (vertical). Notice that the 13th harmonics is not seen, because it has a
node at the selected magnetic pickup position x
m
¼14.8 cm. This missing
harmonic can be understood from Eq. (11). With n¼13, x
m
¼14.8 cm, and
L¼64.3 cm, we have nx
m
/L ¼3.00, so sin(npx
m
=L)¼0, and the pickup sig-
nal of the 13th harmonic should be zero.
40 Am. J. Phys., Vol. 84, No. 1, January 2016 Perov, Johnson, and Perova-Mello 40
The background in the FFT spectrum changes slightly
with frequency, and this fact could lead to errors when taking
the difference of two peak heights located at different fre-
quencies. However, in the range of the harmonics up to the
seventh (analyzed in this work) the change was less than the
noise level as seen in Fig. 3, and was ignored. If analysis of
the eighth and higher harmonics were necessary, then a
background level analysis would be required. The cause of
the background is unknown, but we have noticed that the
change in the background was larger in earlier experiments
with an acoustic guitar compared to the current experiments
with an electric guitar. We speculate that the guitar body
response to the vibrations at different frequencies could be
one of the causes.
The amplitudes bn(in dBV) of the fundamental and sev-
eral higher harmonics of the signal were measured from the
FFT and the measurements were repeated several times for
each pluck location of the string. The ratios V
n
/V
1
of the har-
monic amplitudes (for n¼2–7) of the measured waveforms
were calculated for each harmonic as averages from multiple
measurements using Eq. (15). We can relate this ratio of vol-
tages for the harmonics to the ratio of each harmonic’s am-
plitude as discussed below. The relative amplitudes of
harmonics in the measured signal depend not only on the
amplitudes of harmonics of standing waves in a string, but
also on other factors such as the position of the sensor along
the string, frequency dependences of the sensor and the am-
plifier, the decay times for different harmonics in the string,
and the background.
For quantitative comparison of the measured sensor signal
harmonics, given by B
n
/B
1
, to the calculated (expected) har-
monics of the standing waves, given by b
n
/b
1
, we have taken
into account the factors mentioned above as follows. First, a
magnetic pickup is always placed very close to the string (a
few mm) so the position of the sensor can easily be taken
into account while calculating the harmonic amplitudes of
the standing waves in the string from the FFT analysis of the
measured signal. At the distance x
m
from the bridge, the
string displacement y
n
due to the n
th
harmonic with ampli-
tude B
n
is given by Eq. (2):yn¼Bnsinðnpxn=LÞ. With this,
the relative amplitudes of measured harmonics depend on
the relative displacements of the string at the pickup location
x
m
due to the particular harmonics of the standing wave as
shown in Eq. (13).
Second, the signal voltage (emf) is induced in the
magnetic pickup due to motion of a metal string near
the pickup’s coil. According to Faraday’s Law, the
induced emf is proportional to the time rate of change
of the magnetic flux U. As a result, we can expect that
the contribution of the string displacement y
n
due to the
nth harmonic of the standing wave in the string to the
signal voltage V
n
will be proportional to a frequency of
the harmonic f
n
¼nf
1
. Hence, we can assume that
Vn=V1¼nðyn=y1Þ. Combining this factor with Eq. (13),
the relative amplitudes of harmonics B
n
/B
1
of the stand-
ing wave in the string can be calculated from the rela-
tive amplitudes of the magnetic pickup signal V
n
/V
1
as
Bn
B1
¼Vn
V1
sin pxm=L
ðÞ
nsin npxm=L
ðÞ
:(16)
This equation can be used for quantitative comparison of the
experimental data on the relative amplitudes B
n
/B
1
of the
harmonics of the standing waves to the relative Fourier com-
ponents b
n
/b
1
of the initial shape of the string. The relative
amplitudes of harmonics of the string displacement B
n
/B
1
have been calculated from the relative amplitudes of the har-
monics in the measured signal V
n
/V
1
, and are presented in
Table I.
Using Eq. (12), the Fourier components were calculated
for different locations x
1
of where the string was plucked.
The calculated relative amplitudes b
n
/b
1
are shown in Fig. 4,
together with the experimentally measured amplitudes for
harmonics 2–7. The correlation between the measured ampli-
tudes of the standing waves and the Fourier components of
the initial shape of the string is very good.
As an extension of this experiment, the decay rates of dif-
ferent harmonics can be studied by triggering the oscillo-
scope later in time relative to when the string is plucked. Our
preliminary experiments indicate that the higher harmonics
decay faster compared to the fundamental. This results in the
string sound timbre being more “crisp” initially, to a more
“flat” (or pure) sound (containing mostly the first harmonic),
regardless of where the string is plucked. For the analysis of
the decay rates of different harmonics, a recently published
paper
9
with analysis of the motion of a single Fourier mode
as an example of a transient, free decay of coupled linear
oscillators looks very promising.
