Acoustics
I4
Sound instruments
Strings
A string is a straight,
threadlike, elastic body (intestine, metal), which is stretched
by tension and all points of which apart from its ends are free
to move. When it is displaced from its position of rest (by
striking with a hammer, rubbed with a bow, tugged with a 
finger) and then released, it flings
back, bypasses its position of rest , returns, etc. If its period
is large enough, it gives a sound. The law for the ensuing
vibrations can be explained by the Monochord (Greek: monos = single, Fig. 319) - effectively a
string - the length and tension of which can be changed
measurably and which can be replaced by another string. You
tighten the string by weights, the length of the vibrating part
is determined by the clamps N and U. The tone is only weak in the
absence of a resonance board, but, in return, the Monochord is
clear of vagueness, which would come with it. You find the number
of oscillations per second in dependence on the material, the
dimensions and the tension of the string. It is the larger, that
is, the tone is the higher, the shorter and thinner is the
string, the smaller is the specific weight of the material the
larger the tension. The dependence of the Hertz number n on these parameters is given by the formula of Brook Taylor 1685-1731 1713:
with d the diameter, l the length,
p the specific weight of the string, g·m the weight of the mass causing the
tension .
This formula can
be proved with the Monochord
- apart from
deviations mainly due to the fact that it has been derived for a
row of individual mass points, not for a string, whence it only
takes into account the strain elasticity of the string and not
its bending elasticity. It tells us:
1.
If l is reduced, that is, the string shortened, n increases, that
is, the tone rises; in fact, if the string is shortened to 1/2,
1/3, 1/4 ··· of its length, n becomes 2-, 3-, 4- ··· as
large, that is, the string gives consecutively the harmonic
overtones corresponding to the base tone with n vibrations: The
Octave Fifth of the Octave, the doubled Octave, etc.
2. The same happens , if the tension g·m
is made 4-, 9-, 16- ···times as large as n requires and leaves
everything else unchanged.
3. The same happens when d is smaller, that it,
the string is correspondingly thinner.
4. The periods are under otherwise equal
conditions larger, the tones therefore higher in the case of a
material of less specific weight (smaller p), for example, it is
larger with a catgut than with a steel string, etc.
All of this is
required for the musical instruments: String instruments are
given thinner strings for the high tones, shorter ones for the
deep tones (piano, harpsichord, harp, guitar, zither); on
stringed instruments (they have only 4 strings), one obtains the
higher tones, for which no special strings are available, by
pressing a string with the finger against the finger-board, thus
shortening the vibrating part of the string.
The strings for the deepest tones of the piano and harpsichord are made out of copper,
the specific weight of which is larger than that of steel, etc.
The strings of string instruments are tuned higher or lower by
changes in their tension (by tightening or releasing the screws,
around which the strings are wrapped). Strings cause only
vibration of very small quantities of air and yield therefore
weak tones, so that they would be practically useless, if the
motion of air they cause could not be amplified. One does this by
letting them vibrate on top of very elastic wooden surfaces,
sounding boards, or over wooden boxes with very thin elastic
walls - resonance-boxes. In order that these resonance-boards and
-boxes , in which the air oscillates, will co-vibrate with all
tones, they must meet certain conditions, which - you should now
recall the strange form of string-instruments - are more a result
of experience than theory. The resonance-board of the piano Nernst avoids by
recording the vibrations of the strings by microphones,
amplifying them by electronic valves and then feeding them to
loud speakers. In this way, he overcomes the unharmonic overtones
in the bass, improves the soprano, where it suppresses especially
obnoxious hammer noise, reduces damping and extends fading away.
Flageolet sounds
When a string produces only its base tone without overtones, it vibrates between its two limiting positions(Fig.
320 a) All its point vibrate simultaneously in the same direction
like in the standing, transverse wave. The two points, at which the string is
fixed, are the nodes, the part in between the bulge. You can also
excite the vibration modes of Fig. 320 b, c, d: The points b, g, d then become nodes, and the string
vibrates in two halves, three thirds, four fourth; all points of
the string pass simultaneously through the position of rest, but
the points on neighbouring sides of a node vibrate in opposite
directions (as has been described when dealing with the standing
wave). The string then sounds the octave (b), the fifth of the
octave (c), the next octave (d) - as if it had been shortened to
1/2. 1/3, 1/4. You can demonstrate that the nodes are at rest by
placing on the string v-shaped bits of paper, which will be cast
off except at the nodes.
