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\begin{center}
{\bf THE PHYSICS OF VIBRATING STRINGS}
\vspace{.4in}

Nicholas Giordano \\

Deparment of Physics, Purdue University, West Lafayette, IN 47907 \\
E-mail: $\tt ng@physics.purdue.edu$

\end{center}

\vspace{.2in}

Vibrating strings are key components of many musical
instruments, such as guitars, violins, and pianos.
Hence, if we want to appreciate the physics of these instruments,
we must begin with the physics of a vibrating string.
To a first approximation, all strings are created equal,
as all are described by the wave equation
\begin{equation}
{{\partial ^2 y }\over{\partial t^2}} ~=~
c^2 \: {{\partial ^2 y }\over{\partial x^2}} ~,
\label{wave_equation}
\end{equation}
where $y$ is the displacement of the string from
its equilibrium position, which is a function of position along 
the string, $x$, and time, $t$.
But since guitars sound different from violins, and both sound different
from pianos, we know that there must be more to the
story than (\ref{wave_equation}).  This article will explore some of the
other pieces of physics that are important in musical instruments.
Broadly speaking, these
additional pieces fall into two categories: (i) how the string is
excited, and (ii) small deviations from, or additions to, the
wave equation.

\vspace{.25in}

\noindent{\bf The Wave Equation: A Numerical Approach}

The wave equation (\ref{wave_equation}) can be readily derived from
Newton's second law, and in doing so one finds that the parameter $c$
is equal to $\sqrt {T/\mu}$, where $T$ is the tension and $\mu$
the mass per unit length of the string.  It is also straightforward
to show that solutions to (\ref{wave_equation}) have the form
$y = f(x \pm ct)$, so that $c$ is the wave speed.
To construct a numerical scheme for attacking it,
we discretize $x$ and $t$ in units of $\Delta x$ and $\Delta t$,
so that $y(x,t) \rightarrow y(i \Delta x,n \Delta t) \rightarrow y(i,n)$,
and write the derivatives in finite difference form$^1$
\begin{equation}
{{y(i,n+1)\:+\:y(i,n-1) \: - \: 2 \: y(i,n)} \over {(\Delta t)^2}}
~\approx~ c^2 \: \left [
{{y(i+1,n)\:+\:y(i-1,n) \: - \: 2 \: y(i,n)} \over {(\Delta x)^2}}
\right ] ~.
\label{discrete}
\end{equation}
The $\approx$ symbol is used here to emphasize that
this is only an approximation; there are correction terms which can
be important, as we will see shortly.
Rearranging (\ref{discrete}) gives
\begin{equation}
y(i,n+1) ~=~ 2 \: [ 1\: - \: r^2] \: y(i,n) - y(i,n-1) + r^2 \:
\left [ y(i+1,n) + y(i-1,n) \right ] ~,
\label{algor_1}
\end{equation}
where $r \equiv c \Delta t / \Delta x$.  Thus, if we know 
the string configuration at time steps $n$ and $n-1$, we can calculate the
configuration at step $n+1$.
The initial conditions will depend on how the string is excited, and
we will consider two cases in a moment.
 
Some results obtained with the algorithm (\ref{algor_1})
are shown in Fig.~\ref{plucked_string}.
Here the initial string profile (shown at the top of the
figure) was chosen to be triangular, with the
string at rest, as would be appropriate for a plucked guitar string.
The kink associated with this pluck is seen to split into two separate kinks,
one propagating to the left and one to the right, which reflect from
the ends of the string.  In our simulation
we kept the ends of the string fixed, so the reflections are inverted.

\begin{figure}
\centerline{
\psfig{figure=guitar_calc/profiles/profiles.eps,width=3in}}
\caption{
{\em
Waves propagating on a string with fixed ends.
The string had a length of $0.65~\rm m$ with $c = 200 ~ \rm m/s$, as
would be appropriate for a guitar string.
The simulation used the values 
$\Delta x = 0.001 ~ \rm m$, and $\Delta t = \Delta x / c$.
The initial string profile is given at the top, and successive traces
(moving from top to bottom) show the string at progressively later times.
For clarity, each trace is shifted downwards from the previous one.
}
}
\label{plucked_string}
\end{figure}

