Acoustics

Acoustics

Acoustics is the science that deals with the production, transmission, and reception of sound. Sound may be produced when a material body vibrates; it is transmitted only when there is some material body, called the medium, that can carry the vibrations away from the producing body; it is received when a third material body, attached to some indicating device, is set into vibratory motion by that intervening medium. However, the only vibrations that are considered sound (or sonic vibrations) are those in which the medium vibrates in the same direction as the sound travels, and for which the vibrations are very small. When the rate of vibration is below the range of human hearing, the sound is termed infrasonic; when it is above that range, it is called ultrasonic. (The term supersonic refers to bodies moving at speeds greater than the speed of sound, and is not normally involved in the study of acoustics.)

Production of Sound

There are many examples of vibrating bodies producing sounds. Some are as simple as a string in a violin or piano, or a column of air in an organ pipe or in a clarinet; some are as complex as the vocal chords of a human. Sound may also be caused by a large disturbance which causes parts of a body to vibrate, such as sounds caused by a falling tree.

Vibrations of a String

To understand some of the fundamentals of sound production and propagation it is instructive to first consider the small vibrations of a stretched string held at both ends under tension. While these vibrations are not an example of sound, they do illustrate many of the properties of importance in acoustics as well as in the production of sound. The string may vibrate in a variety of different ways, depending upon whether it is struck or rubbed to set it in motion, and where on the string the action took place. However, its motion can be analyzed into a combination of a large number of simple motions. The simplest, called the fundamental (or the first harmonic), appears in Figure 1, which shows the outermost extensions of the string carrying out this vibration.

The second harmonic is shown in Figure 2; the third harmonic in Figure 3; and so forth (the whole set of harmonics beyond the first are called the overtones). The rate at which these vibrations take place (number of times per second the motion is repeated) is called the frequency, denoted by f (the reciprocal of the frequency, which is the time for one cycle to be competed, is called the period). A single complete vibration is normally termed a cycle, so that the frequency is usually given in cycles per second, or the equivalent modern unit, the hertz (abbreviated Hz). It is characteristic of the stretched string that the second harmonic has a frequency twice that of the fundamental; the third harmonic has a frequency three times that of the fundamental; and so forth. This is true for only a few very simple systems, with most sound-producing systems having a far more complex relationship among the harmonics.

Those points on the string which do not move are called the nodes; the maximum extension of the string (from the horizontal in the Figures) is called the amplitude, and is denoted by A in Figures 1-3. The distance one must go along the string at any instant of time to reach a section having the identical motion is called the wavelength, and is denoted by L in Figures 1-3. It can be seen that the string only contains one-half wavelength of the fundamental, that is, the wavelength of the fundamental is twice the string length. The wavelength of the second harmonic is the length of the string. The string contains one-and-one-half (3/2) wavelengths of the third harmonic, so that its wavelength is two-thirds (2/3) of the length of the string. Similar relationships hold for all the other harmonics.

If the fundamental frequency of the string is called f, and the length of the string is l, it can be seen from the above that the product of the frequency and the wavelength of each harmonic is equal to 2fl. The dimension of this product is a velocity (e.g., feet per second or centimeters per second); detailed analysis of the motion of the stretched string shows that this is the velocity with which a small disturbance on the string would travel down the string.

Vibrations of an Air Column

When air is blown across the entrance to an organ pipe, it causes the air in the pipe to vibrate, so that there are alternate small increases and decreases of the density of the air (condensations and rarefactions). These alternate in space, with the distance between successive condensations (or rarefactions) being the wavelength; they alternate in time, with the frequency of the vibration. One major difference here is that the string vibrates transversely (perpendicular to the length of the string), while the air vibrates longitudinally (in the direction of the column of air). If the pipe is open at both ends, then the density of the air at the ends must be the same as that of the air outside the pipe, while the density inside the pipe can vary above or below that value. Again, as for the vibrations of the string, the density of the air in the pipe can be analyzed into a fundamental and overtones. If the density of the air vibrating in the fundamental mode (of the open pipe) is plotted across the pipe length, the graph is as in Figure 4.

