Longitudinal Vibrations of a Wire—Relative Velocities of
Sound in Brass and Iron—Longitudinal Vibrations of Rods
fixed at One End—Of Rods free at Both Ends—Divisions and
Overtones of Rods vibrating longitudinally—Examination
of Vibrating Bars by Polarized Light—Determination of
Velocity of Sound in Solids—Resonance—Vibrations of
Stopped Pipes: their Divisions and Overtones—Relation
of the Tones of Stopped Pipes to those of Open
Pipes—Condition of Column of Air within a Sounding
Organ-Pipe—Reeds and Reed-Pipes—The Voice—Overtones of
the Vocal Chords—The Vowel Sounds—Kundt’s Experiments—New
Methods of determining the Velocity of Sound
§ 1. _Longitudinal Vibrations of Wires and Rods: Conversion of
Longitudinal into Transverse Vibrations_
We have thus far occupied ourselves exclusively with transversal
vibrations; that is to say, vibrations executed at right angles to
the lengths of the strings, rods, plates, and bells subjected to
examination. A string is also capable of vibrating in the direction of
its length, but here the power which enables it to vibrate is not a
tension applied externally, but the elastic force of its own molecules.
Now this molecular elasticity is much greater than any that we can
ordinarily develop by stretching the string, and the consequence is
that the sounds produced by the _longitudinal vibrations_ of a string
are, as a general rule, much more acute than those produced by its
transverse vibrations. These longitudinal vibrations may be excited by
the oblique passage of a fiddle-bow; but they are more easily produced
by passing briskly along the string a bit of cloth or leather on which
powdered resin has been strewed. The resined fingers answer the same
purpose.
When the wire of our monochord is plucked aside, you hear the sound
produced by its transverse vibrations. When resined leather is rubbed
along the wire, a note much more piercing than the last is heard. This
is due to the longitudinal vibrations of the wire. Behind the table
is stretched a stout iron wire 23 feet long. One end of it is firmly
attached to an immovable wooden tray, the other end is coiled round a
pin fixed firmly into one of our benches. With a key this pin can be
turned, and the wire stretched so as to facilitate the passage of the
rubber. Clasping the wire with the resined leather, and passing the
hand to and fro along it, a rich, loud musical sound is heard. Halving
the wire at its centre, and rubbing one of its halves, the note heard
is the octave of the last: the vibrations are twice as rapid. When the
wire is clipped at one-third of its length and the shorter segment
rubbed, the note is a fifth above the octave. Taking one-fourth of its
length and rubbing as before, the note yielded is the double octave
of that of the whole wire, being produced by four times the number of
vibrations. Thus, in longitudinal as well as in transverse vibrations,
the number of vibrations executed in a given time _is inversely
proportional to the length of the wire_.
And notice the surprising power of these sounds when the wire is rubbed
vigorously. With a shorter length, the note is so acute, and at the
same time so powerful, as to be hardly bearable. It is not the wire
itself which produces this intense sound; it is the wooden tray at
its end to which its vibrations are communicated. And, the vibrations
of the wire being longitudinal, those of the tray, which is at right
angles to the wire, must be transversal. We have here, indeed, an
instructive example of the conversion of longitudinal into transverse
vibrations.
§ 2. _Longitudinal Pulses in Iron and Brass: their Relative Velocities
determined_
Causing the wire to vibrate again longitudinally through its entire
length, my assistant shall at the same time turn the key at the end,
thus changing the tension. You notice no variation of the note. The
longitudinal vibrations of the wire, unlike the transverse ones, are
independent of the tension. Beside the iron wire is stretched a second,
of brass, of the same length and thickness. I rub them both. Their
tones are not the same; that of the iron wire is considerably the
higher of the two. Why? Simply because the velocity of the sound-pulse
is greater in iron than in brass. The pulses in this case pass to and
fro from end to end of the wire. At one moment the wire pushes the
tray at its end; at the next moment it pulls the tray, this pushing
and pulling being due to the passage of the pulse to and fro along the
whole wire. The time required for a pulse to run from one end to the
other _and back_ is that of a complete vibration. In that time the wire
imparts one push and one pull to the wooden tray at its end; the wooden
tray imparts one complete vibration to the air, and the air bends once
in and once out the tympanic membrane. It is manifest that the rapidity
of vibration, or, in other words, the pitch of the note, depends upon
the velocity with which the sound-pulse is transmitted through the
wire.
And now the solution of a beautiful problem falls of itself into our
hands. By shortening the brass wire we cause it to emit a note of
the same pitch as that emitted by the other. You hear both notes now
sounding in unison, and the reason is that the sound-pulse travels
through these 23 feet of iron wire, and through these 15 feet 6 inches
of brass wire, in the same time. These lengths are in the ratio of
11:17, and these two numbers express the relative velocities of sound
in brass and iron. In fact, the former velocity is 11,000 feet, and the
latter 17,000 feet a second. The same method is of course applicable to
many other metals.
[Illustration: FIG. 82.]
When a wire or string, vibrating longitudinally, emits its lowest
note, there is no node whatever upon it; the pulse, as just stated,
runs to and fro along the whole length. But, like a string vibrating
transversely, it can also subdivide itself into ventral segments
separated by nodes. By damping the centre of the wire we make that
point a node. The pulses here run from both ends, meet in the centre,
recoil from each other, and return to the ends, where they are
reflected as before. The note produced is the octave of the fundamental
note. The next higher note corresponds to the division of the wire
into three vibrating segments, separated from each other by two nodes.
The first of these three modes of vibration is shown in Fig. 82, _a_
and _b_; the second at _c_ and _d_; the third at _e_ and _f_; the
nodes being marked by dotted transverse lines, and the arrows in each
case pointing out the direction in which the pulse moves. The rates of
vibration follow the order of the numbers 1, 2, 3, 4, 5, etc., just as
in the case of a wire vibrating transversely.
A _rod_ or _bar_ of wood or metal, with its two ends fixed, and
vibrating longitudinally, divides itself in the same manner as the
wire. The succession of tones is also the same in both cases.
§ 3. _Longitudinal Vibrations of Rods fixed at One End: Musical
Instruments formed on this Principle_
[Illustration: FIG. 83.]
Rods and bars _with one end fixed_ are also capable of vibrating
longitudinally. A smooth wooden or metal rod, for example, with one
of its ends fixed in a vise, yields a musical note, when the resined
fingers are passed along it. When such a note yields its lowest note,
it simply elongates and shortens in quick alternation; there is, then,
no node upon the rod. The pitch of the note is inversely proportional
to the length of the rod. This follows necessarily from the fact that
the time of a complete vibration is the time required for the sonorous
pulse to run twice to and fro over the rod. The first overtone of a
rod, fixed at one end, corresponds to its division by a node at a
point one-third of its length from its free end. Its second overtone
corresponds to a division by two nodes, the highest of which is at
a point one-fifth of the length of the rod from its free end, the
remainder of the rod being divided into two equal parts by the second
node. In Fig. 83, _a_ and _b_, _c_ and _d_, _e_ and _f_, are shown the
conditions of the rod answering to its first three modes of vibration:
the nodes, as before, are marked by dotted lines, the arrows in the
respective cases marking the direction of the pulses.
[Illustration: FIG. 84.]
The order of the tones of a rod fixed at one end and vibrating
longitudinally is that of the odd numbers 1, 3, 5, 7, etc. It is easy
to see that this must be the case. For the time of vibration of _c_ or
_d_ is that of the segment above the dotted line: and the length of
this segment being only one-third that of the whole rod, its vibrations
must be three times as rapid. The time of vibration in _e_ or _f_ is
also that of its highest segment, and as this segment is one-fifth
of the length of the whole rod, its vibrations must be five times as
rapid. Thus the order of the tones must be that of the odd numbers.
Before you, Fig. 84, is a musical instrument, the sounds of which
are due to the longitudinal vibrations of a number of deal rods of
different lengths. Passing the resined fingers over the rods in
succession, a series of notes of varying pitch is obtained. An expert
performer might render the tones of this instrument very pleasant to
you.
§ 4. _Vibrations of Rods free at Both Ends_
[Illustration: FIG. 85.]
Rods _with both ends free_ are also capable of vibrating
longitudinally, and producing musical tones. The investigation of this
subject will lead us to exceedingly important results. Clasping a long
glass tube exactly at its centre, and passing a wetted cloth over one
of its halves, a clear musical sound is the result. A solid glass rod
of the same length would yield the same note. In this case the centre
of the tube is a node, and the two halves elongate and shorten in quick
alternation. M. König, of Paris, has provided us with an instrument
which will illustrate this action. A rod of brass, _a b_, Fig. 85, is
held at its centre by the clamp _s_, while an ivory ball, suspended by
two strings from the points, _m_ and _n_, of a wooden frame, is caused
to rest against the end, _b_, of the brass rod. Drawing gently a bit of
resined leather over the rod near _a_, it is thrown into longitudinal
vibration. The centre, _s_, is a node; but the uneasiness of the
ivory ball shows you that the end, _b_, is in a state of tremor. I
apply the rubber still more briskly. The ball, _b_, rattles, and now
the vibration is so intense that the ball is rejected with violence
whenever it comes into contact with the end of the rod.