\vspace{.25in}

\noindent{\bf The Algorithm: A Closer Look}

The astute reader will recognize that (\ref{algor_1})
bears a strong resemblance to a no-frills Euler algorithm.  While the Euler
method is a very simple and useful for approach for many problems, it
is known to fail miserably for some situations that involve
oscillatory motion, such as a simple 
harmonic oscillator or planetary motion.$^1$
Wave motion is also a type of oscillation, so one might have expected
that an Euler approach would fail here too.
However, the results in Fig.~\ref{plucked_string} suggest that
it works quite nicely.  
Understanding why it works so well provides a useful lesson
in numerical methods.

Many algorithms for dealing with differential equations involve
a discretization procedure and an associated step size.  A typical case
might be the motion of a point mass according to
Newton's second law, for which time would be treated in units
of size $\Delta t$.  Smaller step sizes generally
yield more accurate results (provided that roundoff errors and the like are
not a problem), at the cost of increased computational time.  In our
vibrating string problem we have discretized two variables, so we now
have two step sizes, $\Delta t$ and $\Delta x$, to contend
with.  While one might 
expect that smaller step sizes are always better,
life is not quite this simple.

For the calculation in Fig.~\ref{plucked_string} we chose the
step sizes so that $\Delta x / \Delta t = c$ (i.e., $r = 1$).  For this
special ratio of the step sizes it turns out that
the algorithm (\ref{algor_1}) is {\em exact}.
The reason for this can be appreciated from the physics of the
problem.  The physics, and hence also the wave
equation, are ``local'' in the sense that a particular piece of the
string is affected by, i.e., experiences forces from, only the
adjacent pieces.  Consider now a disturbance
of the string at spatial location $i$.  According
to (\ref{algor_1}) this disturbance will at the next time step
propagate one spatial unit, to $i \pm 1$, then move to $i \pm 2$
at the following time step, etc.
Thus, our algorithm will permit a disturbance, such a 
wave, to propagate at a speed of $\Delta x / \Delta t$.  
It is clearly a good thing if this exactly matches the speed of all solutions.
One should therefore not be surprised that the algorithm works best with
$\Delta x / \Delta t = c$, and it is not hard to show that the corrections
to (\ref{discrete}) vanish for this ratio of the step sizes.
The behavior for other values of this ratio can be appreciated from
similar arguments.  If $r \equiv c \Delta t / \Delta x$ 
is chosen to be less than unity,
the algorithm will permit a disturbance to travel faster than
$c$, which should be cause for concern.  However, it turns out that
the amplitude which actually
travels faster than $c$ is very small.
One can show from (\ref{algor_1}) that 
the disturbance which travels at the maximum speed (allowed by
the algorithm)
of $c / r$ will be smaller than the ``true'' disturbance (the one
which moves with a speed $c$) by an amount
which falls by a factor of $r^2$ after each time step, and thus
becomes rapidly negligible.  Hence, the algorithm will still perform
acceptably (although
it is not exact) when $r < 1$.  In the opposite regime $r > 1$,
a similar argument shows that the fastest component {\em grows}
by a factor of $r^2$ after each time step, and thus things rapidly become 
unstable in this case.
The algorithmic requirement that the parameter $r$ be less than
or equal to unity is known
as the {\em Courant condition}.$^2$  Its physical origin is particularly
easy to appreciate in the context of the wave equation,
and suggests how to adapt this algorithm 
to deal with variants of (\ref{wave_equation}), such as the case 
of a stiff string.$^1$

