Acoustic pressure

A sound wave compresses and dilates the material elements it passes through, generating associated pressure fluctuations. An appropriate sensor (a microphone, for example) placed in the sound field will record a time-varying deviation from the equilibrium pressure found at that point within the fluid. The changing total pressure P measured will vary about the equilibrium pressure P₀ by a small amount called the acoustic pressure, p = P - P₀. The SI unit of pressure is the pascal (Pa), equal to 1 newton per square meter (N/m²). Standard atmospheric pressure (14.7 lb/in.²) is approximately 1 bar = 10⁶ dyne/cm² = 10⁵ Pa. For a typical sound in air, the amplitude of the acoustic pressure may be about 0.1 Pa (one-millionth of an atmosphere); most sounds cause relatively slight perturbations of the total pressure. See Pressure, Pressure measurement, Pressure transducer, Sound pressure

Plane waves

One of the more basic sound waves is the traveling plane wave. This is a pressure wave progressing through the medium in one direction, say the +x direction, with infinite extent in the y and z directions. A two-dimensional analog is ocean surf advancing toward a very long, straight, and even beach. See Wave (physics), Wave equation, Wave motion

A most important plane wave, called harmonie, is the smoothly oscillating monofrequency plane wave described by Eq. (1).

(1) 

(2) 

Description of sound

The characterization of a sound is based primarily on human psychological responses to it. Because of the nature of human perceptions, the correlations between basically subjective evaluations such as loudness, pitch, and timbre and more physical qualities such as energy, frequency, and frequency spectrum are subtle and not necessarily universal.

The strength of a sound wave is described by its intensity. From basic physical principles, the instantaneous rate at which energy is transmitted by a sound wave through unit area is given by the product of acoustic pressure and the component of particle velocity perpendicular to the area. The time average of this quantity is the acoustic intensity. If all quantities are expressed in SI units (pressure amplitude or effective pressure amplitude in Pa, speed of sound in m/s, and density in kg/m³), then the intensity will be in watts per square meter (W/m²). See Sound intensity

Because of the way the strength of a sound is perceived, it has become conventional to specify the intensity of sound in terms of a logarithmic scale with the (dimensionless) unit of the decibel (dB). An individual with unimpaired hearing has a threshold of perception near 10^-12 W/m² between about 2 and 4 kHz, the frequency range of greatest sensitivity. As the intensity of a sound of fixed frequency is increased, the subjective evaluation of loudness also increases, but not proportionally. Rather, the listener tends to judge that every successive doubling of the acoustic intensity corresponds to the same increase in loudness. For sounds lying higher than 4 kHz or lower than 500 Hz, the sensitivity of the ear is appreciably lessened. Sounds at these frequency extremes must have higher threshold intensity levels before they can be perceived, and doubling of the loudness requires smaller changes in the intensity with the result that at higher levels sounds of equal intensities tend to have more similar loudnesses. It is because of this characteristic that reducing the volume of recorded music causes it to sound thin or tinny, lacking both highs and lows of frequency. Since most sound-measuring equipment detects acoustic pressure rather than intensity, it is convenient to define an equivalent scale in terms of the sound pressure level. The intensity level and sound-pressure level are usually taken as identical, but this is not always true. See Decibel

How “high” sound of a particular frequency appears to be is described by the sense of pitch. A few minutes with a frequency generator and a loudspeaker show that pitch is closely related to the frequency. Higher pitch corresponds to higher frequency, with small influences depending on loudness, duration, and the complexity of the waveform. For the pure tones (monofrequency sounds) encountered mainly in the laboratory, pitch and frequency are not found to be proportional. Doubling the frequency less than doubles the pitch. For the more complex waveforms usually encountered, however, the presence of harmonics favors a proportional relationship between pitch and frequency.

