SOUND. —
I. PHYSICAL ORIGIN OF ACOUSTIC STIMULI
The energy adequate to stimulate the auditory apparatus is a particular form of mechanical energy, namely, vibratory energy. A conservative mechanical system, in which elastic potential energy is contained, tends to restore it in the form of kinetic energy; and this, if certain conditions of symmetry are satisfied, can be transformed into potential energy according to a closed cycle of energy exchange, giving rise to a periodic vibratory motion of the material masses of the system around the position of rest. The parasitic dissipations of the energy contained in the system ensure that such a cycle can be repeated only a limited number of times, so that if the vibrating system is not supplied with energy the vibrations are damped out in a more or less long time.The amplitude of the vibrations and the number of vibrations that the system itself performs in the unit of time (frequency) depend on the mechanical quantities involved, in particular on the masses and the restoring forces that continuously draw them back toward the position of rest. A complete vibration per second, i.e., one period per second, is the unit of frequency and is called the Hertz (Hz).
If the vibrating system is situated in an elastic medium (solid, liquid, or gaseous), it is also a “source of vibrations,” inasmuch as its alternating movements are communicated to the medium and transmitted within it as pressure waves. It is in this way that vibrations reach the auditory apparatus through the air, directly setting the tympanic membrane in vibration and, consequently, the structures of the middle and inner ear; a small part of the vibratory energy arrives by direct conduction through solids (e.g., the ground), reaching the inner ear directly by bone conduction, as indeed also happens for a modest fraction of that conducted by the gaseous medium.
The tympanic membrane is sensitive to the “pressure” of the incoming waves, and the vibratory stimulus is adequate only if such pressure lies within certain limits (“threshold of audibility” and “threshold of pain”), while its frequency must also lie within other defined limits (lower and upper limits of the total field of audibility). The increase of stimulus pressure is perceived as an increase in the (subjective) intensity of the sound, while the rise in its frequency is perceived as a rise in the so-called (subjective) pitch of the sound.
Through the outer and middle ear, the mechanical vibrations of acoustic frequency reach the cochlea, where the conditions are created for the excitation of the VIIIth nerve; the nerve impulses travel, undergoing various modifications, through a chain of four neurons and their synapses and junctions to the auditory radiations and the cortical acoustic area. Thus begins the conscious subjective movement during which the sound arises as a result of the higher associative, perceptive, and cognitive processes.
There is no place to speak of sound as present in the environment that surrounds us. In that environment, the vibratory phenomenon has physical reality. It is to be noted that the sensation of sound can arise even in the total absence of vibrations in the external environment, e.g., through the application of electric currents to the auditory apparatus (electrophonic phenomenon); here the electric stimulus is an inadequate but sufficient stimulus.
II. VIBRATING BODIES
Taut strings, metal plates, rods, bells, and almost all objects that surround us are mechanical systems capable of vibrating at acoustic frequencies; even air can become a vibrating body if confined within cavities, as in organ pipes and sound boxes.A marvelous vibrating system is that constituted by the “vocal cords,” better termed “vocal lips.”
III. VIBRATION VELOCITY
The amplitude and frequency of the vibratory motion determine the velocity with which the particles of the vibrating body follow their periodic alternating motion around the position of rest, toward which they are drawn at every instant, as has been said, by the elastic restoring forces.In a tuning fork having a frequency of 100 Hz, excited so that the amplitude of vibration of the end of the prongs is one millimeter, each particle of that end travels in one second 100 times 2 journeys of 1 millimeter each, equal to 200 mm. of travel, with an average velocity therefore of 20 cm. per second, equal to 0.72 km. per hour.
The tympanic membrane, which at a frequency of 3000 Hz can be set in vibration by energy of barely one billionth of a microwatt per square centimeter, has, according to Wilska, a vibration amplitude of 5.2 hundred-billionths of a centimeter; under these conditions, as is easily calculated, it has an average velocity of 3.12 millionths of a millimeter per second, i.e., 12 microns per hour.
