Vibrations of a Rod fixed at Both Ends: its Subdivisions and Corresponding Overtones—Vibrations of a Rod fixed at One End—The Kaleidophone—The Iron Fiddle and Musical Box—Vibrations of a Rod free at Both Ends—The Claque-bois and Glass Harmonica—Vibrations of a Tuning-Fork: its Subdivisions and Overtones—Vibrations of Square Plates—Chladni’s Discoveries—Wheatstone’s Analysis of the Vibrations of Plates—Chladni’s Figures—Vibrations of Disks and Bells—Experiments of Faraday and Strehlke

§ 1. Transverse Vibrations of a Rod fixed at Both Ends

OUR last chapter was devoted to the transverse vibrations of strings. This one I propose devoting to the transverse vibrations of rods, plates, and bells, commencing with the case of a rod fixed at both ends. Its modes of vibration are exactly those of a string. It vibrates as a whole, and can also divide itself into two, three, four, or more vibrating parts. But, for a reason to be immediately assigned, the laws which regulate the pitch of the successive notes are entirely different in the two cases. Thus, when a string divides into two equal parts, each of its halves vibrates with twice the rapidity of the whole; while, in the case of the rod, each of its halves vibrates with nearly three times the rapidity of the whole. With greater strictness, the ratio of the two rates of vibration is as 9 is to 25, or as the square of 3 to the square of 5. In Fig. 52, a a', c c', b b', d d', are sketched the first four modes of vibration of a rod fixed at both ends: the successive rates of vibration, in the four cases bear to each other the following relation:

Number of nodes	0	1	2	3
Number of vibrations	9	25	49	81

the last row of figures being the squares of the odd numbers 3, 5, 7, 9.

In the case of a string, the vibrations are maintained by a tension externally applied; in the case of a rod, the vibrations are maintained by the elasticity of the rod itself. The modes of division are in both cases the same, but the forces brought into play are different, and hence also the successive rates of vibration.

§ 2. Transverse Vibrations of a Rod fixed at One End

Let us now pass on to the case of a rod fixed at one end and free at the other. Here also it is the elasticity of the material, and not any external tension, that sustains the vibrations. Approaching, as usual, sonorous vibrations through more grossly mechanical ones, I fix this long rod of iron, n o, Fig. 53, in a vise, draw it aside, and liberate it. To make its vibrations more evident, its shadow is thrown upon a screen. The rod oscillates as a whole to and fro, between the points p p'. But it is capable of other modes of vibration. Damping it at the point a, by holding it gently there between the finger and thumb, and striking it sharply between a and o, the rod divides into two vibrating parts, separated by a node as shown in Fig. 54. You see upon the screen a shadowy spindle between a and the vise below, and a shadowy fan above a, with a black node between both. The division may be effected without damping a, by merely imparting a sufficiently sharp shock to the rod between a and o. In this case, however, besides oscillating in parts, the rod oscillates as a whole, the partial oscillations being superposed upon the large one.

Fig. 53. Fig. 53.

Fig. 54. Fig. 54.

Fig. 55. Fig. 55.

You notice, moreover, that the amplitude of the partial oscillations depends upon the promptness of the stroke. When the stroke is sluggish, the partial division is but feebly pronounced, the whole oscillation being most marked. But when the shock is sharp and prompt, the whole oscillation is feeble, and the partial oscillations are executed with vigor. If the vibrations of this rod were rapid enough to produce a musical sound, the oscillation of the rod as a whole would correspond to its fundamental tone, while the division of the rod into two vibrating parts would correspond to the first of its overtones. If, moreover, the rod vibrated as a whole and as a divided rod at the same time, the fundamental tone and the overtone would be heard simultaneously. By damping the proper point and imparting the proper shock, we can still further subdivide the rod, as shown in Fig. 55.

§ 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.

Fig. 56. Fig. 56.

Fig. 57. Fig. 57.

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.

