Vibration of Musical Strings—Harmonic Sounds—Nodes—Vibration of Air in
Wind-Instruments—Vibration of Solids—Vibrating
Plates—Bells—Harmony—Sounding Boards—Forced
Vibrations—Resonance—Speaking Machines.
WHEN the particles of elastic bodies are suddenly disturbed by an
impulse, they return to their natural position by a series of
isochronous vibrations, whose rapidity, force, and permanency depend
upon the elasticity, the form, and the mode of aggregation which unites
the particles of the body. These oscillations are communicated to the
air, and on account of its elasticity they excite alternate
condensations and dilatations in the strata of the fluid nearest to the
vibrating body; from thence they are propagated to a distance. A string
or wire stretched between two pins, when drawn aside and suddenly let
go, will vibrate till its own rigidity and the resistance of the air
reduce it to rest. These oscillations may be rotatory, in every plane,
or confined to one plane according as the motion is communicated. In the
piano-forte, where the strings are struck by a hammer at one extremity,
the vibrations probably consist of a bulge running to and fro from end
to end. Different modes of vibration may be obtained from the same
sonorous body. Suppose a vibrating string to give the lowest C of the
pianoforte which is the fundamental note of the string; if it be lightly
touched exactly in the middle, so as to retain that point at rest, each
half will then vibrate twice as fast as the whole, but in opposite
directions; the ventral or bulging segments will be alternately above
and below the natural position of the string, and the resulting note
will be the octave above C. When a point at a third of the length of the
string is kept at rest, the vibrations will be three times as fast as
those of the whole string, and will give the twelfth above C. When the
point of rest is one-fourth of the whole, the oscillations will be four
times as fast as those of the fundamental note, and will give the double
octave; and so on. These acute sounds are called the harmonics of the
fundamental note. It is clear, from what has been stated, that the
string thus vibrating could not give these harmonics spontaneously
unless it divided itself at its aliquot parts into two, three, four, or
more segments in opposite states of vibration separated by points
actually at rest. In proof of this, pieces of paper placed on the string
at the half, third, fourth, or other aliquot points, according to the
corresponding harmonic sound, will remain on it during its vibration,
but will instantly fly off from any of the intermediate points. The
points of rest, called the nodal points of the string, are a mere
consequence of the law of interferences; for, if a rope fastened at one
end be moved to and fro at the other extremity so as to transmit a
succession of equal waves along it, they will be successively reflected
when they arrive at the other end of the rope by the fixed point, and in
returning they will occasionally interfere with the advancing waves;
and, as these opposite undulations will at certain points destroy one
another, the point of the rope in which this happens will remain at
rest. Thus a series of nodes and ventral segments will be produced whose
number will depend upon the tension and the frequency of the alternate
motions communicated to the moveable end. So, when a string fixed at
both ends is put in motion by a sudden blow at any point of it, the
primitive impulse divides itself into two pulses running opposite ways,
which are each totally reflected at the extremities, and, running back
again along the whole length, are again reflected at the other ends. And
thus they will continue to run backwards and forwards, crossing one
another at each traverse, and occasionally interfering, so as to produce
nodes; so that the motion of a string fastened at both ends consists of
a wave or pulse continually doubled back on itself by reflection at the
fixed extremities.
Harmonics generally co-exist with the fundamental sound in the same
vibrating body. If one of the lowest strings of the pianoforte be
struck, an attentive ear will not only hear the fundamental note, but
will detect all the others sounding along with it, though with less and
less intensity as their pitch becomes higher. According to the law of
co-existing undulations, the whole string and each of its aliquot parts
are in different and independent states of vibration at the same time;
and as all the resulting notes are heard simultaneously, not only the
air, but the ear also, vibrates in unison with each at the same instant
(N. 181).
