§ 3 Chladni’s Tonometer: The Iron Fiddle, Musical Box, and the Kaleidophone
And now let us shorten our rod, so as to bring its vibrations into proper relation to our ears. When it is about four inches long, it emits a low musical sound. When further shortened, the tone is higher; and, by continuing to shorten the rod, the speed of vibration is augmented, until finally the sound becomes painfully acute. These musical vibrations differ only in rapidity from the grosser oscillations which a moment ago appealed to the eye.
The increase in the rate of vibrations here observed is ruled by a definite law; the number of vibrations executed at a given time is inversely proportional to the square of the length of the vibrating rod. You hear the sound of this strip of brass, two inches long, as the fiddle-bow is passed over its end. Making the length of the strip one inch, the sound is the double octave of the last one; the rate of vibration is augmented four times. Thus, by doubling the length of the vibrating strip, we reduce its rate of vibration to one-fourth; by trebling the length, we reduce the rate of vibration to one-ninth; by quadrupling the length, we reduce the vibrations to one-sixteenth, and so on. It is plain that, by proceeding in this way, we should finally reach a length where the vibrations would be sufficiently slow to be counted. Or, it is plain that, beginning with a long strip whose vibrations could be counted, we might, by shortening, not only make the strip sound, but also determine the rates of vibration corresponding to its different tones. Supposing we start with a strip 36 inches long, which vibrates once in a second, the strip reduced to 12 inches would, according to the above law, execute 9 vibrations a second; reduced to 6 inches, it would execute 36, to 3 inches, 144; while, if reduced to 1 inch in length, it would execute 1,296 vibrations in a second. It is easy to fill the spaces between the lengths here given, and thus to determine the rate of vibration corresponding to any particular tone. This method was proposed and carried out by Chladni.
A musical instrument may be formed of short rods. Into this common wooden tray a number of pieces of stout iron wire of different lengths are fixed, being ranged in a semicircle. When the fiddle-bow is passed over the series, we obtain a succession of very pleasing notes. A competent performer could certainly extract very tolerable music from a sufficient number of these iron pins. The iron fiddle (violon de fer) is thus formed. The notes of the ordinary musical box are also produced by the vibrations of tongues of metal fixed at one end. Pins are fixed in a revolving, cylinder, the free ends of the tongues are lifted by these pins and then suddenly let go. The tongues vibrate, their length and strength being so arranged as to produce in each particular case the proper rapidity of vibration.
Sir Charles Wheatstone has devised a simple and ingenious optical method for the study of vibrating rods fixed at one end. Attaching light glass beads, silvered within, to the end of a metal rod, and allowing the light of a lamp or candle to fall upon the bead, he obtained a small spot intensely illuminated. When the rod vibrated, this spot described a brilliant line which showed the character of the vibration. A knitting-needle, fixed in a vise with a small bead stuck on to it by marine glue, answers perfectly as an illustration. In Wheatstone’s more complete instrument, which he calls a kaleidophone, the vibrating rods are firmly screwed into a massive stand. Extremely beautiful figures are obtained by this simple contrivance, some of which may now be projected on a magnified scale upon the screen before you.
Fixing the rod horizontally in the vise, a condensed beam is permitted to fall upon the silvered bead, a spot of sunlike brilliancy being thus obtained. Placing a lens in front of the bead, a bright image of the spot is thrown upon the screen, the needle is then drawn aside, and suddenly liberated. The spot describes a ribbon of light, at first straight, but speedily opening out into an ellipse, passing into a circle, and then again through a second ellipse back to a straight line. This is due to the fact that a rod held thus in a vise vibrates not only in the direction in which it is drawn aside, but also at right angles to this direction. The curve is due to the combination of two rectangular vibrations.41 While the rod is thus swinging as a whole, it may also divide itself into vibrating parts. By properly drawing a violin-bow across the needle, this serrated circle, Fig. 56, is obtained, a number of small undulations being superposed upon the large one. You moreover hear a musical tone, which you did not hear when the rod vibrated as a whole only; its oscillations, in fact, were then too slow to excite such a tone. The vibrations which produce these sinuosities, and which correspond to the first division of the rod, are executed with about 6-1/4 times the rapidity of the vibrations of the rod swinging as a whole. Again I draw the bow; the note rises in pitch, the serrations run more closely together, forming on the screen a luminous ripple more minute and, if possible, more beautiful than the last one, Fig. 57. Here we have the second division of the rod, the sinuosities of which correspond to 17-13/36 times its rate of vibration as a whole. Thus every change in the sound of the rod is accompanied by a change of the figure upon the screen.
The rate of vibration of the rod, as a whole, is to the rate corresponding to its first division nearly as the square of 2 is to the square of 5, or as 4:25. From the first division onward the rates of vibration are approximately proportional to the squares of the series of odd numbers 3, 5, 7, 9, 11, etc. Supposing the vibrations of the rod as a whole to number 36, then the vibrations corresponding to this and to its successive divisions would be expressed approximately by the following series of number’s:
36, 225, 625, 1225, 2025, etc.
In Fig. 58, a, b, c, d, e, are shown the modes of division corresponding to this series of numbers. You will not fail to observe that these overtones of a vibrating rod rise far more rapidly in pitch than the harmonics of a string.
Other forms of vibration may be obtained by smartly striking the rod with the finger near its fixed end. In fact, an almost infinite variety of luminous scrolls can be thus produced, the beauty of which may be inferred from the subjoined figures (see next page) first obtained by Sir C. Wheatstone. They may be produced by illuminating the bead with sunlight, or with the light of a lamp or candle. The scrolls, moreover, may be doubled by employing two candles instead of one. Two spots of light then appear, each of which describes its own luminous line when the knitting-needle is set in vibration. In a subsequent lecture we shall become acquainted with Wheatstone’s application of his method to the study of rectangular vibrations.