§ 4. Transverse Vibrations of a Rod free at Both Ends. The Claque-bois and Glass Harmonica

From a rod or bar fixed at one end, we will now pass to rods or bars free at both ends; for such an arrangement has also been employed in music. By a method afterward to be described, Chladni, the father of modern acoustics, determined experimentally the modes of vibration possible to such bars. The simplest mode of division in this case occurs when the rod is divided by two nodes into three vibrating parts. This division is easily illustrated by a flexible box ruler, six feet long. Holding it at about twelve inches from its two ends between the forefinger and thumb of each hand, and shaking it, or causing its centre to be struck, it vibrates, the middle segment forming a shadowy spindle, and the two ends forming fans. The shadow of the ruler on the screen renders the mode of vibration very evident. In this case the distance of each node from the end of the ruler is about one-fourth of the distance between the two nodes. In its second mode of vibration the rod or ruler is divided into four vibrating parts by three nodes. In Fig. 60, 1 and 2, these respective modes of division are shown. Looking at the edge of the ruler 1, the dotted lines cutting a a', b b', show the manner in which the segments bend up and down when the first division occurs, while c c', d d', show the mode of vibration corresponding to the second division. The deepest tone of a rod free at both ends is higher than the deepest tone of a rod fixed at one end in the proportion of 4:25. Beginning with the first two nodes, the rates of vibration of the free bar rise in the following proportion:

Number of nodes		2, 3, 4, 5, 6, 7
Numbers to the squares of which the	}	3, 5, 7, 9, 11, 13
pitch is approximately proportional

Here, also, we have a similarly rapid rise of pitch to that noticed in the last two cases.

For musical purposes the first division only of a free rod has been employed. When bars of wood of different lengths, widths, and depths, are strung along a cord which passes through the nodes, we have the claque-bois of the French, an instrument now before you, A B, Fig. 61. Supporting the cord at one end by a hook k and holding it at the other in the left hand, I run the hammer h along the series of bars, and produce an agreeable succession of musical tones. Instead of using the cord, the bars may rest at their nodes on cylinders of twisted straw; hence the name “straw-fiddle,” sometimes applied to this instrument. Chladni informs us that it is introduced as a play of bells (Glockenspiel) into Mozart’s opera of “Die ZauberflÖte.” If, instead of bars of wood, we employ strips of glass, we have the glass harmonica.

§ 5. Vibrations of a Tuning-fork

From the vibrations of a bar free at both ends it is easy to pass to the vibrations of a tuning-fork, as analyzed by Chladni. Supposing a a, Fig. 62, to represent a straight steel bar, with the nodal points corresponding to its first mode of division marked by the transverse dots. Let the bar be bent to the form b b; the two nodal points still remain, but they have approached nearer to each other. The tone of the bent bar is also somewhat lower than that of the straight one. Passing through various stages of bending, c c, d d, we at length convert the bar into a tuning-fork e e, with parallel prongs; it still retains its two nodal points, which, however, are much closer together than when the bar was straight.

When such a fork sounds its deepest note, its free ends oscillate as in Fig. 63, where the prongs vibrate between the limits b and n, and f and m, and where p and q are the nodes. There is no division of a tuning-fork corresponding to the division of a straight bar by three nodes. In its second mode of division, which corresponds to the first overtone of the fork, we have a node on each prong, and two at the bottom. The principle of Young, referred to at page 155, extends also to tuning-forks. To free the fundamental tone from an overtone, you draw your bow across the fork at the place where the node is required to form the latter. In the third mode of division there are two nodes on each prong and one at the bottom; in the fourth division there are two nodes on each prong and two at the bottom; while in the fifth division there are three nodes on each prong and one at the bottom. The first overtone of the fork requires, according to Chladni, 6-1/4 times the number of vibrations of the fundamental tone.

