An experimental apparatus to study the vibrations of a string can be constructed by mounting a wire between two heavy brass anchors [3]. As shown in Figure (3.1), a screw is used to adjust the position of the anchors, and hence the tension in the wire (string).
Figure 3.1: Schematic of the apparatus used to study the vibrations of a wire
(string).
An alternating sinusoidal current passed through the wire
excites vibrations; this current is usually supplied directly
from a function generator.
An electromagnet,
or large permanent magnet, is placed at the wire's midpoint.
The interaction between this magnetic field and
the magnetic field generated by the wire's alternating current
causes a periodic force to be applied
at the wire's
midpoint.
If a nonmagnetic wire is used, such as tungsten, then both
planar and nonplanar whirling motions are easy to observe. On the other
hand, if a magnetic wire is used, such as steel, then the motion always remains
restricted to a single plane [4]. The use of a magnetic wire introduces an
asymmetry into the system that causes the damping rate
to depend strongly on the direction of oscillation.
A similar asymmetry is seen in the decay rates of violin, guitar, and piano strings. In these musical instruments the string runs over a bridge, which helps to hold the string in place. The bridge damps the motion of the string; however, the damping force applied by the bridge is different in the horizontal and vertical directions [5]. The clamps holding the string in our apparatus are designed to be symmetric, and it is easy to check experimentally that the decay rates in different directions (in the absence of a magnetic field) show no significant variation. Our experimental apparatus can be thought of as the inverse of that found in an electric guitar. There, a magnetic coil is used to detect the motion of a string. In our apparatus, an alternating magnetic field is used to excite motions in a wire.
The horizontal and vertical string displacements are monitored with a pair of inexpensive slotted optical sensors consisting of an LED (light-emitting diode) and a phototransistor in a U-shaped plastic housing [6]. Two optical detectors, one for the horizontal motion and one for the vertical motion, are mounted together in a holder that is fastened to a micropositioner allowing exact placement of the detectors relative to the string. The detectors are typically positioned near the string mounts. This is because the detector's sensitivity is restricted to a small-amplitude range, and the string displacement is minimal close to the string mounts. As shown in Figure 3.2, the string is positioned to obstruct the light from the LED and hence casts a shadow on the surface of the phototransistor.
Figure 3.2: Module used to detect string displacements.
(Adapted from Hanson [6].)
For a small range of the string displacements, the size of this shadow is linearly proportional to the position of the string, and hence also to the output voltage from the photodetector. This voltage is then monitored on an oscilloscope, digitized with a microcomputer, or further processed electronically to construct an experimental Poincaré section as described in section 3.8.1.
Care must be taken to isolate the rig mechanically and acoustically. In our case, we mounted the apparatus on a floating optical table. We also constructed a plastic cover to provide acoustical isolation. The string apparatus is small and easily fits on a desktop. Typical experimental parameters are listed in Table 3.1.
Table 3.1: Parameters for the string apparatus.
Most of our theoretical analysis will be concerned with single-mode oscillations of a string. If we pluck the string near its center, it tends to oscillate in a sinusoidal manner with most of its energy at some primary frequency called the fundamental . A similar plucking effect can be achieved by exciting wire vibrations with the current and the stationary magnetic field. To pluck the string, we switch off the current after we get a large-amplitude string vibration going. The fundamental frequency is recognizable by us as the characteristic pitch we hear when the string is plucked. A large-amplitude (resonant) response is expected when the forcing frequency applied to a string is near to this fundamental. This is the primary resonance of the string, and it is defined by the linear theory as
where 
is the mass per unit length and T is the tension
in the string.
The primary assumption of the linear theory is that the equilibrium length of the string, l, remains unchanged as the string vibrates, that is, l(t) = l, where l(t) is the instantaneous length. In other words, the linear theory assumes that there are no longitudinal oscillations. In developing a simple nonlinear model for the vibrations of a string we must begin to take into account these longitudinal oscillations and the dependence of the string's length on the vibration amplitude.
