The vibrations of a string also depend upon its thickness. Preserving the stretching weight, the length, and the material of the string constant, _the number of vibrations varies inversely as the thickness of the string_. If, therefore, of two strings of the same material, equally long and equally stretched, the one has twice the diameter of the other, the thinner string will execute double the number of vibrations of its fellow in the same time. If one string be three times as thick as another, the latter will execute three times the number of vibrations, and so on.
Finally, the vibrations of a string depend upon the density of the matter of which it is composed. A platinum wire and an iron wire, for example, of the same length and thickness, stretched by the same weight, will not vibrate with the same rapidity. For, while the specific gravity of iron, or in other words its density, is 7·8, that of platinum is 21·5. All other conditions remaining the same, _the number of vibrations is inversely proportional to the square root of the density of the string_. If the density of one string, therefore, be one-fourth that of another of the same length, thickness, and tension, it will execute its vibrations twice as rapidly; if its density be one-ninth that of the other, it will vibrate with three times the rapidity, and so on. The last two laws, taken together, may be expressed thus: _The number of vibrations is inversely proportional to the square root of the weight of the string_.
In the violin and other stringed instruments we avail ourselves of thickness instead of length to obtain the deeper tones. In the piano we not only augment the thickness of the wires intended to produce the bass notes, but we load them by coiling round them an extraneous substance. They resemble horses heavily jockeyed, and move more slowly on account of the greater weight imposed upon the force of tension.
§ 3. _Mechanical Illustrations of Vibrations. Progressive and Stationary Waves. Ventral Segments and Nodes_
These, then, are the four laws which regulate the _transverse_ vibrations of strings. We now turn to certain allied phenomena, which, though they involve mechanical considerations of a rather complicated kind, may be completely mastered by an average amount of attention. And they _must_ be mastered if we would thoroughly comprehend the philosophy of stringed instruments.