You can enforce
these modes of vibration by stretching the string over a sounding
box and placing a sounding tuning fork on it, which has the same
height as the string at the corresponding mode. You can also
provoke the tones by touching the string at a point 1/2, 1/3, 1/4
away from the end of the string and stroking it. You call these
sounds flageolet tones, because they remind one of a flute. The
sounds are similar to each other, because they are free of overtones.
But these modes are only exceptional. As a rule, the
strings of string
instruments do not
vibrate that simply. According to Helmholtz, a
string of a harpsichord, harp, guitar, zither, when plucked at
the point A (Fig. 321), assumes in sequence the forms 1 to 7 of
Fig. 322. At the instant when it is released, it has the form aAc;
but it does not then simply vibrate to and fro between aAc and
aA'c, but between the modes 1 and 7, that is, the
vertical from A to the position of rest ac
moves to and fro along ac; in the vibration of Fig. 321,
it wi\would remain in the same location of ac. Stroked
and tapped strings behave in a similar manner.
It follows from
what has been said earlier, you can understand that these modes of vibration characterize the timbre of string instruments. The
jags and fine dents of 
the modes of vibration (Fig. 323) can
only arise as small waves are superimposed on the longer ones,
that is, when comparatively high overtones join the base tone.
This figure demonstrates already the formation of a dented mode;
it is not difficult to envisage how the superposition of smaller
waves induces these modes.
The forms of
vibrating strings differ depending on whether a string has been
plucked or tapped or stroked; it also differs depending on where these actions were applied. You can
observe them during vibrations with a vibration-microscope (Helmholtz) and photograph them (Raps, O. Krigar-Menzel).
Bars
The string is that
form of a solid, in which it almost alone is useable in musical
instruments. Other forms such as bars, membranes, plates, bells
are only employed occasionally, because they produce overtones,
which are unharmonic to the base tone and thus split the
tone. They are only of interest here as vibrating, but not
sounding bodies. The theory of their oscillations is quite
complicated and we will mention only the most important details.
Bars can vibrate
longitudinally and transversally: Longitudinally, if you rub them
along their length with a cloth, which has been made rough
(colophonium); transversally, if you stroke them like violin
strings or tap them like piano strings. Used in music, they are
made to vibrate transversally (musical box, triangle, celeste
(organ stop)). Their periods depend, like for strings, on their
material and dimensions, and moreover on their mode of support, that is, how and where they are held
fixed during vibration. The period during transversal
oscillations is proportional to a bar's thickness in the plane of
oscillation and inversely proportional to the square of its length - for strings, it is inversely
proportional to its length. In order to double, triple , quadruple
the period of a bar, if it otherwise is not changed, you must
only reduce its length to 1/(2)1/2, 1/(3)1/3,
1/(4)1/4 and not, like for strings, to 1/2, 1/3, 1/4.
A string of length 1000 mm must be shortened to 500,
333.3 or 250 mm, in order to obtain the three overtones of the
base tone of the entire string, a 1000 mm long bar you need only shorten to 707, 577, 500 mm.
The same
bar has, depending on its mode of fixing, a different base
tone: I is deepest, if one end is clamped (vice) and the other
end is free, highest (almost the third octave of the previous
bar), if both ends are
clamped, but also, if both ends are free. If both ends are clamped, it vibrates like a
string,
which sounds its base tone; if the ends
are free, it oscillates as shown by Fig, 324. Two node lines form
(lines, because the bar
represents effectively a multitude of a strings, which lie side
by side), about 1/5 of the bar's length from its ends. However,
the bar can also yield a base tone, which lies in between the ones
alreadydiscussed; in fact, if both ends are only supported, and yet another one, if one
end is fixed and the other end is free or only
supported. In all these cases, the bar oscillated as a unit, that
is, it yields a base tone. But you can also cause it (like a
string) to decompose itself into oscillating sections, separated
by node lines, by forcing certain locations to stay at rest by
holding them firmly. Then also something strange happens: The
parts are not equally long,
whence the nodes are not equidistant (which is quite different
from what happens with strings). Only when both ends are clamped
firmly, that is, when the bar oscillates like a string, the nodes
are equidistant. You
show off the nodes (Chladni) by
spreading on the bar fine, dry sand (you use for this purpose a
bar with a rectangular cross-section); It will roll off the
oscillating sections and collects at the node lines.
Tuning
fork
A bar is not really a musical
instrument; it is only employed in chimes (steel accordion), in the xylophone (wood
accordion) and as triangle. The tuning fork was invented by the
royal trumpeter John Shore
of England's King George I in 1711; it is 
the simplest and most reliable tool
(Fig. 325) and when struck yields a tone of known height. As normal tuning fork, tuned to the Chamber pitch-a, it has
become an indispensable acoustical aid to tuning of musical instruments and choir voices. (In this sense, the
internationally agreed normal tuning fork is something similar to
the internationally agreed normal measure (metre ) for the
measurement of length.)
