The Physics of Music:
Stringed Instruments
James Bernhard
A common way to create a sound wave is by causing an object to
vibrate
The ways that an object can vibrate periodically are called modes
of vibration
In any mode of vibration, points (or regions) with minimal
displacement are called nodes; points with maximal displacement
are called antinodes
Very often, nodes will be points of zero displacement
The amplitude of the sound wave that is produced is determined
by the amplitude of the object’s vibration at antinodes
The normal modes of vibration for an object are a collection of
independent modes of vibration that generate all of the object’s
possible modes of vibration
For a string with fixed endpoints, we state without proof: the
normal modes of vibration are the familiar sinusoidal standing
waves (as in this vibrating string applet )
To explore combinations of normal modes on a string, you can use
the Falstad string applet
For a sneak preview of normal modes on a drumhead, you can look
at Falstad drumhead applet
is an interesting applet where you can explore various
normal modes of vibration for a particular virtual object
Resonata
Some vocabulary:
I
the fundamental mode of vibration is the normal mode with
the lowest frequency
I
the frequency of the fundamental mode is called the
fundamental frequency
I
harmonics are modes of vibration whose frequencies are
integer multiples of the fundamental frequency; the first
harmonic is the fundamental mode
I
overtones are all normal modes of vibration except the
fundamental mode, so in cases where they can be numbered,
their numbers are off by one from the harmonics’ numbers
I
partials are all normal modes of vibration including the
fundamental mode
Let’s figure out the frequencies of the normal modes of vibration of
a string with fixed ends
Recall that
v
,
λ
where v is the wave speed, λ (lambda) is the wavelength, and f is
the frequency
v = λf
which implies f =
The normal modes of vibration of a string with fixed ends have
wavelengths:
2L 2L 2L
λ=
,
,
,....
1
2
3
Putting the last two equations together, we have the normal mode
frequencies:
1
1
1
f =
v, 2 · v, 3 · v, . . . .
2L
2L
2L
To take this one step further, recall that for transverse waves on a
string:
s
T
v=
,
µ
where T is the string tension and µ is the linear mass density
Combining this the previous slide, we find that the frequencies of
the normal modes for a string with fixed endpoints are:
s
s !
s !
1 T
1 T
1 T
, 2·
f =
, 3·
,....
2L µ
2L µ
2L µ
The frequency fn of the n-th normal mode of vibration for a string
with fixed endpoints is:
s
1 T
.
fn = n ·
2L µ
This means that:
I
Increasing the length L of the string decreases the frequencies
of the normal modes
I
Increasing the tension T of the string increases the
frequencies of the normal modes
I
Increasing the thickness of the string (and hence the mass
density µ) decreases the frequencies of the normal modes
None of these should be surprising to stringed instrument players!
Notice that the formula for a string’s normal mode frequencies
gives in the fundamental frequency (f1 , or F ) as:
s
1 T
F =
.
2L µ
If we have two strings with the same mass density µ and tension T
but different lengths L1 and L2 , then their fundamental frequencies
are related by:
p
(1/2L2 ) T /µ
F2
L1
p
=
= .
F1
L2
(1/2L1 ) T /µ
The ratio of the frequencies is the reciprocal of the ratio of the
lengths
For example, if L2 = L1 /2, then
F2
L1
=
= 2,
F1
L1 /2
This means that “stopping” a string at half its length raises the
pitch by an octave
Other intervals can be computed similarly
To produce harmonics on a stringed instrument, place your finger
lightly at a point where you would like to force a node
This will dampen all modes that don’t have nodes at that point
For example, if you place your finger at the point 1/3 of the way
down the string, you’ll dampen all of the normal modes except for
the 3rd, 6th, 9th, . . .
