Some 2500 years ago, Pythagoras (c. 580 BCE - c. 500 BCE) founded a school in Croton, southern Italy. Among his many studies there, he found that when a stretched string was clamped at an intermediate point, the two vibrating segments produced notes that blended harmoniously if their lengths were in the ratio 2:3 (a fifth) or 3:4 (a fourth). Later, Aristotle (384-322 B.C.) extended this discovery to the notes produced by air columns in reed pipes, and his student Aristoxenos interpolated and studied other notes. (He introduced, for example, the notes fa and sol in the ascending fourth from mi to la.
Although these findings became a lasting part of our musical heritage, few advances on the physical basis of music and vibration were made until Marin Mersenne (1588-1648) established the three laws of vibrating strings that now bear his name. In particular, he found how the frequency n (Hz) of vibration of a particular string depends on three quantities: length L, tension force F, and mass per unit length m.
Using Newtonian mechanics, it is possible to deduce the relations among these variables theoretically. Although not difficult, the theory is a little involved and beyond the scope of what we'll be tackling in this lab. Fortunately, however, it is possible for us to deduce much about these relations without detailed physics by using the method of Dimensional Analysis. First used by J. B. J. Fourier (1768 - 1830) in 1822, it is based on the familiar principle that the units on each side of any valid physical equation must be the same. A surprisingly great deal can be learned simply by applying this basic principle.
You can use dimensional analysis to figure out how the frequency n of standing waves on a string should depend on L, F, and m.
Before going on to discuss the experimental arrangement, we review a few
basic points regarding string vibrations. The string in this experiment
should be clamped at both ends; both ends are stationary nodes.
Now it is possible to drive the string at any frequency we choose, but only
at certain discrete frequencies will the vibrations take on a stable,
stationary pattern of standing waves, as illustrated in the figure.
The origin of these standing waves and their distinct frequencies can be understood by recognizing that travelling waves which move down the string in one direction combine with reflected waves moving the opposite way. When the frequency is such that a whole number of half-wavelengths just happen to fit within the length L, the waves reinforce coherently, producing a stationary standing-wave pattern.
The simplest case involves just a single antinode (or maximum of vibration) between the clamped ends. This is called the fundamental mode, and its frequency is called the fundamental frequency, or just the fundamental. For higher frequencies, it is possible to observe overtones, with N antinode "humps" separated by N – 1 stationary nodes.
One end of the string is effectively clamped by the pulley and the tensioning weights. The other is attached to a spindle mounted on the cone of a small loudspeaker. When the speaker is powered by a sine wave from a function generator, it vibrates with a small amplitude and at the frequency set by the function generator. This end, therefore, is not exactly at a node.
Updated 8/30/99 by Peter N. Saeta .