It is easy to elicit the overtones of tuning-forks. Here, for example, is our old series, vibrating respectively 256, 320, 384, and 512 times in a second. In passing from the fundamental tone to the first overtone of each you notice that the interval is vastly greater than that between the fundamental tone and the first overtone of a stretched string. From the numbers just mentioned we pass at once to 1,600, 2,000, 2,400, and 3,200 vibrations a second. Chladni’s numbers, however, though approximately correct, are not always rigidly verified by experiment. A pair of forks, for example, may have their fundamental tones in perfect unison and their overtones discordant. Two such forks are now before you. When the fundamental tones of both are sounded, the unison is perfect; but when the first overtones of both are sounded, they are not in unison. You hear rapid “beats,” which grate upon the ear. By loading one of the forks with wax, the two overtones may be brought into unison; but now the fundamental tones produce loud beats when sounded together. This could not occur if the first overtone of each fork was produced by a number of vibrations exactly 6-1/4 times the rate of its fundamental. In a series of forks examined by Helmholtz, the number of vibrations of the first overtone varied from 5·6 to 6·6 times that of the fundamental.

Starting from the first overtone, and including it, the rates of vibration of the whole series of overtones are as the squares of the numbers 3, 5, 7, 9, etc. That is to say, in the time required by the first overtone to execute 9 vibrations, the second executes 25, the third 49, the fourth 81, and so on. Thus the overtones of the fork rise with far greater rapidity than those of a string. They also vanish more speedily, and hence adulterate to a less extent the fundamental tone by their admixture.

§ 6. Chladni’s Figures

The device of Chladni for rendering these sonorous vibrations visible has been of immense importance to the science of acoustics. Lichtenberg had made the experiment of scattering an electrified powder over an electrified resin-cake, the arrangement of the powder revealing the electric condition of the surface. This experiment suggested to Chladni the idea of rendering sonorous vibrations visible by means of sand strewed upon the surface of the vibrating body. Chladni’s own account of his discovery is of sufficient interest to justify its introduction here:

“As an admirer of music, the elements of which I had begun to learn rather late, that is, in my nineteenth year, I noticed that the science of acoustics was more neglected than most other portions of physics. This excited in me the desire to make good the defect, and by new discovery to render some service to this part of science. In 1785 I had observed that a plate of glass or metal gave different sounds when it was struck at different places, but I could nowhere find any information regarding the corresponding modes of vibration. At this time there appeared in the journals some notices of an instrument made in Italy by the AbbÉ Mazzochi, consisting of bells, to which one or two violin-bows were applied. This suggested to me the idea of employing a violin-bow to examine the vibrations of different sonorous bodies. When I applied the bow to a round plate of glass fixed at its middle it gave different sounds, which, compared with each other, were (as regards the number of their vibrations) equal to the squares of 2, 3, 4, 5, etc.; but the nature of the motions to which these sounds corresponded, and the means of producing each of them at will, were yet unknown to me. The experiments on the electric figures formed on a plate of resin, discovered and published by Lichtenberg, in the memoirs of the Royal Society of GÖttingen, made me presume that the different vibratory motions of a sonorous plate might also present different appearances, if a little sand or some other similar substance were spread over the surface. On employing this means, the first figure that presented itself to my eyes upon the circular plate already mentioned resembled a star with ten or twelve rays, and the very acute sound, in the series alluded to, was that which agreed with the square of the number of diametrical lines.”

§ 7. Vibrations of Square Plates: Nodal Lines

I will now illustrate the experiments of Chladni, commencing with a square plate of glass held by a suitable clamp at its centre. The plate might be held with the finger and thumb, if they could only reach far enough. Scattering fine sand over the plate, the middle point of one of its edges is damped by touching it with the finger-nail, and a bow is drawn across the edge of the plate, near one of its corners. The sand is tossed away from certain parts of the surface, and collects along two nodal lines which divide the large square into four smaller ones, as in Fig. 64. This division of the plate corresponds to its deepest tone.

Fig. 64. Fig. 64.