Harmony consists in an agreeable combination of sounds. When two chords
perform their vibrations in the same time, they are in unison; but, when
their vibrations are so related as to have a common period, after a few
oscillations they produce concord. Thus, when the vibrations of two
strings bear a very simple relation to each other, as where one of them
makes two, three, four, &c., vibrations in the time the other makes one;
or, if it accomplishes three, four, &c., vibrations while the other
makes two, the result is a concord which is the more perfect the shorter
the common period. In discords, on the contrary, the beats are
distinctly audible, which produces a disagreeable and harsh effect,
because the vibrations do not bear a simple relation to one another, as
where one of two strings makes eight vibrations while the other
accomplishes fifteen. The pleasure afforded by harmony is attributed by
Dr. Young to the love of order, and to a predilection for a regular
repetition of sensations natural to the human mind, which is gratified
by the perfect regularity and rapid recurrence of the vibrations. The
love of poetry and dancing he conceives to arise in some degree from the
rhythm of the one and the regularity of the motions in the other.
A blast of air passing over the open end of a tube, as over the reeds in
Pan’s pipes; over a hole in one side, as in the flute; or through the
aperture called a reed, with a flexible tongue, as in the clarinet, puts
the internal column of air into longitudinal vibrations by the alternate
condensations and rarefactions of its particles. At the same time the
column spontaneously divides itself into nodes, between which the air
also vibrates longitudinally, but with a rapidity inversely proportional
to the length of the divisions, giving the fundamental note or one of
its harmonics. The nodes are produced on the principle of interferences
by the reflection of the longitudinal undulations of the air at the ends
of the pipe, as in the musical string, only that in one case the
undulations are longitudinal, and in the other transverse.
A pipe, either open or shut at both ends, when sounded, vibrates entire,
or divides itself spontaneously into two, three, four, &c., segments
separated by nodes. The whole column gives the fundamental note by waves
or vibrations of the same length with the pipe. The first harmonic is
produced by waves half as long as the tube, the second harmonic by waves
a third as long, and so on. The harmonic segments in an open and shut
pipe are the same in number, but differently placed. In a shut pipe the
two ends are nodes, but in an open pipe there is half a segment at each
extremity, because the air at these points is neither rarefied nor
condensed, being in contact with that which is external. If one of the
ends of the open pipe be closed, its fundamental note will be an octave
lower: the air will now divide itself into three, five, seven, &c.,
segments; and the wave producing its fundamental note will be twice as
long as the pipe, so that it will be doubled back (N. 182). All these
notes may be produced separately by varying the intensity of the blast.
Blowing steadily and gently, the fundamental note will sound; when the
force of the blast is increased the note will all at once start up an
octave; when the intensity of the wind is augmented the twelfth will be
heard; and, by continuing to increase the force of the blast, the other
harmonics may be obtained, but no force of wind will produce a note
intermediate between these. The harmonics of a flute may be obtained in
this manner, from the lowest C or D upwards, without altering the
fingering, merely by increasing the intensity of the blast and altering
the form of the lips. Pipes of the same dimensions, whether of lead,
glass, or wood, give the same tone as to pitch under the same
circumstances, which shows that the air alone produces the sound.
Metal springs fastened at one end, when forcibly bent, endeavour to
return to rest by a series of vibrations, which give very pleasing
tones, as in musical boxes. Various musical instruments have been
constructed, consisting of metallic springs thrown into vibration by a
current of air. Among the most perfect of these are Mr. Wheatstone’s
Symphonion, Concertina, and Æolian Organ, instruments of different
effects and capabilities, but all possessing considerable execution and
expression.
The Syren is an ingenious instrument, devised by M. Cagniard de la Tour,
for ascertaining the number of pulsations in a second, corresponding to
each pitch: the notes are produced by jets of air passing through small
apertures, arranged at regular distances in a circle on the side of a
box, before which a disc revolves pierced with the same number of holes.
During a revolution of the disc the currents are alternately intercepted
and allowed to pass as many times as there are apertures in it, and a
sound is produced whose pitch depends on the velocity of rotation.