The statistical errors expected in our measurements
depend on the measurements from the digital oscilloscope
and the location of the magnetic pickup x
m
, and their corre-
sponding errors. The details for the calculations are shown in
the Appendix.
V. CONCLUSION
As a result of this laboratory exercise, students have
the opportunity to understand that harmonic composition
of standing waves in the strings of musical instruments
is strongly dependent on the way the strings are played.
Generally, prior to the lab, students assume that playing
a specific note on any instrument produces oscillations
only at this frequency. Observing visually that other har-
monics are also present and comparing this observation
with how the note sounds when the string is plucked at
different locations allows students to make connections
between the sound waves produced and complex
Table I. Measured average relative amplitudes of harmonics of the sensor
signal recorded when open string #1 (the thinnest one) was plucked at differ-
ent locations x
1
. The string length L¼64.3 cm and the sensor was a distance
14.8 cm from the bridge.
Fret x
1
(cm) x
1
/LB
2
/B
1
B
3
/B
1
B
4
/B
1
B
5
/B
1
B
6
/B
1
B
7
/B
1
1 3.7 0.058 0.544 0.367 0.279 0.155 0.104 0.074
2 7.1 0.110 0.462 0.232 0.163 0.072 0.034 0.009
3 10.4 0.162 0.453 0.220 0.115 0.015 0.018 0.024
4 13.4 0.208 0.392 0.159 0.060 0.024 0.027 0.020
5 16.3 0.253 0.408 0.123 0.032 0.052 0.031 0.009
6 19.0 0.295 0.364 0.052 0.071 0.044 0.013 0.020
7 21.6 0.336 0.297 0.023 0.087 0.027 0.015 0.020
8 24.0 0.373 0.236 0.070 0.101 0.010 0.029 0.014
9 26.3 0.409 0.172 0.094 0.082 0.027 0.024 0.010
10 28.5 0.443 0.110 0.115 0.065 0.034 0.016 0.015
11 30.5 0.464 0.061 0.120 0.045 0.038 0.012 0.018
12 32.5 0.505 0.019 0.128 0.026 0.032 0.015 0.014
41 Am. J. Phys., Vol. 84, No. 1, January 2016 Perov, Johnson, and Perova-Mello 41
mechanical oscillations of the string. Solving the wave
equation with boundary conditions according to the ini-
tial shape of the string allows students to clearly see
how the wave theory is related to the sounding of the
musical instruments.
ACKNOWLEDGMENTS
The authors very much appreciate the comments and
suggestions provided by the reviewers, which helped to
significantly improve both the experiment and its description
and analysis.
APPENDIX: ERROR CALCULATIONS
We obtain measurements of the ratio of amplitudes
as given by Eq. (16) and designate the dbV differen-
ces between the nth and the first harmonic as dnso
that
Fig. 4. Relative amplitudes of harmonics (a) n¼2 through (f) n¼7 of standing waves in a string calculated from Eq. (1) and from the experimentally meas-
ured signals as a function of the pluck location x
1
/L. The points are determined experimentally as B
n
/B
1
from Eqs. (16) and (A9) while the theory (solid) is cal-
culated as b
n
/b
1
from Eq. (12).
42 Am. J. Phys., Vol. 84, No. 1, January 2016 Perov, Johnson, and Perova-Mello 42
dn¼bVnbV1¼20 log Vn
V1
(A1)
and
Vn
V1
¼100:05dn:(A2)
To simplify notation, we designate the ratio of interest as
Rn¼Bn=B1, which depends on the measured quantities dn
and xmso that Rn¼Rnðdn;xmÞ. The squared error in Rnis
thus given by
r2
Rn¼@Rn
@dn

2
r2
dnþ@Rn
@xm

2
r2
xm:(A3)
From Eqs. (16) and (A2) we have
Rn¼Bn
B1
¼100:05dnsin pxm=L
ðÞ
nsin npxm=L
ðÞ
;(A4)
@Rn
@dn
¼sin pxm=L
ðÞ
nsin npxm=L
ðÞ
@
@dn
100:05dn
ðÞ
;(A5)
and
@
@dn
100:05dn
ðÞ
¼0:05ln10 100:05dn
ðÞ
:(A6)
So from Eq. (A5) we get
@Rn
@dn
¼0:05ln10 100:05dn
ðÞ
sin pxm=L
ðÞ
nsin npxm=LðÞ
:(A7)
The last term of Eq. (A3) contains @Rn=@xm, which gives
@Rn
@xm
¼pVn
LV1cos pxm=L
ðÞ
nsin npxm=L
ðÞ

sin pxm=L
ðÞ
cos npxm=L
ðÞ
sin2npxm=L
ðÞ

:(A8)
The error squared is then given by Eq. (A3) with these terms
inserted and noting that Rn¼Bn=B1, giving
r2
Bn=B1¼0:05ln102100:05dn2sinðpxm=LÞ
nsinðnpxm=LÞ2
r2
dn
þp
L100:05dn2cosðpxm=LÞ
nsinðnpxm=LÞ
sinðpxm=LÞcosðnpxm=LÞ
sin2ðnpxm=LÞ2
r2
xm:(A9)
The quantity rxmwas estimated as 3 mm, while the quantity
dnwas measured 5 times for each fret and the mean value
and error in the mean were used for calculations. If several
identical measurements occurred and the error in the mean
was less than 6half the least count (0.625), then the larger
value for rdnwas used (rdn¼0:625=2).
