unplugged: the physics of a acoustic guitar

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Session

IV

April 27,

2012

American University, Washington, DC

12
th

Annual New Millennium Conference

81

UNPLUGGED: THE PHYSICS OF A
ACOUSTIC GUITAR


James Reber

American University

4400 Massachusetts Ave. NW, Washington, DC. 20016
-
8058, jr4063a@student.american.edu


Abstract


The acoustic guitar is one of the world

s most
easily recognized instruments.

This paper discusses the
physics of
the

acoustic guitar

with a focus on the
soundboard
.
The paper

begins with a discussion of
what
sound is and how it travels
,

focusing on the aspects
important to understanding how a guitar works
.
The paper

then

discusses the most common bracing patterns for

nylon
and steel string guitar
s
, and how the final shaping of the
braces affects both the strength and th
e performance of the
instrument. For the shaping of the braces, this paper
discusses the

physics behind the

free plate tuning technique.
Furthermore, t
his paper looks at
experimental bracing
techniques and the reasoning behind the different patterns.



Index Terms


A
coustic
guitar,

bracing

patterns
,

soundboard
,

sound w
aves,


I
NTRODUCTION
&

G
UITAR
D
ESIGN


The modern acoustic guitar is the descendent of
stringed
instruments dating back from some of the earliest
civilizations.

Predecessors to the guitar include instruments
such as the bowl harp and the lute.
Physicist and guitar
builder Dr. Michael Kasha says that t
he
basic

characteristics

of a guitar are
the
f
retted neck, wooden soundboard with ribs
(bracing) and its flat back and curved

sides

[1]
.

While
instruments very similar to the guitar
were being built
in the
16
th

century [2], t
he modern guitar design
didn’t begin

to
appear at the beginning of the 1
9
th

century

[3].
Figure 1
shows the design of the modern guitar.






F
IGURE

1

[4]

T
HE
A
NATOMY OF AN
A
COUSTIC
G
U
I
TAR

All acoustic guitars share similar
anatomy
.

The strings
are fastened to a

guitar at the bridge and the peg head, and
vibrate between the nut
at the end of the neck
and the saddle

on the bridge
.
The twelfth fret is located at exactly the
middle of the strings.
The tuners allow the tensi
on of the
strings to be chan
ged
, and the
bridge transfers the vibrations
to the soundboard
. When a note is played, the strings vibrate
both the top of the guitar as well as

the air inside the body of
the guitar
. These vibrations on the top and in the body of the
guitar produce soun
d
.


S
OUND


In o
rder to understand how a guitar works, one must first
understand how sound
behaves
.

Sound travels in the form of
a wave. There are two types of sound waves, longitudinal
and transverse.

Longitudinal
waves travel parallel to the
source of the wave and trans
verse waves travel
perpendicular to the source of the wave.

Sound is produced
from vibrations through a medium, and travels in the form of
longitudinal waves. The frequency at which the vibrations
occur is referred to as the pitch. In music, pitch
relates

to the
note being played.

The

vibration
s caused from a disturbance such as a
vibrating string
create areas of compression and rarefaction
of the molecules in the medium that the vibration
s

are

traveling through.
Sound is only produced when these
vibration
s are traveling through a medium.
An observer

is

able to hear sounds because these areas of compression and
rarefaction are picked up by
the observer’s

ears, and
translated to the brain.
Figure 2

shows the areas of
compression and rarefaction in the air in a tube caused by
the vibrations from a tuning fork.
Compressions are areas of
with a high density of molecules and rarefactions are areas
with a low density of molecules.





FIGURE 2

[6
]

C
OMPR
ESSIONS AND
R
AREFACTIONS



Session

IV

April 27,

2012

American University, Washington, DC

12
th

Annual New Millennium Conference

82

F
ORCED
V
IBRATIONS


The volume of a sound depends on the amount of air that is
moving due to the vibrations created from the sound source.

