THE PHYSICS OF VIBRATING STRINGS

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

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 (1).
This section of the course begins with a discussion of the wave equation and how
to attack it numerically. This brings out some nice lessons in numerical
stability. It turns out that the Courant condition, which concerns how to best
choose the temporal and spatial grid sizes in a finite difference treatment, can
be easily understood from the physics of the wave equation. We then consider
waves moving on strings in several different contexts, including plucked guitar
strings, and piano strings as they are struck by a (piano) ``hammer.''

To construct a numerical scheme for attacking the wave equation, we
discretize x and t in units of and , so that , and write the derivatives in finite difference form

The symbol is used here to emphasize that this is only an
approximation; there are correction terms which can be important. Rearranging
(2)
gives

where . 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 now consider two cases.

Some results obtained with the algorithm (3)
are shown in Fig. 1.
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.

Figure 1: Waves propagating on
a string with fixed ends. The string had a length of 0.65 m with
c=200 m/s, as would be appropriate for a guitar string. The simulation used
the values , and . 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.

The astute reader will recognize that (3)
bears a strong resemblance to a no-frills Euler algorithm. While the Euler
method is a very simple and useful 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. 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. 1
suggest that it works quite nicely. Understanding why it works so well provides
a useful lesson in numerical methods, and (as was noted above) leads to a
derivation of the Courant condition for the stability of the algorithm.

Once we have obtained the solution for the string displacement as a function
of time, as in the above figure, we can then discuss the sound which is produced
and its associated spectrum. This brings us to several important issues and
techniques, including spectrum analysis, the Fourier transform, and the Fast
Fourier Transform.

We next proceed to the case of a piano. 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 might be 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. However, it is not hard to give a fairly good
approximate approach, which is basically equivalent to the best that has been
done to date in the research literature.

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 by

where z is the amount which the felt is compressed, the exponent describes how the stiffness varies with compression, and
K is a force constant. Here g(x) is a function which
describes how the hammer force is distributed along the string. This can be
accommodated by letting g(x) be a gaussian function centered at
point where the middle of the hammer meets the string, with a full width which
corresponds to the width of the hammer. (Note that (4) has been obtained from
direct experimental measurements.) In a typical case, the hammer meets the
string a distance 1/8 from one end, and has a width of about 1 cm. 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. 2,
which shows the string profile during the initial hammer-string impact
(0.3 ms), just after the hammer falls away from the string (2.1 ms),
and while the string is vibrating freely (3.0 ms). Here we have used string
and hammer parameters appropriate for the note middle C on a piano. The
calculated hammer-string contact time is 2.1 ms, which is quite close to
typical measured values.

Figure 2: Simulation of a piano
string. The string and hammer parameters we used are appropriate for a string
near middle C. The string profile is shown at several different times after
impact with the hammer.

These results can also be compared to those for the plucked guitar string,
and differences in the spectra of guitar and piano tones can be identified and
understood. Other topics of interest can also be addressed, including damping,
and the effect of string stiffness.

Nick Giordano
Mon Sep 8 10:19:18 EST 1997