Fig. 65. Fig. 65.

Fig. 66. Fig. 66.

The signs + and - employed in these figures denote that the two squares on which they occur are always moving in opposite directions. When the squares marked + are above the average level of the plate those marked - are below it; and when those marked - are above the average level those marked + are below it. The nodal lines mark the boundaries of these opposing motions. They are the places of transition from the one motion to the other, and are therefore unaffected by either.

Scattering sand once more over its surface, I damp one of the corners of the plate, and excite it by drawing the bow across the middle of one of its sides. The sand dances over the surface, and finally ranges itself in two sharply-defined ridges along its diagonals, Fig. 65. The note here produced is a fifth above the last. Again damping two other points, and drawing the bow across the centre of the opposite side of the plate, we obtain a far shriller note than in either of the former cases, and the manner in which the plate vibrates to produce this note is represented in Fig. 66.

Thus far plates of glass have been employed held by a clamp at the centre. Plates of metal are still more suitable for such experiments. Here is a plate of brass, 12 inches square, and supported on a suitable stand. Damping it with the finger and thumb of my left hand at two points of its edge, and drawing the bow with my right across a vibrating portion of the opposite edge, the complicated pattern represented in Fig. 67 is obtained.

The beautiful series of patterns shown on page 182 were obtained by Chladni, by damping and exciting square plates in different ways. It is not only interesting but startling to see the suddenness with which these sharply-defined figures are formed by the sweep of the bow of a skilful experimenter.

§ 8. Wheatstone’s Analysis of the Vibrations of Square Plates

And now let us look a little more closely into the mechanism of these vibrations. The manner in which a bar free at both ends divides itself when it vibrates transversely has been already explained. Rectangular pieces of glass or of sheet metal—the glass strips of Fig. 69. Fig. 69. the harmonica, for example—also obey the laws of free rods and bars. In Fig. 69 is drawn a rectangle a, with the nodes corresponding to its first division marked upon it, and underneath it is placed a figure showing the manner in which the rectangle, looked at edgewise, bends up and down when it is set in vibration.42 For the sake of plainness the bending is greatly exaggerated. The figures b and c indicate that the vibrating parts of the plate alternately rise above and fall below the average level of the plate. At one moment, for example, the centre of the plate is above the level and its ends below it, as at b; while at the next moment its centre is below and its two ends above the average level, as at c. The vibrations of the plate consist in the quick successive assumption of these two positions. Similar remarks apply to all other modes of division.

Now suppose the rectangle gradually to widen, till it becomes a square. There then would be no reason why the nodal lines should form parallel to one pair of sides rather than to the other. Let us now examine what would be the effect of the coalescence of two such systems of vibrations.

To keep your conceptions clear, take two squares of glass and draw upon each of them the nodal lines belonging to a rectangle. Draw the lines on one plate in white, and on the other in black; this will help you to keep the plates distinct in your mind as you look at them. Now lay one square upon the other so that their nodal lines shall coincide, and then realize with perfect mental clearness both plates in a state of vibration. Let us assume, in the first instance, that the vibrations of the two plates are concurrent; that the middle segment and the end segments of each rise and fall together; and now suppose the vibrations of one plate transferred to the other. What would be the result? Evidently vibrations of a double amplitude on the part of the plate which has received this accession. But suppose the vibrations of the two plates, instead of being concurrent, to be in exact opposition to each other—that when the middle segment of the one rises the middle segment of the other falls—what would be the consequence of adding them together? Evidently a neutralization of all vibration.