When an object that is vibrating is held against another
object, the second object will
also vibrate. This is called
forced vibration. If the second object has a larger surface
area, the vibrations will move more air, making the volume
louder
.
One of the main physics principles that apply to an
acoustic guitar is forced vibrations.
Forced vib
rations occur
when a vibrating object forces another object to vibrate.
Hewitt uses the example of a tuning fork placed on a table.
The vibrating tuning fork forces the table to vibrate at the
same frequency
. The vibrating table moves more air
molecules th
an the tuning fork alone, and t
herefore produces
a higher volume

[7]
. On a guitar, the vibration of the strings
is transferred to the soundboard through the bridge, forcing
the soundboard to vibrate at the same frequency as the
string
, which produces sound
.


H
OW A
G
UITAR
F
UNCTIONS


There are two different kinds of guitars: classical and folk.
The main difference

between a classical and a folk guitar

is
the type of strings used. Classical guitars use nylon strings
and folk guitars use steel strings. Both ty
pes of guitars are
tuned to the same frequencies, but the nylon strings have a
much lower density than steel strings, and therefore the
tension on the soundboard of the guitar is much less for a
classical guitar than a folk guitar
.

The bracing for a steel
string guitar must be much stronger than the bracing of a
classical guitar in order to handle the increased force on the
soundboard due to the steel strings.

When a note is played on the guitar, the

string vibrates
back and fourth producing a wave.

The frequency of the note
is determined by the velocity of the

wave on the

string
divided by the wavelength, shown in (1).




(1)




A guitar has six strings, each one of a different thickness and
tuned to a different frequency.
The strings must be different
thicknesses because the velocity of a wave on a string
depends on the tension
,

and the linear mass density of the
string, shown

in (2)
, where


is the velocity,

is the tension
and

is the linear mass density
.
The thicker strings have a
lower frequency, and the thinner strings have a higher
frequency.

This is because thicker strings have a higher
linear mass density, which reduces the velocity of the
wave
on the
string

for a given tension, and the result is a lower
frequency.

Linear mass density is expressed in (3)
, where

is the
mass of the string and


is the length of the string
.





(2)




(3)



Regardless of whether or not a guitar is strung with
nylon or steel strings, the frequency of the open strings
,

or
the frequency produced when a string is not fretted,

is not
changed. Depending on the size of
the instrument (and
therefore the wavelength of the string), the velocity will
change.
For a given frequency, a long wavelength means

a
lower velocity, and a short wavelength means a higher
velocity. The strings on all guitars in standard tuning are
tuned to the same frequencies, despite the length of the
strings. Small guitars are tuned to the same frequencies as
larger guitars.
Table 1

shows the pitch as well as the
frequencies of a guitar in standard tuning
. The number after
the letter name of the pitch refers to the octave at which the
pitch is heard

[8]
.


TABLE 1

G
UITAR
S
TRING
N
OTES AND
F
REQUENCIES


String

Pitch

Frequency

1

E2

82 H
z

2

A2

110 Hz

3

D3

147 Hz

4

G3

196 Hz

5

B3

243 Hz

6

E4

330 Hz


The tension on the bridge of the guitar due to the strings
varies

depending on the material of the strings due to the
change in the linear mass density. Table 2 compares the
tension of

the most common gauge of

n
ylon and steel
strings
. The
tensions

shown in Table 2

are based on the most
common scale length of each style of guitar; 25.6 for nylon,
25.4 for steel.


TABLE 2
[9]

N
YLON AND
S
TEEL
S
TRING
T
ENSIONS


String

Nylon (Normal Tension)


Steel (Light Gauge)

1
st

E

14 Lbs

25.1 Lbs

2
nd

A

15 Lbs

28.4 Lbs

3
rd

D

15.6 Lbs

29.5 Lbs

4
th

G

12.1 Lbs

29.4 Lbs

5
th

B

11.6 Lbs

23.3 Lbs

6
th

E

15.3 Lbs

23.3 Lbs

Total

83.6 Lbs

159 Lbs


In order to change the note being played, the player
changes
the wavelength of the string by shortening the
length of the string

with their fingers
. The velocity of the
wave on the string remains constant, and by shortening the
length of the string the frequency increases

[10
]
.