§ 5. _Fracture of Glass Tube by Sonorous Vibrations_
When the wetted cloth is passed over the surface of a glass tube
the film of liquid left behind by the cloth is seen forming narrow
tremulous rings all along the rod. Now this shivering of the liquid is
due to the shivering of the glass underneath it, and it is possible
so to augment the intensity of the vibration that the glass shall
actually go to pieces. Savart was the first to show this. Twice in this
place I have repeated this experiment, sacrificing in each case a fine
glass tube 6 feet long and 2 inches in diameter. Seizing the tube at
its centre C, Fig. 86, I swept my hand vigorously to and fro along C
D, until finally the half most distant from my hand was shivered into
annular fragments. On examining these it was found that, narrow as they
were, many of them were marked by circular cracks indicating a still
more minute subdivision.
In this case also the rapidity of vibration is inversely proportional
to the length of the rod. A rod of half the length vibrates
longitudinally with double the rapidity, a rod of one-third the length
with treble the rapidity, and so on. The time of a complete vibration
being that required by the pulse to travel to and fro over the rod, and
that time being directly proportional to the length of the rod, the
rapidity of vibration must, of necessity, be in the inverse proportion.
This division of a rod by a single node at its centre corresponds to
the deepest tone produced by its longitudinal vibration. But, as in
all other cases hitherto examined, such rods can subdivide themselves
further. Holding the long glass rod _a e_, Fig. 87, at a point _b_,
midway between its centre and one of its ends, and rubbing its short
section, _a b_, with a wet cloth, the point _b_ becomes a node, a
second node, _d_, being formed at the same distance from the opposite
end of the rod. Thus we have the rod divided into three vibrating
parts, consisting of one whole ventral segment, _b d_, and two half
ones, _a b_ and _d e_. The sound corresponding to this division of the
rod is the octave of its fundamental note.
[Illustration: FIG. 86.]
[Illustration: FIG. 87.]
You have now a means of checking me. For, if the second mode of
division just described produces the octave of the fundamental note,
and if a rod of half the length produces the same octave, then the
whole rod held at a point one-fourth of its length from one of its ends
ought to emit the same note as the half rod held in the middle. When
both notes are sounded together they are heard to be identical in pitch.
[Illustration: FIG. 88.]
Fig. 88, _a_ and _b_, _c_ and _d_, _e_ and _f_, shows the three first
divisions of a rod free at both ends and vibrating longitudinally. The
nodes, as before, are marked by transverse dots, the direction of the
pulses being shown by arrows. The order of the tones is that of the
numbers, 1, 3, 4, etc.
§ 6. _Action of Sonorous Vibrations on Polarized Light_
When a tube or rod vibrating longitudinally yields its fundamental
tone, its two ends are in a state of free vibration, the glass
there suffering neither strain nor pressure. The case at the centre
is exactly the reverse; here there is no vibration, but a quick
alternation of tension and compression. When the sonorous pulses meet
at the centre they squeeze the glass; when they recoil they strain it.
_Thus while at the ends we have the maximum of vibration, but no change
of density, at the centre we have the maximum changes of density, but
no vibration._
We have now cleared the way for the introduction of a very beautiful
experiment made many years ago by Biot, but never, to my knowledge,
repeated on the scale here presented to you. The beam from our electric
lamp, L, Fig. 89, being sent through a prism, B, of bi-refracting spar,
a beam of polarized light is obtained. This beam impinges on a second
prism of spar, _n_, but, though both prisms are perfectly transparent,
the light which has passed through the first cannot get through the
second. By introducing a piece of glass between the two prisms, and
subjecting the glass to either strain or pressure, the light is enabled
to pass through the entire system.
[Illustration: FIG. 89.]
I now introduce between the prisms B and _n_ a rectangle, _s s′_, of
plate glass, 6 feet long, 2 inches wide, and one-third of an inch
thick, which is to be thrown into longitudinal vibration. The beam
from L passes through the glass at a point near its centre, which is
held in a vise, _c_, so that when a wet cloth is passed over one of
the halves, _c s′_, of the strip, the centre will be a node. During
its longitudinal vibration the glass near the centre is, as already
explained, alternately strained and compressed; and this successive
strain and pressure so changes the condition of the light as to
enable it to pass through the second prism. The screen is now dark;
but on passing the wetted cloth briskly over the glass a brilliant
disk of light, three feet in diameter, flashes out upon the screen.
The vibration quickly subsides, and the luminous disk as quickly
disappears, to be, however, revived at will by the passage of the
wetted cloth along the glass.
The light of this disk appears to be continuous, but it is really
intermittent, for it is only when the glass is under strain or pressure
that the light can get through. In passing from strain to pressure,
and from pressure to strain, the glass is for a moment in its natural
state, which, if it continued, would leave the screen dark. But the
impressions of brightness, due to the strains and pressures, remain
far longer upon the retina than is necessary to abolish the intervals
of darkness; hence the screen appears illuminated by a continuous
light. When the glass rectangle is shifted so as to cause the beam
of polarized light to pass through it close to its end, _s_, the
longitudinal vibrations of the glass have no effect whatever upon the
polarized beam.
Thus, by means of this subtile investigator, we demonstrate that, while
the centre of the glass, where the vibration is _nil_, is subjected to
quick alternations of strain and pressure, the ends of the rectangle,
where the vibration is a maximum, suffer neither.[44]
§ 7. _Vibrations of Rods of Wood: Determination of Relative Velocities
in Different Woods_
Rods of wood and metal also yield musical tones when they vibrate
longitudinally. Here, however, the rubber employed is not a wet cloth,
but a piece of leather covered with powdered resin. The resined fingers
also elicit the music of the rods. The modes of vibration here are
those already indicated, the pitch, however, varying with the velocity
of the sonorous pulse in the respective substances. When two rods of
the same length, the one of deal, the other of Spanish mahogany, are
sounded together, the pitch of the one is much lower than that of
the other. Why? Simply because the sonorous pulses pass more slowly
through the mahogany than through the deal. Can we find the relative
velocity of sound through both? With the greatest ease. We have only to
carefully shorten the mahogany rod till it yields the same note as the
deal one. The notes, rendered approximate by the first trials, are now
identical. Through this rod of mahogany 4 feet long, and through this
rod of deal 6 feet long, the sound-pulse passes in the same time, and
these numbers express the relative velocities of sound through the two
substances.
Modes of investigation, which could only be hinted at in our earlier
lectures, are now falling naturally into our hands. When in the first
lecture the velocity of sound in air was spoken of, many possible
methods of determining this velocity must have occurred to your minds,
because here we have miles of space to operate upon. Its velocity
through wood or metal, where such distances are out of the question,
is determined in the simple manner just indicated. From the notes
which they emit when suitably prepared, we may infer with certainty
the _relative_ velocities of sound through different solid substances;
and determining the ratio of the velocity in any one of them to
its velocity in air, we are able to draw up a table of absolute
velocities. But how is air to be introduced into the series? We shall
soon be able to answer this question, approaching it, however, through
a number of phenomena with which, at first sight, it appears to have no
connection.
RESONANCE
§ 8. _Experiments with Resonant Jars. Analysis and Explanation_
The series of tuning-forks now before you have had their rates of
vibration determined by the siren. One, you will remember, vibrates
256 times in a second, the length of its sonorous wave being 4 feet 4
inches. It is detached from its case, so that when struck against a pad
you hardly hear it. When held over this glass jar, A B, Fig. 90, 18
inches deep, you still fail to hear the sound of the fork. Preserving
the fork in its position, I pour water with the least possible noise
into the jar. The column of air underneath the fork shortens, the
sound augments in intensity, and when the water has reached a certain
level it bursts forth with extraordinary power. A greater quantity of
water causes the sound to sink, and become finally inaudible, as at
first. By pouring the water carefully out, a point is reached where the
reinforcement of the sound again occurs. Experimenting thus, we learn
that there is one particular length of the column of air which, when
the fork is placed above it, produces a maximum augmentation of the
sound. This reinforcement of the sound is named _resonance_.
Operating in the same way with all the forks in succession, a column of
air is found for each, which yields a maximum resonance. These columns
become shorter as the rapidity of vibration increases. In Fig. 91 the
series of jars is represented, the number of vibrations to which each
resounds being placed above it.
[Illustration: FIG. 90.]
[Illustration: FIG. 91.]
What is the physical meaning of this very wonderful effect? To solve
this question we must revive our knowledge of the relation of the
motion of the fork itself to the motion of the sonorous wave produced
by the fork. Supposing a prong of this fork, which executes 256
vibrations in a second, to vibrate between the points _a_ and _b_, Fig.
92, in its motion from _a_ to _b_ the fork generates half a sonorous
wave, and as the length of the whole wave emitted by this fork is 4
feet 4 inches, at the moment the prong reaches _b_ the foremost point
of the sonorous wave will be at C, 2 feet 2 inches distant from the
fork. The motion of the wave, then, is vastly greater than that of
the fork. In fact, the distance _a b_ is, in this case, not more than
one-twentieth of an inch, while the wave has passed over a distance
of 26 inches. With forks of lower pitch the difference would be still
greater.
[Illustration: FIG. 92.]
Our next question is, what is the length of the column of air which
resounds to this fork? By measurement with a two-foot rule it is found
to be 13 inches. But the length of the wave emitted by the fork is 52
inches; hence _the length of the column of air which resounds to the
fork is equal to one-fourth of the length of the sound-wave produced by
the fork_. This rule is general, and might be illustrated by any other
of the forks instead of this one.