\vspace{.25in}

\noindent{\bf The Tone of a Plucked Guitar String}

The initial string profile considered in Fig.~\ref{plucked_string}
is appropriate for a plucked guitar string.
To understand the nature of the tone produced by our numerical
guitar, we must consider how this string vibration is converted into
sound.  Since the radius of a typical string is much smaller than
the wavelength of any sound it might produce,
the amount of sound generated by the direct interaction of
a string with the surrounding air is negligibly small.$^3$
With an acoustic guitar, sound is produced by the vibration of the body
of the instrument. 
One end of the string is terminated at the bridge, which is attached
to the top plate of the guitar (see Fig.~\ref{guitar_bridge}).  
The force from the string on the
bridge causes motion of the entire guitar body which in turn produces sound.
To treat this problem in full would require that we model the
vibrations of the body.  This is more than we can tackle here,
so we will make the rather drastic 
assumption that both the body motion and
the sound pressure amplitude are proportional to the force on the bridge
in the direction perpendicular to the plane of the top plate of the
guitar (the plate to which the bridge is attached).  
The basic idea is that the top plate acts like a large speaker,
so it is the perpendicular motion that produces sound most efficiently.
This force is equal to the component of the force from
the string in the perpendicular direction, 
which is just the tension in the string 
times $dy/dx$ (the slope of the string) at the bridge.  
This quantity can be readily
calculated from the simulation, and its behavior
for our plucked guitar string is shown in Fig.~\ref{guitar_bridge}.

\begin{figure}
\centerline{\hbox{
\psfig{figure=guitar_calc/f_t_edited.eps,width=3.25in}
\psfig{figure=guitar_calc/fft.eps,width=3.25in}
}}
\caption{
{\em
Left: force on the bridge our numerical guitar as a function
of time from the simulation in Fig.~\protect{\ref{plucked_string}}.
The inset shows a schematic of an acoustic guitar.
Right: power spectrum of the bridge force.
}
}
\label{guitar_bridge}
\end{figure}

The power spectrum of the bridge force 
is also shown in Fig.~\ref{guitar_bridge}.  As expected, we see
a harmonic pattern of peaks at frequencies $n f_1$, where 
$f_1 \approx 150 ~ \rm Hz$ is the fundamental and $n$ is an integer.
The force on the bridge is essentially a square wave (which should be no
surprise given Fig.~\ref{plucked_string}), so it has substantial
strength at high frequencies, and this causes the amplitudes of the
harmonics to fall rather slowly with $n$.
Note also that these amplitudes do not vary monotonically with $n$, and
some are extremely weak (such as the $n = 10$ harmonic, which would be
near $1500 ~ \rm Hz$).  This can be understood in terms of the
symmetry of the initial string profile, as discussed in Ref.~1.

\vspace{.25in}

\noindent{\bf Strings and Hammers: The Nature of a Piano Tone}

Our calculation for a plucked guitar string did not really require a
numerical solution; it could have been obtained from the original
plucked waveform with a little Fourier analysis.  However, the situation
for a piano makes a simulation imperative.
A piano string is set into motion by the blow from a ``hammer,''
which is actually just a wooden mallet covered with a compressible layer 
of felt.
While our first urge as physicists is to treat the felt as a simple
spring describable by Hooke's law, it turns out that
life is not this simple.  Experiments have found that
the restoring force for felt
depends on the rate at which it is compressed,
and the preceding compression history.  Because of this hysteresis,
a full, general treatment of the piano hammer problem 
has not yet been worked out.  Here we will describe an approximate
approach, which is basically
equivalent to the best that has been done to date.$^4$

If we ignore the hysteresis, the force of a piano hammer on a string
is given approximately by
$K z ^p$, where $z$ is the amount
that the felt is compressed.  The exponent has a very un-Hooke-like
value of $p \sim 2.5$, although it depends somewhat on
the way the felt is processed.  We can add the hammer to the simulation
by treating it as a mass which strikes our numerical string
at a certain location, with the hammer--string interaction
force given above.  To implement this scheme, we rewrite
the wave equation (\ref{wave_equation})
in a form which emphasizes the connection with Newton's second law
\begin{equation}
\Delta m \: {{\partial ^2 y }\over{\partial t^2}} ~=~
c^2 \: \Delta m \: {{\partial ^2 y }\over{\partial x^2}} ~
+ ~ F_{hammer} ~.
\label{hammer}
\end{equation}
Here the left hand side is the mass times the acceleration for
an element of the string of length $\Delta x$, and mass
$\Delta m = \rho_{string} \Delta x \pi r^2$, where $r$ is the radius
of the string.  The first term on the right is the restoring force
due to neighboring elements of the string, and the last term is the
force due to the hammer.  This force is given by
\begin{equation}
F_{hammer} ~=~ g(x) \: K \: z^p ~,
\label{hammer_force}
\end{equation}
where $g(x)$ is a function which describes how the hammer force is
distributed along the string.  We will take the hammer strike ``point''
to be at a location $1/8$ from one end of the string (this is the location
used in most pianos), but since a hammer is not a ``point'' object, there
will be some force at neighboring locations as well.  This can be accommodated
by letting $g(x)$ be a gaussian function centered at $1/8$ of the distance
from one end, with a full width of $8 ~ \rm mm$ (which is also a typical
value).