The wavelength for a string on a
guitar depends on the
scale length of a guitar, or the distance from the nut

near the
headstock of the guitar

to the
saddle

on the bridge,

with the
Session

IV

April 27,

2012

American University, Washington, DC

12
th

Annual New Millennium Conference

83

twelfth fret being in the middle
.
The first harmonic, or

t
he
fundamental, shown in F
igure
3

shows the motion

of a
plucked open string

(vibrating between the nut and the
saddle)
, which is half the wavelength of the wave on the
string. This is known as the first harmonic.
Harmonics are
whole number multiples of the fundamental frequency. The
second harmonic is for
med
on a guitar
by crea
ting a node at
the twelfth fret

and is twice the frequency of the open string
.
Creating a node at the fifth or seventeenth fret forms the
third harmonic and the frequency is three times higher than
the open string

[11
]
.




FIGURE

3

[12
]

W
AVES ON A
G
UITAR
S
TRING


There are two different
p
henomena

that make the
conversion of the mechanical energy from the player
plucking a string to sound energy more efficient
, and
therefore louder
.

Dr. Wolfe notes that a guitar does not
amplify the sound fro
m the vibrations of the strings
.

First,
when the string vibrates above the sound hole of the guitar,
the vibrations of the strings create

areas of compressions and
rarefaction in the air around t
he sound

hole
.

These
vibrations

compress

the
air inside the body of the guitar, which raises
the internal pressure. The air is then forced out due to the
high pressure. This is referred to as Helmholtz resonance.

The vibration of the air inside the body of

a guitar

mostly
affects the lower frequencies,
so a guitar with a smaller body
would produce softer low frequencies.
This becomes
apparent when looking at the violin family of instruments.
The lower pitched instruments such as the cello or bass have
large
r bodies than the violin of viola.

The
other

way that a guitar converts the mechanical
energy to sound energy is through the vibration of the top of
the guitar. The top or soundboard is designed to vibrate, and
because of its large surface area, the vibra
tions move more
air than the string alone

could
.
The vibrating soundboard

is
an example of forced vibrations.

The strings vibrate against
the bridge, which forces the soundboard to vibrate.
The
soundboard projects the higher frequencies of a guitar

into
the air around the guitar
. The more surface area of the
soundboard, the louder the
produced
frequencies are [13]
.

The combination of the low frequencies

projected

from the
vibration of the air inside the body of the guitar as well as
the high
frequen
cies

projected from the vibration of the
soundboard gives the guitar its unique sound.



T
HE
O
VERTONE
S
ERIES


Every instrument
’s

sound

quality or timbre is

different

due
to the

instrument’s unique

overtone series.
When a note is
played on a guitar, the pit
ch that is heard is the fundamental
frequency.
There is also a c
ombination of other frequencies,
known as partial
tones

or harmonics
, which

are emitted

along with the fundamental frequency.

The volumes of these
partial tones affect the timbre

or tone

of the instrument.
The
amount of partial tones and the volume of each partial tone

for a note
make every instrument

sound different
.
The
combination of these partial tones is known as the overtone
series

[14].
Figure 4 shows
an

example
of the amplitude of

the different frequencies produced from a single note played
on a guitar, with the first peak being the fundamental
frequency
.

The x
-
axis
shows

the frequency
of each overtone
and the y
-
axis shows the intensity of each of those
frequencies.




FIGURE

4

[15]

G
UITAR
O
VERTONE
S
ERIES


The overtone series is unique for not only every
instrument, but also every guitar. There are several
components that affect the overtone series of a guitar,

from
the woods used for the instrument
to how the strings are
plucked
.

Arguably the leading factor that affects the
overtone series of a guitar is the bracing on the underside of
the soundboard.