[Illustration: FIG. 93.]
Let the prong, vibrating between the limits _a_ and _b_, be placed over
its resonant jar, A B, Fig. 93. In the time required by the prong to
move from _a_ to _b_, the condensation it produces runs down to the
bottom of the jar, is there reflected, and, as the distance to the
bottom and back is 26 inches, the reflected wave will reach the fork at
the moment when it is on the point of returning from _b_ to _a_. The
rarefaction of the wave is produced by the retreat of the prong from
_b_ to _a_. This rarefaction will also run to the bottom of the jar and
back, overtaking the prong just as it reaches the limit, _a_, of its
excursion. It is plain from this analysis that the vibrations of the
fork are perfectly synchronous with the vibrations of the aërial column
A B; and in virtue of this synchronism the motion accumulates in the
jar, spreads abroad in the room, and produces this vast augmentation of
the sound.
When we substitute for the air in one of these jars a gas of different
elasticity, we find the length of the resounding column to be
different. The velocity of sound through coal-gas is to its velocity
in air about as 8:5. Hence, to synchronize with our fork, a jar filled
with coal-gas must be deeper than one filled with air. I turn this
jar, 18 inches long, upside down, and hold close to its open mouth our
agitated tuning-fork. It is scarcely audible. The jar, with air in it,
is 5 inches too deep for this fork. Let coal-gas now enter the jar. As
it ascends the note at a certain point swells out, proving that for the
more elastic gas a depth of 18 inches is not too great. In fact, it
is not great enough; for if too much gas be allowed to enter the jar
the resonance is weakened. By suddenly turning the jar upright, still
holding the fork close to its mouth, the gas escapes, and at the point
of proper admixture of gas and air the note swells out again.[45]
§ 9. _Reinforcement of Bell by Resonance_
This fine, sonorous bell, Fig. 94, is thrown into intense vibration
by the passage of a resined bow across its edge. You hear its sound,
pure, but not very forcible. When, however, the open mouth of this
large tube, which is closed at one end, is brought close to one of the
vibrating segments of the bell, the tone swells into a musical roar.
As the tube is alternately withdrawn and advanced, the sound sinks and
swells in this extraordinary manner.
[Illustration: FIG. 94.]
The second tube, open at both ends, is capable of being lengthened
and shortened by a telescopic slider. When brought near the vibrating
bell, the resonance is feeble. On lengthening the tube by drawing out
the slider at a certain point, the tone swells out as before. If the
tube be made longer, the resonance is again enfeebled. Note the fact,
which shall be explained presently, that the open tube which gives the
maximum resonance is exactly twice the length of the closed one. For
these fine experiments we are indebted to Savart.
§ 10. _Expenditure of Motion in Resonance_
With the India-rubber tube employed in our third chapter it was found
necessary to time the impulses properly, so as to produce the various
ventral segments. I could then feel that the muscular work performed,
when the impulses were properly timed, was greater than when they were
irregular. The same truth may be illustrated by a claret-glass half
filled with water. Endeavor to move your hand to and fro, in accordance
with the oscillating period of the water: when you have thoroughly
established synchronism, the work thrown upon the hand apparently
augments the weight of the water. So likewise with our tuning-fork;
when its impulses are timed to the vibrations of the column of air
contained in this jar, its work is greater than when they are not so
timed. As a consequence of this the tuning-fork comes sooner to rest
when it is placed over the jar than when it is permitted to vibrate
either in free air, or over a jar of a depth unsuited to its periods of
vibration.[46]
Reflecting on what we have now learned, you would have little
difficulty in solving the following beautiful problem: You are provided
with a tuning-fork and a siren, and are required by means of these two
instruments to determine the velocity of sound in air. To solve this
problem you lack, if anything, the mere power of manipulation which
practice imparts. You would first determine, by means of the siren, the
number of vibrations executed by the tuning-fork in a second; you would
then determine the length of the column of air which resounds to the
fork. This length multiplied by 4 would give you, approximately, the
wave-length of the fork, and the wave-length multiplied by the number
of vibrations in a second would give you the velocity in a second.
Without quitting your private room, therefore, you could solve this
important problem. We will go on, if you please, in this fashion,
making our footing sure as we advance.
§ 11. _Resonators of Helmholtz_
[Illustration: FIG. 94_a_.]
Helmholtz has availed himself of the principle of resonance in
analyzing composite sounds. He employs little hollow spheres, called
_resonators_, one of which is shown in Fig. 94_a_. The small projection
_b_, which has an orifice, is placed in the ear, while the sound-waves
enter the hollow sphere through the wide aperture at _a_. Reinforced
by the resonance of such a cavity, and rendered thereby more powerful
than its companions, a particular note of a composite clang may be in a
measure isolated and studied alone.
ORGAN-PIPES
§ 12. _Principles of Resonance applied to Organ-Pipes_
Thus disciplined we are prepared to consider the subject of
organ-pipes, which is one of great importance. Before me on the table
are two resonant jars, and in my right hand and my left are held two
tuning-forks. I agitate both, and hold them over this jar. One of them
only is heard. Held over the other jar, the other fork alone is heard.
Each jar selects that fork whose periods of vibration synchronize with
its own. And instead of two forks suppose several of them to be held
over the jar; from the confused assemblage of pulses thus generated,
the jar would select and reinforce that one which corresponds to its
own period of vibration.
When I blow across the open mouth of the jar, or, better still, for
the jar is too wide for this experiment, when I blow across the
open end of a glass tube, _t u_, Fig. 95, of the same length as the
jar, a fluttering of the air is thereby produced, an assemblage of
pulses at the open mouth of the tube being generated. And what is the
consequence? The tube selects that pulse of the flutter which is in
synchronism with itself, and raises it to a musical sound. The sound,
you perceive, is precisely that obtained when the proper tuning-fork is
placed over the tube. The column of air within the tube has, in this
case, virtually created its own tuning-fork; for by the reaction of its
pulses upon the sheet of air issuing from the lips it has compelled
that sheet to vibrate in synchronism with itself, and made it thus act
the part of the tuning-fork.
[Illustration: FIG. 95.]
Selecting for each of the other tuning-forks a resonant tube, in every
case, on blowing across the open end of the tube, a tone is produced
identical in pitch with that obtained through resonance.
When different tubes are compared, the rate of vibration is found to
be inversely proportional to the length of the tube. These three tubes
are 24, 12, and 6 inches long, respectively. I blow gently across the
24-inch tube, and bring out its fundamental note; similarly treated,
the 12-inch tube yields the octave of the note of the 24-inch. In like
manner the 6-inch tube yields the octave of the 12-inch. It is plain
that this must be the case; for, the rate of vibration depending on
the distance which the pulse has to travel to complete a vibration,
if in one case this distance be twice what it is in another, the rate
of vibration must be twice as slow. In general terms, the rate of
vibration is inversely proportional to the length of the tube through
which the pulse passes.
§ 13. _Vibrations of Stopped Pipes: Modes of Division: Overtones_
But that the current of air should be thus able to accommodate itself
to the requirements of the tube, it must enjoy a certain amount of
_flexibility_. A little reflection will show you that the power of
the reflected pulse over the current must depend to some extent on
the force of the current. A stronger current, like a more powerfully
stretched string, requires a great force to deflect it, and when
deflected vibrates more quickly. Accordingly, to obtain the fundamental
note of this 24-inch tube, we must blow very gently across its open
end; a rich, full, and forcible musical tone is then produced. With
a little stronger blast the sound approaches a mere rustle; blowing
stronger still, a tone is obtained of much higher pitch than the
fundamental one. This is the first overtone of the tube, to produce
which the column of air within it has divided itself into two vibrating
parts, with a node between them. With a still stronger blast a still
higher note is obtained. The tube is now divided into three vibrating
parts, separated from each other by two nodes. Once more I blow
with sudden strength; a higher note than any before obtained is the
consequence.
In Fig. 96 are represented the divisions of the column of air
corresponding to the first three notes of a tube stopped at one end.
At _a_ and _b_, which correspond to the fundamental note, the column
is undivided; the bottom of the tube is the only node, and the pulse
simply moves up and down from top to bottom, as denoted by the arrows.
In _c_ and _d_, which correspond to the first overtone of the tube, we
have one nodal surface shown by dots at _x_, against which the pulses
abut, and from which they are reflected as from a fixed surface. This
nodal surface is situated at one-third of the length of the tube from
its open end. In _e_ and _f_, which correspond to the second overtone,
we have two nodal surfaces, the upper one, _x′_, of which is at
one-fifth of the length of the tube from its open end, the remaining
four-fifths being divided into two equal parts by the second nodal
surface. The arrows, as before, mark the directions of the pulses.
[Illustration: FIG. 96.]
We have now to inquire into the relation of these successive notes to
each other. The space from node to node has been called all through
“a ventral segment”; hence the space between the middle of a ventral
segment and a node is a semi-ventral segment. You will readily bear in
mind the law that _the number of vibrations is directly proportional
to the number of semi-ventral segments_ into which the column of air
within the tube is divided. Thus, when the fundamental note is sounded,
we have but a single semi-ventral segment, as at _a_ and _b_. The
bottom here is a node, and the open end of the tube, where the air
is agitated, is the middle of a ventral segment. The mode of division
represented in _c_ and _d_ yields three semi-ventral segments; in _e_
and _f_ we have five. The vibrations, therefore, corresponding to this
series of notes, augment in the proportion of the series of odd numbers
1:3:5. Could we obtain still higher notes, their relative rates of
vibration would continue to be represented by the odd numbers 7, 9, 11,
13, etc.