IV. PROPAGATION VELOCITY
Similar calculations can be made for the particles of the medium (gaseous, liquid, or solid) in which the vibratory motion is propagated. In air, for example, the velocity of the particles reaches at most fractions of a millimeter per second, since, although the motion is rapid, their displacement around the position of rest is always extremely small.The velocity, however, with which the vibratory motion is propagated (commonly indicated by the term “velocity of sound”) depends on the physical characteristics of the medium, more precisely on its density and its modulus of elasticity, quantities in turn related to various factors such as, for example, temperature and pressure. In air the velocity of propagation is 331.45 meters per second at zero degrees centigrade under normal atmospheric pressure of 760 mm. of Hg., and it increases by 0.607 m. per second for each degree of temperature rise, so that for it a mean value of 340 m. per second may be assumed in relation to the average ambient temperature at our latitudes. In liquid media the velocity is greater (in distilled water, for example, it is 1439 m./sec. at 5° C), and in solids it is several times greater still.
The propagation velocity does not depend, within wide limits of approximation, on the frequency of the mechanical vibrations (at least within the limits of the acoustic range); a strict dependence between frequency and velocity would render almost impossible the distant listening of words and music, since tones of different pitch would arrive with different delays at the listener’s ear. The propagation velocity is not perceived as such by the ear, but we are accustomed to appreciate it indirectly in the fact, for example, that we hear the sound with greater or lesser delay (in relation to distance) from the instant in which we see the act that generated the mechanical vibrations. Moreover, this velocity is of primary importance in determining the direction from which an acoustic-frequency vibration comes, being related to the differences of time and phase of perception at the two ears.
The vibratory displacement of the particles takes place in the same direction as the propagation; in this sense there do not exist in the medium waves as such, i.e., contrary to what one is inclined to imagine by comparison with the model offered by waves on the surface of water, in which the motion of each particle takes place in a direction perpendicular to the direction of propagation of the disturbance on the surface.
In the propagation of elastic waves in a homogeneous medium, therefore, there exist only compressed and rarefied zones that succeed one another in orderly fashion, spaced at equal intervals in the direction of propagation of the elastic disturbance. The length of this interval takes the name of “wavelength” and is inversely proportional to the frequency. At 100 periods per second with a velocity of 340 m. per second, each wave is 3.4 m. long; at 1000 periods per second, with the same velocity, each wave is 34 cm. long.
MOTI SEMPLICI E MOTI COMPOSTI: TIMBRO
The mechanical vibrations of bodies at acoustic frequencies are periodic movements similar to pendular motion, but they do not achieve the same regularity and simplicity unless, in rare cases, particular conditions of structural simplicity and symmetry are met (as, for example, in a tuning fork). In such cases, the course of the vibratory motion over time can be represented by a sinusoidal function of the type \( y = \sin x \).
A vibrating elastic body of any shape, however, vibrates by subdividing into various vibrating portions (composite vibratory motion), so that vibrations of various frequencies and amplitudes are produced simultaneously. These frequencies may or may not have simple multiple relationships (harmonic ratios) between them. The component with a frequency double that of the fundamental frequency is called the second harmonic, that with triple the frequency the third harmonic, and so on. Any composite motion can always be decomposed into a series of simple vibratory motions, i.e., sinusoidal ones, which, coexisting, determine the composite motion itself. The search for such components is commonly known as "analysis of sound."
It is the "composition" of the vibratory motion that corresponds to the subjective appreciation of timbre. The timbre varies as a result of changes in the number, frequency, and amplitude of the vibrations that make up the complex vibration. Differences in phase are of lesser importance.
When the component vibrations are not in harmonic ratios but are completely independent of any mathematical ratio law—especially when the vibratory motion decays rapidly—the listener defines the auditory sensation more as noise than as sound. A typical example is provided by the "Savart wheel," namely, a toothed wheel against which a piece of cardboard can be pressed. As the wheel turns, each tooth causes the cardboard to oscillate, producing a well-defined noise; but if the successive impacts on the teeth follow rapidly, the auditory sensation may be better defined as a sound whose pitch depends on the number of impacts per second, i.e., on the speed and number of teeth on the wheel.