Instead of placing the plates so that their nodal lines coincide, set these lines at right angles to each other. That is to say, push A over A', Fig. 70. In these figures the letter P means positive, indicating, in the section where it occurs, a motion of the plate upward; while N means negative, indicating, where it occurs, a motion downward. You have now before you a kind of check pattern, as shown in the third square, consisting of a square s in the middle, a smaller square b at each corner, and four rectangles at the middle portions of the four sides. Let the plates vibrate, and let the vibrations of their corresponding sections be concurrent, as indicated by the letters P and N; and then suppose the vibrations of one of them transferred to the other. What must result? A moment’s reflection will show you that the big middle square s will vibrate with augmented energy; the same is true of the four smaller squares b, b, b, b, at the four corners; but you will at once convince yourselves that the vibrations in the four rectangles are in opposition, and that where their amplitudes are equal they will destroy each other. The middle point of each side of the plate of glass would therefore be a point of rest; the points where the nodal lines of the two plates cross each other would also be points of rest. Draw a line through every three of these points and you will obtain a second square inscribed in the first. The sides of this square are lines of no motion.

We have thus far been theorizing. Let us now clip a square plate of glass at a point near the centre of one of its edges, and draw the bow across the adjacent corner of the plate. When the glass is homogeneous, a close approximation to this inscribed square is obtained. The reason is that when the plate is agitated in this manner the two sets of vibrations which we have been considering actually coexist in the plate, and produce the figure due to their combination.

Again, place the squares of glass one upon the other exactly as in the last case; but now, instead of supposing them to concur in their vibrations, let their corresponding sections oppose each other: that is, let A cover A', Fig. 71. Then it is manifest that on superposing the vibrations the middle point of our middle square must be a point of rest; for here the vibrations are equal and opposite. The intersections of the nodal lines are also points of rest, and so also is every corner of the plate itself, for here the added vibrations are also equal and opposite. We have thus fixed four points of rest on each diagonal of the square. Draw the diagonals, and they will represent the nodal lines consequent on the superposition of the two vibrations.

These two systems actually coexist in the same plate when the centre is clamped and one of the corners touched, while the fiddle-bow is drawn across the middle of one of the sides. In this case the sand which marks the lines of rest arranges itself along the diagonals. This, in its simplest possible form, is Sir C. Wheatstone’s analysis of these superposed vibrations.

§ 9. Vibrations of Circular Plates

Passing from square plates to round ones, we also obtain various beautiful effects. This disk of brass is supported horizontally upon an upright stand: it is blackened, and fine white sand is scattered lightly over it. The disk is capable of dividing itself in various ways, and of emitting notes of various pitch. I sound the lowest fundamental note of the disk by touching its edge at a certain point and drawing the bow across the edge at a point 45° distant from the damped one. You hear the note and you see the sand. It quits the four quadrants of the disk, and ranges itself along two of the diameters, Fig. 72, A (next page). When a disk divides itself thus into four vibrating segments, it sounds its deepest note. I stop the vibration, clear the disk, and once more scatter sand over it. Damping its edge, and drawing the bow across it at a point 30° distant from the damped one, the sand immediately arranges itself in a star. We have here six vibrating segments, separated from each other by their appropriate nodal lines, Fig. 72, B. Again I damp a point, and agitate another nearer to the damped one than in the last instance; the disk divides itself into eight vibrating segments with lines of sand between them, Fig. 72, C. In this way the disk may be subdivided into ten, twelve, fourteen, sixteen sectors, the number of sectors being always an even one. As the division becomes more minute the vibrations become more rapid, and the pitch consequently more high. The note emitted by the sixteen segments into which the disk is now divided is so acute as to be almost painful to the ear. Here you have Chladni’s first discovery. You can understand his emotion on witnessing this wonderful effect, “which no mortal had previously seen.” By rendering the centre of the disk free, and damping appropriate points of the surface, nodal circles and other curved lines may be obtained.

The rate of vibration of a disk is directly proportional to its thickness, and inversely proportional to the square of its diameter. Of these three disks two have the same diameter, but one is twice as thick as the other; two of them are of the same thickness, but one has half the diameter of the other. According to the law just enunciated, the rules of vibration of the disks are as the numbers 1, 2, 4. When they are sounded in succession, the musical ears present can testify that they really stand to each other in the relation of a note, its octave, and its double octave.