It is evident that this _must_ be the law of succession. For the time
of vibration in _c_ or _d_ is that of a stopped tube of the length
_x y_; but this length is one-third of the length of the whole tube,
consequently its vibrations must be three times as rapid. The time
of vibration in _e_ or _f_ is that of a stopped tube of the length
_x′ y′_, and inasmuch as this length is one-fifth that of the whole
tube, its vibrations must be five times as rapid. We thus obtain the
succession 1, 3, 5; if we pushed matters further we should obtain the
continuation of the series of odd numbers.
[Illustration: FIG. 97.]
And here it is once more in your power to subject my statements to an
experimental test. Here are two tubes, one of which is three times
the length of the other. I sound the fundamental note of the longest
tube, and then the next note above the fundamental. The vibrations of
these two notes are stated to be in the ratio of 1:3. This latter note,
therefore, ought to be of precisely the same pitch as the fundamental
note of the shorter of the two tubes. When both tubes are sounded their
notes are identical.
It is only necessary to place a series of such tubes of different
lengths thus together to form that ancient instrument, Pan’s pipes, P
P′, Fig. 97 (page 223), with which we are so well acquainted.
The successive divisions, and the relation of the overtones of a rod
fixed at one end (described in page 205), are plainly identical with
those of a column of air in a tube stopped at one end, which we have
been here considering.
§ 14. _Vibrations of Open Pipes: Modes of Division: Overtones_
From tubes closed at one end, and which, for the sake of brevity,
may be called stopped tubes, we now pass to tubes open at both ends,
which we shall call open tubes. Comparing, in the first instance, a
stopped tube with an open one of the same length, we find the note of
the latter an octave higher than that of the former. This result is
general. To make an open tube yield the same note as a closed one, it
must be twice the length of the latter. And, since the length of a
closed tube sounding its fundamental note is one-fourth of the length
of its sonorous wave, the length of an open tube is one-half that of
the sonorous wave that it produces.
It is not easy to obtain a sustained musical note by blowing across the
end of an open glass tube; but a mere puff of breath across the end
reveals the pitch to the disciplined ear. In each case it is that of a
closed tube half the length of the open one.
There are various ways of agitating the air at the ends of pipes and
tubes, so as to throw the air-columns within them into vibration. In
organ-pipes this is done by blowing a thin sheet of air against a sharp
edge. You will have no difficulty in understanding the construction
of an open organ-pipe, from this model, Fig. 98, one side of which has
been removed so that you may see its inner parts. Through the tube _t_
the air passes from the wind-chest into the chamber, C, which is closed
at the top, save a narrow slit, _e d_, through which the compressed air
of the chamber issues. This thin air-current breaks against the sharp
edge, _a b_, and there produces a fluttering noise, and the proper
pulse of this flutter is converted by the resonance of the pipe above
into a musical sound. The open space between the edge, _a b_, and the
slit below it is called the _embouchure_. Fig. 99 represents a stopped
pipe of the same length as that shown in Fig. 98, and hence producing a
note an octave lower.
[Illustration: FIG. 98.]
[Illustration: FIG. 99.]
Instead of a fluttering sheet of air, a tuning-fork whose rate of
vibration synchronizes with that of the organ-pipe may be placed at
the embouchure, as at A B, Fig. 100. The pipe will resound. Here,
for example, are four open pipes of different lengths, and four
tuning-forks of different rates of vibration. Striking the most slowly
vibrating fork, and bringing it near the embouchure of the longest
pipe, the pipe _speaks_ powerfully. When blown into, the same pipe
yields a tone identical with that of the tuning-fork. Going through all
the pipes in succession, we find in each case that the note obtained
by blowing into the pipe is exactly that produced when the proper
tuning-fork is placed at the embouchure. Conceive now the four forks
placed together near the same embouchure; we should have pulses of four
different periods there excited; but out of the four the pipe would
select only one. And if four hundred vibrating forks could be placed
there instead of four, the pipe would still make the proper selection.
This it also does when for the pulses of tuning-forks we substitute
the assemblage of pulses created by the current of air when it strikes
against the sharp upper edge of the embouchure.
[Illustration: FIG. 100.]
The heavy vibrating mass of the tuning-fork is practically uninfluenced
by the motion of the air within the pipe. But this is not the case when
air itself is the vibrating body. Here, as before explained, the pipe
creates, as it were, its own tuning-fork, by compelling the fluttering
stream at its embouchure to vibrate in periods answering to its own.
[Illustration: FIG. 101.]
The condition of the air within an open organ-pipe, when its
fundamental note is sounded, is that of a rod free at both ends, held
at its centre, and caused to vibrate longitudinally. The two ends
are places of vibration, the centre is a node. Is there any way of
_feeling_ the vibrating air-column so as to determine its nodes and
its places of vibration? The late excellent William Hopkins has taught
us the following mode of solving this problem: Over a little hoop is
stretched a thin membrane, forming a little tambourine. The front of
this organ-pipe, P P′, Fig. 101, is of glass, through which you can see
the position of any body within it. By means of a string, the little
tambourine, _m_, can be raised or lowered at pleasure through the
entire length of the pipe. When held above the upper end of the pipe,
you hear the loud buzzing of the membrane. When lowered into the pipe,
it continues to buzz for a time; the sound becoming gradually feebler,
and finally ceasing totally. It is now in the middle of the pipe, where
it cannot vibrate, because the air around it is at rest. On lowering it
still further, the buzzing sound instantly recommences, and continues
down to the bottom of the pipe. Thus, as the membrane is raised and
lowered in quick succession, during every descent and ascent, we have
two periods of sound separated from each other by one of silence. If
sand were strewed upon the membrane, it would dance above and below,
but it would be quiescent at the centre. We thus prove experimentally
that, when an organ-pipe sounds its fundamental note, it divides itself
into two semi-ventral segments separated by a node.
[Illustration: FIG. 102.]
What is the condition of the air at this node? Again, that of the
middle of a rod, free at both ends, and yielding the fundamental note
of its longitudinal vibration. The pulses reflected from both ends
of the rod, or of the column of air, meet in the middle, and produce
compression; they then retreat and produce rarefaction. Thus, while
there is no vibration in the centre of an organ-pipe, the air there
undergoes the greatest changes of density. At the two ends of the pipe,
on the other hand, the air-particles merely swing up and down without
sensible compression or rarefaction.
If the sounding pipe were pierced at the centre, and the orifice
stopped by a membrane, the air, when condensed, would press the
membrane outward, and, when rarefied, the external air would press
the membrane inward. The membrane would therefore vibrate in unison
with the column of air. The organ-pipe, Fig. 102, is so arranged that
a small jet of gas, _b_, can be lighted opposite the centre of the
pipe, and there acted upon by the vibrations of a membrane. Two other
gas-jets, _a_ and _c_, are placed nearly midway between the centre and
the two ends of the pipe. The three burners, _a_, _b_, _c_, are fed in
the following manner: through the tube, _t_, the gas enters the hollow
chamber, _e d_, from which issue three little bent tubes, shown in the
figure, each communicating with a capsule closed underneath by the
membrane. This is in direct contact with the air of the organ-pipe.
From the three capsules issue the three little burners, with their
flames, _a_, _b_, _c_.
Blowing into the pipe so as to sound its fundamental note, the three
flames are agitated, but the central one is most so. Lowering the
flames so as to render them very small, and blowing again, the central
flame, _b_, is extinguished, while the others remain lighted. The
experiment may be performed half a dozen times in succession; the
sounding of the fundamental note always quenches the middle flame.
By blowing more sharply into the pipe, it is caused to yield its first
overtone. The middle node no longer exists. The centre of the pipe is
now a place of maximum vibration, while two nodes are formed midway
between the centre and the two ends. But if this be the case, and if
the flame opposite the node be always blown out, then, when the first
overtone of this pipe is sounded, the two flames _a_ and _c_ ought
to be extinguished, while the central flame remains lighted. This is
the case. When the first harmonic is sounded the two nodal flames are
infallibly extinguished, while the flame _b_ in the middle of the
ventral segment is not sensibly disturbed.
There is no theoretic limit to the subdivision of an organ-pipe,
either stopped or open. In stopped pipes we begin with 1 semi-ventral
segment, and pass on to 3, 5, 7, etc., semi-ventral segments, the
number of vibrations of the successive notes augmenting in the same
ratio. In open pipes we begin with 2 semi-ventral segments, and pass on
to 4, 6, 8, 10, etc., the number of vibrations of the successive notes
augmenting in the same ratio; that is to say, in the ratio 1:2:3:4:5,
etc. When, therefore, we pass from the fundamental tone to the first
overtone in an open pipe, we obtain the octave of the fundamental. When
we make the same passage in a stopped pipe, we obtain a note a fifth
above the octave. No intermediate modes of vibration are in either
case possible. If the fundamental tone of a stopped pipe be produced
by 100 vibrations a second, the first overtone will be produced by 300
vibrations, the second by 500, and so on. Such a pipe, for example,
cannot execute 200 or 400 vibrations in a second. In like manner the
open pipe, whose fundamental note is produced by 100 vibrations a
second, cannot vibrate 150 times in a second, but passes, at a jump, to
200, 300, 400, and so on.
