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 stringinstruments 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, drysand 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.