S
OUNDBOARD
B
RACING


The purpose of the bracing of a guitar is to both provide
support against the soundboard warping due to the
tension of
the strings as well as help transfer the vibrations of the
strings to the soundboard.

Ideally, the bracing transfers the
vibration of the strings to the entire soundboard of the
instrument.

In about 1850 Antonio Torres of Spain introduced a
fan
ned bracing pattern on his nylon string guitars.
His
Session

IV

April 27,

2012

American University, Washington, DC

12
th

Annual New Millennium Conference

84

bracing pattern provided sufficient strength to the
soundboard of the guitar as well as enhanced the tone of the
instrument.
This pattern has remained
one of
th
e most
common bracing techniques

used for c
lassical guitars
to this
day

[15
]
.
Figure
5

shows the Torres bracing pattern.





FIGURE

5

[1
6
]

T
ORRES
B
RACING
P
ATTERN


Steel guitar strings began to become available by the
early 20
th

century and proved to provide louder volumes than
nylon strings. While Torres’ pattern was sufficient for nylon
strings, the increased tension of
the steel strings
was too high

and caused the soundboard to warp
.
To compensate for this,
guitar builders bega
n using an X

b
racing pattern
,

which
provided more support
for the soundboard. Christian Martin
who founded

C.F. Martin guitars

in the 1830’s was an
innovator of the X bracing pattern
[
17]
. Figure

6 shows
Martin’s X bracing pattern.






FIGURE

6

[1
8
]

M
ARTIN
X

B
RACING


Martin’s X bracing pattern has remained the standard
for steel string guitar bracing while Torres’ fan bracing
pattern has remained the standard for nylon string guitars.


V
IBRATIONS ON THE SOU
NDBOARD

OF A
G
U
ITAR


The soundboard of a
guitar is designed to oscillate due to the
vibrations of the strings. The more that the soundboard is
able to flex, the more volume the instrument produces

because of the higher amount of air being vibrated
.
A

soundboard with no bracing would be much louder

than a
soundboard with bracing

because the soundboard would be
free to vibrate
, but the tension of the strings would cause the
wood to warp, making the instrument unplayable.
G
uitar
builder

Bert Eendebak
exp
lains that the structural
requirements of a guitar harm the musical quality of the
instrument
[
18].


The soundboard of a guitar oscillates in different
patterns depending on the frequency of the note being
played.
One way to visualize these patterns is wit
h

Chladni

figures
.

Chladni figures provide a visual reference for where
the nodes are located.
When the soundboard is vibrating at a
certain frequency, there are areas that on the soundboard that
do not vibrate due to standing waves, or stationary waves.
S
tanding waves occur when two opposing waves of the same
wavelength and amplitude meet and cancel
.

When a material
such as sand

or some kind of power

is placed on a plate (or
soundboard) that is vibrating

at a certain frequency
, it is
attracted t
o the areas

that are not moving, or nodes [19].
Figure 7 shows and example of two nodes

marked

N

.





FIGURE

7

[20
]

N
ODES


The nodes are formed on a guitar soundboard

when
waves

encounter

each other and cancel.
Even if one note is
plucked
on a guitar string, the

interference of the vibra
tions
on the soundboard
still form nodes. Figure 8

shows how the
interference between waves of the same wavelength forms
nodes. The green and blue lines represent two interfering
waves and the red line shows the amplitude of the r
esulting
waveform.





FIGURE

8

[21
]

N
ODES FORMED BY
W
AVE
I
NTERFERENCE


Figure 8 shows nodes created from waves traveling in one
dimension
. Waves
on the soundboard of a guitar travel in
Session

IV

April 27,

2012

American University, Washington, DC

12
th

Annual New Millennium Conference

85

two dimensions
.

Figure 9 shows the pattern of the nodes
formed on a guitar soundboard at 77 Hz, 375 Hz and 511
Hz.
These patterns are examples of Chladni patterns on the
soundboard of a guitar.
The black lines show

the areas of no
vibration or the nodes on the soundboard.