[Illustration: FIG. 103.]
In open pipes, as in stopped ones, the number of vibrations executed in
the unit of time is inversely proportional to the length of the pipe.
This follows from the fact, already dwelt upon so often, that the time
of a vibration is determined by the distance which the sonorous pulse
has to travel to complete a vibration.
In Fig. 103, _a_ and _b_ (at the bottom) represent the division of an
open pipe corresponding to its fundamental tone; _c_ and _d_ represent
the division corresponding to its first, _e_ and _f_ the division
corresponding to its second overtone, the dots marking the nodes. The
distance _m n_ is one-half, _o p_ is one-fourth, and _s t_ is one-sixth
of the whole length of the pipe. But the pitch of _a_ is that of a
stopped pipe equal in length to _m n_; the pitch of _c_ is that of a
stopped pipe of the length _o p_; while the pitch of _e_ is that of a
stopped pipe of the length _s t_. Hence, as these lengths are in the
ratio of 1/2:1/4:1/6, or as 1:1/2:1/3, the rates of vibration must be
as the reciprocals of these, or as 3:2:1. From the mere inspection,
therefore, of the respective modes of vibration, we can draw the
inference that the succession of tones of an open pipe must correspond
to the series of natural numbers.
The pipe _a_, Fig. 103, has been purposely drawn twice the length of
_a_, Fig. 93 (p. 215). It is perfectly manifest that to complete a
vibration the pulse has to pass over the same distance in both pipes,
and hence that the pitch of the two pipes must be the same. The open
pipe, _a n_, consists virtually of two stopped ones, with the central
nodal surface at _m_ as their common base. This shows the relation of a
stopped pipe to an open one to be that which experiment establishes.
§ 15. _Velocity of Sound in Gases, Liquids, and Solids determined by
Musical Vibrations_
We have already learned that the _relative_ velocities of sound in
different solid bodies may be determined from the notes which they
emit when thrown into longitudinal vibration. It was remarked at the
time that to draw up a table of _absolute_ velocities we only required
the accurate comparison of the velocity in any one of those solids
with the velocity in air. We are now in a condition to supply this
comparison. For we have learned that the vibrations of the air in an
organ-pipe open at both ends are executed precisely as those of a rod
free at both ends. Any difference of rapidity, therefore, between the
vibrations of a rod and of an open organ-pipe of the same length must
be due solely to the different velocities with which the sonorous
pulses are propagated through them. Take therefore an organ-pipe of a
certain length, emitting a note of a certain pitch, and find the length
of a rod of pine which yields the same note. This length would be ten
times that of the organ-pipe, which would prove the velocity of sound
in pine to be ten times its velocity in air. But the absolute velocity
in air is 1,090 feet a second; hence the absolute velocity in pine is
10,900 feet a second, which is that given in our first chapter (p. 74).
To the celebrated Chladni we are indebted for this beautiful mode of
determining the velocity of sound in solid bodies.
We had also in our first lecture a table of the velocities of sound in
other gases than air. I am persuaded that you could tell me, after due
reflection, how this table was constructed. It would only be necessary
to find a series of organ-pipes which, when filled with the different
gases, yield the same note; the lengths of these pipes would give the
relative velocities of sound through the gases. Thus we should find the
length of a pipe filled with hydrogen to be four times that of a pipe
filled with oxygen, yielding the same note, and this would prove the
velocity of sound in the former to be four times its velocity in the
latter.
But we had also a table of velocities through various liquids. How was
it constructed? By forcing the liquids through properly constructed
organ-pipes, and comparing their musical tones. Thus, in water it
requires a pipe a little better than four feet long to produce the
note of an air-pipe one foot long; and this proves the velocity of
sound in water to be somewhat more than four times its velocity in
air. My object here is to give you a clear notion of the way in
which scientific knowledge enables us to cope with these apparently
insurmountable problems. It is not necessary to go into the niceties
of these measurements. You will, however, readily comprehend that all
the experiments with gases might be made with the same organ-pipe, the
velocity of sound in each respective gas being immediately deduced from
the pitch of its note. With a pipe of constant length the pitch, or, in
other words, the number of vibrations, would be directly proportional
to the velocity. Thus, comparing oxygen with hydrogen, we should find
the note of the latter to be the double octave of that of the former,
whence we should infer the velocity of sound in hydrogen to be four
times its velocity in oxygen. The same remark applies to experiments
with liquids. Here also the same pipe may be employed throughout, the
velocities being inferred from the notes produced by the respective
liquids.
In fact, the length of an open pipe being, as already explained,
one-half the length of its sonorous wave, it is only necessary to
determine, by means of the siren, the number of vibrations executed
by the pipe in a second, and to multiply this number by twice the
length of the pipe, in order to obtain the velocity of sound in the
gas or liquid within the pipe. Thus, an open pipe 26 inches long and
filled with air executes 256 vibrations in a second. The length of its
sonorous wave is twice 26 inches, or 4-1/3 feet: multiplying 256 by
4-1/3 we obtain 1,120 feet per second as the velocity of sound through
air of this temperature. Were the tube filled with carbonic-acid gas,
its vibrations would be slower: were it filled with hydrogen, its
vibrations would be quicker; and applying the same principle, we should
find the velocity of sound in both these gases.
So likewise the length of a solid rod free at both ends, and sounding
its fundamental note, is half that of the sonorous wave in the
substance of the solid. Hence we have only to determine the rate of
vibration of such a rod, and multiply it by twice the length of the
rod, to obtain the velocity of sound in the substance of the rod. You
can hardly fail to be impressed by the power which physical science has
given us over these problems; nor will you refuse your admiration to
that famous old investigator, Chladni, who taught us how to master them
experimentally.
REEDS AND REED-PIPES
The construction of the siren and our experiments with that instrument
are, no doubt, fresh in your recollection. Its musical sounds are
produced by the cutting up into puffs of a series of air-currents.
The same purpose is effected by a vibrating reed, as employed in the
accordion, the concertina, and the harmonica. In these instruments it
is not the vibrations of the reed itself which, imparted to the air,
and transmitted through it to our organs of hearing, produce the music;
the function of the reed is _constructive_, not _generative_; it molds
into a series of discontinuous puffs that which without it would be a
continuous current of air.
[Illustration: FIG. 104.]
Reeds, if associated with organ-pipes, sometimes command, and are
sometimes commanded by, the vibrations of the column of air. When they
are stiff they rule the column; when they are flexible the column rules
them. In the former case, to derive any advantage from the air-column,
its length ought to be so regulated that either its fundamental tone
or one of its overtones shall correspond to the rate of vibration of
the reed. The metal reed commonly employed in organ-pipes is shown in
Fig. 104, A and B, both in perspective and in section. It consists
of a long and flexible strip of metal, V V, placed in a rectangular
orifice, through which the current of air enters the pipe. The manner
in which the reed and pipe are associated is shown in Fig. 105. The
front, _b c_, of the space containing the flexible tongue is of
glass, so that you may see the tongue within it. A conical pipe, A
B, surmounts the reed.[47] The wire _w i_, shown pressing against the
reed, is employed to lengthen or shorten it, and thus to vary within
certain limits its rate of vibration. At one time in the practice of
music the reed closed the aperture by simply falling against its sides;
every stroke of the reed produced a tap, and these linked themselves
together to an unpleasant, screaming sound, which materially injured
that of the associated organ-pipe. This was mitigated, but not removed,
by permitting the reed to strike against soft leather; but the reed
now employed is the _free reed_, which vibrates to and fro between
the sides of the aperture, almost, but not quite, filling it. In this
way the unpleasantness referred to is avoided. When reed and pipe
synchronize perfectly, the sound is most pure and forcible; a certain
latitude, however, is possible on both sides of perfect synchronism.
But if the discordance be pushed too far, the pipe ceases to be of any
use. We then obtain the sound due to the vibrations of the reed alone.
[Illustration: FIG. 105.]
Flexible wooden reeds, which can accommodate themselves to the
requirements of the pipes above them, are also employed in organ-pipes.
Perhaps the simplest illustration of the action of the reed commanded
by its aërial column is furnished by a common wheaten straw. At about
an inch from a knot, at _r_, I bury my penknife in this straw, _s r′_,
Fig. 106, to a depth of about one-fourth of the straw’s diameter, and,
turning the blade flat, pass it upward toward the knot, thus raising
a strip of the straw nearly an inch in length. This strip, _r r′_,
is to be our reed, and the straw itself is to be our pipe. It is now
eight inches long. When blown into, it emits this decidedly musical
sound. When cut so as to make its length six inches, the pitch is
higher; with a length of four inches, the pitch is higher still; and
with a length of two inches, the sound is very shrill indeed. In these
experiments the reed was compelled to accommodate itself throughout to
the requirements of the vibrating column of air.
[Illustration: FIG. 106.]
