Acoustics
I1 Generation, propagation and perception of sound
Physiological acoustics. Physical acoustics
The theory of sound belongs to Physics as well
as to Physiology, because the cause, which under ordinary
conditions evokes sound sensations, can be reduced to a strange
kind of motion, an investigation of
which is indispensable for insight into the nature of sound
sensation. Sound sensation arises by stimulation of the
auditory nerve. Every stimulation of the auditory nerve causes it;
according to Helmholtz , it is in effect "the ear's peculiar mode of
reaction to external stimulation". The stimulus, which is to
be considered as normal arises through oscillations of an elastic
membrane (ear drum), which closes the inner end of the auditory
canal. It acts with the aid of the adjacent auditory ossicles and the endolymph (auditory fluid), into which extend the ends
of the auditory nerve, on the auditory nerve. This stimulation of the nerve generates the sensation of sound.
The ear drum starts to vibrate through the oscillations of the
air in the auditory canal, which are caused by the oscillations
of the body, which we consider to be the source of a sound. Its oscillations are transmitted to the air in the
auditory canal through the air in between them. Corresponding to
this process, we must consider the oscillations, arising from the
source of the sound and being transferred through the air to the
ear, as the source of the sound sensation. In a physical sense,
we call this the motion of the sound or, briefly, acoustics.
Hence we have in acoustics two areas: Physiological acoustics, dealing with sound sensations, and physical acoustics, dealing with sound propagation. It is
impossible to strictly separate these two regimes, because the
ear is the natural, although also not the only means, for the
examination of the motion.
We experience many different
sound sensations such as noise and musical sounds. Their difference 
is so huge, that we must draw the conclusion that they
have different physical foundations. You can generate noise by means of musical sounds: The
simultaneous tuning of
instruments in an orchestra generates an auditory sensation at
the border between noise
and music. In fact, resonators display every
sound as a mixture of musical sounds. Hence we view a
musical sound as the simpler one of the two auditory sensations. Scott's phono-autograph ( similar to the phonograph of Edison) demonstrates that, although they have the same origin
as sound oscillations, noise
and musical sounds
differ objectively; it records the vibrations graphically. It
consists of a membrane (like our ear drum), to which a pen is
attached, and a drum with paper covered by soot, which the pen
touches. The drum rotates and shifts along its axle like a nut on
a screw. If the membrane oscillates due to a sound, the pen draws
a curve in the soot and thus displays the oscillations of the
membrane. All vibrations of musically sounding bodies have the
same form and can be compared with those of a pendulum, while
this is not the case for noise.
A sounding body, a vibrating body
In many cases, it is easily verified that sounding bodies really oscillate. Although you cannot see single oscillations of a vibrating string, you see that it looks like Fig. 304, that is, it moves to and fro between the arcs a and b. Gradually, the image becomes flatter until it returns to the position c of the silent string at rest, You can discern that a sounding body is in motion by silencing it - for example, a sounding string, a bell, a tuning fork - by touching it with your fingers. You can also make it display the individual oscillations: You can make a tuning fork write down its oscillations, photograph the oscillations of a string, make visible the oscillations of the air in an organ pipe by means of a flame, etc. In most cases, the air carries to us the oscillations of a sounding body, but there exist also sound sensations, in which it does not take part. You can hear a sounding tuning fork or a ticking watch, if you hold it with your teeth; if you place your ear to the ground, you can hear the noise of a distant railway train, motor car, etc. while it is not yet audible. In the first example, the oscillations come to the ear through the bones of the skull, in the second case through the ground. In fact, sound is transmitted by all bodies, in which elastic vibrations can be maintained and propagated - solids, fluids and gases. However, in most cases, the air transmits it. You can convince yourself of this by activating an electrical bell inside a glass dome, being evacuated by a pump; the sound becomes softer and softer and eventually disappears.
Propagation of sound in the air
Sound is spread through the air
as longitudinal waves cause by oscillating bodies. You can
reproduce the compactions and rarefactions of the air 
photographically (Charles Venon Boys, Mach, Salcher). Fig. 305 demonstrates how sound waves spread about the centre of
their generation.