§ 10. Strehlke and Faraday’s Experiments: Deportment of Light Powders

The actual movement of the sand toward the nodal lines may be studied by clogging the sand with a semi-fluid substance. When gum is employed to retard the motion of the particles, the curves which they individually describe are very clearly drawn upon the plates. M. Strehlke has sketched these appearances, and from him the patterns A, B, C, Fig. 73, are borrowed.

Fig. 74. Fig. 74.

Fig. 75. Fig. 75.

Fig. 76. Fig. 76.

An effect of vibrating plates which long perplexed experimenters is here to be noticed. When with the sand strewed over a plate a little fine dust is mingled, say the fine seed of lycopodium, this light substance, instead of collecting along the nodal lines, forms little heaps at the places of most violent motion. It is heaped at the four corners of the plate, Fig. 74, at the four sides of the plate, Fig. 75, and lodged between the nodal lines of the plate, Fig. 76. These three figures represent the three states of vibration illustrated in Figs. 64, 65, and 66. The dust chooses in all cases the place of greatest agitation. Various explanations of this effect had been given, but it was reserved for Faraday to assign its extremely simple cause. The light powder is entangled by the little whirlwinds of air produced by the vibrations of the plate: it cannot escape from the little cyclones, though the heavier sand particles are readily driven through them. When, therefore, the motion ceases, the light powder settles down at the places where the vibration was a maximum. In vacuo no such effect is observed: here all powders, light and heavy, move to the nodal lines.

§ 11. Vibration of Bells: Means of rendering them visible

The vibrating segments and nodes of a bell are similar to those of a disk. When a bell sounds its deepest note, the coalescence of its pulses causes it to divide into four vibrating segments, separated from each other by four nodal lines, which run up from the sound-bow to the Fig. 77. Fig. 77. crown of the bell. The place where the hammer strikes is always the middle of a vibrating segment; the point diametrically opposite is also the middle of such a segment. Ninety degrees from these points, we have also vibrating segments, while at 45 degrees right and left of them we come upon the nodal lines. Supposing the strong, dark circle in Fig. 77 to represent the circumference of the bell in a state of quiescence, then when the hammer falls on any one of the segments a, c, b, or d, the sound-bow passes periodically through the changes indicated by the dotted lines. At one moment it is an oval, with a b for its longest diameter; at the next moment it is an oval, with c d for its longest diameter. The changes from one oval to the other constitute, in fact, the vibrations of the bell. The four points n, n, n, n, where the two ovals intersect each other, are the nodes. As in the case of a disk, the number of vibrations executed by a bell in a given time varies directly as the thickness, and inversely as the square of the bell’s diameter.

Like a disk, also, a bell can divide itself into any even number of vibrating segments, but not into an odd number. By damping proper points in succession the bell can be caused to divide into 6, 8, 10, and 12 vibrating parts. Beginning with the fundamental note, the number of vibrations corresponding to the respective divisions of a bell, as of a disk, is as follows:

Number of divisions		4, 6, 8, 10, 12
Numbers the squares of which express the	}	2, 3, 4, 5, 6
rates of vibration

Thus, if the vibrations of the fundamental tone be 40, that of the next higher tone will be 90, the next 160, the next 250, the next 360, and so on. If the bell be thin, the tendency to subdivision is so great that it is almost impossible to bring out the pure fundamental tone without the admixture of the higher ones.