The clarinet is a reed-pipe. It has a single broad tongue, with which
a long, cylindrical tube is associated. The reed-end of the instrument
is grasped by the lips, and by their pressure the slit between the reed
and its frame is narrowed to the required extent. The overtones of a
clarinet are different from those of a flute. A flute is an open pipe,
a clarinet a stopped one, the end at which the reed is placed answering
to the closed end of the pipe. The tones of a flute follow the order
of the natural numbers 1, 2, 3, 4, etc., or of the even numbers 2,
4, 6, 8, etc.; while the tones of a clarinet follow the order of the
odd numbers 1, 3, 5, 7, etc. The intermediate notes are supplied by
opening the lateral orifices of the instrument. Sir C. Wheatstone
was the first to make known this difference between the flute and
clarinet, and his results agree with the more thorough investigations
of Helmholtz. In the hautboy and bassoon we have two reeds inclined
to each other at a sharp angle, with a slit between them, through
which the air is urged. The pipe of the hautboy is _conical_, and its
overtones are those of an open pipe—different, therefore, from those
of a clarinet. The pulpy end of a straw of green corn may be split by
squeezing it, so as to form a double reed of this kind, and such a
straw yields a musical tone. In the horn, trumpet, and serpent, the
performer’s lips play the part of the reed. These instruments are
formed of long, conical tubes, and their overtones are those of an
open organ-pipe. The music of the older instruments of this class was
limited to their overtones, the particular tone elicited depending on
the force of the blast and the tension of the lips. It is now usual to
fill the gaps between the successive overtones by means of keys, which
enable the performer to vary the length of the vibrating column of air.
§ 16. _The Voice_
The most perfect of reed instruments is the organ of voice. The vocal
organ in man is placed at the top of the trachea or wind-pipe, the
head of which is adjusted for the attachment of certain elastic bands
which almost close the aperture. When the air is forced from the lungs
through the slit which separates these _vocal chords_, they are thrown
into vibration; by varying their tension, the rate of vibration is
varied, and the sound changed in pitch. The vibrations of the vocal
chords are practically unaffected by the resonance of the mouth, though
we shall afterward learn that this resonance, by reinforcing one or the
other of the tones of the vocal chords, influences in a striking manner
the quality of the voice. The sweetness and smoothness of the voice
depend on the perfect closure of the slit of the glottis at regular
intervals during the vibration.
[Illustration: FIG. 107.]
The vocal chords may be illuminated and viewed in a mirror, placed
suitably at the back of the mouth. Varied experiments of this kind
have been executed by Sig. Garcia.[48] I once sought to project
the larynx of M. Czermak upon a screen in this room, but with only
partial success. The organ may, however, be viewed directly in the
laryngoscope; its motions, in singing, speaking, and coughing, being
strikingly visible. It is represented at rest in Fig. 107. The
roughness of the voice in colds is due, according to Helmholtz, to
mucous flocculi, which get into the slit of the glottis, and which are
seen by means of the laryngoscope. The squeaking falsetto voice, with
which some persons are afflicted, Helmholtz thinks, may be produced
by the drawing aside of the mucous layer which ordinarily lies under
and loads the vocal chords. Their edges thus become sharper and their
weight less; while, their elasticity remaining the same, they are
shaken into more rapid tremors. The promptness and accuracy with which
the vocal chords can change their tension, their form, and the width of
the slit between them, to which must be added the elective resonance of
the cavity of the mouth, render the voice the most perfect of musical
instruments.
[Illustration: FIG. 108.]
The celebrated comparative anatomist, John Müller, imitated the
action of the vocal chords by means of bands of India-rubber. He
closed the open end of a glass tube by two strips of this substance,
leaving a slit between them. On urging air through the slit, the bands
were thrown into vibration, and a musical tone produced. Helmholtz
recommends the form shown in Fig. 108, where the tube, instead of
ending in a section at right angles to its axis, terminates in two
oblique sections, over which the bands of India-rubber are drawn.
The easiest mode of obtaining sounds from reeds of this character is
to roll round the end of a glass tube a strip of thin India-rubber,
leaving about an inch of the substance projecting beyond the end of
the tube. Taking two opposite portions of the projecting rubber in the
fingers, and stretching it, a slit is formed, the blowing through which
produces a musical sound, which varies in pitch, as the sides of the
slit vary in tension.
§ 17. _Vowel Sounds_
The formation of the vowel sounds of the human voice excited long ago
philosophic inquiry. We can distinguish one vowel sound from another,
while assigning to both the same pitch and intensity. What, then, is
the quality which renders the distinction possible? In the year 1779
this was made a prize question by the Academy of St. Petersburg, and
Kratzenstein gained the prize for the successful manner in which he
imitated the vowel sounds by mechanical arrangements. At the same time
Von Kempelen, of Vienna, made similar and more elaborate experiments.
The question was subsequently taken up by Mr. Willis, who succeeded
beyond all his predecessors in the experimental treatment of the
subject. The true theory of vowel sounds was first stated by Sir C.
Wheatstone, and quite recently they have been made the subject of
exhaustive inquiry by Helmholtz. You will find little difficulty in
comprehending their origin.
Mounted on the acoustic bellows, without any pipe associated with it,
when air is urged through its orifice, a free reed speaks in this
forcible manner. When upon the frame of the reed a pyramidal pipe
is fixed, you notice a change in the sound; and by pushing my flat
hand over the open end of the pipe, the similarity between the sound
produced and that of the human voice is unmistakable. Holding the palm
of the hand over the end of the pipe so as to close it altogether, and
then raising the hand twice in quick succession, the word “mamma” is
heard as plainly as if it were uttered by an infant. For this pyramidal
tube I now substitute a shorter one, and with it make the same
experiment. The “mamma” now heard is exactly such as would be uttered
by a child with a stopped nose. Thus, by associating with a vibrating
reed a suitable pipe, we can impart to the sound the qualities of the
human voice.
In the organ of voice, the reed is formed by the vocal chords, and
associated with this reed is the resonant cavity of the mouth,
which can so alter its shape as to resound, at will, either to the
fundamental tone of the vocal chords or to any of their overtones. With
the aid of the mouth, therefore, we can _mix together_ the fundamental
tone and the overtones of the voice in different proportions. Different
vowel sounds are due to different admixtures of this kind. Striking
one of this series of tuning-forks, and placing it before my mouth, I
adjust the size of that cavity until it resounds forcibly to the fork.
Then, without altering in the least the shape or size of my mouth, I
urge air through the glottis. The vowel sound “U” (_oo_ in hoop) is
produced, and no other. I strike another fork, and, placing it in front
of the mouth, adjust the cavity to resonance. Then removing the fork
and urging air through the glottis, the vowel sound “O,” and it only,
is heard. I strike a third fork, adjust my mouth to it, and then urge
air through the larynx; the vowel sound _ah!_ and no other, is heard.
In all these cases the vocal chords have been in the same constant
condition. They have generated throughout the same fundamental tone and
the same overtones, the changes of sound which you have heard being due
solely to the fact that different tones in the different cases were
reinforced by the resonance of the mouth. Donders first proved that the
mouth resounded differently for the different vowels.
In the formation of the different vowel sounds the resonant cavity of
the mouth undergoes, according to Helmholtz, the following changes:
For the production of the sound “U” (_oo_ in hoop), the lips must
be pushed forward, so as to make the cavity of the mouth as deep as
possible, and the orifice of the mouth, by the contraction of the lips,
as small as possible. This arrangement corresponds to the deepest
resonance of which the mouth is capable. The fundamental tone itself of
the vocal chords is here reinforced, while the higher tones retreat.
The vowel “O” requires a somewhat wider opening of the mouth. The
overtones which lie in the neighborhood of the middle _b_ of the
soprano come out strongly in the case of this vowel.
When “Ah” is sounded, the mouth assumes the shape of a funnel, widening
outward. It is thus tuned to a note an octave higher than in the case
of the vowel “O.” Hence, in sounding “Ah,” those overtones are most
strengthened which lie near the higher _b_ of the soprano. As the mouth
is in this case wide open, all the other overtones are also heard,
though feebly.
In sounding “A” and “E,” the hinder part of the mouth is deepened,
while the front of the tongue rises against the gums and forms a tube;
this yields a higher resonance-tone, rising gradually from “A” to “E,”
while the hinder hollow space yields a lower resonance-tone, which is
deepest when “E” is sounded.
These examples sufficiently illustrate the subject of vowel sounds. We
may blend in various ways the elementary tints of the solar spectrum,
producing innumerable composite colors by their admixture. Out of
violet and red we produce purple, and out of yellow and blue we produce
white. Thus also may elementary sounds be blended so as to produce all
possible varieties of clang-tint. After having resolved the human voice
into its constituent tones, Helmholtz was able to imitate these tones
by tuning-forks, and, by combining them appropriately together, to
produce the sounds of all the vowels.
§ 18. _Kundt’s Experiments: New Modes of determining Velocity of Sound_
Unwilling to interrupt the continuity of our reasonings and experiments
on the sound of organ-pipes, and their relations to the vibrations of
solid rods, I have reserved for the conclusion of this discourse some
reflections and experiments which, in strictness, belong to an earlier
portion of the chapter. You have already heard the tones, and made
yourselves acquainted with the various modes of division of a glass
tube, free at both ends, when thrown into longitudinal vibration. When
it sounds its fundamental tone, you know that the two halves of such
a tube lengthen and shorten in quick alternation. If the tube were
stopped at its ends, the closed extremities would throw the air within
the tube into a state of vibration; and if the velocity of sound in air
were equal to its velocity in glass, the air of the tube would vibrate
in synchronism with the tube itself. But the velocity of sound in air
is far less than its velocity in glass, and hence, if the column of air
is to synchronize with the vibrations of the tube, it can only do so
by dividing itself into vibrating segments of a suitable length. In an
investigation of great interest, recently published in “Poggendorff’s
Annalen,” M. Kundt, of Berlin, has taught us how these segments may
be rendered visible. Into this six-foot tube is introduced the light
powder of lycopodium, being shaken all over the interior surface. A
small quantity of the powder clings to that surface. Stopping the ends
of the tube, holding its centre by a fixed clamp, and sweeping a wet
cloth briskly over one of its halves, instantly the powder, which a
moment ago clung to its interior surface, falls to the bottom of the
tube in the forms shown in Fig. 109, the arrangement of the lycopodium
marking the manner in which the column of air has been divided. Every
node here is encircled by a ring of dust, while from node to node the
dust arranges itself in transverse streaks along the ventral segments.