The process is not as simple as it has been described. In Fig 284, we looked at a single straight line of mass particles, which were initially at rest. Now we are concerned with the air space around a sound source and with displaced particles. However, our conceptions must nevertheless be correct, because the observations agree essentially with the conclusions of the theory.
According to v = (e/d)1/2, the velocity of propagation of sound through a substance depends only on its density d and elasticity e. Hence it must be independent of the wave length, that is, the same for all wave lengths. This experience is also confirmed by the simultaneous sound of several, simultaneously propagating sounds; for example, that of an accord, which sounds unchanged independent of its distance from its source. However, longer waves belong to lower sounds; if short and long waves would propagate at different speeds, we could not receive the high tones of an accord simultaneously with the lower ones. We would hear a broken up accord, one tone after another.
This formula teaches us that
our concepts are justified. If you substitute for e and d
numbers, which refer to a given material, you obtain the velocity
of sound passing through them. For solids
and fluids, there is good agreement between the
computed and observed velocities; it is so good that you can
employ an observed velocity v with the known density d,
in order to control measurements of the elasticity coefficient e.
In air, the speed of sound at 0º C is 331 m/sec, but according
to the formula it should
be 
279.4 m/sec.
This disagreement is only apparent ( Laplace 1818):
The elasticity of air is influenced by the temperature changes,
which a sound wave causes by intermittent compactions and
rarefactions. Every compaction increases the temperature, every
rarefaction reduces it. These differences in the temperature
cannot adjust quickly, as the oscillations pass, and increase the
differences in the pressure, that is, the elastic forces, on
which the propagation of sound depends. Laplace has computed this effect and has also obtained
agreement of experiments and the theory of sound propagation for
gases. According to his theory, you must multiply the formula v
= (e/d)1/2 by (cp/cv)1/2,
where cp and cv
are the specific heats of the gas at constant pressure and
constant volume.
If surface waves are generated on water (Fig. 306) around a and they meet the wall AB, they are reflected by the wall. There arises a new system of waves which develops out of them. It is completely like the first and spreads out towards the side from which the waves came. In a physical sense, the reflected system of waves has no centre of generation, but in a geometrical sense it is clear, that the waves appear to be as if they had a centre. The system of waves, reflected by the wall AB, spreads out around this apparent centre a' just as the initial system has spread about a.
Air waves behave just as water waves: They are reflected by each obstacle (some of them turn out to be effective, although not expected a priori (Latin = not deductively), for example, a cloud!) and form a reflected system of waves, which behaves just as the initial one which generated it. Also this reflected system of waves has no real centre of excitation, but its impression on your ear corresponds to the reflected water wave system you could see. As in that case the eye constructed the centre of excitation a', in this case the ear does so: It has the impression of the presence of a new source of waves, which seems to lie behind the obstacle.
Ordinary reflection phenomena of sound are reverberation and echo (Greek: hcw = sound). Reverberation occurs in every closed room. However, you only become conscious of it when it disturbs you, for example, when the sound reflected by the walls lengthens spoken words and thereby makes them unclear.The acoustics of a room depends essentially on reverberation, but the many reflections of the sound - from one wall to another, from objects in the room, etc. - complicate achievement of good acoustics and make it accidental. Most disturbing is reverberation in large rooms with a curved ground plan and smooth, uninterrupted walls. In concert halls and theatres, arches, parapets, etc. tend to reduce these perturbations. However, you cannot go too far with these interruptions of walls, for reverberation is required in support of a sound, as you can confirm since a public speaker outside, where the sound travels away, has greater difficulties in making himself understood than in a closed room.
The sound, reflected by walls, can support the first tone as it returns to the source before it has ceased to sound. (The distance of the walls from the source of sound in relation to the velocity of sound and the duration of the first sound contribute here.) If it only arrives at the source of the sound as the first tone is finished, it obviously cannot reinforce it - but it can double its duration and, if it is clearly separated from it, repeat it. These clearly audible repeats of a tone by its reverberation are called echoes. In contrast to reverberation, echo really gives the impression of the existence of a second source of sound, especially when it repeats several syllables of musical sounds. However, one can then only be concerned with a short sequence of words or tones, because the distance between the source of the sound and the reflecting wall imposes a limit - moreover, this happens only with very loud tones, because in the case of a long distance the sound becomes inaudible.