The tuning fork arises out of a
bar with free ends. Such a bar, when it gives the base tone, has two nodes, each a little more than
1/5 of its length from its ends. If you bend it, the nodes come
closer together; in the tuning fork, they they are the ends of
the curved section. The prongs oscillate ( almost like two bars
with fixed and free ends each) towards each other and away from
each other. The arc between the nodes of the fork changes in
the process, as a result of its elasticity, its curvature and oscillated therefore in the
direction of the longitudinal axis of the fork; its handle does
the same. If its handle is placed on a sounding board, the
sounding tuning fork forces the resonance board to vibrate
strongly and reinforce its sound.
The period of a tuning fork
depends in a complicated manner on the length and thickness of
the prongs and the density and elasticity of its material (mostly
steel); it decrease its period slightly at rising temperature
(about 0.0001 per degree). The Physical-Technical Government
Institute in Berlin tunes normal tuning forks to the chamber-tone
a o 435 Hertz at 15º C.
Plates
and membranes
Plates and membranes oscillate always
transversally; they must be treated here separately (Our language
is here not strict in that very thin, leaf like plates are also
referred to as membranes)! They are interrelated like bars and
strings: Plates (and bars) have already elasticity through their
strength, in the cymbal and the tom-tom. Membranes (and strings)
receive elasticity first of all through tension, as in the
kettle-drum, the drum, the tambourine; of course, also out ear
drum, which closes the outer auditory channel. (It is a very
shallow funnel; because it is very irregularly constructed and
not uniformly strained, it resonates with all possible
vibrations, but well
damped by the
auditory ossicles. If that were not the case, we would hear
tones, which follow each other quickly, blended with each other.)
Vibrations of plates (metal,
wood, glass), which are supported at one point, are of special
theoretical interest. They will oscillate when stroked by a bow (Chladni 1787). As in the case of bars, their Hertz number depends how fixed they are held and the relative
position of the point of excitation from the point of support.
Hence the same plate can yield different base tones, since these
may be altered in many different ways. However, in the process,
the plate vibrates never as a whole, but always only in parts,
separated by nodal lines. The variety of possible states of
vibration becomes even larger, if one does not let still one point of the plate
vibrate., say, by touching it. Since then this point stays at
rest, one node line must pass through it.
Chladni has made the node lines visible by spreading fine, dry
sand on a
horizontal supported plate. The sand rolls off the vibrating
parts and stays put on the nodal lines, since they are at rest.
In this way arise Chladni's sound figures. Fig. 326 displays one
and the same disk; the location of stroking is always at b,
that of holding it at c, that of touching it at a.
You can also make visible the bulges
of plate vibrations (Felix
Savart 1791-1841 1827), if
you employ instead of sand very fine powder, the best being
lycopodium (Fig. 327). According to Faraday, the
strong upwards motion of the bulges pushes the air and with it
the light powder upwards; as it moves back, the plate dilutes the
air over these locations, so that air streams to it from all
sides and accumulates the powder there.
Technically most important are the very thin
plates in microphones, in telephones and in the very similar
loud-speakers. Here a big difference arises, when the plate as a
whole moves to and fro perpendicularly to its plane without
deformation, like a piston (piston ,membrane) or when it bends
during vibration (deforms). In Fig. 569, the disks represent
piston membranes. The real telephone and microphone membranes
deform and therefore tend to eigen (German:
own) vibrations and hence to modify the transmitted sound. The
membranes of radio telephones and microphones , condensator
telephones and microphones are almost free of it and transmit the
sound true to its nature; however, they yield only very weak
voice signals and require amplification. As a rule, the
condensator telephone is a a plate condensator, one plate of
which is a thin telephone membrane. It vibrates due to the
alternating current in the condensator, which is provoked by
induction due to voice signals. Condensator telephones, which are
used in place of microphones on the speech side, are called
condensator microphones. The ordinary telephone service employs
due to their greater robustness exclusively grain microphones.
The
electro-magnetic post telephone has a circular plate of iron,
about 0.2 - 0.4 mm thick and 50 mm in diameter (the diameter of
the sound opening is about 10 mm). The electro-dynamic
loudspeaker of Riegger has a square, extremely thin plate made out of
corrugated dural-aluminium 20x20 to 50x50 cm and more. It depends
on (Figs. 328/329) the interaction of a magnetic field and a
conductor with alternating current: A meander formed, vertical
band, through which flows an alternating current, vibrates in a
suitably arranged magnet system and the leaf fixed to it
oscillates with it. The leaf, positioned between strips of felt,
moves readily and is effectively a piston membrane.