[Illustration: FIG. 109.]
You will have little difficulty in seeing that we perform here, with
air, substantially the same experiment as that of M. Melde with a
vibrating string. When the string was too long to vibrate as a whole,
it met the requirements of the tuning-fork to which it was attached
by dividing into ventral segments. Now, in all cases, the length from
a node to its next neighbor is half that of the sonorous wave: how
many such half-waves then have we in our tube in the present instance?
Sixteen (the figure shows only four of them). But the length of our
glass tube vibrating thus longitudinally is also half that of the
sonorous wave _in glass_. Hence, in the case before us, with the same
rate of vibration, the length of the semi-wave in glass is sixteen
times the length of the semi-wave in air. In other words, the velocity
of sound in glass is sixteen times its velocity in air. Thus, by a
single sweep of the wet rubber, we solve a most important problem.
But, as M. Kundt has shown, we need not confine ourselves to air.
Introducing any other gas into the tube, a single stroke of our wet
cloth enables us to determine the relative velocity of sound in that
gas and in glass. When hydrogen is introduced, the number of ventral
segments is less than in air; when carbonic acid is introduced, the
number is greater.
From the known velocity of sound in air, coupled with the length of
one of these dust segments, we can immediately deduce the number of
vibrations executed in a second by the tube itself. Clasping a glass
tube at its centre and drawing my wetted cloth over one of its halves,
I elicit this shrill note. The length of every dust segment, now
within the tube, is 3 inches. Hence the length of the aërial sonorous
wave corresponding to this note is 6 inches. But the velocity of
sound in air of our present temperature is 1,120 feet per second; a
distance which would embrace 2,240 of our sonorous waves. This number,
therefore, expresses the number of vibrations per second executed by
the glass tube now before us.
Instead of damping the centre of the tube, and making it a nodal point,
we may employ any other of its subdivisions. Laying hold of it, for
example, at a point midway between its centre and one of its ends, and
rubbing it properly, it divides into three vibrating parts, separated
by two nodes. We know that in this division the note elicited is the
octave of that heard when a single node is formed at the middle of the
tube; for the vibrations are twice as rapid. If therefore we divide the
tube, having air within it, by two nodes instead of one, the number of
ventral segments revealed by the lycopodium dust will be thirty-two
instead of sixteen. The same remark applies, of course, to all other
gases.
Filling a series of four tubes with air, carbonic acid, coal-gas,
and hydrogen, and then rubbing each so as to produce two nodes, M.
Kundt found the number of dust segments formed within the tube in the
respective cases to be as follows:
Air 32 dust segments
Carbonic acid 40 ”
Coal-gas 20 ”
Hydrogen 9 ”
Calling the velocity in air unity, the following fractions express the
ratio of this velocity to those in the other gases:
32
Carbonic acid —— = 0·8
40
32
Coal-gas —— = 1·6
20
32
Hydrogen —— = 3·56
9
[Illustration: FIG. 110.]
Referring to a table introduced in our first chapter, we learn that
Dulong by a totally different mode of experiment found the velocity in
carbonic acid to be 0.86, and in hydrogen 3·8 times the velocity in
air. The results of Dulong were deduced from the sounds of organ-pipes
filled with the various gases; but here, by a process of the utmost
simplicity, we arrive at a close approximation to his results. Dusting
the interior surfaces of our tubes, filling them with the proper
gases, and sealing their ends, they may be preserved for an indefinite
time. By properly shaking one of them at any moment, its inner surface
becomes thinly coated with the dust; and afterward a single stroke
of the wet cloth produces the division from which the velocity of
sound in the gas may be immediately inferred. Savart found that a
spiral nodal line is formed round a tube or rod when it vibrates
longitudinally, and Seebeck showed that this line was produced, not
by longitudinal, but by secondary transverse vibrations. Now this
spiral nodal line tends to complicate the division of the dust in our
present experiments. It is, therefore, desirable to operate in a manner
which shall altogether avoid the formation of this line; M. Kundt has
accomplished this, by exciting the longitudinal vibrations in one tube,
and producing the division into ventral segments in another, into which
the first fits like a piston. Before you is a tube of glass, Fig. 110,
seven feet long, and two inches internal diameter. One end of this
tube is filled by the movable stopper _b_. The other end, K K, is also
stopped by a cork, through the centre of which passes the narrower tube
A _a_, which is firmly clasped at its middle by the cork, K K. The end
of the interior tube, A _a_, is also closed with a projecting stopper,
_a_, almost sufficient to fill the larger tube, but still fitting into
it so loosely that the friction of _a_ against the interior surface is
too slight to interfere practically with its vibrations. The interior
surface between _a_ and _b_ being lightly coated with the lycopodium
dust, a wet cloth is passed briskly over A K; instantly the dust
between _a_ and _b_ divides into a number of ventral segments. When
the length of the column of air, _a b_, is equal to that of the glass
tube, A _a_, the number of ventral segments is sixteen. When, as in
the figure, _a b_ is only one-half the length of A _a_, the number of
ventral segments is eight.
But here you must perceive that the method of experiment is capable
of great extension. Instead of the glass tube, A _a_, we may employ a
rod of any other solid substance—of wood or metal, for example, and
thus determine the relative velocity of sound in the solid and in air.
In the place of the glass tube, for example, a rod of brass of equal
length may be employed. Rubbing its external half by a resined cloth,
it divides the column _a b_ into the number of ventral segments proper
to the metal’s rate of vibrations. In this way M. Kundt operated with
brass, steel, glass, and copper, and his results prove the method to
be capable of great accuracy. Calling, as before, the velocity of
sound in air unity, the following numbers expressive of the ratio of
the velocity of sound in brass to its velocity in air were obtained in
three different series of experiments:
1st experiment 10·87
2d experiment 10·87
3d experiment 10·86
The coincidence is here extraordinary. To illustrate the possible
accuracy of the method, the length of the individual dust segments was
measured. In a series of twenty-seven experiments, this length was
found to vary between 43 and 44 millimètres (each millimètre 1/25th of
an inch), never rising so high as the latter and never falling so low
as the former. The length of the metal rod, compared with that of one
of the segments capable of this accurate measurement, gives us at once
the velocity of sound in the metal, as compared with its velocity in
air.
Three distinct experiments, performed in the same manner on steel,
gave the following velocities, the velocity through air, as before,
being regarded as unity:
1st experiment 15·34
2d experiment 15·33
3d experiment 15·34
Here the coincidence is quite as perfect as in the case of brass.
In glass, by this new mode of experiment, the velocity was found to be
15·25.[49]
Finally, in copper the velocity was found to be
11·96.
[Illustration: FIG. 111.]
These results agree extremely well with those obtained by other
methods. Wertheim, for example, found the velocity of sound in steel
wire to be 15·108; M. Kundt finds it to be 15·34: Wertheim also found
the velocity in copper to be 11·17; M. Kundt finds it to be 11·96. The
differences are not greater than might be produced by differences in
the materials employed by the two experimenters.
The length of the aërial column may or may not be an exact multiple
of the wave-length, corresponding to the rod’s rate of vibration. If
not, the dust segments usually take the form shown in Fig. 111. But
if, by means of the stopper, _b_, the column of air be made an exact
multiple of the wave-length, then the dust quits the vibrating segments
altogether, and forms, as in Fig. 112, little isolated heaps at the
nodes.
§ 19. _Explanation of a Difficulty_
And here a difficulty presents itself. The stopped end _b_ of the tube
Fig. 110 is, of course, a place of no vibration, where in all cases a
nodal dust-heap is formed; but, whenever the column of air was an exact
multiple of the wave-length, M. Kundt always found a dust-heap close to
the end _a_ of the vibrating rod also. Thus the point from which all
the vibration emanated seemed itself to be a place of no vibration.
[Illustration: FIG. 112.]
This difficulty was pointed out by M. Kundt, but he did not attempt its
solution. We are now in a condition to explain it. In Lecture III. it
was remarked that in strictness a node is not a place of no vibration;
that it is a place of _minimum_ vibration; and that, by the addition of
the minute pulses which the node permits, vibrations of vast amplitude
may be produced. The ends of M. Kundt’s tube are such points of minimum
motion, the lengths of the vibrating segments being such that, by the
coalescence of direct and reflected pulses, the air at a distance of
half a ventral segment from the end of the tube vibrates much more
vigorously than that at the end of the tube itself. This addition of
impulses is more perfect when the aërial column is an exact multiple
of the wave-length, and hence it is that, in this case, the vibrations
become sufficiently intense to sweep the dust altogether away from
the vibrating segments. The same point is illustrated by M. Melde’s
tuning-forks, which, though they are the sources of all the motion, are
themselves nodes.