Propagation of sound

Plane waves are a considerable simplification of an actual sound field. The sound radiated from a source (such as a loudspeaker, a hand clap, or a voice) must spread outward much like the widening circles from a pebble thrown into a lake. A simple model of this more realistic case is a spherical source vibrating uniformly in all directions with a single frequency of motion. The sound field must be spherically symmetric with an amplitude that decreases with increasing distance from the source, and the fluid elements must have particle velocities that are directed radially.

Not all sources radiate their sound uniformly in all directions. When someone is speaking in an unconfined space, for example an open field, a listener circling the speaker hears the voice most well defined when the speaker is facing the listener. The voice loses definition when the speaker is facing away from the listener. Higher frequencies tend to be more pronounced in front of the speaker, whereas lower frequencies are perceived more or less uniformly around the speaker.

Diffraction

It is possible to hear but not see around the corner of a tall building. However, higher-frequency sound (with shorter wavelength) tends to bend or “spill” less around edges and corners than does sound of lower frequency. The ability of a wave to spread out after traveling through an opening and to bend around obstacles is termed diffraction. This is why it is often difficult to shield a listener from an undesired source of noise, like blocking aircraft or traffic noise from nearby residences. Simply erecting a brick or concrete wall between source and receiver is often an insufficient remedy, because the sounds may diffract around the top of the wall and reach the listeners with sufficient intensity to be distracting or bothersome. See Acoustic noise, Diffraction

Rays

Since the speed of sound varies with the local temperature (and pressure, in other than perfect gases), the speed of a sound wave can be a function of position. Different portions of a sound wave may travel with different speeds of sound.

Each small element of a surface of constant phase traces a line in space, defining a ray along which acoustic energy travels. The sound beam can then be viewed as a ray bundle, like a sheaf of wheat, with the rays distributed over the cross-sectional area of the surface of constant phase. As the major lobe spreads with distance, this area increases and the rays are less densely concentrated. The number of rays per unit area transverse to the propagation path measures the energy density of the sound at that point.

It is possible to use the concept of rays to study the propagation of a sound field. The ray paths define the trajectories over which acoustic energy is transported by the traveling wave, and the flux density of the rays measures the intensity to be found at each point in space. This approach, an alternative way to study the propagation of sound, is approximate in nature but has the advantage of being very easy to visualize.

Reflection and transmission

If a sound wave traveling in one fluid strikes a boundary between the first fluid and a second, then there may be reflection and transmission of sound. For most cases, it is sufficient to consider the waves to be planar. The first fluid contains the incident wave of intensity I_i and reflected wave of intensity I_r; the second fluid, from which the sound is reflected, contains the transmitted wave of intensity I_t. The directions of the incident, reflected, and transmitted plane sound waves may be specified by the grazing angles Θ_i, Θ_r, and Θ_t (measured between the respective directions of propagation and the plane of the reflecting surface). See Reflection of sound

Absorption

When sound propagates through a medium, there are a number of mechanisms by which the acoustic energy is converted to heat and the sound wave weakened until it is entirely dissipated. This absorption of acoustic energy is characterized by a spatial absorption coefficient for traveling waves. See Sound absorption

in the broad sense, the vibratory motion of the particles in anelastic medium, propagating as waves in gaseous, fluid, or solid mediums; in the narrow sense, a phenomenon that is perceived by a special sensory organ in humans and animals. Humans hear sound having a frequency between 16 and 20,000 hertz (Hz). The physical concept of sound includes both audible and inaudible sound. Sound with a frequency below 16 Hz is called infrasound; with a frequency above 20,000 Hz, ultrasound. Very high-frequency elastic waves in the range from 10⁹ to 10¹²–10¹³ Hz are called hyper-sound. The infrasonic frequency region has virtually no lower limit; infrasonic vibrations are encountered in nature at frequencies of tenths and hundredths of a hertz. The frequency range of hypersonic waves is limited at the top by physical factors that characterize the atomic and molecular structure of the medium: the length of an elastic wave must be substantially greater than the free path length of molecules in gases and greater than the interatomic distances in fluids and solids. As a result, hypersound cannot propagate in air at a frequency of 10⁹ Hz or higher or in solids at a frequency higher than 10¹²–10¹³” Hz.