A glass or metallic rod, when struck at one end, or rubbed in the
direction of its length with a wet finger, vibrates longitudinally, like
a column of air, by the alternate condensation and expansion of its
constituent particles, producing a clear and beautiful musical note of a
high pitch, on account of the rapidity with which these substances
transmit sound. Rods, surfaces, and, in general, all undulating bodies,
resolve themselves into nodes. But in surfaces the parts which remain at
rest during their vibrations are lines which are curved or plane
according to the substance, its form, and the mode of vibration. If a
little fine dry sand be strewed over the surface of a plate of glass or
metal, and if undulations be excited by drawing the bow of a violin
across its edge, it will emit a musical sound, and the sand will
immediately arrange itself in the nodal lines, where alone it will
accumulate and remain at rest, because the segments of the surface on
each side will be in different states of vibration, the one being
elevated while the other is depressed; and, as these two motions meet in
the nodal lines, they neutralise one another. These lines vary in form
and position with the part where the bow is drawn across, and the point
by which the plate is held. The motion of the sand shows in what
direction the vibrations take place. If they be perpendicular to the
surface, the sand will be violently tossed up and down till it finds the
points of rest. If they be tangential, the sand will only creep along
the surface to the nodal lines. Sometimes the undulations are oblique,
or compounded of both the preceding. If a bow be drawn across one of the
angles of a square plate of glass or metal held firmly by the centre,
the sand will arrange itself in two straight lines parallel to the sides
of the plate, and crossing in the centre so as to divide it into four
equal squares, whose motions will be contrary to each other. Two of the
diagonal squares will make their excursions on one side of the plate,
while the other two make their vibrations on the other side of it. This
mode of vibration produces the lowest tone of the plate (N. 183). If the
plate be still held by the centre, and the bow applied to the middle of
one of the sides, the vibrations will be more rapid, and the tone will
be a fifth higher than in the preceding case: now the sand will arrange
itself from corner to corner, and will divide the plate into four equal
triangles, each pair of which will make their excursions on opposite
sides of the plate. The nodal lines and pitch vary not only with the
point where the bow is applied, but with the point by which the plate is
held, which being at rest necessarily determines the direction of one of
the quiescent lines. The forms assumed by the sand in square plates are
very numerous, corresponding to all the various modes of vibration. The
lines in circular plates are even more remarkable for their symmetry,
and upon them the forms assumed by the sand may be classed in three
systems. The first is the diametrical system, in which the figures
consist of diameters dividing the circumference of the plate into equal
parts, each of which is in a different state of vibration from those
adjacent. Two diameters, for example, crossing at right angles, divide
the circumference into four equal parts; three diameters divide it into
six equal parts; four divide it into eight, and so on. In a metallic
plate, these divisions may amount to thirty-six or forty. The next is
the concentric system, where the sand arranges itself in circles, having
the same centre with the plate; and the third is the compound system,
where the figures assumed by the sand are compounded of the other two,
producing very complicated and beautiful forms. Galileo seems to have
been the first to notice the points of rest and motion in the
sounding-board of a musical instrument; but to Chladni is due the whole
discovery of the symmetrical forms of the nodal lines in vibrating
plates (N. 184). Professor Wheatstone has shown, in a paper read before
the Royal Society in 1833, that all Chladni’s figures, and indeed all
the nodal figures of vibrating surfaces, result from very simple modes
of vibration oscillating isochronously, and superposed upon each other;
the resulting figure varying with the component modes of vibration, the
number of the superpositions, and the angles at which they are
superposed. For example, if a square plate be vibrating so as to make
the sand arrange itself in straight lines parallel to one side of the
plate, and if, in addition to this, such vibrations be excited as would
have caused the sand to form in lines perpendicular to the first had the
plate been at rest, the combined vibrations will make the sand form in
lines from corner to corner (N. 185).
