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_
[Illustration: FIG. 52.]
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.
[Illustration: FIG. 53.]
[Illustration: FIG. 54.]
[Illustration: 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.
[Illustration: FIG. 56.]
[Illustration: 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.
[Illustration: FIG. 58.]
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.
[Illustration: FIG. 59.]
§ 4. _Transverse Vibrations of a Rod free at Both Ends. The Claque-bois
and Glass Harmonica_
[Illustration: FIG. 60.]
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.
[Illustration: FIG. 61.]
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.
[Illustration: FIG. 62.]
[Illustration: FIG. 63.]
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.
[Illustration: FIG. 64.]
[Illustration: FIG. 65.]
[Illustration: 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.
[Illustration: FIG. 67.]
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.
[Illustration: FIG. 68.]
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_
[Illustration: FIG. 69.]
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 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.
[Illustration: FIG. 70.]
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.
[Illustration: FIG. 71.]
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.
[Illustration: FIG. 72.]
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.
[Illustration: FIG. 73.]
[Illustration: FIG. 74.]
[Illustration: FIG. 75.]
[Illustration: 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_
[Illustration: FIG. 77.]
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
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 }
rates of vibration } 2, 3, 4, 5, 6
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.
[Illustration: FIG. 78.]
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]
[Illustration: FIG. 79.]
[Illustration: 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.
[Illustration: FIG. 81.]
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.