In this case, when you play the string, the resulting frequencies are
3f1 , 6f1 , 9f1 , . . . , where f1 is the fundamental frequency of the
string
Since these are all multiples of 3f1 , we perceive 3f1 as the
fundamental frequency of the note being played when we
determine the pitch
The timbre is thinner though because the amplitudes are not as
large as if we had actually plucked a string with that fundamental
frequency
Similarly, if k/n is a fraction in lowest terms, touching the string
lightly k/n of the way down the string gives a harmonic whose
timbre is thinner and whose frequency is nf1
Harmonics produced this way form an acoustically pure interval
relative to the pitch of the open string (not an equal tempered
interval)
Many of you are already familiar with how to produce harmonics,
and we will also experiment with harmonics in the lab
We now turn our attention to the violin (as a representative of
bowed stringed instruments)
“Violin” comes from the Italian word violono, meaning “little viola”
Developed by the Cremona school of violin makers in Northern
Italy, founded by Andrea Amati (d. 1580)
His descendants and pupils continued to develop the construction
of the violin over the next 150 years or so (e.g., Antonio Stradivari
and Giuseppe Guarneri)
The components of a violin are shown and described at
Hans Johansson’s website (among many other places)
We won’t go into all of these, only some of the most important
ones for producing sound
The strings are tuned to G3 , D4 , A4 , and E5 (where middle C is
C4 )
A range of at least 2 octaves above the top-most string has good
effect
The top string is normally of fine steel wire (but sometimes nylon
or twisted gut)
Bottom strings are usually of gut overwound with fine wire (silver
wire on really good strings) to increase their linear mass density µ
High mass density is good not just to make the pitch lower but so
that a high enough tension can be supported for a strong tone
The body consists of a front plate and a back plate, joined by ribs
on the sides
Two F-holes are cut into the top plate for resonance purposes (to
be described later)
The sound post is almost below the bridge and the E string, and is
held in position by friction
It transmits vibrations between the plates, and its position and fit
are very important to the instrument’s tone
Below the other side of the bridge, the bass bar runs longitudinally
below the top plate
This distributes the pressure and increases the speed that
vibrations are transmitted at
The bridge is not attached, but is held in place by the tension of
the strings, its treble foot being most firmly in place
The bridge rocks primarily in its own plane
Over time, the violin neck has actually lengthened some, and
pitches are being tuned sharper, so the strings have greater tension
To support this tension, a heavier bass bar is used to support the
increased force that the bridge exerts
These changes have helped the instrument acquire a larger tone for
modern symphony orchestras
The bow is strung with resined horsehair, the resin helping the bow
grip the string
The stick on the bow bends inward toward the horsehair (in
contrast to early bows, which bent outward)
This helps the bow respond like a stiff spring despite its light
construction, which makes it better for producing strong attacks
and articulations
The Discovery Channel has an excellent video of a violin bow in
slow motion at Time Warp violin
There is an interesting commentary on this video by John Hartge
at violinist.com
The tone quality of a violin is determined primarily by the
resonance characteristics of the wooden body and the contained
air, which are coupled in complex ways
If you want to make a practice violin with practically no sound,
have just a wooden frame but no enclosed air, as on practiceviolins.com
(or search for ”Travel Violin”)
If you want a different way to amplify the sound, try a strohviolin
or strohcello, as in this video by Francois Sarhan
Air enclosed in a container with an opening will resonate with a
certain frequency that can be determined (approximately) by
blowing over the opening
In a violin, this resonance frequency is called the main air
resonance of the instrument
Wider f-holes would cause air to enter and escape more quickly,
and so would lead to an increase in the main air resonance
An approximate formula for the main air resonance f is:
√
4
A
f = 0.27c √ ,
V
where c is the speed of sound, A is the area of each f-hole, and V
is the enclosed volume of air
Typical values for these are A = .0005 m2 and V = .00184 m3 (and
c = 343 m/s), which puts the main air resonance at about a D4
The way that a violin produces sound involves not only vibration of
the enclosed air, but vibration of the body as well (and these two
are coupled)
The body also has a resonance frequency, called the main wood
resonance, which has been found to be close to A4 on good violins
Some violins (but much more so violas and cellos) have a wolf note
at the main wood resonance
At the wolf note frequency, instead of the sound’s frequency being
determined entirely by the string resonance, the vibrations of the
body dominate off and on, giving rise to an unstable beating
Also, glue and varnish may also play some acoustical role