I will now repeat before you a homely, but an instructive experiment. This common jug, when a fiddle-bow is drawn across its edge, divides into four vibrating segments exactly like a bell. The jug is provided with a handle; and you are to notice the influence of this handle upon the tone. When the fiddle-bow is drawn across the edge at a point diametrically opposite to the handle, a certain note is heard. When it is drawn at a point 90° from the handle, the same note is heard. In both these cases the handle occupies the middle of a vibrating segment, loading that segment by its weight. But I now draw the bow at an angular distance of 45° from the handle; the note is sensibly higher than before. The handle in this experiment occupies a node; it no longer loads a vibrating segment, and hence the elastic force, having to cope with less weight, produces a more rapid vibration. Chladni executed with a teacup the experiment here made with a jug. Now bells often exhibit round their sound-bows an absence of uniform thickness tantamount to the want of symmetry in the case of our jug; and we shall learn subsequently that the intermittent sound of many bells, noticed more particularly when their tones are dying out, is produced by the combination of two distinct rates of vibration, which have this absence of uniformity for their origin.

There are no points of absolute rest in a vibrating bell, for the nodes of the higher tones are not those of the fundamental one. But it is easy to show that the various parts of the sound-bow, when the fundamental tone is predominant, vibrate with very different degrees of intensity. Suspending a little ball of sealing-wax a, Fig. 78 (next page), by a string, and allowing it to rest gently against the interior surface of an inverted bell, it is tossed to and fro when the bell is thrown into vibration. But the rattling of the sealing-wax ball is far more violent when it rests against the vibrating segments than when it rests against the nodes. Permitting the ivory bob of a short pendulum to rest in succession against a vibrating segment and against a node of the “Great Bell” of Westminster, I found that in the former position it was driven away five inches, in the latter only two inches and three-quarters, when the hammer fell upon the bell.

Could the “Great Bell” be turned upside down and filled with water, on striking it the vibrations would express themselves in beautiful ripples upon the liquid surface. Similar ripples may be obtained with smaller bells, or even with finger and claret glasses, but they would be too minute for our present purpose. Filling a large hemispherical glass with water, and passing the fiddle-bow across its edge, large crispations immediately cover its surface. When the bow is vigorously drawn, the water rises in spray from the four vibrating segments. Projecting, by means of a lens, a magnified image of the illuminated water-surface upon the screen, pass the bow gently across the edge of the glass, or rub the finger gently along the edge. You hear this low sound, and at the same time observe the ripples breaking, as it were, in visible music over the four sectors of the figure.

You know the experiment of Leidenfrost which proves that, if water be poured into a red-hot silver basin, it rolls about upon its own vapor. The same effect is produced if we drop a volatile liquid, like ether, on the surface of warm water. And, if a bell-glass be filled with ether or with alcohol, a sharp sweep of the bow over the edge of the glass detaches the liquid spherules, which, when they fall back, do not mix with the liquid, but are driven over the surface on wheels of vapor to the nodal lines. The warming of the liquid, as might be expected, improves the effect. M. Melde, to whom we are indebted for this beautiful experiment, has given the drawings, Figs. 79 and 80, representing what occurs when the surface is divided into four and into six vibrating parts. With a thin wineglass and strong brandy the effect may also be obtained.43

Fig. 79. Fig. 79.

Fig. 80. Fig. 80.

The glass and the liquid within it vibrate here together, and everything that interferes with the perfect continuity of the entire mass disturbs the sonorous effect. A crack in the glass passing from the edge downward extinguishes its sounding power. A rupture in the continuity of the liquid has the same effect. When a glass containing a solution of carbonate of soda is struck with a bit of wood, you hear a clear musical sound. But when a little tartaric acid is added to the liquid, it foams, and a dry, unmusical collision takes the place of the musical sound. As the foam disappears, the sonorous power returns, and now that the liquid is once more clear, you hear the musical ring as before.

The ripples of the tide leave their impressions upon the sand over which they pass. The ripples produced by sonorous vibrations have been proved by Faraday competent to do the same. Attaching a plate of glass to a long flexible board, and pouring a thin layer of water over the surface of the glass, on causing the board to vibrate its tremors chase the water into a beautiful mosaic of ripples. A thin stratum of sand strewed upon the plate is acted upon by the water, and carved into patterns, of which Fig. 81 is a reduced specimen.