M. Savart’s experiments on the vibrations of flat glass rulers are
highly interesting. Let a lamina of glass 27^{in}·56 long, 0·59 of an
inch broad, and 0·06 of an inch in thickness, be held by the edges in
the middle, with its flat surface horizontal. If this surface be strewed
with sand, and set in longitudinal vibration by rubbing its under
surface with a wet cloth, the sand on the upper surface will arrange
itself in lines parallel to the ends of the lamina, always in one or
other of two systems (N. 186). Although the same one of the two systems
will always be produced by the same plate of glass, yet among different
plates of the preceding dimensions, even though cut from the same sheet
side by side, one will invariably exhibit one system, and the other the
other, without any visible reason for the difference. Now, if the
positions of these quiescent lines be marked on the upper surface, and
if the plate be turned so that the lower surface becomes the upper one,
the sand being strewed, and vibrations excited as before, the nodal
lines will still be parallel to the ends of the lamina, but their
positions will be intermediate between those of the upper surface
(N. 187). Thus it appears that all the motions of one half of the
thickness of the lamina, or ruler, are exactly contrary to those of the
corresponding points of the other half. If the thickness of the lamina
be increased, the other dimensions remaining the same, the sound will
not vary, but the number of nodal lines will be less. When the breadth
of the lamina exceeds the 0·6 of an inch, the nodal lines become curved,
and are different on the two surfaces. A great variety of forms are
produced by increasing the breadth and changing the form of the surface;
but in all it appears that the motions in one half of the thickness are
opposed to those in the other half.
M. Savart also found, by placing small paper rings round a cylindrical
tube or rod, so as to rest upon it at one point only, that, when the
tube or rod is continually turned on its axis in the same direction, the
rings slide along during the vibrations, till they come to a quiescent
point, where they rest. By tracing these nodal lines he discovered that
they twist in a spiral or corkscrew round rods and cylinders, making one
or more turns according to the length; but at certain points, varying in
number according to the mode of vibration of the rod, the screw stops,
and recommences on the other side, though it is turned in a contrary
direction; that is, on one side it is a right-handed screw, on the other
a left (N. 188). The nodal lines in the interior surface of the tube are
perfectly similar to those in the exterior, but they occupy intermediate
positions. If a small ivory ball be put within the tube, it will follow
these nodal lines when the tube is made to revolve on its axis.
All solids which ring when struck, such as bells, drinking glasses,
gongs, &c., have their shape momentarily and forcibly changed by the
blow, and from their elasticity, or tendency to resume their natural
form, a series of undulations take place, owing to the alternate
condensations and rarefactions of the particles of solid matter. These
have also their harmonic tones, and consequently nodes. Indeed,
generally, when a rigid system of any form whatever vibrates either
transversely or longitudinally, it divides itself into a certain number
of parts which perform their vibrations without disturbing one another.
These parts are at every instant in alternate states of undulation; and,
as the points or lines where they join partake of both, they remain at
rest, because the opposing motions destroy one another.
The air, notwithstanding its rarity, is capable of transmitting its
undulations when in contact with a body susceptible of admitting and
exciting them. It is thus that sympathetic undulations are excited by a
body vibrating near insulated tended strings, capable of following its
undulations, either by vibrating entire, or by separating themselves
into their harmonic divisions. If two chords equally stretched, of which
one is twice or three times longer than the other, be placed side by
side, and if the shorter be sounded, its vibrations will be communicated
by the air to the other, which will be thrown into such a state of
vibration that it will be spontaneously divided into segments equal in
length to the shorter string. When a tuning-fork receives a blow and is
made to rest upon a piano-forte during its vibration, every string
which, either by its natural length or by its spontaneous subdivisions,
is capable of executing corresponding vibrations, responds in a
sympathetic note. The same effect will be produced by the stroke of a
bell near a piano or harp. Some one or other of the notes of an organ
are generally in unison with one of the panes or with the whole sash of
a window, which consequently resounds when those notes are sounded. A
peal of thunder has frequently the same effect. The sound of very large
organ-pipes is generally inaudible till the air be set in motion by the
undulations of some of the superior accords, and then the sound becomes
extremely energetic. Recurring vibrations occasionally influence each
other’s periods. For example, two adjacent organ-pipes nearly in unison
may force themselves into concord; and two clocks, whose rates differed
considerably when separate, have been known to beat together when fixed
to the same wall, and one clock has forced the pendulum of another into
motion, when merely standing on the same stone pavement. These forced
oscillations, which correspond in their periods with those of the
exciting cause, are to be traced in every department of physical
science. Several instances of them have already occurred in this work.
Such are the tides, which follow the sun and moon in all their motions
and periods. The nutation of the earth’s axis also, which corresponds
with the period, and represents the motion of the nodes of the moon, is
again reflected back to the moon, and may be traced in the nutation of
the lunar orbit. And, lastly, the acceleration of the moon’s mean motion
represents the action of the planets on the earth reflected by the sun
to the moon.