Basic characteristics. An important characteristic of sound is its spectrum, which is produced by expanding the sound into simple harmonic vibrations (so-called frequency sound analysis). The spectrum is continuous if the energy of the sound vibrations is distributed continuously over a fairly broad frequency range; it is a linear spectrum if it has a set of discrete frequency components. Sound having a continuous spectrum is perceived as a noise like the rustling of leaves in the wind or the sounds of mechanisms in operation. A musical sound has a linear spectrum with multiple frequencies; the fundamental frequency determines the pitch of the sound as perceived by hearing, and the set of harmonic components distinguishes its timbre. The spectrum of speech sounds contains formants, which are stable groups of frequency components corresponding to certain phonetic elements. The energy parameter of sound vibrations is the sound intensity—the energy carried by the sound wave through a unit surface, perpendicular to the direction of propagation, per unit time. Sound intensity is a function of the sound pressure amplitude, as well as of the properties of the medium and the shape of the wave. The subjective parameter, which is associated with the intensity of the sound, is loudness, which is dependent on frequency. The human ear is most sensitive in the frequency range from 1 to 5 kHz. In this region the threshold of audibility—that is, the intensity of the weakest audible sounds—is on the order of 10^-12 watts per sq m (W/m²), and the corresponding sound pressure is 10^-5 new-tons per sq m (N/m² ). The upper limit of intensity for sound perception by the human ear is called the threshold of pain; it is slightly dependent on frequency in the audible range and is approximately equal to 1 W/m². Much greater intensities (up to 104 kW/m²) are achieved in ultrasonic technology.

Sound sources. Sound sources are phenomena that produce a local variation in pressure or mechanical stress. Vibrating solid bodies, such as the cones in loudspeakers, the diaphragms in telephones, and the strings and sounding boards in musical instruments, are the most common sound sources; in the ultrasonic frequency range the sources may be plates and rods made from piezoelectric or magnetostrictive materials. Vibrations of limited volumes of the medium itself—for example, in organ pipes, wind instruments, and whistles—may also be sound sources. The vocal apparatus of humans and animals is a complex vibratory system. The vibration of sound sources may be excited by a blow (in the case of a bell) or by plucking (in the case of a string); a self-excited vibration mode can be maintained in such objects by a current of air (in wind instruments). Electroacoustic transducers, in which mechanical vibrations are produced by converting electrical current oscillations of the same frequency, are a broad class of sound sources. In nature, sound is produced when air flows around solid bodies because of the creation and breakaway of vortices (for example, when the wind blows over wires, pipes, and the crests of ocean waves). Low-frequency and infrasonic sounds are produced by explosions and avalanches. The sources of acoustic noise include machines and mechanisms used in technology, as well as gas and water jets. The study of industrial, transportation, and aerodynamic sources of noise is receiving a great deal of attention in view of their harmful effects on the human body and industrial equipment.

Sound receivers. Sound receivers pick up sound energy and convert it into other forms. The hearing apparatus of humans and animals is in this category. In technology, electroacoustic transducers are generally used for the reception of sound: microphones in air, hydrophones in water, and geophones in the earth’s crust. In addition to such transducers, which reproduce the time dependence of the sound signal, there are receivers that measure the parameters of sound waves averaged with respect to time—for example, the Raleigh disk and the acoustic radiometer.

Propagation of sound waves. The propagation of sound waves is characterized primarily by the speed of sound. Longitudinal waves propagate in gaseous and fluid mediums (the direction of the particles’ vibratory motion coincides with the direction of propagation of the wave) at a velocity determined by the compressibility and density of the medium. The speed of sound in dry air at a temperature of 0°C is 330 m/sec, and in fresh water at 17°C it is 1,430 m/sec. In addition to longitudinal waves, transverse waves, for which the direction of the vibrations is perpendicular to the direction of propagation of the wave, as well as surface waves (Rayleigh waves), may propagate in solids. For most metals the velocity of longitudinal waves ranges from 4,000 to 7,000 m/sec; the velocity of transverse waves, between 2,000 and 3,500 m/sec.