SUMMARY OF CHAPTER IV

A rod fixed at both ends and caused to vibrate transversely divides itself in the same manner as a string vibrating transversely.

But the succession of its overtones is not the same as those of a string, for while the series of tones emitted by the string is expressed by the natural numbers 1, 2, 3, 4, 5, etc., the series of tones emitted by the rod is expressed by the squares of the odd numbers 3, 5, 7, 9, etc.

A rod fixed at one end can also vibrate as a whole, or can divide itself into vibrating segments separated from each other by nodes.

In this case the rate of vibration of the fundamental tone is to that of the first overtone as 4:25, or as the square of 2 to the square of 5. From the first division onward the rates of vibration are proportional to the squares of the odd numbers 3, 5, 7, 9, etc.

With rods of different lengths the rate of vibration is inversely proportional to the square of the length of the rod.

Attaching a glass bead silvered within to the free end of the rod, and illuminating the bead, the spot of light reflected from it describes curves of various forms when the rod vibrates. The kaleidophone of Wheatstone is thus constructed.

The iron fiddle and the musical box are instruments whose tones are produced by rods, or tongues, fixed at one end and free at the other.

A rod free at both ends can also be rendered a source of sonorous vibrations. In its simplest mode of division it has two nodes, the subsequent overtones correspond to divisions by 3, 4, 5, etc., nodes. Beginning with its first mode of division, the tones of such a rod are represented by the squares of the odd numbers 3, 5, 7, 9, etc.

The claque-bois, straw-fiddle, and glass harmonica are instruments whose tones are those of rods or bars free at both ends, and supported at their nodes.

When a straight bar, free at both ends, is gradually bent at its centre, the two nodes corresponding to its fundamental tone gradually approach each other. It finally assumes the shape of a timing-fork which, when it sounds its fundamental note, is divided by two nodes near the base of its two prongs into three vibrating parts.

There is no division of a tuning-fork by three nodes.

In its second mode of division, which corresponds to the first overtone of the fork, there is a node on each prong and two others at the bottom of the fork.

The fundamental tone of the fork is to its first overtone approximately as the square of 2 is to the square of 5. The vibrations of the first overtone are, therefore, about 6-1/4 times as rapid as those of the fundamental. From the first overtone onward the successive rates of vibration are as the squares of the odd numbers 3, 5, 7, 9, etc.

We are indebted to Chladni for the experimental investigation of all these points. He was enabled to conduct his inquiries by means of the discovery that, when sand is scattered over a vibrating surface, it is driven from the vibrating portions of the surface, and collects along the nodal lines.

Chladni embraced in his investigations plates of various forms. A square plate, for example, clamped at the centre, and caused to emit its fundamental tone, divides itself into four smaller squares by lines parallel to its sides.

The same plate can divide itself into four triangular vibrating parts, the nodal lines coinciding with the diagonals. The note produced in this case is a fifth above the fundamental note of the plate.

The plate may be further subdivided, sand-figures of extreme beauty being produced; the notes rise in pitch as the subdivision of the plate becomes more minute.

These figures may be deduced from the coalescence of different systems of vibration.

When a circular plate clamped at its centre sounds its fundamental tone, it divides into four vibrating parts, separated by four radial nodal lines.

The next note of the plate corresponds to a division into six vibrating sectors, the next note to a division into eight sectors; such a plate can divide into any even number of vibrating sectors, the sand-figures assuming beautiful stellar forms.

The rates of vibration corresponding to the divisions of a disk are represented by the squares of the numbers 2, 3, 4, 5, 6, etc. In other words, the rates of vibration are proportional to the squares of the numbers representing the sectors into which the disk is divided.

When a bell sounds its deepest note it is divided into four vibrating parts separated from each other by nodal lines, which run upward from the sound-bow and cross each other at the crown.

It is capable of the same subdivisions as a disk; the succession of its tones being also the same.

The rate of vibration of a disk or bell is directly proportional to the thickness and inversely proportional to the square of the diameter.

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