In consequence of the facility with which the air communicates
undulations, all the phenomena of vibrating plates may be exhibited by
sand strewed on paper or parchment, stretched over a harmonica glass or
large bell-shaped tumbler. In order to give due tension to the paper or
vellum, it must be wetted, stretched over the glass, gummed round the
edges, allowed to dry, and varnished over, to prevent changes in its
tension from the humidity of the atmosphere. If a circular disc of glass
be held concentrically over this apparatus, with its plane parallel to
the surface of the paper, and set in vibration by drawing a bow across
its edge, so as to make sand on its surface take any of Chladni’s
figures, the sand on the paper will assume the very same form, in
consequence of the vibrations of the disc being communicated to the
paper by the air. When the disc is removed slowly in a horizontal
direction, the forms on the paper will correspond with those on the
disc, till the distance is too great for the air to convey the
vibrations. If the disc while vibrating be gradually more and more
inclined to the horizon, the figures on the paper will vary by degrees;
and, when the vibrating disc is perpendicular to the horizon, the sand
on the paper will form into straight lines parallel to the surface of
the disc, by creeping along it instead of dancing up and down. If the
disc be made to turn round its vertical diameter while vibrating, the
nodal lines on the paper will revolve, and exactly follow the motion of
the disc. It appears, from this experiment, that the motions of the
aërial molecules in every part of a spherical wave, propagated from a
vibrating body as a centre, are parallel to each other, and not
divergent like the radii of a circle. When a slow air is played on a
flute near this apparatus, each note calls up a particular form in the
sand, which the next note effaces, to establish its own. The motion of
the sand will even detect sounds that are inaudible. By the vibrations
of sand on a drum-head the besieged have discovered the direction in
which a counter-mine was working. M. Savart, who made these beautiful
experiments, employed this apparatus to discover nodal lines in masses
of air. He found that the air of a room, when thrown into undulations by
the continued sound of an organ-pipe, or by any other means, divides
itself into masses separated by nodal curves of double curvature, such
as spirals, on each side of which the air is in opposite states of
vibration. He even traced these quiescent lines going out at an open
window, and for a considerable distance in the open air. The sand is
violently agitated where the undulations of the air are greatest, and
remains at rest in the nodal lines. M. Savart observed, that when he
moved his head away from a quiescent line towards the right the sound
appeared to come from the right, and when he moved it towards the left
the sound seemed to come from the left, because the molecules of air are
in different states of motion on each side of the quiescent line.
A musical string gives a very feeble sound when vibrating alone, on
account of the small quantity of air set in motion; but when attached to
a sounding-board, as in the harp and piano-forte, it communicates its
undulations to that surface, and from thence to every part of the
instrument; so that the whole system vibrates isochronously, and by
exposing an extensive undulating surface, which transmits its
undulations to a great mass of air, the sound is much reinforced. The
intensity is greatest when the vibrations of the string or sounding body
are perpendicular to the sounding-board, and least when they are in the
same plane with it. The sounding-board of the piano-forte is better
disposed than that of any other stringed instrument, because the hammers
strike the strings so as to make them vibrate at right angles to it. In
the guitar, on the contrary, they are struck obliquely, which renders
the tone feeble, unless when the sides, which also act as a
sounding-board, are deep. It is evident that the sounding-board and the
whole instrument are agitated at once by all the superposed vibrations
excited by the simultaneous or consecutive notes that are sounded, each
having its perfect effect independently of the rest. A sounding-board
not only reciprocates the different degrees of pitch, but all the
nameless qualities of tone. This has been beautifully illustrated by
Professor Wheatstone in a series of experiments on the transmission
through solid conductors of musical performances, from the harp, piano,
violin, clarinet, &c. He found that all the varieties of pitch, quality,
and intensity are perfectly transmitted with their relative gradations,
and may be communicated, through conducting wires or rods of very
considerable length, to a properly disposed sounding-board in a distant
apartment. The sounds of an entire orchestra may be transmitted and
reciprocated by connecting one end of a metallic rod with a
sounding-board near the orchestra, so placed as to resound to all the
instruments, and the other end with the sounding-board of a harp, piano,
or guitar, in a remote apartment. Professor Wheatstone observes, “The
effect of this experiment is very pleasing; the sounds, indeed, have so
little intensity as scarcely to be heard at a distance from the
reciprocating instrument; but, on placing the ear close to it, a
diminutive band is heard in which all the instruments preserve their
distinctive qualities, and the pianos and fortes, the crescendos and
diminuendos, their relative contrasts. Compared with an ordinary band
heard at a distance through the air, the effect is as a landscape seen
in miniature beauty through a concave lens, compared with the same scene
viewed by ordinary vision through a murky atmosphere.”