If we now write the derivatives in (\ref{hammer}) in their
finite difference forms, we can derive a recurrence
relation very similar to (\ref{algor_1}).
This can be used
to calculate the motion of the string, and a simple
Euler method can be employed for the hammer.  The
strategy then is to begin at $t=0$ with the string in its undisplaced
state ($y=0$ everywhere) and give the hammer some initial velocity.
When the two meet, the hammer felt is compressed by an amount equal
to the difference between the hammer position and the string position
at the strike point, resulting in an
interaction force which acts on both the string and the hammer.  This
causes the string to move, and the hammer to eventually rebound.
Some results from such a simulation are given in Fig.~\ref{piano},
which shows the string profile during the initial hammer--string 
impact  ($0.3 ~ \rm ms$),
just after the hammer falls away from the string ($2.1 ~ \rm ms$),
and while the string is vibrating freely ($3.0 ~ \rm ms$).
Here we have used string and hammer parameters appropriate for
the note middle C on a piano.$^4$  The calculated hammer--string contact
time is $2.1 ~ \rm ms$, which is within a 
few percent of a typical measured value (which is perhaps
better agreement than we should have expected!).  
As with a guitar, this string vibration leads to sound through the motion
it causes at the bridge.  (In a piano, the strings are attached to a
bridge which is in turn supported by the soundboard, which, roughly
speaking, plays the role of the top plate of the guitar, as sketched
in Fig.~\ref{guitar_bridge}.)
The force on the bridge of our numerical
piano has a rather more complex form than
we found for a guitar, demonstrating
the crucial importance of the method used to excite the string.
This is also reflected in the spectrum of our piano string signal, 
as the high harmonics are somewhat weaker than found in the guitar spectrum.
This is one reason why the two instruments sound so different (but 
certainly not the only reason).

\begin{figure}
\centerline{\hbox{
\psfig{figure=piano/piano_prof_edited.eps,width=2.25in}
\psfig{figure=piano/piano_f_t.eps,width=2.25in}
\psfig{figure=piano/piano_fft.eps,width=2.25in}
}}
\caption{
{\em
Simulation of a piano string.  The string and hammer parameters$^4$
we used are appropriate for a string near middle C. 
Left: string profiles at several different times.
Middle: force on the bridge as a function of time.
Right: spectrum of the bridge force.
}
}
\label{piano}
\end{figure}

\vspace{.25in}

\noindent{\bf Damping and Nonlinearity}

The string vibrations we have encountered so far lack an important
feature, as they do not decay with time.  Our simulations to this point
have not included any energy loss mechanisms, which is certainly not
very realistic.
For guitar and piano strings
we know that the sound typically decays within a few seconds, and the
nature of this decay is very important for our perception of these tones. 
There are several effects which contribute to the damping of
string vibrations in these instruments.  We will now consider two
damping mechanisms which illustrate different physics.  