During the propagation of waves of great amplitude, the compression phase propagates at a higher velocity than the rarefaction phase, so that the sinusoidal shape of the wave becomes gradually distorted and the sound wave is converted into a shock wave. In many cases sound dispersion is observed, that is, the velocity of propagation is a function of the frequency. Sound dispersion leads to a change in the shape of complex acoustic signals, including a number of harmonic components, and, in particular, to the distortion of sound pulses.

The phenomena of interference and diffraction, which are typical of all types of waves, can occur during the propagation of sound waves. When the size of obstacles and the inhomogeneities of the medium are large compared to the wavelength, the propagation of sound obeys the usual laws of reflection and refraction for waves and may be dealt with from the viewpoint of geometric acoustics.

During the propagation of a sound wave in a given direction, gradual attenuation occurs—that is, its intensity and amplitude decrease. Knowledge of the laws of attenuation is of practical importance in determining the maximum propagation distance for an acoustic signal. Attenuation depends on a number of factors, which become manifest to a greater or lesser degree according to the characteristics of the sound itself (primarily its frequency) and the properties of the medium. All of these factors may be classified in two large groups. The first group includes factors associated with the laws of wave propagation in the medium. Thus, in the case of propagation in an infinite medium, the intensity of a sound from a source of finite size decreases inversely as the square of the distance. Inhomogeneity of the medium’s properties causes scattering of the wave in various directions, thus weakening it in the original direction, as is the case with sound scattered by bubbles in water, by the agitated surface of an ocean, and by atmospheric turbulence; high-frequency ultrasound is scattered in polycrystalline metals and by dislocations in crystals. The propagation of sound in the atmosphere and in the ocean is affected by temperature and pressure distribution and by the force and velocity of the wind. These factors cause bending of the sound rays—that is, refraction—which in particular accounts for the fact that sound is audible farther with the wind than against it. The distribution of the speed of sound with depth in the ocean explains the existence of the so-called underwater sound channel, in which extremely long-range sound propagation is observed: for example, the sound of an explosion propagates for more than 5,000 km.

The second group of factors determining sound attenuation is associated with the physical processes in a substance, including the irreversible transformation of sound energy into other forms, mainly heat—that is, the absorption of sound, which is caused by the viscosity and thermal conductivity of the medium (“classical absorption”)—and the transformation of sound energy into the energy of intramolecular processes (molecular or relaxation absorption). Sound absorption increases markedly with frequency. Therefore, high-frequency ultrasound and hypersound usually propagate only over very short distances, most often no more than several centimeters. Infrasonic waves, which are distinguished by low absorption and weak scattering, are propagated the farthest in the atmosphere, in water, and in the earth’s crust. At high ultrasonic and hypersonic frequencies additional absorption occurs in solids as a result of the interaction of the waves with the thermal vibrations of the crystal lattice, with electrons, and with light waves. Under certain conditions this interaction can produce “negative absorption,” or the amplification of sound waves.

The importance of sound waves, and consequently their study (in acoustics), is extremely great. Since ancient times, sound has served as a means of communication and signaling. The study of all of its characteristics has made possible the development of more advanced data transmission systems, an increase in the range of signaling systems, and the creation of improved musical instruments. Sound waves are virtually the only form of signals that propagate in water, where they are used for submarine communications, navigation, and echolocation. Low-frequency sound is a tool for the study of the earth’s crust. The practical application of ultrasound has created ultrasonics, an entire branch of modern technology. Ultrasound is used for monitoring and measurement (particularly in flaw detection), as well as for operations on substances (ultrasonic cleaning, mechanical treatment, and welding). High-frequency sound waves, particularly hypersound, are an important means for research in solid-state physics.