However, the situation changes if these two aspects can be neglected as, for example, in the case when the sound is reflected by concave mirror surfaces, which are located with respect to each other like in Fig. 307. The ticking of a clock fades mostly away at distances of 1 - 2 m, but you can hear it much further away, if the clock is located at the focal point A of one mirror and your ear - or better the end of a ear-trumpet - at the focal point of another mirror B. Then the reflected sound does not return to its source A and cannot perturb the initial sound. The sequence of tones radiating from the source can then be endless - like the ticking of a clock - without collision with echoes. All paths of the sound of the clock to the mirror and back from there pass through B, the entire reflected sound reaches B. Therefore the tone is heard at B, even if its very soft at A like the ticking of a clock.
In vaults of elliptic shape (whispering galleries), the sound arrives from one focus of the ellipse at the other; even quite softly spoken words, which come from one focus, are clearly heard at the other focus - but not at other points.
Since several years (the reader should be reminded that this was written in 1935), reflected sound waves are employed on ships to sound the depth of water, in oceanography even of the deep ocean. The sonic depth sounder of Alexander Behm 1880-1952 determines for this purpose the number of seconds between the launch of a sharp sound under water and the arrival of its echo from the bottom of the sea. Using this time difference and the known velocity of sound in water (1435 m/sec in fresh water at 8ºC), the water depth is computed.
You can produce longitudinal waves in every material, transverse ones in every solid. In order to produce musical sounds (note that we are only interested in these), almost only solid bodies are employed in the form of strings (gut, metal); sometimes also bars are used (triangle, tuning fork), bells, plates (basin) and membranes (drum), but less frequently than strings. The tones generated by oscillations of columns of liquids (siren, and organ pipe) are of little practical importance. Of greater importance are the oscillations of air columns as generators of sound in all wind-instruments, the organ and the larynx. The mode of generation of the oscillations is so characteristic for the instruments themselves that Helmholtz subdivides them according to their mode of excitation:
1. By striking
(piano, harp, guitar, pizzicato of string instrument);
2. By stroking with a bow (string instruments);
3. By blowing against a sharp edge (flute and
organ-pipes);
4. By blowing against elastic tongues (all other
blow-instruments, tongue pipes of organs and larynx).
You know how the instruments are stroked. Blowing against air columns is described later on . It is obvious that an air column is made to oscillate in a manner different from that required by a solid, which can be activated and moved by mechanical gadgets (bow, hammer, finger). However, whatever are the material and form of the instruments - the common mode of excitation of all of them is resonance, that is, by making them oscillate.
A body oscillates by resonance whenever it is activated by waves, sent out by another oscillating body. Just as a body floating on top of water dances as waves reach it, a body in air oscillates as sound waves reach it. However, this comparison does not cover all details of resonance.
In order that a body surrounded by air can be made to oscillate by means of air waves, because each of these has only very little energy, one condition must be met. We will explain this by a comparison: A heavy body, suspended like a pendulum - a swing - can already be made to swing by a very weak push. Initially, these oscillations are very small. But if you repeat the push always as the swing during its motion moves again in the direction, in which it received the first push - in other words, if the excitations reoccur in the same sense and speed as the oscillations of the swing - their effects will add and the amplitude increase. Even a small child can cause in this manner a heavy swing to oscillate strongly. However, obviously, the vibrations cannot grow unless the directions of motion and excitation coincide.
Moreover: If the direction of the pushes coincides with the direction of the swinging motion, it will be sufficient, if the rate of the pushes is half or a quarter of that of the oscillations of the swing, that is, when it receives a new push at every second or third cycle.