FIGURE

9

[22
]

N
ODES ON A GUITAR


Chladni figures can be
helpful

in guitar construction.
Guitar builders can use the patterns to determine how the
bracing on the soundboard needs to be altered.
Altering the
bracing on the soundboard until a desired

chladni pattern is
found is referred to as free plate tuning.


F
REE
P
LATE
T
UNING


While the patterns formed on the soundboard depend on the
shape and material of the soundboard, the

patterns
can be
useful for the final shaping of the braces.
While the
main
purpose of the braces is to provide structural support for the
soundboard, they also

help

transmit the vibration of the
strings to the entire soundboard.
T
he braces must be shaped
in such a way that the stiffness to mass ratio of the
soundboard is the

same in all directions.

One method of ensuring that the stiffness to mass ratio
is
consistent along the soundboard is by using the ring
-
and
-
a
-
half method.

In order to use this
method
, the soundboard is
vibrated at different frequencies until a Chladni
figure

forms
where the nodes form a ring below the sound hole, and half a
ring above the sound hole
. The frequency at which this
pattern occurs varies depending on the soundboard being
tested

[23]
.
Figure
10 shows a pattern very close to the ring
-
and
-
a
-
hal
f pattern.




FIGURE

10

[24
]

R
ING
-
AND
-
A
-
H
ALF
P
ATTERN

Once the frequency that forms the ring
-
and
-
a
-
half
pattern is found, the braces on the soundboard can be shaped
to improve the
pattern.

The goal is to have a perfect circle
below the sound hole where the

bridge is attached as well as
a perfectly curved line above the sound

hole [25]. By
changing the height and width of the braces, the resistance
against
flexing

is

ch
anged.

A thinner brace will flex more
than a thick brace.

Equation (4
) shows the relations
hip of
how height and width

of a brace

affect
s

the resistance
against bowing
, where


is the resistance,

is the width of
the brace and

is the height of the brace
[26]
.





(4
)


When constructing the soundboard for a guitar, t
he
braces are shaved down, reducing the resistance against flex,
until the ring
-
and
-
a
-
half pattern is perfect.
When t
he
ring
-
and
-
a
-
half
pattern is
perfect,
the stiffness to mass ratio of the
soundboard is isotropic, or
consistent

across the entire
soundboard

[27]
.
The thickness of the bracing also has effect
on the tone

or timbre

of a guitar. Guitars with heavy bracing
have better tone, but less volume. This is due to the fact that
the heavy bracing doesn’t allow the soundboard to vibrate as
well, therefore moving less air.
Lighter bracing does not
transmit the vibration of the st
rings to the soundboard as
efficiently as thick bracing, but
it allows the soundboard to
vibrate more

because it is less stiff
, producing more volume
[28].


E
XPERIMENTAL
B
RACING
P
ATTERNS


While the X bracing pattern for steel string guitars and the
fan
-
bracing

pattern for nylon string guitars have been the
most commonly practiced bracing techniques, some guitar
builders have experimented with alternative
bracing patterns.
O
ne of

these
alternative patterns

is the Kasha bracing pattern

for nylon string

or classical guitars
.

In the 1960’s,
Dr.
Michael Kasha created an asymmetrical bracing pattern
that
he based on Chladni figures

for circular plates.
His goal was
to create a guitar that produced more volume

and better tone

than
classical
guitars with
the
traditional fan bracing.

Figure
11 shows Kasha’s bracing pattern.




FIGURE

11

[29
]

T
HE
K
ASHA
S
OUNDBOARD


Session

IV

April 27,

2012

American University, Washington, DC

12
th

Annual New Millennium Conference

86

Kasha’s design addressed the issue of heavy bracing
producing good tone
but low volume and light bracing
producing high volume but poor tone.
The low
er notes on a
guitar have more amplitude

due to their increased mass
, and
therefore force the soundboard to vibr
ate more than
the
higher notes make it vibrate
.
Kasha’s
bracing pattern

simultaneously achieves the ideal conditions of heavy
bracing for low notes and light bracing for high notes [30].