An experiment of Helmholtz’s is here capable of instructive
application. Upon the string of the sonometer described in our third
lecture I place the iron stem of this tuning-fork, which executes 512
complete vibrations in a second. At present you hear no augmentation of
the sound of the fork; the string remains quiescent. But on moving the
fork along the string, at the number 33, a loud, swelling note issues
from the string. At this particular tension the length 33 exactly
synchronizes with the vibrations of the fork. By the intermediation
of the string, therefore, the fork is enabled to transfer its motion
to the sonometer, and through it to the air. The sound continues as
long as the fork vibrates, but the least movement to the right or to
the left from this point causes a sudden fall of the sound. Tightening
the string, the note disappears; for it requires a greater length of
this more highly tensioned string to respond to the fork. But, on
moving the fork further away, at the number 36 the note again bursts
forth. Tightening still more, 40 is found to be the point of maximum
power. When the string is slackened, it must, of course, be shortened
in order to make it respond to the fork. Moving the fork now toward
the end of the string, at the number 25 the note is found as before.
Again, shifting the fork to 35, nothing is heard; but, by the cautious
turning of the key, the point of synchronism, if I may use the term,
is moved further from the end of the string. It finally reaches the
fork, and at that moment a clear, full note issues from the sonometer.
In all cases, before the exact point is attained, and immediately in
its vicinity, we hear “beats,” which, as we shall afterward understand,
are due to the coalescence of the sound of the fork with that of the
string, when they are nearly, but not quite, in unison with each other.
In these experiments, though the fork was the source of all the motion,
_the point on which it rested was a nodal point_. It constituted
the comparatively fixed extremity of the wire, whose vibrations
synchronized with those of the fork. The case is exactly analogous to
that of the hand holding the India-rubber tube, and to the tuning-fork
in the experiments of M. Melde. It is also an effect precisely the same
in kind as that observed by M. Kundt, where the part of the column of
air in contact with the end of his vibrating rod proved to be a node
instead of the middle of a ventral segment.
ADDENDUM REGARDING RESONANCE
The resonance of caves and of rocky inclosures is well known. Bunsen
notices the thunder-like sound produced when one of the steam jets
of Iceland breaks out near the mouth of a cavern. Most travellers in
Switzerland have noticed the deafening sound produced by the fall of
the Reuss at the Devil’s Bridge. The sound heard when a hollow shell
is placed close to the ear is a case of resonance. Children think
they hear in it the sound of the sea. The noise is really due to the
reinforcement of the feeble sounds with which even the stillest air is
pervaded, and also in part to the noise produced by the pressure of the
shell against the ear itself. By using tubes of different lengths, the
variation of the resonance with the length of the tube may be studied.
The channel of the ear itself is also a resonant cavity. When a poker
is held by two strings, and when the fingers of the hands holding the
poker are thrust into the ears on striking the poker against a piece
of wood, a sound is heard as deep and sonorous as that of a cathedral
bell. When open, the channel of the ear resounds to notes whose periods
of vibration are about 3,000 per second. This has been shown by
Helmholtz, and Madame Seiler has found that dogs which howl to music
are particularly sensitive to the same notes. We may expect from Mr.
Francis Galton interesting results in connection with this subject.
SUMMARY OF CHAPTER V
When a stretched wire is suitably rubbed, in the direction of its
length, it is thrown into longitudinal vibrations: the wire can either
vibrate as a whole or divide itself into vibrating segments separated
from each other by nodes.
The tones of such a wire follow the order of the numbers 1, 2, 3, 4,
etc.
The _transverse_ vibrations of a rod fixed at both ends do not follow
the same order as the transverse vibrations of a stretched wire; for
here the forces brought into play, as explained in Lecture IV., are
different. But the longitudinal vibrations of a stretched wire do
follow the same order as the longitudinal vibrations of a rod fixed at
both ends, for here the forces brought into play are the same, being in
both cases the elasticity of the material.
A rod fixed at one end vibrates longitudinally as a whole, or it
divides into two, three, four, etc., vibrating parts, separated from
each other by nodes. The order of the tones of such a rod is that of
the odd numbers 1, 3, 5, 7, etc.
A rod free at both ends can also vibrate longitudinally. Its lowest
note corresponds to a division of the rod into two vibrating parts by
a node at its centre. The overtones of such a rod correspond to its
division into three, four, five, etc., vibrating parts, separated from
each other by two, three, four, etc., nodes. The order of the tones of
such a rod is that of the numbers 1, 2, 3, 4, 5, etc.
We may also express the order by saying that while the tones of a rod
fixed at both ends follow the order of the odd numbers 1, 3, 5, 7,
etc., the tones of a rod free at both ends follow the order of the even
numbers 2, 4, 6, 8, etc.
At the points of maximum vibration the rod suffers no change of
density; at the nodes, on the contrary, the changes of density reach
a maximum. This may be proved by the action of the rod upon polarized
light.
Columns of air of definite length resound to tuning-forks of definite
rates of vibration.
The length of a tube filled with air, and closed at one end, which
resounds to a fork is one-fourth of the length of the sonorous wave
produced by the fork.
This resonance is due to the synchronism which exists between the
vibrating period of the fork and that of the column of air.
By blowing across the mouth of a tube closed at one end, we produce a
flutter of the air, and some pulse of this flutter may be raised by the
resonance of the tube to a musical sound.
The sound is the same as that obtained when a tuning-fork, whose rate
of vibration is that of the tube, is placed over the mouth of the tube.
When a tube closed at one end—a stopped organ-pipe, for example—sounds
its lowest note, the column of air within it is undivided by a node.
The overtones of such a column correspond to its division into parts,
like those of a rod fixed at one end and vibrating longitudinally. The
order of its tones is that of the odd numbers 1, 3, 5, 7, etc. That
this must be the order follows from the manner in which the column is
divided.
In organ-pipes the air is agitated by causing it to issue from a narrow
slit, and to strike upon a cutting edge. Some pulse of the flutter thus
produced is raised by the resonance of the pipe to a musical sound.
When, instead of the aërial flutter, a tuning-fork of the proper rate
of vibration is placed at the embouchure of an organ-pipe, the pipe
_speaks_ in response to the fork. In practice, the organ-pipe virtually
creates its own tuning-fork, by compelling the sheet of air at its
embouchure to vibrate in periods synchronous with its own.
An open organ-pipe yields a note an octave higher than that of a closed
pipe of the same length. This relation is a necessary consequence of
the respective modes of vibration.
When, for example, a stopped organ-pipe sounds its deepest note, the
column of air, as already explained, is undivided. When an open pipe
sounds its deepest note, the column is divided by a node at its centre.
The open pipe in this case virtually consists of two stopped pipes with
a common base. Hence it is plain that the fundamental note of an open
pipe must be the same as that of a stopped pipe of half its length.
The length of a stopped pipe is one-fourth that of the sonorous wave
which it produces, while the length of an open pipe is one-half that of
its sonorous wave.
The order of the tones of an open pipe is that of the even numbers 2,
4, 6, 8, etc., or of the natural numbers 1, 2, 3, 4, etc.
In both stopped and open pipes the number of vibrations executed in a
given time is inversely proportional to the length of the pipe.
The places of maximum vibration in organ-pipes are places of minimum
changes of density; while at the places of minimum vibration the
changes of density reach a maximum.
The velocities of sound in gases, liquids, and solids may be inferred
from the tones which equal lengths of them produce; or they may be
inferred from the lengths of these substances which yield equal tones.
Reeds, or vibrating tongues, are often associated with vibrating
columns of air. They consist of flexible laminæ, which vibrate to
and fro in a rectangular orifice, thus rendering intermittent the
air-current passing through the orifice.
The action of the reed is the same as that of the siren.
The flexible wooden reeds sometimes associated with organ-pipes are
compelled to vibrate in unison with the column of air in the pipe;
other reeds are too stiff to be thus controlled by the vibrating air.
In this latter case the column of air is taken of such a length that
its vibrations synchronize with those of the reed.
By associating suitable pipes with reeds we impart to their tones the
qualities of the human voice.
The vocal organ in man is a reed instrument, the vibrating reed in this
case being elastic bands placed at the top of the trachea, and capable
of various degrees of tension.
The rate of vibration of these vocal chords is practically uninfluenced
by the resonance of the mouth; but the mouth, by changing its shape,
can be caused to resound to the fundamental tone, or to any of the
overtones of the vocal chords.
By the strengthening of particular tones through the resonance of the
mouth, the clang-tint of the voice is altered.
The different vowel-sounds are produced by different admixtures of the
fundamental tone and the overtones of the vocal chords.
When the solid substance of a tube stopped at one, or at both ends, is
caused to vibrate longitudinally, the air within it is also thrown into
vibration.
By covering the interior surface of the tube with a light powder,
the manner in which the aërial column divides itself may be rendered
apparent. From the division of the column the velocity of sound in
the substance of the tube, compared with its velocity in air, may be
inferred.
Other gases may be employed instead of air, and the velocity of sound
in these gases, compared with its velocity in the substance of the
tube, may be determined.
The end of a rod vibrating longitudinally may be caused to agitate a
column of air contained in a tube, compelling the air to divide itself
into ventral segments. These segments may be rendered visible by light
powders, and from them the velocity of sound in the substance of the
vibrating rod, compared with its velocity in air, may be inferred.
In this way the relative velocities of sound in all solid substances
capable of being formed into rods, and of vibrating longitudinally, may
be determined.