The application of this model to the swinging of an elastic body is obvious. The swing is the body, which is to sound, say, a string. The pushes are arriving air waves, sent by the swinging body, say, a sounding organ pipe, to the string. The rate, at which they follow in sequence, that is, their number per second (Hertz's number), is the number of oscillations, the height of the sound of the pipe. For the string to begin to oscillate in this manner, it must be able to swing at the rate, at which the air waves arrive, that is, the tone, which it can give, must be exactly as high as that of the source of sound or its tone must be an overtone of the tone of the exciting source - must oscillate twice, three times, ··· as fast as the source of sound, so that it receives a push at each second, third, ··· oscillation.
This is the condition for the sounding of a source of sound by resonance. If two strings have been tuned to the same tone and one of them is struck, the other will also sound. If you raise from a string of a piano its damper (by pressing the key down slowly), so that it can oscillate freely, and you sing into the piano the tone, which it would send out if struck, it will reproduce this tone. Given two tuning forks with the same tone, if you strike one of them, the other will also sound. If you hold a sounding tuning fork or bell over an air column, for example, an organ pipe, which would when blown at give the same sound, the pipe will also sound.
In these cases, the wave generated a sound, while hitherto if only transmitted it. The difference is: Here the wave is received, say, swallowed, and it rises again as a tone of the resonating source of a tone, while before is could spread out freely. It can also happen that a sound oscillation is swallowed without reappearing as a new tone; it is then really swallowed and its energy is converted into heat. We will come across a counterpart in the Theory of Light: Under special circumstances, light can, after being swallowed, rearise as new light or also only be converted into heat, but also here both can occur, if its stopped from spreading by a body.
Naturally, easily moved bodies can be made to resonate by a smaller number of impacts, that is, a tone of shorter duration, than bodies which are less movable; a string, membrane, a column of air is more readily excited than a bell, a plate, a tuning fork. Evident contrasts are, on the one hand, a stretched membrane, a body of very little mass, which begins to oscillate almost simultaneously with the tone, causing it to vibrate - on the other hand, a tuning fork, a body of great mass, which for resonance requires a tone of longer duration.
This difference in the mass has another consequence: An oscillating body of less mass has less energy while it oscillates, but it also loses it faster (through friction by the surrounding air and internal friction) after the exciting tone has ceased. However, if a body loses the oscillations, aroused by a push, after a very few cycles, the amplitude decreases already noticeably after the first cycle. Hence it is also excited by impacts, the rate of which is not completely in agreement with the rate of its own oscillations. Hence, if the consecutive impacts deviate somewhat from the rate of the body's eigen-oscillations (German: eigen = own), they do not collide noticeably with these. In other words, The eigen-vibrations do not have then a substantial role and the body can also be excited by other tones.
Similar considerations apply to bodies, which are not readily excited and maintain their eigen-vibrations for a long period: The must be tuned exactly to the tone, the sound waves of which they receive. Two extremes are the membrane in Rudolf König's apparatus and the tuning fork: The former resonates at every tone, the latter only at one tone and even not when the exciting tone deviates by a few cycles per second from that of the tuning fork. Fundamentally speaking, for our hearing apparatus, the property of a weakly stretched membrane is important - to resonate with each tone. to start with it simultaneously and to end with it. The ear drum corresponds in this respect to the membrane in König's apparatus, whence its oscillations also do not collide with those it receives from the sound source - it adjusts to every tone. In practice, co-oscillation is employed in the resonance boxes and sound-boards of string instruments.
The resonator of Helmholtz is indispensable for tone analysis. It is a hollow glass or brass sphere with two diametrically opposite openings (Fig. 308). The air inside the sphere is made to vibrate and to transmit its vibrations almost instantly to the ear drum. For this purpose, the rim of the one opening is formed into a cylinder, facing the incoming wave, the rim of the other opening into a cone, which you press tightly into your ear. Such a sphere has a natural tone, the height of which depends on its size (at 70 mm, the tone c2). When there enter the waves of a tone of the same height as that of the resonator, the sphere resonates and blasts into the ear: The arriving tone is reinforced by making the air in the resonator oscillate and instantly enters the ear. The main point is: The resonator allows only this one tone to sound for the ear. If a mixture of sounds contains this tone, it sounds, otherwise it is silent. Speaking now mechanically: It filters only this one tone out of the cocktail of tones and sends it to the ear. (It performs for the ear, what a coloured piece of glass does for the eye. A glass of a certain red colour only transmits light of this colour. If you place it between your eye and the source of light, your eye only receives light, if light of this colour is radiated by the light source.) Hence resonators allow to analyse sounds and to determine whether a given sound contains a given tone or not.