A bell-shaped body oscillates like a plate, a
church tower bell like a circular plate, fixed at the centre.
Such a plate yields the deepest sound, when two of its diameters
are nodal lines, that is, if it oscillates in four quadrants -
just like a bell, when it has four nodal lines, which run from
its head towards its rim, so that the body of the bell also
oscillates in four quadrants; the second overtone occurs, if it
vibrates in six, the third, if it vibrates in eight parts.
Longitudinally vibrating solid
bodies
Strings can be made to oscillate longitudinally
by rubbing them along their length with a cloth, made rough by
colophonium, or of they are stroked with a bow at a very sharp
angle. The tones are shrill and much higher than than those of
transverse oscillations. They are only of theoretical interest,
for music they are to be avoided. For this reason, the bow should
be moved as much at a right angle to the string as possible.
Also longitudinal oscillations of bars are
only of interest theoretically; they are important for the
measurement of the velocity of sound in solids, namely of the
material of the bar itself, and for the computation of its
coefficient of elasticity. They are generated by rubbing the bars
along their length - metal and wooden bars with a cloth made
rough by colophonium, glass bars with a wet cloth or moistened
fingers - or by tapping their ends with a hammer. Also during
longitudinal oscillations, the states of oscillation of a bar
differ depending on whether its ends are free or not.
Assuming the
bar with free ends yields its base tone, it vibrates then as a
unit, like a string as a unit at its base tone. However, there is
a great difference between them, because the string vibrates transversally, the bar longitudinally. Just realize that, if
you tap the end of a bar longitudinally, then a longitudinal wave
starts off from that end, is reflected at the other end and
interferes, while it returns, with the wave coming in the
opposite direction. This is how a longitudinal
wave is formed. Its centre must
be a point of rest - a node -, bulges form at both ends, that is, the motion
near the centre is weakest, but alternates between compaction and
rarefaction; at the ends, occurs strong motion, but no change in
density. The bar cannot assist us by its appearance, because we
cannot see the to and fro motion of its points as we could see
the up and down of the string. However, if we plot the distances,
by which the individual points deviate from their p[position of
rest, as in Fig. 327,
transversally to the bar's position of rest, we can obtain the
image of a wave and understand the process. The curves a,
b, c of Fig. 330 thus show the deviations of
the individual points of a bar: a and b at the
instant, when the points of the bar simultaneously turn, c
at the instant, at which they also simultaneously pass the
position of rest. Figs. 331 a, b, c
show correspondingly a string at its base tone.
You see that the bar, as it produces its
base tone, forms along its entire length the half wave length of
its base tone. If you measure its length and find the height of
its base tone, you can determine the velocity, at which the
longitudinal wave propagates, since
v = l/t = wave length/period.
The wave length l is equal to twice the length of the bar and the
period t follows from the relation n ·
t = 1, where n is the Hertz number of
the tone, so that also v = l · n.
The height of the tone yields n as well as t. Melde
obtained for a 1400 mm long steel bar a base tone, the height of
which (measured with a mono-chord) was 1872 Hertz, whence v=2800mm·1872
= = 5249m, for a 1574 mm long glass bar a base tone with 1696
oscillations, whence v = 5339 m. Since the
velocity of propagation v is related to the modulus of
elasticity e by v = (e/d)1/2,
you can compute e from the velocity of propagation and
the density.
The bar with one fixed and
one free end must behave differently, because there can arise at
a fixed end only a node, at a free end a bulge. The
fixed-free-bar vibrates at its base tone as a whole, whence all
its points form a quarter wave, because a bulge and its next node are
always a quarter wave apart. Thus such a bar has a base tone,
which is by one octave deeper than that of the free-free-bar.
Air columns
You can compare
the longitudinal vibrations of bars with the air columns in
tubes, which are open at both ends or open at one end and covered
at the other end. If the tubes contain mechanisms, by which the
air can be made to vibrate, they are called pipes - open or open-closed ones. Beside string instruments, they
are the most used musical instruments which comprise all
wind-instruments and the organ as well as the larynx.
In the tube, the
air vibrates in standing longitudinal waves. The standing wave
is formed like that of a longitudinally tapped bar: A wave runs
from the one end along the tube, is reflected at the other end
and interferes with the one meeting it. Where are the nodes and the bulges?
