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 v¼ﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ

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