Helmholtz wrote: "If you close the one ear and place a resonator into the other ear, you hear most of the tones, which are generated in the neighbourhood, much more dampened than usually; however, if the eigen-tone of the resonator is present, it blasts strongly into your ear. Hence everybody, even those with not musically trained or badly working ears, are in a position, to identify the given tone, even if it is rather weak, from among a large number of tones; sometimes you can even identify the tone of the resonator in the whistling of the wind, in the rattling of cart wheels, in the rushing of the water as you come to the surface."
Sensitive flames - gas-flames - which change their shape, as they are visited by waves of certain tones, are very sensitive resonators. In order to become sensitive, they must meet certain conditions of gas pressure, form of the burner, size of the gas conduct, etc. For example, under such suitable conditions, the flame a of Fig. 309 changes to b, when at some distance from it a whistle is blown or an anvil is hit with a hammer. The vowel flame (as Tyndal called it, because the different vowels affect it differently) - a flame, which under suitable conditions is 60 cm long, reduces its length at the slightest hit with a hammer on an anvil to about 15.5 cm, contracts at the rattle of a bunch of keys and reacts to the softest ticking of a clock.
The knowledge of the resonance phenomenon yields us an understanding of our sensing of tones: The action of our auditory organ leads us directly to resonance. The organ inside the ear - named after its discoverer in 1852 the tunnel of Corti - reminds us of a musical instrument, which, just as the piano has for every tone a string; the membrana basilaris is a membrane with parallel, lengthwise stretched fibres, comparable with a piano's strings, which are not very close sidewards, so that everyone can oscillate on its own as a result of its tension. On the membrana basilaris (basic membrane) stand Corti's arches, rigidly joined to it. Helmholtz wrote: "The entire arrangement convinces one that Corti's organ is an apparatus, suitable for receiving the oscillations of the basic membrane and to oscillate itself; however, with present day knowledge, it cannot be ascertained in what manner the oscillations occur." The auditory nerve fibres end at Corti's arches. Helmholtz continues: "Hence the essential result of our description of the ear can be summarised by saying that we have found the auditory nerves everywhere linked to special, partly elastic, partly rigid gadgets, which can resonate under the influence of external oscillations and then probably shake and stimulate the mass of nerves." People assume that Corti's arches through their linkage to corresponding fibres of the basic membrane are tuned effectively to individual, differently high tones and resonate with these tones. According to this hypothesis, every simple tone is sensed by a special nerve fibre, and conversely differently high tones stimulate different nerve fibres; hence, since the overtones have different heights, a sound with various overtones stimulates simultaneously different nerve fibres, that is, is also sensed by several different nerve fibres, that is, by a assortment of differently high, but simple tones. Accordingly, this hypothesis yields an acceptable explanation for an ear's decomposition of a sound into simple tones.
The oscillations of the
membrane of a telephone, phonograph, gramophone and
phono-autograph are resonances
of membranes, which respond to every
tone and therefore also transmit every tone and
reproduce it, although not everyone with the same clarity. The
phonograph is based on the same idea as the phono-autograph, but it is
much more. It also draws the curves of oscillations, but - and
this is the decisive point - it can from the curves reproduce the
sounds, whence it becomes a speaking and singing machine. The
phonograph (Fig. 310) was conceived and
realized by Edison. Its essential components are: A
cylindrical barrel, the surface of which is covered by a
specially prepared, completely uniform waxlike layer C,
a membrane m with a saphire tip, facing the barrel,
where it rests on the barrel's cover. The barrel rotates at a
uniform speed about an axle and shifts along like a nut on a
screw. When the membrane presses the saphire tip against the wax
layer and the barrel rotates, the tip cuts a screwed trace of a
certain depth around the barrel. If the membrane oscillates under
the influence of a source of sound, the tip cuts at different
depths into the wax and the corresponding profile of the groove
in the wax reflects the state of the oscillations (Fig. 311). Up till now, that is, as long as the
membrane oscillates under the influence of a sound, the
phonograph is a receiver and sound recorder. However, if after
removal of the source of sound the barrel is brought back to its
starting position, then the sapphire tip is returned to the
groove and the barrel is turned again, the tip travels up an down
along the grove and this causes the membrane, to which it is
fixed, to execute the same oscillations which it performed as a
result of the sound. This process reproduces the sound, earlier
recorded. In a gramophone, one uses today (1935) circular disks
instead of the cylindrical barrels.