a)
Electronic mail: pperov@suffolk.edu; Permanent address: Suffolk
University, 8 Ashburton Pl, Boston, Massachusetts 02108
b)
Electronic mail: wjohnson@suffolk.edu
c)
Electronic mail: nperova@gmail.com
1
M. C. Wittmann, “The object coordination class applied to wavepulses:
analyzing student reasoning in wave physics,” Int. J. Sci. Educ. 24(1),
97–118 (2002).
2
M. C. Wittmann and E. F. Redish, “Understanding and affecting student
reasoning about sound waves,” Int. J. Sci. Educ. 25(8), 991–1013
(2003).
3
J. L. Sandoval and A. V. Porta. “Fourier analysis for vibrating string’s pro-
file using optical detection,” Am. J. Phys. 53(12), 1195–1203 (1985).
4
S. B. Whitfield and K. B. Flesch, “An experimental analysis of a vibrating
guitar string using high-speed photography,” Am. J. Phys. 82(1), 102–109
(2014).
5
E. Kashy, D. A. Johnson, J. McIntyre, and S. L. Wolfe, “Transverse stand-
ing waves in a string with free ends,” Am. J. Phys. 65(4), 310–313 (1997).
6
W. Haerbell, “Laboratory exercises on oscillation modes of pipes,” Am. J.
Phys. 77(3), 204–208 (2009).
7
M. L. Boas, Mathematical Methods in the Physical Sciences, 2nd ed. (John
Wiley & Sons, Hoboken, NJ, 1983), p. 554.
8
Neo-D magnetic sound-hole pickup, Fishman Transducers, Inc.,
<http://www.fishman.com/products/details.asp?id¼49>.
9
D. Politzer. “The plucked string: an example of non-normal dynamics,”
Am. J. Phys. 83(5), 395–402 (2015).
43 Am. J. Phys., Vol. 84, No. 1, January 2016 Perov, Johnson, and Perova-Mello 43
  • Article
    Full-text available
    This article describes a novel and interdisciplinary context for learning about real-world sound waves. The 'song' of the plainfin midshipman fish consists of an acoustical wave that is periodic but not sinusoidal. This acoustical signal is the focus of active research in sound source localization by fishes, the effects of hormones on hearing systems and the elucidation of neural mechanisms involved in social acoustic communication. In this paper, we introduce the reproductive biology and bioacoustics of the midshipman fish. We describe the use of the advertisement 'song' of the male fish to visualize and interpret the dramatically different displacement, velocity and acceleration waveforms, to explore the roles of pressure and particle motion in production and detection of acoustical waves and to apply Fourier analysis to understand the implications of the frequency spectrum in the production, transmission and reception of sound in an aquatic environment.
  • Article
    We develop a low-cost and simple experiment to visualise Fourier’s synthesis using a short, soft, and light plastic coiled spring oscillating in a horizontal plane, and a basic camera (120 fps). It is shown that the spring obeys a linear wave differential equation, as gravitational influence is neglected. A nonlinear criterion is evaluated to determine if magnitudes of the parameters in the initial conditions satisfy the linear wave equation. Our setup promotes some desirable characteristics that make Fourier’s synthesis experiments feasible, visual, and enlightening: (i) it requires few, common, and cheap resources, and the experiment can be carried out even in a high-school laboratory; (ii) since the spring’s tension is small (∼1 N, on average), the frequencies of normal modes are low (close to 2 Hz), and therefore, it is possible to record the oscillations just with the camera and extract a considerable number of position and time data in just one cycle; (iii) when the video is loaded in the Tracker free software, it can be reproduced in slow motion. Since the frequencies involved are low, an interesting and instructive temporal sequence of images of the spring displaying the typical trapezoidal shape appears clearly; (iv) the tools associated with the Tracker software tools can yield the relevant oscillation parameters, such as the damping constant, amplitudes, frequencies, and phases; and (v) it is possible to carry out superposition of a snapshot of the spring in Tracker at any time, and to draw the related Fourier synthesis graphs. The visual match between the shape of the spring and the theoretical graph is remarkable, and can be enhanced by adding the damping term.