The "zero value" at the ends denotes the fact that the density at the ends of the pipe must be the same as outside the pipe (the ambient density), while inside the pipe the density varies above and below that value with the frequency of the fundamental, with a maximum (and minimum) at the center. The density plot for the fundamental looks just like that for the fundamental of the vibrating stretched string (Figure 1). In the same manner, plots of the density for the various overtones would look like those of the string overtones. The frequency of the fundamental can be calculated from the fact that the velocity, which is analogous to that found for vibrations of the string, is the velocity with which sound travels in the air, usually denoted by c. Since the wavelength of the fundamental is twice the pipe length, its frequency is c/2 l, where l is the length of the organ pipe. (While the discussion here is in terms of the density variations in the air, these are accompanied by small variations in the air pressure, and small motions of the air itself. At places of increased density the pressure is increased; where the pressure is changing rapidly, the air motion is greatest.) When a musician blows into the mouthpiece of a clarinet, the air rushing past the reed causes it to vibrate which then causes the column of air in the clarinet to vibrate in a manner similar to, but more complicated than, the motion of the organ pipe. These vibrations (as for all vibrations) can also be analyzed into harmonics. By opening and closing the keyholes in the clarinet, different harmonics of the clarinet are made to grow louder or softer causing different tones to be heard.

Sound Production in General

Thus, the production of sound depends upon the vibration of a material body, with the vibration being transmitted to the medium that carries the sound away from the sound producer. The vibrating violin string, for example, causes the body of the violin to vibrate; the "back-and-forth" motion of the parts of the body of the violin causes the air in contact with it to vibrate. That is, small variations in the density of the air are produced by the motion of the violin body, and these are carried forth into the air surrounding the violin. As the sound is carried away, the small variations in air density are propagated in the direction of travel of the sound.

Sounds from humans, of course, are produced by forcing air across the vocal cords, which causes them to vibrate. The various overtones are enhanced or diminished by the size and shape of the various cavities in the head (the sinuses, for example), as well as the placement of the tongue and the shape of the mouth. These factors cause specific wavelengths, of all that are produced by the vocal cords, to be amplified differently so that different people have their own characteristic voice sounds. These sounds can be then controlled by changing the placement of the tongue and the shape of the mouth, producing speech. The frequencies usually involved in speech are from about 100 to 10,000 Hz. However, humans can hear sounds in the frequency range from about 20 to 18,000 Hz. These outer limits vary from person to person, with age, and with the loudness of the sound. The density variations (and corresponding pressure variations) produced in ordinary speech are extremely small, with ordinary speech producing less than one-millionth the power of a 100 watt light bulb! In the sonic range of frequencies (those produced by humans), sounds are often produced by loudspeakers, devices using electronic and mechanical components to produce sounds. The sounds to be transmitted are first changed to electrical signals by a microphone (see Reception of sounds, below), for example, or from an audio tape or compact disc; the frequencies carried by the electrical signals are those to be produced as the sound signals. In the simplest case, the wires carrying the electrical signals are used to form an electromagnet which attracts and releases a metal diaphragm. This, in turn, causes the variations in the density in the air adjacent to the diaphragm. These variations in density will have the same frequencies as were in the original electrical signals.

Ultrasonic vibrations are of great importance in industry and medicine, as well as in investigations in pure science. They are usually produced by applying an alternating electric voltage across certain types of crystals (quartz is a typical one) that expand and contract slightly as the voltage varies; the frequency of the voltage then determines the frequency of the sounds produced.

Transmission of Sound

In order for sound to travel between the source and the receiver there must be some material between them that can vibrate in the direction of travel (called the propagation direction). (The fact that sound can only be transmitted by a material medium means that an explosion outside a spaceship would not be heard by its occupants!) The motion of the sound-producing body causes density variations in the medium (see Figure 5, which schematically shows the density variations associated with a sound wave), which move along in the direction of propagation. The transmission of sounds in the form of these density variations is termed a wave since these variations are carried forward without significant change, although eventually friction in the air itself causes the wave to dissipate. (This is analogous to a water wave in which the particles of water vibrate up and down, while the "wave" propagates forward.) Since the motion of the medium at any point is a small vibration back and forth in the direction in which the wave is proceeding, sound is termed a longitudinal wave. (The water wave, like the violin string, is an example of a transverse wave.) The most usual medium of sound transmission is air, but any substance that can be compressed can act as a medium for sound propagation. A fundamental characteristic of a wave is that it carries energy and momentum away from a source without transporting matter from the source.