The ear possesses remarkable powers of analysis and can distinguish the individual components in a complex sound, appreciating their pitch and intensity. This analysis occurs as if the ear were composed of a complete and very numerous series of resonators and should, according to the acoustic law of Ohm, be independent of the phase relationships of the component tones. This is true within the limits of a first approximation, since strictly speaking the ear is not a perfect analyzer; in fact, finite zones of the basilar membrane, to which recent research by Bekesy attributes this function, respond to a lesser degree even to stimuli of frequencies close to those for which they are specifically tuned. While this ensures, on the one hand, a response to stimuli of any frequency (within the acoustic range), it also gives rise, on the other hand, to the possibility that, with simultaneous stimulation by two tones of slightly different frequency, periodic reinforcement and weakening of the sound may occur at the rate of the difference between the two frequencies (subjective beats).
Changes in timbre due to phase shifts of the individual components of a complex tone can occur through phenomena of interference combination with auricular harmonics; that is, the effective stimulus applied to the cochlear organ is altered, and consequently the subjective appreciation differs. Significant changes in timbre result from considerable reductions in the amplitude of the total auditory field due to affections of the auditory organ in the numerous forms of deafness.
PSYCHOLOGICAL ATTRIBUTES OF SOUND
The principal attributes are intensity and pitch, which the subject translates into judgments of "loud" and "soft" and of "high" and "low," respectively. Subjective intensity corresponds, for a given vibratory stimulus, to the acoustic pressure at the level of the tympanic membrane and thus to the amplitude of vibration, in general to the number of ergs transmitted through a square centimeter every second. Pitch, on the other hand, corresponds to the frequency of the vibratory stimulus. These two correspondences are not, however, as strict and unambiguous as was once believed and as is still found in many treatises: both pitch and intensity are each functions of the two variables, acoustic pressure and frequency; so that, for example, it is possible to induce changes in pitch by modifying the intensity of a sound. Such changes are small in percentage but account for many phenomena in physiological acoustics.
The acoustic pressure level of a vibratory stimulus can be expressed by the number of decibels (dB) that indicates by how much it exceeds the value of the reference threshold, i.e., the minimum acoustic pressure capable of generating an auditory sensation, a threshold that, by convention, is assumed relative to the frequency of 1000 Hz. On this scale:
\[
\begin{array}{ll}
\text{zero dB} & 10^{-16} \text{ watt cm}^{-2} \\
\text{°} & 0.000204 \text{ dyne cm}^{-2} \\
\text{°} & 73.8 \text{ dB below 1 dyne per cm}^2.
\end{array}
\]
These reference levels represent the auditory threshold for a normal ear but do not constitute starting values for a strictly defined "scale of intensity" (subjective). The auditory field extends from 16 Hz to 19,000 Hz on average for normal hearing; that is, between wavelengths of 21.3 m and 1.8 cm, respectively.
OTHER PSYCHOLOGICAL ATTRIBUTES OF SOUND
In addition to timbre, which has already been mentioned, there are other attributes; it is noted that to define a new one, it is sufficient that the organism's reaction to variations in the stimulus be systematic and that the corresponding values be capable of being plotted on a graph. Rich in 1916 clearly stated the fact that sounds have a subjective character of "width" or "extension" (volume) that, according to his findings, depends on pitch. Halverson in 1924 emphasized the dependence of volume on intensity. Stevens in 1934 demonstrated that it is possible to make two sounds appear equal in volume even though they differ in pitch and pre-existing intensity, because if volume is to remain constant, the higher sound must have greater intensity. Thus, to keep volume constant, starting from a vibratory stimulus of frequency 900 Hz and pressure at the tympanic membrane 55 dB above threshold, an increase of 100 Hz in frequency must be followed by an increase of 10 dB in pressure.
Many subjects also tend to attribute to pure sounds the subjective evaluation of "compactness" or "density," and this has been the subject of detailed research by Stevens (1934). Density has an inverse course to volume, in the sense that an increase in frequency is compensated by a decrease in pressure. Thus, to keep density constant, an increase in frequency from 500 to 600 Hz in a vibratory stimulus that produces a pressure of 60 dB above threshold at the tympanic membrane must be compensated by a decrease of 4 dB in pressure.