One such
mechanism is associated with air drag on the string.  It turns out that
Stokes considered this problem nearly 150 years ago, and calculated
the viscous drag on a cylinder moving through a fluid.$^5$  
He was interested in
rather slowly moving objects, and therefore only considered the
drag force to order linear in the velocity.  There are also terms
which are of higher order in $v$, and it turns out that
the velocity of a typical piano string is large enough that the
force from the air varies more nearly
as the square of the velocity.  
This order $v^2$ contribution to the drag 
force is associated with the kinetic energy imparted to
the air as it is pushed about by the string, and is a well
known result$^1$ for objects moving at speeds of typically $1-30 ~ \rm m/s$.
Interestingly, while the effect of the viscous force calculated by Stokes
has been discussed for a stringed instrument,
it appears that no one has ever estimated the (larger)
effect of the drag force which is quadratic in $v$.
This simulation is straightforward, we only need to add
a damping term to the wave equation.  
It is most convenient conceptually to use the Newton's second law approach
of (\ref{hammer}) and simply add the damping force to that equation
of motion.  This yields
\begin{equation}
\Delta m \: {{\partial ^2 y }\over{\partial t^2}} ~=~
c^2 \: \Delta m \: {{\partial ^2 y }\over{\partial x^2}} ~
+ ~ F_{hammer}  ~+~ F_{drag} ~,
\label{damping_eqn}
\end{equation}
where the last term on the right is the
force due to air drag on an element of the string.  It has
the form$^1$ $F_{drag} \approx - C \rho_{air} A v^2$, where $\rho_{air}$
is the density of air, $A = 2 r \Delta x$ 
is the cross-sectional area of an element
of the string (which determines how much air is displaced by the string), 
and $C$ is a constant which has a value near $1/2$.
We have repeated the simulation of Fig.~\ref{piano} with this drag force
included.
The resulting slow decay of the string vibration can be seen from
the root-mean-square (rms) vibration amplitude as a function of time, which
is shown in Fig.~\ref{damping}.
We see that this calculated tone persists for
many seconds, which is much longer than found with a real piano.
Not surprisingly, we will see in a moment that another damping mechanism is
more important (i.e., leads to a faster decay).  Even so,
an interesting feature of the damping due to air drag is that it is
nonlinear, which makes a numerical approach necessary.
This nonlinearity causes different frequencies to
be damped at different rates, so the spectrum of the tone will
change with time.  It also causes the decay to be nonexponential.

\begin{figure}
\centerline{
\psfig{figure=piano/damp_edited.eps,width=2.25in}}
\caption{
{\em
Damping of a piano string: rms vibration amplitude as a function
of time.  The string and hammer parameters are
the same as in Fig.~\protect{\ref{piano}}.
}
}
\label{damping}
\end{figure}

While we have already touched very briefly on
the force of the string on the bridge,
and its role in sound production, we have so far assumed in our
simulations that the
bridge is perfectly rigid.  Of course, this cannot be strictly true,
since it must move at least a little
if any sound is to be generated.  This means that
our simulation should allow the bridge end of the
string to move.  The work done by the string on the bridge will cause a
transfer of energy to the bridge, and hence will
be a mechanism for energy loss.
A complete simulation would treat the vibrations of
the bridge and the board on which it is mounted.  This is a very
ambitious calculation, which has been carried out for a guitar,$^6$ but
not (yet) for a piano.  Here we will treat the bridge motion in a
simple, but reasonably accurate way.  

To proceed, we need an equation of motion for the bridge.  One's first
thought might be to treat it as
a simple mass, or a mass connected to a spring,
but a better approach involves a quantity known as the mechanical
impedance, $Z$.  As with the electrical impedance, $Z$ is very useful
in discussions of mechanical response in the frequency domain.
If the force on the bridge has a single frequency, one can write
\begin{equation}
v_{bridge} ~ = ~ {{F_{bridge} } \over { Z }} ~.
\label{imped}
\end{equation}
In general, $Z$ is complex, which is just another way of saying that
the force on the bridge and its velocity are not necessarily in phase with
one another. 
$Z$ is also a function of frequency,
so (\ref{imped}) is not very useful as an equation of motion in the
time domain.  However, it turns out that the behavior of $Z$ for  
a real piano can be used to our advantage.
Experiments$^7$ show that
at frequencies from about 100~Hz up to several kHz, $Z$ for a piano
is approximately constant (e.g., independent of frequency), 
with a value near $10^3 ~ \rm kg/s$.  
If we make now make the {\em assumption}
that $Z$ is real, $v_{bridge}$ and $F_{bridge}$ will always be in phase,
and the two will be simply proportional to each other, even in the
time domain.  Hence, with these assumptions we can employ (\ref{imped})
as an equation of motion for $v_{bridge}$, with this also being the
velocity of one end of the string (in all of our simulations to this point,
the ends of the string have been held rigidly).
We note again that since
the force on, and velocity of, the bridge are in phase, energy will
be transferred from the string to the bridge.
Adding bridge motion to our model gives the result shown in Fig.~\ref{damping}.
The string decay is seen to be exponential, with a decay time of about 1~s.  
This is in good accord with the decay time found for real piano tones.$^3$