S
UMMARY


There are many elements that go into producing a quality
guitar.
Both classical and folk guitar builders strive to
produce
instru
ments with good tone as well as
powerful

volume.
While there is a lot to be said about the materials
used in constructing a guitar, one can argue that the bracing
is one of the most important aspects.

Guitar builders, also
known as

l
uthiers
,

are constantly

making adjustments to the
standard X bracing and fan
-
bracing patterns in order to
create an instrument that creates ideal conditions for sound
to travel.
The design of a guitar is based on several physics
concepts that can be

utilized

to produce an instru
ment of the
highest quality.


R
EFERENCES


[1]

Kasha, M,

"A New Look at The History of the Classic Guitar",
Guitar

Review
,

30 Aug.,
1968, pp.3
-
12.

[2]
Tyler, J, “The Renaissance Guitar 1500
-
1600”
Early Music
, Vol. 3,
No.

4, Oct., 1975, pp.
341
-
347

[3]
Guy, P,

"A Brief History of the Guitar."
Guitar Handbook
. Paul
Guy

Guitars, 2001. Web. 27 Feb.
2012.

<http://www.guyguitars.com/eng/handbook/BriefHistory.html>.

[4]

Wolfe, J,

"How Does a Guitar Work?"
How a Guitar Works
.
University

New South Whales. Web. 29 Feb.
2012.

<http://www.phys.unsw.edu.au/music/guitar/guitarintro.html>.

[5
] Ref. 4

[6]

Hewitt, Paul G,

Conceptual Physics
, 11th ed. San Francisco:
Pearson

Addison Wesley, 2010.

[7
] Ref. 6
.

[8
]
Parker, B. R,

Good Vibrations: The Physics of Music
.
Baltimore:

Johns Hopkins UP, 2009.
p
p.

160
-
163


[9]
Just Strings
, Web. 29 Feb. 20
12. <http://www.juststrings.com
>
.

[10
] Ref. 8
.

[11
]
Ref. 6
.

[
12
] Ref. 6
.

[13
] Ref
.

4
.

[14] Ref
.

6
.

[15]
Hokin, S. "The Physics of Everyday Stuff: The Guitar."
Bsharp.org
.

2012. Web. 29 Feb. 2012.
<
http://www.bsharp.org/physics/guitar
>.

[15
] Ref
.

3
.

[16
]
Usher, T
.

"The Spanish Guitar in the Nineteenth and
Twentieth

Centuries."
The Galpin Socie
ty Journal

9 (1956): 5
-
36.
JSTOR
. Web.

15 Feb. 2012
.

[17] Ref. 3
.

[18]
Eendebak,
B,

"The Soundboard."
Design of a Classical
Guitar
.

2011. Web. 01 Mar.
2012.

<http://www.designofaclassicalguitar.com/soundboard.php>.


[
19
]

Wolfe, J,

"How Does a Guitar Work?"
Chladni Patterns for
Guitar

Plates
. University
New South Whales. Web. 29 Feb.
2012.


<http://www.phys.unsw.edu.au/music/guitar/guitarintro.html>.

[20]
Ref
.

19
.

[21] Ref
.
19
.

[22] Ref
.

19
.

[23]

Johnston, J. E. "The Theory Behind Free Plate Tuning Using
Chladni

Mode Patterns."
Jack Johnston Guitar Maker
. 27 Feb. 2011.
Web. 01

Mar.
2012.

<http://jackjohnstonguitarmaker.com/TheTheoryBehindFreePlat
eTuni

ngUsingChla
dniModePatterns.aspx>.

[24] Ref. 19
.

[25] Ref. 23
.

[26] Ref. 19
.

[27] Ref. 23
.

[28
]
Perlmeter, A,

"Redesigning the Guitar."
Science News

98.8/9
(1970):


180
-
81.
JSTOR
. Web. 15 Feb. 2012.

[29] Ref. 28
.

[30] Ref. 28
.