Every one is aware of the reinforcement of sound by the resonance of
cavities. When singing or speaking near the aperture of a wide-mouthed
vessel, the intensity of some one note in unison with the air in the
cavity is often augmented to a great degree. Any vessel will resound if
a body vibrating the natural note of the cavity be placed opposite to
its orifice, and be large enough to cover it, or at least to set a large
portion of the adjacent air in motion. For the sound will be alternately
reflected by the bottom of the cavity and the undulating body at its
mouth. The first impulse of the undulating substance will be reflected
by the bottom of the cavity, and then by the undulating body, in time to
combine with the second new impulse. This reinforced sound will also be
twice reflected in time to conspire with the third new impulse; and, as
the same process will be repeated on every new impulse, each will
combine with all its echoes to reinforce the sound prodigiously.
Professor Wheatstone, to whose ingenuity we are indebted for so much new
and valuable information on the theory of sound, has given some very
striking instances of resonance. If one of the branches of a vibrating
tuning-fork be brought near the embouchure of a flute, the lateral
apertures of which are stopped so as to render it capable of producing
the same sound as the fork, the feeble and scarcely audible sound of the
fork will be augmented by the rich resonance of the column of air within
the flute, and the tone will be full and clear. The sound will be found
greatly to decrease by closing or opening another aperture; for the
alteration in the length of the column of air renders it no longer fit
perfectly to reciprocate the sound of the fork. This experiment may be
made on a concert flute with a C tuning-fork. But Professor Wheatstone
observes, that in this case it is generally necessary to finger the
flute for B, because, when blown into with the mouth, the under-lip
partly covers the embouchure, which renders the sound about a semitone
flatter than it would be were the embouchure entirely uncovered. He has
also shown, by the following experiment, that any one among several
simultaneous sounds may be rendered separately audible. If two bottles
be selected, and tuned by filling them with such a quantity of water as
will render them unisonant with two tuning-forks which differ in pitch,
on bringing both of the vibrating tuning-forks to the mouth of each
bottle alternately, in each case that sound only will be heard which is
reciprocated by the unisonant bottle.
Several attempts have been made to imitate the articulation of the
letters of the alphabet. About the year 1779, MM. Kratzenstein of St.
Petersburg, and Kempelen of Vienna, constructed instruments which
articulated many letters, words, and even sentences. Mr. Willis of
Cambridge has adapted cylindrical tubes to a reed, whose length can be
varied at pleasure by sliding joints. Upon drawing out a tube while a
column of air from the bellows of an organ is passing through it, the
vowels are pronounced in the order, _i_, _e_, _a_, _o_, _u_. On
extending the tube, they are repeated after a certain interval, in the
inverted order, _u_, _o_, _a_, _e_, _i_. After another interval they are
again obtained in the direct order, and so on. When the pitch of the
reed is very high, it is impossible to sound some of the vowels, which
is in perfect correspondence with the human voice, female singers being
unable to pronounce _u_ and _o_ in their high notes. From the singular
discoveries of M. Savart on the nature of the human voice, and the
investigations of Mr. Willis on the mechanism of the larynx, it may be
presumed that ultimately the utterance or pronunciation of modern
languages will be conveyed, not only to the eye, but also to the ear of
posterity. Had the ancients possessed the means of transmitting such
definite sounds, the civilised world would still have responded in
sympathetic notes at the distance of many ages.