For a piano and also for an acoustic guitar, damping due
to bridge motion is more important than the effect due to air drag, but
this may not
be true for all instruments.  For example, the end supports in
an electric guitar are much more rigid than those of an acoustic
instrument, so air damping may be important in that case.
We should also mention that many other effects contribute to the decay 
of piano and guitar tones.
In fact, this decay is often {\em not} purely exponential, so 
our model is certainly not complete (although it is a reasonable start).

This article has
touched on only a few of the interesting problems which
arise with vibrating strings in musical instruments.  
Some of these problems
involve what may seem, at first glance, to be fairly small modifications of
the basic wave equation (\ref{wave_equation}).  However, they
have a very substantial impact on the
sound which is produced.  After all, most people have no
trouble distinguishing a guitar from a piano, and even two pianos
may not sound much alike.  Hence, a careful
treatment of these ``small'' modifications is essential; in a sense, they
are the problem. 

\vspace{.25in}

\noindent{\bf Suggestions For Further Study}

\begin{enumerate}

\item
Repeat the calculation of Fig.~\ref{plucked_string}, but let the
ends of the string be completely free (this would not be
particularly relevant for a vibrating string, but might be
applicable to a wind instrument).  
You should now observe that the reflected waves are not inverted.

\item
In the calculation of Fig.~\ref{plucked_string} we assumed that the
string was held fixed prior to $t=0$, and released from
rest.  This produced left and right going kinks, i.e., excitations.  Set up
initial conditions which produce an excitation which travels in only
one direction.  Hint: you will need to specify the proper initial
velocity.

\item
In all of our simulations we have assumed that the string is perfectly
flexible.  All real strings have some
stiffness, although it is small for the strings used in
musical instruments.  This leads to a modified wave equation
\begin{equation}
{{\partial ^2 y }\over{\partial t^2}} ~=~
c^2 \: \left [ {{\partial ^2 y }\over{\partial x^2}} ~ - 
~ \epsilon \: {{\partial ^4 y }\over{\partial x^4}} \right ] ~,
\label{stiff_wave_equation}
\end{equation}
where the stiffness parameter $\epsilon$ depends on the Young's modulus
and radius of the string, and other factors.  
While $\epsilon$ is usually ``small'' (a typical value
is $\epsilon \sim 1.5 \times 10^{-5}$), stiffness has an important effect,
as it adds dispersion to the system.  That is,
the wave speed now increases slightly as the frequency is increased, so
the string no longer produces a purely ``harmonic'' spectrum.
This dispersion also upsets the stability of our algorithm,
making it necessary
to use $r$ parameters less than unity in a simulation.  
Calculate the spectrum of a vibrating string described 
by (\ref{stiff_wave_equation}), and determine how much the peaks in
the spectrum are shifted from a perfect harmonic sequence.
Further details of this calculation (and, in particular, how to
deal with the fourth derivative in (\ref{stiff_wave_equation}))
are given in Refs.~1, 3, and 4.

\item
To do a better job of modeling 
an acoustic guitar, one must give a more realistic treatment of the
bridge.  A full treatment of this motion would involve the vibrations of
the body of the guitar, and given its complicated geometry,
this is a very difficult problem.$^6$  However, to a rough first
approximation one can model the bridge as an oscillator
with several resonant frequencies.  For simplicity, assume that
it is a single oscillator with a resonant frequency 
of $100 ~ \rm Hz$ with a $Q$ of $20$.
Add this to the simulation, and observe how it affects the spectrum.
Hint: use the resonant frequency and $Q$ of the bridge to derive
the effective mass, spring constant, and viscous damping parameter 
of the equivalent oscillator (some parameter values are given in Ref.~6).