Strength, height and colour of tone
Tones differ from each other by strength, height and colour. The meaning of strength and height is well known. One calls tone colour the individual timbre of a tone, by which the ear can distinguish, for example, a violin from an organ or a human voice - also one person from another. Whereby differ the oscillations of a strong tone from those of a weak tone? Experience tells us: We hear a tone, which spreads unimpeded, more strongly or weakly, depending on our distance from the source of the sound; but we hear unchanged height and colour. Only our distance from the source of a tone influences its strength. The sounding instrument is the source of the oscillations, which propagate to our ear drum. At a greater distance from their source, the oscillations are smaller than closer by - just as water waves become lower the further they have spread. The physical cause for the subsidence of the sound sensation at greater distance from the source must therefore be sought in the decrease of the amplitude. The fact that the strength of a sound indeed depends on the amplitude can be heard and seen in the case of an oscillating string (Fig. 304). While you cannot see individual oscillations, because the oscillating string is blurred - a band which is wider at the centre - your hear that the tone becomes softer the narrower is the band.
We must distinguish between the
strength of the sound sensation and the strength of the motion of 
the sound. The
first is a psychic, the second a physical process. The exploration of their
interrelationship is a task of Psychology (the basic, psycho-physical Law of Fechner and Werner). The strength of a sound sensation
naturally grows with that of the sound motion. We will only deal
with the latter, the physical strength of the sound.
The change of the strength of a sound with the distance from its source in a homogeneous, isotropic medium like air is as follows: The source supplies the kinetic energy of its oscillations (the physical measure of the strength of sound) to its immediate neighbourhood and the wave motion spreads it throughout the space as spherical waves which have their centre at the source of the sound. (we will assume that none of the energy of the source gets lost, that is, that the entire energy arrives on such a spherical surface.) The larger this surface, over which this energy is distributed, the less of it is available per unit surface area. Spherical surfaces with the radii 2, 3, ··· , n have 4, 9, ··· , n2 as many surface area units as the spherical surface with unit radius. Consequently, an area of 1 cm2, which is hit perpendicularly by the sound wave, receives at distances of 2, 3, ··· , n metres only 1/4, 1/9, ··· , n2 of the energy, which it receives at a distance of 1 m, that is, the strength of a sound is inversely proportional to the square of the distance from the source of the sound.
You can measure the strength of the sound (for
example, of loud speakers, E. Meyer) 
with the disk
of Rayleigh: A very small,very thin,
circular disk, which is suspended by a very thin thread and
protected from air draft (sheet mica, about 5 mm diameter, 0.05
mm thick; wollaston wire 10 cm long, 3 - 5
m diameter) and inclined at an angle to the direction
of the sound; as soon as sound waves arrive, it tends to position
itself perpendicularly to that
direction (measurement by telescope, mirror and
scale). The turning moment at the disk is proportional to the
strength of the sound. You compute from it the amplitude of the
oscillation of the pressure of the waves in dyn/cm².
The measurement of the amplitude also employs a link to an aneroid manometer with a resonator (vibration manometer of Wien 1866-1938). The conical opening of the resonator (Fig. 308) is widened and air tight sealed by the membrane, which replaces the ear drum. Its vibrations are viewed in a mirror, linked to it; it widens the image of a light line into a band, the width of which is proportional to the amplitude of the pressure, the square of which yields a measure of the strength of the sound. Conversion of a sound into another form of energy (thermal, electrical), if the process is known qualitatively, allows indirect measurement of the strength of sound.
