HANDBOOK OF ACOUSTICS - online book

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ON THE VIBRATIONS OF STRINGS.               89
giving the higher note, that is, the diameters of the wires being as 1:2, the vibration numbers of the tones produced are as 2 : 1. Illustrations of this law can be found in many musical instruments; thus, the second string of the violin being of the same length as the first, must be thicker in order that it may give a deeper tone.
To prove the third law, stretch a string on the Sonometer with a weight of, say 161bs, and note the pitch of the resulting tone. Now stretch the same string with weights of 25fbs, 36tbs, and 64Ibs successively, and observe the pitch of each tone. Calling the tone produced by the tension of 161bs (d), those produced by the tensions of 25, 36, and 641bs will be found to be (n), (s), and (d1), respectively. Now we have already ascertained, that the vibration numbers of d, PI, 8, d1, are as 4 : 5 : 6 : 8 and these numbers are the square roots of 16, 25, 36, and 64. Examples of the application of this law are to be met with in the tuning of all stringed instruments. The viohnist, harpist, or pianoforte tuner stretches his strings still more to sharpen, and relaxes the tension to flatten them.
The fourth law can be proved by stretching two strings of different densities, but of the same length and thickness, by the same weight. Now, by means of the movable bridge, gradually shorten the vibrating part of the heavier string, till it gives a note of the same pitch as the whole length of the lighter one. Now measure the length Z of the lighter string and the length Zx of the vibrating portion of the heavier one; it will be found that
D1 and D being the densities of the heavier and lighter string respectively. These densities can be ascertained by weighing equal lengths of the two strings. Let N be the number of vibrations per second performed by the length Z\ of the heavier string, then if N{ be the number performed by the whole length Z of the same string, we know by the first law that
therefore from (I)
Now as N also denotes the number of vibrations per second per­formed by the lighter string, this proves the law. As illustrations ■of the application of this law to musical instruments, the weighting of the lowest strings of the pianoforte by coiling wire round them.