\item
A notable feature of the hammer force (\ref{hammer_force}) is that
it is quite non-linear; that is, the exponent $p$
is far from the Hooke-like value of unity.  
Physically this means that the felt becomes effectively stiffer, and the
force larger, the more the felt is compressed.  It also has a very
important effect on the piano tone.  Explore this by comparing the
spectrum obtained with a small hammer velocity to that found with
a large hammer velocity (values of $0.5$ and $4 ~ \rm m/s$ are good,
and realistic, choices).  You should find that the louder note contains
more strength in the high harmonics, as compared with the softer note.
Hence, a loud note does not just have a larger volume, but its spectrum
(i.e., timbre) is also different.  Explain this in terms of the nonlinearity
of the hammer force.
It is also interesting to evaluate your calculated tones by
listening to them.  It is a reasonable first approximation to
assume that the sound pressure is simply
proportional to the bridge velocity.

\item
In all of our discussions to this point we have considered the motion
of only a single string.  However, guitars and pianos contain many strings,
and in the case of pianos most notes involve three strings.
Since the strings are all in contact with the bridge, it is the
bridge motion which mediates the interaction between strings.
Add a second string to the calculation associated with
(\ref{imped}) and Fig.~\ref{damping}.  Use a hammer to excite one string, and
calculate how much motion is induced in the other string.
It is also interesting to examine how the vibrational amplitudes of both
strings decay with time.  Try to relate your results to the
behavior of coupled oscillators.$^8$

\end{enumerate}

\noindent{\bf Acknowledgments}

I thank P. Muzikar for many insightful comments and
critical suggestions, A. Tubis for
encouraging my interest in the physics of musical instruments, and
H. A. Conklin, Jr, and B. E. Richardson for helpful correspondence.
The program examples used in this article are available at 
$\tt www.physics.purdue.edu/ \sim ng/comp\_phys.html$, which is
the WEB site for Ref~1.  Many related programs can
also be found there.
Portions of this work were supported by the National Science Foundation
through grant PHY-9722031.

\vspace{.25in}

\section*{References}

\begin{enumerate}

%1
\item
N. Giordano, {\em Computational Physics} (Prentice-Hall, Upper Saddle
River, NJ, 1997).  This book gives more details
of many of the derivations in this article.

%2
\item
W. H. Press, B. P. Flannery, S. A. Teukolsky, and W. T. Vetterling,
{\em Numerical Recipes}, Cambridge University Press, 1986.  Chapter
17 discusses the problem of numerical stability with regards to the
wave equation.

%3
\item
N. H. Fletcher and T. D. Rossing, {\em The Physics of Musical Instruments,}
(Springer-Verlag, New York, 1991).

%4
\item
D. E. Hall, ``Piano string excitation. VI: Nonlinear modeling,''
J. Acoust. Soc. Am. {\bf 92}, 95 (1992);
A. Chaigne and A. Askenfelt, ``Numerical simulations of piano strings. I.
``Physical model for a struck string using finite difference
methods,'' J. Acoust. Soc. Am. {\bf 95}, 1112 (1994);
``Numerical simulations of piano strings. II.
Comparisons with measurements and systematic exploration of some hammer-string
parameters,'' J. Acoust. Soc. Am. {\bf 95}, 1631 (1994).

%5
\item
G. G. Stokes, ``On the effect of internal friction of fluids on the motion
of pendulums,'' Proc. Cambridge Phil. Soc. {\bf 1}, 104 (1850).

%6
\item
O. Christensen and B. B. Vistisen, ``Simple model for low-frequency
guitar function,'' J. Acoust. Soc. Am. {\bf 68}, 758 (1980).
B. E. Richardson and M. Brooke, 
``Modes of vibration and radiation fields of guitars,''
Proc. Inst. Acoust. {\bf 15(3)}, 686 (1993).

%7
\item
H. A. Conklin, Jr., ``Design and tone in the mechanoacoustic piano. Part II.
Piano structure,'' J. Acoust. Soc. Am. {\bf 100}, 695 (1996).
N. Giordano, to be published.

%8
\item
See G. Weinreich, ``Coupled Piano Strings,''
J. Acoust. Soc. Am. {\bf 62}, 1474 (1977), for a nice discussion of this
problem.

\end{enumerate}

\end{document}