  • Article
    Full-text available
    Motion of a single Fourier mode of the plucked string is an example of transient, free decay of coupled, damped oscillators. It shares the rarely discussed features of the generic case, e.g., possessing a complete set of non-orthogonal eigenvectors and no normal modes, but it can be analyzed and solved analytically by hand in an approximation that is appropriate to musical instruments' plucked strings.
  • Article
    We use high-speed photography (1200 frames/s) to investigate the vibrational motion of a plucked guitar string over several cycles. We investigate the vibrational pattern for plucking the string at two different locations along the string's length, and with different initial amplitudes. The vibrational patterns are then compared to a standing wave model of the string vibrations. We find excellent agreement between the observed vibrational patterns and the model for small-initial-amplitude displacement of the string. For larger amplitude displacements, the qualitative behavior of the string's vibrational pattern differs significantly from the small-amplitude displacement. This behavior may be due to the presence of inharmonicity, as suggested by its incorporation into the model calculations.
  • Article
    This paper describes an improved lab setup to study the vibrations of air columns in pipes. Features of the setup include transparent pipes which reveal the position of a movable microphone inside the pipe; excitation of pipe modes with a miniature microphone placed to allow access to the microphone stem for open, closed, or conical pipes; and sound insulation to avoid interference between different setups in a student lab. The suggested experiments on the modes of open, closed, and conical pipes, the transient response of a pipe, and the effect of pipe diameter are suitable for introductory physics laboratories, including laboratories for nonscience majors and music students, and for more advanced undergraduate laboratories. For honors students or for advanced laboratory exercises, the quantitative relation between the resonance width and damping time constant is of interest.
  • Article
    In this paper, we describe an experimental method to analyze the vibration of a string fixed at both ends using the symmetry between time and space behavior. An optoelectronic sensor detects the motion of a point directly displayed on an oscilloscope. The plot of this motion as a function of time is associated with the space behavior of the whole string. The Fourier spectrum of the obtained shape is used to compare experimental results with theoretical predictions. This detection method may also be used to analyze damping effects. The experimental part is intended for intermediate and sophomore university students, but it may be realized for high school students just as a demonstrative experiment.
  • Article
    A method to excite and visualize transverse standing waves on a vibrating string with one or both ends free is described. A stretched coil extension spring serves as the flexible vibrating string. A free end is achieved by attaching one end to a long thin steel wire that stretches the spring and thus provides tension with a nearly inertia-free suspension. The apparatus can be used as a qualitative lecture demonstration or as a quantitative experiment where it can serve as an instructive example of the application of boundary conditions.
  • Article
    Scitation is the online home of leading journals and conference proceedings from AIP Publishing and AIP Member Societies
  • Article
    Full-text available
    Detailed investigations of student reasoning show that students approach the topic of wave physics using both event-like and object-like descriptions of wavepulses, but primarily focus on object properties in their reasoning. Student responses to interview and written questions are analysed using diSessa and Sherin's coordination class model which suggests that student use of specific reasoning resources is guided by possibly unconscious cues. Here, the term reasoning resources is used in a general fashion to describe any of the smaller grain size models of reasoning (p-prims, facets of knowledge, intuitive rules, etc) rather than theoretically ambiguous (mis)conceptions. Student applications of reasoning resources, including one previously undocumented, are described. Though the coordination class model is extremely helpful in organising the research data, problematic aspects of the model are also discussed. Comment: 20 pages, 8 figures, 27 references
  • Article
    Full-text available
    Student learning of sound waves can be helped through the creation of group-learning classroom materials whose development and design rely on explicit investigations into student understanding. We describe reasoning in terms of sets of resources, i.e. grouped building blocks of thinking that are commonly used in many different settings. Students in our university physics classes often used sets of resources that were different from the ones we wish them to use. By designing curriculum materials that ask students to think about the physics from a different view, we bring about improvement in student understanding of sound waves. Our curriculum modifications are specific to our own classes, but our description of student learning is more generally useful for teachers. We describe how students can use multiple sets of resources in their thinking, and raise questions that should be considered by both instructors and researchers. Comment: 23 pages, 4 figures, 3 tables, 28 references, 7 notes. Accepted for publication in the International Journal of Science Education