It is not easy to indicate through what mechanisms the perception of volume and density is achieved. It is obvious that the fact that the responses are differentiated implies a different neural and cortical excitation. Thus, it can be thought that density is linked to the density of nervous excitation in the cortex, while volume (Boring, 1926) would depend on the "width" of the cortical area involved.
The possibility of judging sounds as “brilliant” and “dull” has led some to posit brilliance as a further attribute of sound. Yet this attribute does not seem to possess an independent existence, since already in 1929 Toland indicated it as closely bound up with pitch, while Stevens, on the basis of more rigorous analysis, regards it as identical with density. Stevens in fact found that varying degrees of brilliance can be produced only by varying timbre; hence brilliance is more a subjective appraisal of timbre than a new attribute of sound.
Other important attributes of sound are the “musical” ones: namely, that two or more sounds of different frequency may be judged consonant or dissonant whether produced simultaneously or successively, and that their relative interval may be judged equal, larger, or smaller than another, leading to the establishment of musical scales; and, in conjunction with judgments about the time intervals between one sound and another, to harmony, counterpoint, orchestration—in a word, to music in general.
VIII. REFLECTION
When the vibratory motion of air particles, in the course of its propagation, encounters an elastic surface such as a slab of marble or a smooth wall, it is reflected according to laws similar to those governing the reflection of light from a mirror. Under particular environmental conditions, this reflection gives rise to an echo.The vibratory motion that reaches the ear is always different from that produced by the source, owing to the various and successive reflections from the walls of the room and the objects within II. There thus arises a persistence of acoustic energy in the environment: the phenomenon known as reverberation, which is always present unless one is operating in a free field—that is, in a theoretical space where reflecting walls either do not exist or are infinitely distant, or where such walls, though present at finite distance, are completely absorbent and reflect none of the incident energy. The existence of absorbent materials such as glass wool or mineral wool makes it possible to construct so-called “dead” rooms in which the energy reflected from the walls is only a small percentage of the incident energy. No wall, however, is perfectly reflective; yet there are materials, such as marble, that are sufficiently reflective to allow the construction of “reverberant” rooms for purposes of acoustic investigation. The proper balance of reflective and absorbent materials is of the greatest importance for the acoustics of auditoriums and theaters.
IX. RESONANCE
A material system can be set into oscillation by the action of external periodic forces (forced vibration). In such a case the period and amplitude of the vibration are not those proper to the system when it vibrates on its own, but those belonging to the external forces, though partly modified by the presence of the material system on which they act.The vibrations of the latter may become very large when the system’s own period coincides with that of the external forces; in this case the system enters into resonance and even a small force can generate considerable oscillations. Thus two pendulums oscillating with the same period mutually stimulate each other to oscillate if placed near one another; and if one is at rest while the other is set in motion, the first will soon be seen to oscillate as well, owing to the rhythmic impulses it receives through the air. A similar experiment can be performed with two tuning forks of the same frequency; if one is struck and placed near the other, the second will be observed to vibrate. The same effect can be demonstrated with two musical instruments, for example two violins whose strings have been tuned in unison and placed in the same room even several meters apart: if the strings of the first violin are plucked one by one and then damped with the fingers, the corresponding strings of the second will respond in orderly fashion—provided they are excited by plucking rather than by bowing or other means that might alter the frequency of the string being rubbed.
Air enclosed within a cavity can also resonate; the air in a tube, for instance, resonates at a frequency dependent on the tube’s length. More complexly, the air in the cavities known as “sound boxes” in musical instruments resonates. These cavities—typical examples are those of violins—have been empirically designed so that the air within them resonates as uniformly as possible across the wide range of frequencies produced by the instrument; given their complex shape, different lengths of air column and different masses of air are encountered in different directions. Of great importance for acoustic applications are the “Helmholtz resonators,” hollow spheres fitted with two diametrically opposed openings through which the internal air communicates with the ambient air and with the listener’s ear. The internal air resonates at a frequency determined by the diameter of the cavity.