Since the speed of sound in air is about 331 meters per second (about 1,088 feet per second), human speech involves wavelengths from about 3.3 meters to 3.3 centimeters (from about 11 feet to 1.3 inches). Thus, the wavelengths of speech are of the size of ordinary objects, unlike light, whose wavelengths are extremely small compared to items that are part of everyday life. Because of this, sound does not ordinarily cast "acoustic shadows" but, because its wavelengths are so large, can be transmitted around ordinary objects. For example, if a light is shining on a person, and a book is placed directly between them, the person will no longer be able to see the light (a shadow is cast by the book on the eyes of the observer). However, if one person is speaking to another, then placing a book between them will hardly affect the sounds heard at all; the sound waves are able to go around the book to the observer's ears. On the other hand, placing a high wall between a highway and houses can greatly decrease the sounds of the traffic noises if the dimensions of the wall (height and length) are large compared with the wavelength of the traffic sounds. Thus, sound waves (as for all waves) tend to "go around" (e.g., ignore the presence of) obstacles which are small compared with the wavelength of the wave; and are reflected by obstacles which are large compared with the wavelength. For obstacles of approximately the same size as the wavelength, waves exhibit a very complex behavior known as diffraction, in which there are enhanced and diminished values of the wave amplitude, but which is too complicated to be described here in detail.

The speed of sound in a gas is proportional to the square root of the pressure divided by the density. Thus, helium, which has a much lower density than air, transmits sound at a greater speed than air. If a person breathes some helium, the characteristic wavelengths are still determined by the shape of the mouth, but the greater sound speed causes the speech to be emitted at a higher frequency-thus the "Donald Duck" sounds from someone who speaks after taking a breath of helium from a balloon.

In general, the speed of sound in liquids is greater than in gases, and greater still in solids. In sea water, for example, the speed is about 1,447 meters per second (about 4,750 feet per second); as for a gas, the speed increases as the pressure increases, and as the density decreases. Typical speeds of sound in solids are 5,000 meters per second, but vary considerably from one solid to another.

Reception of Sound

By far the most important sound receiver in use is the human ear. While the details of the workings of the ear are too complicated to be described here, the process consists of the sound wave (the density variations carried through the air) striking the eardrum, causing it to vibrate with the same set of frequencies as had been carried in the wave. This vibration is transmitted through a set of bones in the ear to a liquid-filled chamber; small hairs in the liquid are set into motion and excite nerves which then transmit these frequencies to the brain.

In the sonic range of frequencies, the microphone, a device using electrical and mechanical components, is the common method of receiving sounds. One simple form is to have a diaphragm as one plate of an electrical condenser. When the diaphragm vibrates under the action of a sound wave, the current in the circuit varies due to the varying capacitance of the condenser. This varying current can then be used to activate a meter or oscilloscope or, after suitable processing, make an audio tape or some such permanent record.

Applications

The applications of acoustical devices are far too numerous to describe; one only has to look around our homes to see some of them: telephones, radios and television sets, compact disc players and tape recorders; even clocks that "speak" the time! Probably one of the most important from the human point of view is the hearing aid, a miniature microphone-amplifier-loudspeaker that is designed to enhance whatever range of frequencies a person finds difficulty hearing.

However, one of the first large-scale "industrial" uses of sound propagation was by the military in World War I, in the detection of enemy submarines by means of sonar (for sound navigation and ranging). This was further developed during the period between then and World War II, and since then. The ship hunting submarine has a sound source and receiver projecting from the ship's hull that can be used for either listening or in an echo-ranging mode; the source and receiver are directional, so that they can send and receive an acoustic signal from only a small range of directions at one time. In the listening mode of operation, the operator tries to determine what are the sources of any noise that might be heard: the regular beat of an engine heard underwater can tell that an enemy might be in the vicinity. In the echo-ranging mode, a series of short bursts of sound is sent out, and the time for the echo to return is noted; that time interval multiplied by the speed of sound in water indicates (twice) the distance to the reflecting object. Since the sound source is directional, the direction in which the object lies is also known. This is now such a well developed method of finding underwater objects that commercial versions are available for fishermen to hunt for schools of fish.

Ultrasonic sources, utilizing pulses of frequencies in the many millions of cycles per second (and higher!), are now used for inspecting metals for flaws. The small wavelengths make the pulses liable to reflection from any imperfections in a piece of metal. Castings may have internal cracks which will weaken the structure; welds may be imperfect, possibly leading to failure of a metal-to-metal joint; metal fatigue may produce cracks in areas impossible to inspect byeye. The use of ultrasonic inspection techniques is increasingly important for failure prevention in bridges, aircraft, and pipelines, to name just a few.

The use of ultrasonics in medicine is also of growing importance. The detection of kidney stones or gallstones is routine, as is the imaging of fetuses to detect suspected birth defects, cardiac imaging, blood flow measurements, and so forth.

Thus, the field of acoustics covers a vast array of different areas of use, and they are constantly expanding. Acoustics in the communications industry, in various phases of the construction industries, in oil field exploration, in medicine, in the military, and in the entertainment industry, all attest to the growth of this field and to its continuing importance in the future.

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