In the bulges, the motion is
strong, but neither compaction nor rarefaction, but at the node
is rest and a strong variation of pressure. If a standing wave
forms in a tube, the vibration process at the tube's end
must take another form when it is open and when it is covered. At
the open end, the air inside the tube
borders on the outside air; every difference in pressure must be
equalized immediately, whence there must be a bulge. At a covered
end, the air cannot move longitudinally, whence there must
be a node. We see that a standing wave in a tube
must always have a bulge at an open end, a node at a closed end;
in a tube, both ends of which are open, you must
have a bulge at each end, if one end is closed, the other open,
you must have a node and a bulge respectively. Figs. 332/333
present two equally long tubes, one open, the other covered, as
well as transversally represented longitudinal waves. The images a
present the motion, when there are in the open tube only two
end bulges (at the centre in between a node) and in the covered
tube only a bulge at one end, a node at the other end. However,
you can imagine that there can be more nodes and bulges in a
tube, because the conditions at open and closed ends can then
also be met, when the oscillations proceed as in the images b,
c, d, that is, there are also bulges between those at
the ends.
What is the
significance of this increase in the numbers of bulges and nodes? Two neighbouring , that is separated by
one bulge, nodes contain in between half a wave length, a half
wave. If there exist only two bulges at the ends this means that
the oscillation in the tube forms half a wave of the length of
the tube - obviously the longest possible half wave. The
corresponding tone is therefore the deepest possible one, the base
tone. (Since we know the wave lengths of the individual
tones, we can predict what length a tube must have for a given
base tone: It must be half as long as this wave.) The
introduction of one bulge in between the end bulges
(Fig. 332 b) means conversion of that longest half wave
into two waves, each of which is half as long as that of
the base tone. An introduction of two bulges (Fig. 332 c)
causes conversion into three half waves, one third as long as
that of the base tone, etc. Like the string half wave (Fig. 320 a) was subdivided by nodes, the tube half wave of the base tone
(Fig. 332 a) is also subdivided by nodes. Like the
string, as it vibrates undivided, yields the base tone, but when it is
subdivided into 2, 3, ··· n sections, tones with 2-,
3- ··· n time as large Hertz numbers (the sequence of
harmonic overtones) so does the open tube, that is, it can also
give apart from the base tone the entire sequence of harmonic overtones.
It is different for the covered tube! There must always be a node at the
covered end, a bulge at the open end. If only the end
bulge is present and only the end node,
the standing longitudinal oscillation in the tube is given
by Fig. 333 a. However, you can imagine that
also in the covered tube there are more nodes and bulges, because
the conditions that a node must lie at its closed end and a bulge
at its open end are also met, when as in Figs. 333 b, c,
d, there are nodes and bulges . What means the increase of bulges and
nodes in the closed tube? A node and a neighbouring bulge always bound a quarter
of a wave length, a quarter wave. If there exists besides
the end node and the end bulge no other node in the tube, it
means that the oscillation in the tube forms a quarter wave, the
length of which equals the length of the tube - it is obviously
the longest quarter wave which can occur
in the tube, that is, it is the base tone. Hence we can predict,
as in the case of the tube open at both ends, the length of the
tube closed at one end for a given base tone: It must be the
fourth of the length of the wave of this tone. Note: the covered
tube only a fourth
of the length, the open one
half the length. Hence follows the result:
An open tube, which is to give the same base tone as an open tube
of given length, is therefore only half as long as the open tube
and an open tube with a wall at its middle (Fig. 334) does not
change its base tone. Experiments show that this is true. The
tone only becomes softer - and this is readily understood, since
the wall takes the place of a node, that is, a location without
motion, which behaves like the covered and of a closed tube.
An introduction of
one bulge and node between two ends (Fig. 333 b)
means: Conversion of that longest quarter wave into three, each
of which is 1/3 as long as that of the base tone, that is, the Hertz number is multiplied by three. An introduction of two
bulges and two nodes means: Conversion of that longest
quarter wave into five waves, each of which is 1/5 as long as
that of the base tone, that is, the Hertz number is multiplied by 5, etc. Hence also
the covered air column can either oscillate as a unit or
subdivided by nodes. But while in the open tube you can achieve every
integral multiple of the Hertz number of that of the base
tone, in the covered tube only every odd numbered
multiple (1-, 3-. 5-,··· is possible, that is, only
odd-numbered overtones of the base tone. This is quite natural,
because the 2-, 4-, 6 fold would demand waves from one, two,
three half waves in the tube, that is, waves with bulges at both ends of the tube, which is impossible In a
tube covered at one end.
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